Finite Group Databasebeta

← #29698517

Subgroups — #29698517

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
1C2×#16839882C211yesmaximal, characteristic
2#21454700C211yesmaximal, characteristic
3#21454692C211yesmaximal, characteristic
4#21454416C211yesmaximal, characteristic, complemented
5#21446601C211yesmaximal, characteristic, complemented
6#21446596C211yesmaximal, characteristic, complemented
7#21446052C211yesmaximal, characteristic, complemented
8#21446049C211yesmaximal, characteristic, complemented
9#21446045C211yesmaximal, characteristic, complemented
10#21446042C211yesmaximal, characteristic, complemented
11#21445984C211yesmaximal, characteristic, complemented
12#21445883C211yesmaximal, characteristic, complemented
13#21445879C211yesmaximal, characteristic, complemented
14#21445705C211yesmaximal, characteristic, complemented
15C61×#2912C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840242C2211yescharacteristic
19#16840242C2211yescharacteristic
20#16839882C418yescomplemented
21#16839872—21no—
22#16839867—21no—
23#16839862—21no—
24#16839860—21no—
25C4×#11758202—21no—
26C4×#11758201—21no—
27C4×#11758200—21no—
28C4×#11758199—21no—
29C2×#14519713C2211yescharacteristic
30C2×#14519712C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839805C2211yescharacteristic
33#16839801—21no—
34#16839801—21no—
35#16839799—21no—
36C2×#14519711C2211yescharacteristic
37C2×#14519710C2211yescharacteristic
38#107×D61—21no—
39C2×#14519708C2211yescharacteristic
40C2×#14519704C2211yescharacteristic
41C2×#14519703C2211yescharacteristic
42#16839763C2211yescharacteristic
43#16839763C2211yescharacteristic
44#16839761C2211yescharacteristic
45#16839761C2211yescharacteristic
46#16839718C2211yescharacteristic
47#16839718C2211yescharacteristic
48C2×#14519687C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839627C2211yescharacteristic
51#16839531C2211yescharacteristic
52#16839531C2211yescharacteristic
53#16839530C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839521C2211yescharacteristic
56#16839521C2211yescharacteristic
57C2×#131×C61C2211yescharacteristic
58C61×#520C2211yescharacteristic
59C61×#520C2211yescharacteristic
60C61×#428—21no—
61C61×#428—21no—
62C61×#428C2211yescharacteristic
63C61×#428C2211yescharacteristic
64C61×#427—21no—
65C61×#427C2211yescharacteristic
66C2×#106×C61—21no—
67C2×#106×C61C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69#14519797C2×C414yes—
70#14519797Q814yes—
71C2×#11758202D411yescharacteristic
72C2×#11758200D411yescharacteristic
73C2×#11758201D411yescharacteristic
74C2×#11758199D411yescharacteristic
75#14519783—22no—
76#14519783C2×C414yescomplemented
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758194C2311yescharacteristic
80C2×#11758189—21no—
81C22×#92418C2311yescharacteristic
82C2×C8×D61—21no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×D244—21no—
86C4×D244—21no—
87#14519737—41no—
88#14519737—41no—
89#14519735—21no—
90#14519735—21no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—41no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519713—22no—
98#14519712C2×C414yescomplemented
99#14519711—21no—
100#14519711—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519704C2×C414yescomplemented
110#14519703C2×C414yescomplemented
111#14519701—21no—
112#14519701—21no—
113#14519700—21no—
114#14519700—21no—
115#14519698—21no—
116#14519698—21no—
117#14519697—21no—
118#14519697—21no—
119#14519688—21no—
120#14519688—21no—
121#14519687C2×C414yescomplemented
122C4×#92418—21no—
123C4×#92418—21no—
124#14519684—41no—
125C2×#41×C61C2311yescharacteristic
126#131×C61—22no—
127#131×C61C2×C414yescomplemented
128C2×M16×C61—21no—
129C2×M16×C61C2311yescharacteristic
130C22×C8×C61—21no—
131C22×C8×C61C2311yescharacteristic
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—41no—
137C4×D4×C61—41no—
138#117×C61—21no—
139#117×C61—21no—
140#117×C61C2311yescharacteristic
141#117×C61C2311yescharacteristic
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61C2311yescharacteristic
145C2×C4⋊C4×C61C2311yescharacteristic
146#107×C61—21no—
147#107×C61—21no—
148#107×C61D411yescharacteristic
149#107×C61D411yescharacteristic
150#106×C61—21no—
151#106×C61—21no—
152#106×C61—21no—
153#106×C61—21no—
154#106×C61D411yescharacteristic
155#106×C61D411yescharacteristic
156#41×C61—21no—
157#41×C61—21no—
158#41×C61C22×C412yes—
159#41×C61C2×Q812yes—
160C2×Q8×C61C2×D411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—41no—
167C22×C4×C61—41no—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173#11758222—21no—
174#11758222—21no—
175#11758222C22×C412yes—
176#11758222C2×Q812yes—
177C2×D244C2×D411yescharacteristic
178C2×C4×D61—21no—
179C2×C4×D61—41no—
180C2×C4×D61—41no—
181C2×Dic122C2×D411yescharacteristic
182M16×C61—21no—
183M16×C61—24no—
184M16×C61C22×C412yes—
185C2×C8×C61—21no—
186C2×C8×C61—24no—
187C2×C8×C61C22×C412yes—
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D411yescharacteristic
190C2×C8×C61#4112yes—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—41no—
196C4⋊C4×C61—41no—
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D411yescharacteristic
201C22⋊C4×C61—21no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—41no—
204C22⋊C4×C61—41no—
205C42×C61—21no—
206C42×C61—21no—
207C42×C61—41no—
208C42×C61—41no—
209#11758202C4⋊C414yescomplemented
210#11758201C4⋊C414yescomplemented
211#11758200C4⋊C414yescomplemented
212#11758199C4⋊C414yescomplemented
213#11758198—41no—
214#11758198—41no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219#11758196—41no—
220#11758196—41no—
221#11758196—82no—
222C4×Dic61—21no—
223C4×Dic61—41no—
224C4×Dic61—41no—
225C4×Dic61C22×C412yes—
226#11758194—21no—
227#11758194—21no—
228#11758194—21no—
229#11758194—21no—
230#11758194—21no—
231#11758194C22×C412yes—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418C22×C412yes—
236C2×#92418C2×D411yescharacteristic
237C2×#92418C2×D411yescharacteristic
238C2×#92418C2×Q812yes—
239#11758189—24no—
240C8×D61—24no—
241C23×C61—21no—
242C23×C61H(5,𝔽2)11yescharacteristic
243C22×D61—21no—
244Q8×C61—21no—
245Q8×C61C2×C4⋊C412yes—
246D4×C61—22no—
247D4×C61—22no—
248D4×C61—21no—
249D4×C61C2×C4⋊C412yes—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—81no—
260C2×C4×C61—81no—
261C2×C4×C61—161no—
262C2×C4×C61—161no—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×Q811yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C22⋊D61—22no—
270C22⋊D61—22no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—81no—
278C2×Dic61—81no—
279D244—21no—
280D244C2×C4⋊C412yes—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—161no—
284C4×D61—161no—
285Dic122—21no—
286Dic122C2×C4⋊C412yes—
287C8×C61—22no—
288C8×C61C4×D412yes—
289C8×C61C4×D412yes—
290#92418—22no—
291#92418—41no—
292#92418—41no—
293#92418C2×C4⋊C412yes—
294#92418C2×C4⋊C412yes—
295#22693—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61#52011yescharacteristic
302C22×C61D4211yescharacteristic
303C22×C61#55411yescharacteristic
304C2×D61—22no—
305C2×D61—41no—
306C2×D61—41no—
307C4×C61—22no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C22×C4⋊C411yescharacteristic
315C4×C61C2×C4×D411yescharacteristic
316C4×C61#54711yescharacteristic
317C4×C61D4×Q811yescharacteristic
318Dic61—22no—
319Dic61—42no—
320Dic61—81no—
321Dic61—81no—
322C2×#444—611no—
323#2952—611no—
324#2951—611no—
325#2913—611no—
326#2912D6111yescharacteristic, complemented
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×C61D4×C4⋊C411yescharacteristic
341C2×C61#328811yescharacteristic
342C2×C61#328911yescharacteristic
343D61—42no—
344C2×#135—611no—
345C2×#131—611no—
346C2×#131C2×D6111yescharacteristic
347#520—611no—
348#520—611no—
349#520C2×D6111yescharacteristic
350#520C2×D6111yescharacteristic
351#444—618no—
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×#108—611no—
362#428—21no—
363#428—21no—
364#428—611no—
365#428—611no—
366#428—1221no—
367#428—1221no—
368#428C2×D6111yescharacteristic
369#428C2×D6111yescharacteristic
370#427—21no—
371#427—1221no—
372#427C2×D6111yescharacteristic
373C2×#107—611no—
374C2×#107—611no—
375C2×#107—1221no—
376C2×#106—21no—
377C2×#106—611no—
378C2×#106C2×D6111yescharacteristic
379C2×#104—611no—
380C2×#104—611no—
381#417—611no—
382#417—611no—
383#417—611no—
384#417—611no—
385#405—611no—
386#405—611no—
387C2×#97—611no—
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×#41—611no—
398C2×#41C22×D6111yescharacteristic
399#135—614no—
400#135—614no—
401C2×Q16—611no—
402C2×SD16—611no—
403C2×SD16—611no—
404C2×D8—611no—
405#131—22no—
406#131—614no—
407#131—1222no—
408#131C4×D6114yescomplemented
409C2×M16—21no—
410C2×M16—611no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16—1221no—
414C2×M16C22×D6111yescharacteristic
415C22×C8—21no—
416C22×C8—611no—
417C22×C8—1221no—
418C22×C8C22×D6111yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—41no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—2441no—
430C4×D4—2441no—
431#117—21no—
432#117—21no—
433#117—1221no—
434#117—1221no—
435#117C22×D6111yescharacteristic
436#117C22×D6111yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—2441no—
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C42—611no—
447#108—614no—
448#108—1222no—
449#107—21no—
450#107—21no—
451#107—1221no—
452#107—1221no—
453#107—1221no—
454#107—1221no—
455#107—1221no—
456#107—1221no—
457#107C22⋊D6111yescharacteristic
458#107C22⋊D6111yescharacteristic
459#106—21no—
460#106—21no—
461#106—21no—
462#106—21no—
463#106—1221no—
464#106—1221no—
465#106—1221no—
466#106—1221no—
467#106C22⋊D6111yescharacteristic
468#106C22⋊D6111yescharacteristic
469#104—614no—
470#104—614no—
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—614no—
480#97—1221no—
481#97—1221no—
482C4×C8—1221no—
483C4×C8—1221no—
484#95—2441no—
485#41—21no—
486#41—21no—
487#41—612no—
488#41—612no—
489#41—1221no—
490#41—1221no—
491#41C2×C4×D6112yes—
492#41Q8×D6112yes—
493C2×Q8—611no—
494C2×Q8D4×D6111yescharacteristic
495C2×D4—21no—
496C2×D4—611no—
497C2×D4—1221no—
498C2×D4D4×D6111yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—2441no—
509C22×C4—2441no—
510C22×C4C23×D6111yescharacteristic
511Q16—614no—
512SD16—614no—
513SD16—614no—
514D8—614no—
515M16—21no—
516M16—24no—
517M16—612no—
518M16—1221no—
519M16—1224no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16—1221no—
524M16C2×C4×D6112yes—
525C2×C8—21no—
526C2×C8—24no—
527C2×C8—612no—
528C2×C8—611no—
529C2×C8—611no—
530C2×C8—612no—
531C2×C8—1221no—
532C2×C8—1224no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8C2×C4×D6112yes—
536C2×C8#1175822212yes—
537C2×C8C2×C22⋊D6111yescharacteristic
538C2×C8C2×C22⋊D6111yescharacteristic
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—4882no—
550C4⋊C4—2441no—
551C4⋊C4—2441no—
552C4⋊C4C2×C22⋊D6111yescharacteristic
553C4⋊C4C2×C22⋊D6111yescharacteristic
554C4⋊C4C2×C22⋊D6111yescharacteristic
555C4⋊C4C2×C22⋊D6111yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—2441no—
561C22⋊C4—2441no—
562C42—21no—
563C42—21no—
564C42—41no—
565C42—41no—
566C42—612no—
567C42—1221no—
568C42—2441no—
569C42—2441no—
570C23—21no—
571C23—1221no—
572C23#1451982211yescharacteristic
573Q8—21no—
574Q8—612no—
575Q8—1221no—
576Q8C4⋊C4×D6112yes—
577D4—22no—
578D4—22no—
579D4—21no—
580D4—612no—
581D4—1222no—
582D4—1222no—
583D4—1221no—
584D4C4⋊C4×D6112yes—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—161no—
597C2×C4—161no—
598C2×C4—1221no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—9761no—
603C2×C4—9761no—
604C2×C4—4881no—
605C2×C4—4881no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C22×C22⋊D6111yescharacteristic
613C2×C4C2×Q8×D6111yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C2×C4#1451982511yescharacteristic
616C8—22no—
617C8—612no—
618C8—612no—
619C8—1222no—
620C8—2441no—
621C8—2441no—
622C8C4×C22⋊D6112yes—
623C8C4×C22⋊D6112yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—1222no—
630C22—2441no—
631C22—2441no—
632C22#1684024311yescharacteristic
633C22D4×C22⋊D6111yescharacteristic
634C22#1684046711yescharacteristic
635C4—22no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—1222no—
643C4—4881no—
644C4—4881no—
645C4—2442no—
646C4C2×C4⋊C4×D6111yescharacteristic
647C4C2×C4×C22⋊D6111yescharacteristic
648C4#1684041311yescharacteristic
649C4Q8×C22⋊D6111yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—2442no—
654C2C4⋊C4×C22⋊D6111yescharacteristic
655C2#2145758111yescharacteristic
656C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2912C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#2912D61
hypocenterC61#22693
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
4. #21454416C2semidirect
5. #21446601C2semidirect
6. #21446596C2semidirect
7. #21446052C2semidirect
8. #21446049C2semidirect
9. #21446045C2semidirect
10. #21446042C2semidirect
11. #21445984C2semidirect
12. #21445883C2semidirect
13. #21445879C2semidirect
14. #21445705C2semidirect
15. C61×#2912C2semidirect
20. #16839882C4semidirect
76. #14519783C2×C4semidirect
98. #14519712C2×C4semidirect
109. #14519704C2×C4semidirect
110. #14519703C2×C4semidirect
121. #14519687C2×C4semidirect
127. #131×C61C2×C4semidirect
209. #11758202C4⋊C4semidirect
210. #11758201C4⋊C4semidirect
211. #11758200C4⋊C4semidirect
212. #11758199C4⋊C4semidirect
326. #2912D61semidirect
396. C61#22693semidirect
408. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398822
#214547002
#214546922
#214544162
#214466012
#214465962
#214460522
#214460492
#214460452
#214460422
#214459842
#214458832
#214458792
#214457052
C61×#29122
#2269361