Finite Group Databasebeta

← #29696724

Subgroups — #29696724

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459397C211yesmaximal, characteristic
2#21454770C211yesmaximal, characteristic, complemented
3#21448296C211yesmaximal, characteristic, complemented
4#21446798C212yesmaximal, complemented
5#21446764C211yesmaximal, characteristic, complemented
6#21446761C212yesmaximal, complemented
7#21445838C212yesmaximal
8#21445836C211yesmaximal, characteristic
9#21445628C212yesmaximal
10C4×#14519705C211yesmaximal, characteristic
11C61×#1994C211yesmaximal, characteristic, complemented
12#16840492C411yescharacteristic, complemented
13#16840489C412yescomplemented
14#16840487C411yescharacteristic, complemented
15#16840473C411yescharacteristic, complemented
16#16840470—21no—
17#16840468C412yescomplemented
18#16840466—22no—
19#16840427C411yescharacteristic, complemented
20#16840425—21no—
21#16840316—21no—
22#16840316—22no—
23#16840315C2212yes—
24C4×#11758222C2211yescharacteristic
25#16840230C2212yes—
26C2×C4×Dic122C2211yescharacteristic
27C4×Q8×D61—21no—
28#16839891—22no—
29#16839889—21no—
30#16839886—21no—
31#16839785C2212yes—
32C2×#14519705C2211yescharacteristic
33C2×#14519705C2211yescharacteristic
34C2×#14519703C2212yes—
35C4×#11758194C2211yescharacteristic
36M16×Dic61C2212yes—
37#16839668—21no—
38#16839660—21no—
39#16839646—22no—
40C4×#11758196C2212yes—
41C4×C4⋊Dic61C2211yescharacteristic
42#16839608C2212yes—
43#16839523C2212yes—
44#16839522C2211yescharacteristic
45#16839522C2211yescharacteristic
46#16839521C2212yes—
47#16839521C2212yes—
48C61×#533C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C43×C61C2211yescharacteristic
51C61×#339C2212yes—
52C61×#339C2212yes—
53C2×#11758222C2311yescharacteristic
54C22×Dic122D411yescharacteristic
55C2×Q8×D61—21no—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58#14519776—22no—
59#14519776—21no—
60#14519776C2×C412yes—
61#14519776C2×C412yes—
62#14519775—21no—
63#14519775—22no—
64#14519768C2×C412yes—
65#14519768C2×C412yes—
66#14519767—22no—
67#14519767—22no—
68#14519767—22no—
69#14519757—21no—
70#14519757—22no—
71#14519755—22no—
72#14519755—22no—
73#14519755—22no—
74#14519754—22no—
75#14519754—22no—
76#14519753—21no—
77#14519753—22no—
78Q8×Dic61—21no—
79C4×Dic122—22no—
80C4×Dic122—22no—
81C4×Dic122C2×C411yescharacteristic
82C4×Dic122C2×C411yescharacteristic
83C4×Dic122C2×C412yes—
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122D411yescharacteristic
87C4×Dic122D411yescharacteristic
88C4×Dic122D412yes—
89#14519742—21no—
90#14519742—22no—
91#14519742C2×C412yes—
92#14519742C2×C412yes—
93C4×D244C2×C411yescharacteristic
94C4×D244C2×C411yescharacteristic
95#14519739C2×C412yes—
96#14519739C2×C412yes—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—21no—
99#14519737—22no—
100#14519737—22no—
101#14519736C2311yescharacteristic
102#14519734—21no—
103#14519734—22no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196D412yes—
108C4⋊C4×D61—21no—
109C4⋊C4×D61—22no—
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2312yes—
112C42×D61—21no—
113C42×D61—22no—
114#14519706—22no—
115#14519706—22no—
116#14519705—22no—
117#14519705Q814yescomplemented
118#14519703—24no—
119#14519688—21no—
120#14519687—22no—
121C4×#92418—21no—
122C8×Dic61—22no—
123#14519684—21no—
124#14519684—22no—
125#14519684—22no—
126#14519684—22no—
127C2×M16×C61C2312yes—
128C4⋊Q8×C61C2×C411yescharacteristic
129C4⋊Q8×C61C2×C411yescharacteristic
130#125×C61C2×C412yes—
131C22⋊Q8×C61—21no—
132C22⋊Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×Q8×C61—21no—
135#117×C61C2312yes—
136C2×C42×C61—21no—
137C2×C42×C61—21no—
138C2×C42×C61C2311yescharacteristic
139C2×C42×C61C2311yescharacteristic
140C2×C42×C61C2311yescharacteristic
141C4⋊C8×C61—21no—
142C4⋊C8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—21no—
151C22×Dic61—22no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154Q8×D61—44no—
155#11758222—21no—
156#11758222—21no—
157#11758222C2×Q812yes—
158#11758222#4112yes—
159C2×D244C22×C411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—42no—
168C2×Dic122C22⋊C411yescharacteristic
169C2×Dic122C22⋊C411yescharacteristic
170C2×Dic122C22⋊C412yes—
171C2×Dic122C22×C411yescharacteristic
172M16×C61—22no—
173M16×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C412yes—
180C4⋊C4×C61C22×C412yes—
181C22⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187C42×C61—42no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×Q812yescomplemented
193C22⋊Dic61—21no—
194#11758198—21no—
195#11758198—21no—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—42no—
201C4⋊Dic61C22⋊C411yescharacteristic
202C4⋊Dic61C22⋊C411yescharacteristic
203C4⋊Dic61C22⋊C412yes—
204C4⋊Dic61C22×C411yescharacteristic
205C4⋊Dic61C22×C411yescharacteristic
206#11758196—21no—
207#11758196—21no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—44no—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218#11758196C22⋊C412yes—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61—22no—
225C4×Dic61—22no—
226C4×Dic61C22×C412yes—
227C4×Dic61C22×C412yes—
228C4×Dic61C2×D412yes—
229C4×Dic61C2×D412yes—
230#11758194—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194C2×Q812yescomplemented
234C2×#92418—21no—
235C2×#92418—21no—
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238Q8×C61—42no—
239Q8×C61—81no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
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×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C22⋊D61—22no—
261C22⋊D61—22no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61C2×C22⋊C412yes—
272C2×Dic61C2×C22⋊C412yes—
273C2×Dic61C2×C22⋊C412yes—
274D244—21no—
275D244C4×Q812yes—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—44no—
279C4×D61—42no—
280C4×D61—82no—
281C4×D61—81no—
282Dic122—21no—
283Dic122—22no—
284Dic122—22no—
285Dic122—42no—
286Dic122—44no—
287Dic122—82no—
288Dic122C4×Q812yes—
289C8×C61—42no—
290#92418—22no—
291#92418—41no—
292#22529—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61D4×Q811yescharacteristic
298C2×D61—22no—
299C2×D61—41no—
300C2×D61—41no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61#42112yes—
309C4×C61C2×C4×Q811yescharacteristic
310C4×C61#51911yescharacteristic
311C4×C61#54811yescharacteristic
312C4×C61D4×Q811yescharacteristic
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—22no—
317Dic61—22no—
318Dic61—41no—
319Dic61—41no—
320Dic61—82no—
321Q8×#41—611no—
322Q8×M16—611no—
323#2295—611no—
324#1995—612no—
325#1994D6111yescharacteristic, complemented
326#1986—612no—
327#1986—611no—
328#1806—611no—
329#1806—612no—
330#1759—612no—
331C4×#104—611no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61Q8×C22⋊C411yescharacteristic
335C2×C61C2×#42111yescharacteristic
336C2×C61#290111yescharacteristic
337D61—42no—
338Q82—611no—
339#557—611no—
340#557—612no—
341#554—611no—
342#554—612no—
343#554—1222no—
344#554—1221no—
345D4×Q8—611no—
346D4×Q8—1221no—
347#533—612no—
348#533—1222no—
349#533—1221no—
350#533C2×D6111yescharacteristic
351C4×#41—612no—
352C4×#41—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8—1221no—
355#446—1221no—
356#446—1222no—
357C8×Q8—1221no—
358#423—612no—
359#423C2×D6111yescharacteristic
360C2×#104—611no—
361C2×#104—612no—
362C2×#104—611no—
363C4×M16—612no—
364C4×M16—611no—
365#391—1221no—
366#389—1221no—
367#387—1222no—
368C4×C4⋊C4—612no—
369C4×C4⋊C4—611no—
370#376—612no—
371C43C2×D6111yescharacteristic
372#339—612no—
373#339—612no—
374#339—612no—
375#339—611no—
376#339—611no—
377#339C2×D6112yes—
378#339C2×D6112yes—
379C61#2252911yescharacteristic, complemented, Sylow
380C2×#41—611no—
381C22×Q8—611no—
382C22×Q8—1221no—
383C2×M16—611no—
384C2×M16—611no—
385C2×M16C22×D6112yes—
386C4⋊Q8—612no—
387C4⋊Q8—612no—
388C4⋊Q8—1221no—
389C4⋊Q8—1222no—
390C4⋊Q8—1222no—
391C4⋊Q8—1221no—
392C4⋊Q8C4×D6111yescharacteristic
393C4⋊Q8C4×D6111yescharacteristic
394#125—612no—
395#125—612no—
396#125—1222no—
397#125—1222no—
398#125—1222no—
399#125C4×D6112yes—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8—1222no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—611no—
414C4×Q8—611no—
415C4×Q8—612no—
416C4×Q8—611no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—612no—
420C4×Q8—612no—
421C4×Q8—611no—
422C4×Q8—611no—
423C4×Q8—1222no—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×D4—612no—
429C4×D4—612no—
430C4×D4—611no—
431C4×D4—611no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4—1221no—
436#117—611no—
437#117—1222no—
438#117—1221no—
439#117C22×D6112yes—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—612no—
449C2×C42—612no—
450C2×C42—1222no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453C2×C42C22×D6111yescharacteristic
454C2×C42C22×D6111yescharacteristic
455C4⋊C8—21no—
456C4⋊C8—21no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459#104—614no—
460#104—1222no—
461#104—1224no—
462#97—1222no—
463#97—1221no—
464C4×C8—1222no—
465C4×C8—1221no—
466#95—1222no—
467#95—1222no—
468#95—1222no—
469#95—1221no—
470#41—612no—
471#41—612no—
472#41—1221no—
473#41—1221no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—611no—
477C2×Q8—611no—
478C2×Q8—612no—
479C2×Q8—611no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—2442no—
483C2×Q8—2444no—
484C2×D4—611no—
485C2×D4—1221no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1222no—
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499M16—22no—
500M16—22no—
501M16—612no—
502M16—1221no—
503M16—1221no—
504M16—1221no—
505C2×C8—22no—
506C2×C8—1221no—
507C2×C8—1221no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—612no—
513C4⋊C4—611no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—611no—
519C4⋊C4—611no—
520C4⋊C4—611no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2444no—
532C4⋊C4—2442no—
533C4⋊C4C2×C4×D6112yes—
534C4⋊C4C2×C4×D6112yes—
535C22⋊C4—22no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—42no—
546C42—42no—
547C42—612no—
548C42—612no—
549C42—612no—
550C42—612no—
551C42—1222no—
552C42—1222no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—1222no—
557C42—1222no—
558C42C2×C4×D6111yescharacteristic
559C42C2×C4×D6111yescharacteristic
560C42D4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C42Q8×D6112yescomplemented
563C23—1221no—
564C23C2×D4×D6111yescharacteristic
565Q8—42no—
566Q8—81no—
567Q8—612no—
568Q8—1221no—
569Q8—1222no—
570Q8—1222no—
571Q8—4882no—
572Q8—2442no—
573Q8—2444no—
574D4—612no—
575D4—1222no—
576D4—1222no—
577D4—1221no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1221no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—4881no—
598C2×C4—4882no—
599C2×C4—2444no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4C22⋊C4×D6111yescharacteristic
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×Q8×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4#1451982411yescharacteristic
615C2×C4#1451982511yescharacteristic
616C8—42no—
617C8—1222no—
618C8—2441no—
619C22—21no—
620C22—21no—
621C22—1222no—
622C22—2441no—
623C22—2441no—
624C22C2×C22⋊C4×D6111yescharacteristic
625C22#1684042211yescharacteristic
626C22#1684042511yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—1222no—
635C4—1222no—
636C4—1222no—
637C4—1222no—
638C4—1222no—
639C4—4882no—
640C4—2441no—
641C4—2441no—
642C4#1683977712yes—
643C4C4×Q8×D6111yescharacteristic
644C4#1684023811yescharacteristic
645C4#1684042011yescharacteristic
646C4#1684042711yescharacteristic
647C2—21no—
648C2—21no—
649C2—2442no—
650C2#2144865311yescharacteristic
651C2C2×#1683977711yescharacteristic
652C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1994C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1994D61
hypocenterC61#22529
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
2. #21454770C2semidirect
3. #21448296C2semidirect
4. #21446798C2semidirect
5. #21446764C2semidirect
6. #21446761C2semidirect
11. C61×#1994C2semidirect
12. #16840492C4semidirect
13. #16840489C4semidirect
14. #16840487C4semidirect
15. #16840473C4semidirect
17. #16840468C4semidirect
19. #16840427C4semidirect
117. #14519705Q8semidirect
192. C42×C61C2×Q8semidirect
233. #11758194C2×Q8semidirect
325. #1994D61semidirect
379. C61#22529semidirect
562. C42Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#214593972
#214547702
#214482962
#214467982
#214467642
#214467612
#214458382
#214458362
#214456282
C4×#145197052
C61×#19942
#2252961