Finite Group Databasebeta

← #29698521

Subgroups — #29698521

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454703C211yesmaximal, characteristic
2#21454695C211yesmaximal, characteristic
3#21454419C211yesmaximal, characteristic, complemented
4#21454414C211yesmaximal, characteristic, complemented
5#21446605C211yesmaximal, characteristic, complemented
6#21446600C211yesmaximal, characteristic, complemented
7#21446054C211yesmaximal, characteristic, complemented
8#21446054C211yesmaximal, characteristic, complemented
9#21446047C211yesmaximal, characteristic, complemented
10#21446047C211yesmaximal, characteristic, complemented
11#21445987C211yesmaximal, characteristic, complemented
12#21445883C211yesmaximal, characteristic, complemented
13#21445878C211yesmaximal, characteristic, complemented
14#21445703C211yesmaximal, characteristic, complemented
15C2×C61×#444C211yesmaximal, characteristic
16#16840245C2211yescharacteristic
17#16840245C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16839874—21no—
21#16839869—21no—
22#16839863—21no—
23#16839863—21no—
24Q16×Dic61—21no—
25SD16×Dic61—21no—
26SD16×Dic61—21no—
27D8×Dic61—21no—
28C2×#14519713C2211yescharacteristic
29#16839804—21no—
30#16839804—21no—
31#16839804—21no—
32#16839804—21no—
33#16839804C2211yescharacteristic
34#16839804C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839800—21no—
38#16839800—21no—
39#16839800C2211yescharacteristic
40C2×#14519711—21no—
41C2×#14519711C2211yescharacteristic
42C2×#14519711C2211yescharacteristic
43C2×#14519709—21no—
44C2×#14519709C2211yescharacteristic
45C2×#14519709C2211yescharacteristic
46#16839763C2211yescharacteristic
47#16839763C2211yescharacteristic
48#16839763C2211yescharacteristic
49#16839763C2211yescharacteristic
50#16839717C2211yescharacteristic
51#16839717C2211yescharacteristic
52#16839632C2211yescharacteristic
53#16839628C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C2×#135×C61C2211yescharacteristic
61C2×#131×C61C2211yescharacteristic
62C2×#131×C61C2211yescharacteristic
63C61×#444C418yescomplemented
64C2×#108×C61C2211yescharacteristic
65C2×#104×C61C2211yescharacteristic
66C2×#104×C61C2211yescharacteristic
67C2×#97×C61C2211yescharacteristic
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70Q8×Dic61—21no—
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61—21no—
74D4×Dic61—21no—
75D4×Dic61—21no—
76D4×Dic61—21no—
77D4×Dic61—21no—
78D4×Dic61—41no—
79D4×Dic61—41no—
80D4×Dic61—41no—
81D4×Dic61—41no—
82#14519736—21no—
83#14519736—21no—
84#14519736—21no—
85#14519736—21no—
86#14519736C2311yescharacteristic
87#14519736C2311yescharacteristic
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—41no—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97#14519713—22no—
98#14519711—21no—
99#14519711—21no—
100#14519711—21no—
101#14519711—21no—
102#14519711—21no—
103#14519711—21no—
104#14519711—21no—
105#14519711—21no—
106#14519711D411yescharacteristic
107#14519711D411yescharacteristic
108#14519709—21no—
109#14519709—21no—
110#14519709—21no—
111#14519709—21no—
112#14519709—21no—
113#14519709—21no—
114#14519709—21no—
115#14519709—21no—
116#14519709D411yescharacteristic
117#14519709D411yescharacteristic
118#14519701—21no—
119#14519701—21no—
120#14519701—21no—
121#14519701—21no—
122#14519698—21no—
123#14519698—21no—
124#14519698—21no—
125#14519698—21no—
126#14519687—21no—
127#14519687—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130#14519684—41no—
131C2×#41×C61C2311yescharacteristic
132C2×#41×C61C2311yescharacteristic
133#135×C61C2×C414yes—
134#135×C61Q814yes—
135C2×Q16×C61D411yescharacteristic
136C2×SD16×C61D411yescharacteristic
137C2×SD16×C61D411yescharacteristic
138C2×D8×C61D411yescharacteristic
139#131×C61—22no—
140#131×C61—22no—
141#131×C61C2×C414yescomplemented
142#131×C61C2×C414yescomplemented
143C2×M16×C61—21no—
144C2×M16×C61—21no—
145C2×M16×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C22×C8×C61—21no—
148C22×C8×C61—21no—
149C22×C8×C61C2311yescharacteristic
150C22×C8×C61C2311yescharacteristic
151C2×C42×C61C2311yescharacteristic
152#108×C61C2×C414yescomplemented
153#104×C61C2×C414yescomplemented
154#104×C61C2×C414yescomplemented
155#97×C61C2×C414yescomplemented
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159#41×C61—21no—
160#41×C61C22×C412yes—
161#41×C61C22×C412yes—
162#41×C61C2×Q812yes—
163#41×C61C2×Q812yes—
164C2×Q8×C61C2×D411yescharacteristic
165C2×Q8×C61C2×D411yescharacteristic
166C2×D4×C61—21no—
167C2×D4×C61—21no—
168C2×D4×C61C2×D411yescharacteristic
169C2×D4×C61C2×D411yescharacteristic
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61C2411yescharacteristic
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61—21no—
178C22×Dic61—41no—
179C22×Dic61—41no—
180C22×Dic61—41no—
181C22×Dic61—41no—
182Q16×C61C4⋊C414yescomplemented
183SD16×C61C4⋊C414yescomplemented
184SD16×C61C4⋊C414yescomplemented
185D8×C61C4⋊C414yescomplemented
186M16×C61—21no—
187M16×C61—21no—
188M16×C61—24no—
189M16×C61—24no—
190M16×C61C22×C412yes—
191M16×C61C22×C412yes—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—24no—
195C2×C8×C61—24no—
196C2×C8×C61C22×C412yes—
197C2×C8×C61C22×C412yes—
198C2×C8×C61C2×D411yescharacteristic
199C2×C8×C61C2×D411yescharacteristic
200C2×C8×C61C2×D411yescharacteristic
201C2×C8×C61C2×D411yescharacteristic
202C2×C8×C61C2×Q812yes—
203C2×C8×C61#4112yes—
204C42×C61—21no—
205C42×C61C22×C412yes—
206C22⋊Dic61—21no—
207C22⋊Dic61—21no—
208C22⋊Dic61—41no—
209C22⋊Dic61—41no—
210C22⋊Dic61—41no—
211C22⋊Dic61—41no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—41no—
221C4⋊Dic61—41no—
222C4⋊Dic61—41no—
223C4⋊Dic61—41no—
224C4⋊Dic61—82no—
225C4⋊Dic61C2×D411yescharacteristic
226C4⋊Dic61C2×D411yescharacteristic
227C4⋊Dic61C2×D411yescharacteristic
228C4⋊Dic61C2×D411yescharacteristic
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61—41no—
232C4×Dic61—41no—
233C4×Dic61—41no—
234C4×Dic61—41no—
235#11758194—21no—
236#11758194—21no—
237#11758194—21no—
238#11758194—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C23×C61—21no—
242C23×C61—21no—
243C23×C61H(5,𝔽2)11yescharacteristic
244Q8×C61—21no—
245Q8×C61—21no—
246Q8×C61C2×C4⋊C412yes—
247Q8×C61C2×C4⋊C412yes—
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—22no—
251D4×C61—22no—
252D4×C61—21no—
253D4×C61—21no—
254D4×C61C2×C4⋊C412yes—
255D4×C61C2×C4⋊C412yes—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×Q811yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—81no—
280C2×Dic61—81no—
281C2×Dic61—81no—
282C2×Dic61—81no—
283C2×Dic61—161no—
284C2×Dic61—161no—
285C2×Dic61—161no—
286C2×Dic61—161no—
287C8×C61—22no—
288C8×C61—22no—
289C8×C61C2×C4⋊C412yes—
290C8×C61C2×C4⋊C412yes—
291C8×C61C4×D412yes—
292C8×C61C4×D412yes—
293#92418—41no—
294#92418—41no—
295#22693—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61#52011yescharacteristic
305C22×C61D4211yescharacteristic
306C22×C61#55411yescharacteristic
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61C22×C4⋊C411yescharacteristic
311C4×C61C2×C4×D411yescharacteristic
312C4×C61#54711yescharacteristic
313C4×C61D4×Q811yescharacteristic
314Dic61—41no—
315Dic61—41no—
316Dic61—41no—
317Dic61—41no—
318Dic61—81no—
319Dic61—81no—
320Dic61—81no—
321Dic61—81no—
322C2×#444D6111yescharacteristic
323#2952—611no—
324#2951—611no—
325#2913—611no—
326#2912—611no—
327#1952—611no—
328#1951—611no—
329#1860—611no—
330#1860—611no—
331#1859—611no—
332#1859—611no—
333#1847—611no—
334#1816—611no—
335#1815—611no—
336#1773—611no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—42no—
340C2×C61—42no—
341C2×C61D4×C4⋊C411yescharacteristic
342C2×C61#328811yescharacteristic
343C2×C61#328911yescharacteristic
344C2×#135C2×D6111yescharacteristic
345C2×#131C2×D6111yescharacteristic
346C2×#131C2×D6111yescharacteristic
347#520—611no—
348#520—611no—
349#520—611no—
350#520—611no—
351#444Dic6118yescomplemented
352#442—1221no—
353#441—1221no—
354#440—1221no—
355#440—1221no—
356C4×Q16—1221no—
357C4×SD16—1221no—
358C4×SD16—1221no—
359C4×D8—1221no—
360C2×#108—611no—
361C2×#108C2×D6111yescharacteristic
362#428—611no—
363#428—611no—
364#428—611no—
365#428—611no—
366#428—1221no—
367#428—1221no—
368#428—1221no—
369#428—1221no—
370#427—611no—
371#427—1221no—
372#427—1221no—
373C2×#107—611no—
374C2×#107—611no—
375C2×#107—1221no—
376C2×#106—611no—
377C2×#106—611no—
378C2×#106—1221no—
379C2×#104C2×D6111yescharacteristic
380C2×#104C2×D6111yescharacteristic
381#417—611no—
382#417—611no—
383#417—611no—
384#417—611no—
385#405—611no—
386#405—611no—
387C2×#97C2×D6111yescharacteristic
388#384—611no—
389#382—611no—
390#343—611no—
391#343—611no—
392#343—611no—
393#343—611no—
394#339—611no—
395#339—611no—
396C61#2269311yescharacteristic, complemented, Sylow
397C2×#41C22×D6111yescharacteristic
398C2×#41C22×D6111yescharacteristic
399#135Dic12214yes—
400#135C2×Dic6114yes—
401C2×Q16D24411yescharacteristic
402C2×SD16D24411yescharacteristic
403C2×SD16D24411yescharacteristic
404C2×D8D24411yescharacteristic
405#131—22no—
406#131—22no—
407#131C2×Dic6114yescomplemented
408#131C2×Dic6114yescomplemented
409C2×M16—21no—
410C2×M16—21no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16C22×D6111yescharacteristic
414C2×M16C22×D6111yescharacteristic
415C22×C8—21no—
416C22×C8—21no—
417C22×C8C22×D6111yescharacteristic
418C22×C8C22×D6111yescharacteristic
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—2441no—
428C4×D4—2441no—
429C4×D4—2441no—
430C4×D4—2441no—
431#117—611no—
432#117—611no—
433#117—1221no—
434#117—1221no—
435#117—1221no—
436#117—1221no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—2441no—
446C2×C42C22×D6111yescharacteristic
447#108—1222no—
448#108C2×Dic6114yescomplemented
449#107—611no—
450#107—611no—
451#107—1221no—
452#107—1221no—
453#107—1221no—
454#107—1221no—
455#107—1221no—
456#107—1221no—
457#107—1221no—
458#107—1221no—
459#106—611no—
460#106—611no—
461#106—1221no—
462#106—1221no—
463#106—1221no—
464#106—1221no—
465#106—1221no—
466#106—1221no—
467#106—1221no—
468#106—1221no—
469#104C2×Dic6114yescomplemented
470#104C2×Dic6114yescomplemented
471#103—1221no—
472#103—1221no—
473#103—1221no—
474#103—1221no—
475#102—1221no—
476#102—1221no—
477#102—1221no—
478#102—1221no—
479#97—1221no—
480#97—1221no—
481#97C2×Dic6114yescomplemented
482C4×C8—1221no—
483C4×C8—1221no—
484#95—2441no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C2×Dic12212yes—
490#41C2×Dic12212yes—
491#41C22×Dic6112yes—
492#41C22×Dic6112yes—
493C2×Q8C2×D24411yescharacteristic
494C2×Q8C2×D24411yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×D24411yescharacteristic
498C2×D4C2×D24411yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—2441no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4—2441no—
510C22×C4C23×D6111yescharacteristic
511Q16C4⋊Dic6114yescomplemented
512SD16C4⋊Dic6114yescomplemented
513SD16C4⋊Dic6114yescomplemented
514D8C4⋊Dic6114yescomplemented
515M16—21no—
516M16—21no—
517M16—24no—
518M16—24no—
519M16—1221no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16C22×Dic6112yes—
524M16C22×Dic6112yes—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8C2×Dic12212yes—
532C2×C8C2×D24411yescharacteristic
533C2×C8C2×D24411yescharacteristic
534C2×C8D4×D6111yescharacteristic
535C2×C8D4×D6111yescharacteristic
536C2×C8#1175822412yes—
537C2×C8C22×Dic6112yes—
538C2×C8C22×Dic6112yes—
539C4⋊C4—611no—
540C4⋊C4—611no—
541C4⋊C4—611no—
542C4⋊C4—611no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—4882no—
552C4⋊C4—2441no—
553C4⋊C4—2441no—
554C4⋊C4—2441no—
555C4⋊C4—2441no—
556C22⋊C4—1221no—
557C22⋊C4—1221no—
558C22⋊C4—2441no—
559C22⋊C4—2441no—
560C22⋊C4—2441no—
561C22⋊C4—2441no—
562C42—21no—
563C42—1221no—
564C42—1221no—
565C42—2441no—
566C42—2441no—
567C42—2441no—
568C42—2441no—
569C42C22×Dic6112yes—
570C23—21no—
571C23—21no—
572C23#1451982311yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×C4⋊Dic6112yes—
576Q8C2×C4⋊Dic6112yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×C4⋊Dic6112yes—
584D4C2×C4⋊Dic6112yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—9761no—
599C2×C4—9761no—
600C2×C4—9761no—
601C2×C4—9761no—
602C2×C4—4881no—
603C2×C4—4881no—
604C2×C4—4881no—
605C2×C4—4881no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C23×Dic6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C22×D24411yescharacteristic
613C2×C4C22×Dic12211yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4#1451982411yescharacteristic
616C8—22no—
617C8—22no—
618C8—2441no—
619C8—2441no—
620C8C2×C4⋊Dic6112yes—
621C8C2×C4⋊Dic6112yes—
622C8D4×Dic6112yes—
623C8D4×Dic6112yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1684024511yescharacteristic
633C22D4×D24411yescharacteristic
634C22#1684046911yescharacteristic
635C4—22no—
636C4—22no—
637C4—42no—
638C4—4881no—
639C4—4881no—
640C4—4881no—
641C4—4881no—
642C4—2441no—
643C4—2441no—
644C4—2441no—
645C4—2441no—
646C4C22×C4⋊Dic6111yescharacteristic
647C4C2×D4×Dic6111yescharacteristic
648C4#1684041611yescharacteristic
649C4D4×Dic12211yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×C4⋊Dic6111yescharacteristic
655C2#2145758611yescharacteristic
656C2#2145759511yescharacteristic
subgroupG/H
centerC22D4×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#444C2
socleC22×C61D42
prime radicalC2×#135×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#444D61
hypocenterC61#22693
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#444

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21454419C2semidirect
4. #21454414C2semidirect
5. #21446605C2semidirect
6. #21446600C2semidirect
7. #21446054C2semidirect
8. #21446054C2semidirect
9. #21446047C2semidirect
10. #21446047C2semidirect
11. #21445987C2semidirect
12. #21445883C2semidirect
13. #21445878C2semidirect
14. #21445703C2semidirect
63. C61×#444C4semidirect
141. #131×C61C2×C4semidirect
142. #131×C61C2×C4semidirect
152. #108×C61C2×C4semidirect
153. #104×C61C2×C4semidirect
154. #104×C61C2×C4semidirect
155. #97×C61C2×C4semidirect
182. Q16×C61C4⋊C4semidirect
183. SD16×C61C4⋊C4semidirect
184. SD16×C61C4⋊C4semidirect
185. D8×C61C4⋊C4semidirect
351. #444Dic61semidirect
396. C61#22693semidirect
407. #131C2×Dic61semidirect
408. #131C2×Dic61semidirect
448. #108C2×Dic61semidirect
469. #104C2×Dic61semidirect
470. #104C2×Dic61semidirect
481. #97C2×Dic61semidirect
511. Q16C4⋊Dic61semidirect
512. SD16C4⋊Dic61semidirect
513. SD16C4⋊Dic61semidirect
514. D8C4⋊Dic61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214547032
#214546952
#214544192
#214544142
#214466052
#214466002
#214460542
#214460472
#214459872
#214458832
#214458782
#214457032
C2×C61×#4442
#2269361