Finite Group Databasebeta

← #29698518

Subgroups — #29698518

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×#16839885C211yesmaximal, characteristic
2#21454702C211yesmaximal, characteristic
3#21454693C211yesmaximal, characteristic
4#21454415C211yesmaximal, characteristic, complemented
5#21454410C211yesmaximal, characteristic, complemented
6#21446602C211yesmaximal, characteristic, complemented
7#21446597C211yesmaximal, characteristic, complemented
8#21446051C211yesmaximal, characteristic, complemented
9#21446051C211yesmaximal, characteristic, complemented
10#21446044C211yesmaximal, characteristic, complemented
11#21446044C211yesmaximal, characteristic, complemented
12#21445884C211yesmaximal, characteristic, complemented
13#21445879C211yesmaximal, characteristic, complemented
14#21445703C211yesmaximal, characteristic, complemented
15C61×#1847C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840243C2211yescharacteristic
20#16840243C2211yescharacteristic
21#16840243C2211yescharacteristic
22#16840243C2211yescharacteristic
23#16839885C418yescomplemented
24#16839873—21no—
25#16839868—21no—
26#16839861—21no—
27#16839861—21no—
28#16839856—21no—
29#16839849—21no—
30#16839849—21no—
31#16839844—21no—
32C2×#14519713C2211yescharacteristic
33C2×#14519713C2211yescharacteristic
34#16839803—21no—
35#16839803—21no—
36#16839802C2211yescharacteristic
37#16839802C2211yescharacteristic
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839801—21no—
41#16839801—21no—
42#16839800C2211yescharacteristic
43#16839799—21no—
44#16839797—21no—
45#107×D61—21no—
46#106×D61—21no—
47C2×#14519705C2211yescharacteristic
48C2×#14519705C2211yescharacteristic
49#16839761C2211yescharacteristic
50#16839761C2211yescharacteristic
51#16839761C2211yescharacteristic
52#16839761C2211yescharacteristic
53#16839717C2211yescharacteristic
54#16839717C2211yescharacteristic
55#16839531C2211yescharacteristic
56#16839531C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839531C2211yescharacteristic
59#16839522C2211yescharacteristic
60#16839522C2211yescharacteristic
61C2×#107×C61C2211yescharacteristic
62C2×#107×C61C2211yescharacteristic
63C2×#106×C61C2211yescharacteristic
64C2×#106×C61C2211yescharacteristic
65C2×#97×C61C2211yescharacteristic
66C61×#384C2211yescharacteristic
67C61×#382C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69C2×#11758222C2311yescharacteristic
70#14519799C2×C414yes—
71#14519799Q814yes—
72C2×Dic244D411yescharacteristic
73C2×#11758190D411yescharacteristic
74C2×#11758190D411yescharacteristic
75C2×D488D411yescharacteristic
76#14519783—22no—
77#14519783—22no—
78#14519783C2×C414yescomplemented
79#14519783C2×C414yescomplemented
80C2×#11758194C2311yescharacteristic
81C2×#11758194C2311yescharacteristic
82C2×#11758194C2311yescharacteristic
83C2×#11758194C2311yescharacteristic
84C2×#11758189—21no—
85C2×#11758189—21no—
86C2×C8×D61—21no—
87C2×C8×D61—21no—
88#14519742—21no—
89#14519742—21no—
90#14519742—21no—
91#14519742—21no—
92#14519739—21no—
93#14519739—21no—
94#14519739—21no—
95#14519739—21no—
96C4×C22⋊D61—41no—
97C4×C22⋊D61—41no—
98C4×C22⋊D61—41no—
99C4×C22⋊D61—41no—
100#14519736C2311yescharacteristic
101#14519736C2311yescharacteristic
102#14519735—21no—
103#14519735—21no—
104#14519735—21no—
105#14519735—21no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61—21no—
109C4⋊C4×D61—21no—
110#14519713—22no—
111#14519713C2×C414yescomplemented
112#14519711D411yescharacteristic
113#14519711D411yescharacteristic
114#14519710—21no—
115#14519710—21no—
116#14519710—21no—
117#14519710—21no—
118#14519709D411yescharacteristic
119#14519709D411yescharacteristic
120#14519708—21no—
121#14519708—21no—
122#14519708—21no—
123#14519708—21no—
124#14519705C2×C414yescomplemented
125#14519705C2×C414yescomplemented
126#14519700—21no—
127#14519700—21no—
128#14519700—21no—
129#14519700—21no—
130#14519697—21no—
131#14519697—21no—
132#14519697—21no—
133#14519697—21no—
134#14519687—21no—
135#14519687—21no—
136C8×Dic61—21no—
137C8×Dic61—21no—
138C22×C8×C61C2311yescharacteristic
139C22×C8×C61C2311yescharacteristic
140C2×C4⋊C4×C61—41no—
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C4⋊C4×C61C2311yescharacteristic
144C2×C4⋊C4×C61C2311yescharacteristic
145C2×C42×C61C2311yescharacteristic
146#107×C61—21no—
147#107×C61—21no—
148#107×C61—21no—
149#107×C61—21no—
150#106×C61—21no—
151#106×C61—21no—
152#106×C61—21no—
153#106×C61—21no—
154#97×C61C2×C414yescomplemented
155#95×C61—41no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—21no—
163C2×C22⋊D61—21no—
164#11758222—21no—
165#11758222—21no—
166#11758222—21no—
167#11758222—21no—
168#11758222C22×C412yes—
169#11758222C22×C412yes—
170#11758222C2×Q812yes—
171#11758222C2×Q812yes—
172C2×D244C2×D411yescharacteristic
173C2×D244C2×D411yescharacteristic
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—41no—
177C2×C4×D61—41no—
178C2×C4×D61—41no—
179C2×C4×D61—41no—
180C2×Dic122C2×D411yescharacteristic
181C2×Dic122C2×D411yescharacteristic
182C2×C8×C61C22×C412yes—
183C2×C8×C61C22×C412yes—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×Q812yes—
189C2×C8×C61#4112yes—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—82no—
199C42×C61—21no—
200C42×C61C22×C412yes—
201C22⋊Dic61—21no—
202C22⋊Dic61—21no—
203#11758198—41no—
204#11758198—41no—
205#11758198—41no—
206#11758198—41no—
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209C4⋊Dic61C2×D411yescharacteristic
210C4⋊Dic61C2×D411yescharacteristic
211#11758196—41no—
212#11758196—41no—
213#11758196—41no—
214#11758196—41no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—41no—
218C4×Dic61—41no—
219C4×Dic61—41no—
220C4×Dic61—41no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194—21no—
225#11758194—21no—
226#11758194—21no—
227#11758194C22×C412yes—
228#11758194C22×C412yes—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233Dic244C4⋊C414yescomplemented
234D488C4⋊C414yescomplemented
235#11758190C4⋊C414yescomplemented
236#11758190C4⋊C414yescomplemented
237#11758189—24no—
238#11758189—24no—
239C8×D61—24no—
240C8×D61—24no—
241C23×C61H(5,𝔽2)11yescharacteristic
242C22×D61—21no—
243C22×D61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—81no—
252C2×C4×C61—81no—
253C2×C4×C61—81no—
254C2×C4×C61—81no—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C22⋊D61—22no—
262C22⋊D61—22no—
263C22⋊D61—22no—
264C22⋊D61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271D244—21no—
272D244—21no—
273D244C2×C4⋊C412yes—
274D244C2×C4⋊C412yes—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—161no—
280C4×D61—161no—
281C4×D61—161no—
282C4×D61—161no—
283Dic122—21no—
284Dic122—21no—
285Dic122C2×C4⋊C412yes—
286Dic122C2×C4⋊C412yes—
287C8×C61C2×C4⋊C412yes—
288C8×C61C2×C4⋊C412yes—
289C8×C61C4×D412yes—
290C8×C61C4×D412yes—
291#92418—22no—
292#92418—22no—
293#92418—41no—
294#92418—41no—
295#22693—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61#52011yescharacteristic
299C22×C61D4211yescharacteristic
300C22×C61#55411yescharacteristic
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C4×C61—42no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C22×C4⋊C411yescharacteristic
313C4×C61C2×C4×D411yescharacteristic
314C4×C61#54711yescharacteristic
315C4×C61D4×Q811yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—41no—
319Dic61—41no—
320Dic61—41no—
321Dic61—41no—
322C2×#444—611no—
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#1847D6111yescharacteristic, complemented
334#1816—611no—
335#1815—611no—
336#1773—611no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61D4×C4⋊C411yescharacteristic
340C2×C61#328811yescharacteristic
341C2×C61#328911yescharacteristic
342D61—42no—
343D61—42no—
344C2×#135—611no—
345C2×#131—611no—
346C2×#131—611no—
347#520—611no—
348#520—611no—
349#520—611no—
350#520—611no—
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—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—1221no—
374C2×#107C2×D6111yescharacteristic
375C2×#107C2×D6111yescharacteristic
376C2×#106—1221no—
377C2×#106C2×D6111yescharacteristic
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×#97C2×D6111yescharacteristic
388#384C2×D6111yescharacteristic
389#382C2×D6111yescharacteristic
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×#41—611no—
399#135—614no—
400#135—614no—
401C2×Q16—611no—
402C2×SD16—611no—
403C2×SD16—611no—
404C2×D8—611no—
405#131—614no—
406#131—614no—
407#131—1222no—
408#131—1222no—
409C2×M16—611no—
410C2×M16—611no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16—1221no—
414C2×M16—1221no—
415C22×C8—1221no—
416C22×C8—1221no—
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—41no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447#108—614no—
448#108—1222no—
449#107—21no—
450#107—21no—
451#107—21no—
452#107—21no—
453#107—611no—
454#107—611no—
455#107—1221no—
456#107—1221no—
457#107—1221no—
458#107—1221no—
459#106—21no—
460#106—21no—
461#106—21no—
462#106—21no—
463#106—611no—
464#106—611no—
465#106—1221no—
466#106—1221no—
467#106—1221no—
468#106—1221no—
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—1221no—
480#97—1221no—
481#97C4×D6114yescomplemented
482C4×C8—1221no—
483C4×C8—1221no—
484#95—41no—
485#41—612no—
486#41—612no—
487#41—612no—
488#41—612no—
489#41—1221no—
490#41—1221no—
491#41—1221no—
492#41—1221no—
493C2×Q8—611no—
494C2×Q8—611no—
495C2×D4—611no—
496C2×D4—611no—
497C2×D4—1221no—
498C2×D4—1221no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—2441no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4—2441no—
510C22×C4C23×D6111yescharacteristic
511Q16—614no—
512SD16—614no—
513SD16—614no—
514D8—614no—
515M16—612no—
516M16—612no—
517M16—1221no—
518M16—1221no—
519M16—1224no—
520M16—1224no—
521M16—1221no—
522M16—1221no—
523M16—1221no—
524M16—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1224no—
528C2×C8—1224no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8C2×C4×D6112yes—
532C2×C8C2×C4×D6112yes—
533C2×C8D4×D6111yescharacteristic
534C2×C8D4×D6111yescharacteristic
535C2×C8D4×D6111yescharacteristic
536C2×C8D4×D6111yescharacteristic
537C2×C8Q8×D6112yes—
538C2×C8#1175822612yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—82no—
548C4⋊C4—611no—
549C4⋊C4—611no—
550C4⋊C4—611no—
551C4⋊C4—611no—
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—
569C42C2×C4×D6112yes—
570C23—1221no—
571C23—1221no—
572C23#1451982211yescharacteristic
573Q8—612no—
574Q8—612no—
575Q8—1221no—
576Q8—1221no—
577D4—612no—
578D4—612no—
579D4—1222no—
580D4—1222no—
581D4—1222no—
582D4—1222no—
583D4—1221no—
584D4—1221no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—9761no—
607C2×C4—9761no—
608C2×C4—9761no—
609C2×C4—9761no—
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C2×Q8×D6111yescharacteristic
614C2×C4C2×#1175822611yescharacteristic
615C2×C4#1451982511yescharacteristic
616C8—1222no—
617C8—1222no—
618C8—2441no—
619C8—2441no—
620C8C4⋊C4×D6112yes—
621C8C4⋊C4×D6112yes—
622C8#1451973912yes—
623C8#1451973912yes—
624C22—21no—
625C22—21no—
626C22—1222no—
627C22—1222no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22—2441no—
632C22#1684024311yescharacteristic
633C22#1684039711yescharacteristic
634C22#1684047311yescharacteristic
635C4—42no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4—1222no—
641C4—1222no—
642C4—2441no—
643C4—2441no—
644C4—2441no—
645C4—2441no—
646C4C2×C4⋊C4×D6111yescharacteristic
647C4C2×#1451973911yescharacteristic
648C4#1684041511yescharacteristic
649C4#1684042711yescharacteristic
650C2—21no—
651C2—21no—
652C2—2442no—
653C2—2442no—
654C2#2144871911yescharacteristic
655C2#2145758511yescharacteristic
656C2#2145759411yescharacteristic
subgroupG/H
centerC22#16840397
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC22×C61D42
prime radicalC2×#14519799C22
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#22693
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #21454415C2semidirect
5. #21454410C2semidirect
6. #21446602C2semidirect
7. #21446597C2semidirect
8. #21446051C2semidirect
9. #21446051C2semidirect
10. #21446044C2semidirect
11. #21446044C2semidirect
12. #21445884C2semidirect
13. #21445879C2semidirect
14. #21445703C2semidirect
15. C61×#1847C2semidirect
23. #16839885C4semidirect
78. #14519783C2×C4semidirect
79. #14519783C2×C4semidirect
111. #14519713C2×C4semidirect
124. #14519705C2×C4semidirect
125. #14519705C2×C4semidirect
154. #97×C61C2×C4semidirect
233. Dic244C4⋊C4semidirect
234. D488C4⋊C4semidirect
235. #11758190C4⋊C4semidirect
236. #11758190C4⋊C4semidirect
333. #1847D61semidirect
396. C61#22693semidirect
481. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398852
#214547022
#214546932
#214544152
#214544102
#214466022
#214465972
#214460512
#214460442
#214458842
#214458792
#214457032
C61×#18472
#2269361