Finite Group Databasebeta

← #29696915

Subgroups — #29696915

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459414C211yesmaximal, characteristic
2#21448297C211yesmaximal, characteristic, complemented
3#21446738C211yesmaximal, characteristic, complemented
4#21446738C212yesmaximal, complemented
5#21446650C212yesmaximal, complemented
6#21446646C211yesmaximal, characteristic, complemented
7#21445836C212yesmaximal, complemented
8#21445836C211yesmaximal, characteristic, complemented
9#21445632C212yesmaximal, complemented
10C4×#14519703C211yesmaximal, characteristic, complemented
11Q8×M16×C61C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic, complemented
13#16840497C412yescomplemented
14#16840491C411yescharacteristic, complemented
15#16840485C411yescharacteristic, complemented
16#16840485C412yescomplemented
17#16840460—22no—
18#16840460—21no—
19#16840434C411yescharacteristic, complemented
20#16840423—21no—
21#16840312C2211yescharacteristic
22#16840312C2212yes—
23#16840299—21no—
24#16840299—22no—
25C4×#11758226—21no—
26C4×#11758222C2212yes—
27C4×#11758222C2211yescharacteristic
28C2×#14519703C2212yes—
29C2×#14519703C2212yes—
30C2×#14519703C2211yescharacteristic
31C2×#14519703C2211yescharacteristic
32C2×#14519703C2212yes—
33C2×#14519703C2212yes—
34C2×#14519703C2212yes—
35#16839669—21no—
36#16839661—21no—
37#16839648—22no—
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2212yes—
40#16839608C2212yes—
41C42×Dic61C2211yescharacteristic
42#16839523C2211yescharacteristic
43#16839523C2212yes—
44#16839523C2211yescharacteristic
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#446—22no—
48C8×Q8×C61—21no—
49C61×#423C2211yescharacteristic
50C61×#423C2212yes—
51C4×M16×C61C2212yes—
52C4×M16×C61C2211yescharacteristic
53C2×#11758226—21no—
54C2×#11758222C2312yes—
55C2×#11758222C2311yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519777C2×C412yes—
58#14519774C2×C411yescharacteristic
59#14519774C2×C412yes—
60#14519772—22no—
61#14519772—22no—
62#14519772—22no—
63#14519768—22no—
64#14519768—21no—
65#14519766—21no—
66#14519766—22no—
67#14519765C2×C412yes—
68#14519765C2×C412yes—
69#14519765C2×C412yes—
70#14519762—22no—
71#14519762—22no—
72#14519762—22no—
73#14519753—21no—
74#14519753—22no—
75#14519752—22no—
76#14519752—21no—
77#14519751—21no—
78#14519751—22no—
79#14519751—21no—
80#14519751—22no—
81Q8×Dic61—21no—
82C4×Dic122C2×C412yes—
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C412yes—
85C4×Dic122C2×C411yescharacteristic
86C4×D244—21no—
87C4×D244—22no—
88C4×D244—22no—
89C4×D244—21no—
90C4×D244C2×C412yes—
91C4×D244C2×C411yescharacteristic
92C4×D244C2×C412yes—
93C4×D244C2×C411yescharacteristic
94#14519739—22no—
95#14519739—21no—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—22no—
100#14519736C2311yescharacteristic
101#14519735—22no—
102#14519735—21no—
103#14519734—22no—
104#14519734—21no—
105C2×C4⋊Dic61—22no—
106C2×C4⋊Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—21no—
113C42×D61—22no—
114#14519703—24no—
115#14519703—22no—
116#14519703—24no—
117#14519703—22no—
118#14519703—42no—
119#14519703—42no—
120#14519703—42no—
121#14519684—22no—
122#14519684—22no—
123#14519684—21no—
124#14519684—22no—
125C22×Q8×C61D411yescharacteristic
126C2×M16×C61C2312yes—
127C2×M16×C61C2311yescharacteristic
128C2×M16×C61C2311yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×Q8×C61C2×C412yes—
131C4×Q8×C61C2×C411yescharacteristic
132C4×Q8×C61D411yescharacteristic
133C4×Q8×C61D411yescharacteristic
134C4×Q8×C61D412yes—
135C2×C4⋊C4×C61D412yes—
136C2×C4⋊C4×C61D411yescharacteristic
137C2×C42×C61C2311yescharacteristic
138C2×C42×C61C2312yes—
139C4⋊C8×C61—21no—
140C4⋊C8×C61—22no—
141C4⋊C8×C61—22no—
142C4⋊C8×C61—21no—
143#97×C61—22no—
144#97×C61—21no—
145C4×C8×C61—21no—
146C4×C8×C61—22no—
147C2×Q8×C61—42no—
148C2×Q8×C61C22⋊C411yescharacteristic
149C2×Q8×C61C22⋊C411yescharacteristic
150C2×Q8×C61C22⋊C412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—22no—
161#11758226—44no—
162#11758222—22no—
163#11758222—21no—
164#11758222—22no—
165#11758222—21no—
166#11758222—22no—
167#11758222—21no—
168#11758222—22no—
169#11758222—21no—
170C2×D244—22no—
171C2×D244—21no—
172C2×D244—22no—
173C2×D244—21no—
174C2×D244C22×C412yes—
175C2×D244C22×C411yescharacteristic
176C2×C4×D61—22no—
177C2×C4×D61—21no—
178C2×C4×D61—22no—
179C2×C4×D61—21no—
180C2×C4×D61—21no—
181C2×C4×D61—22no—
182C2×Dic122C22×C412yes—
183C2×Dic122C22×C411yescharacteristic
184M16×C61—21no—
185M16×C61—22no—
186M16×C61—21no—
187M16×C61—22no—
188M16×C61—21no—
189M16×C61#4112yescomplemented
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—22no—
193C4⋊C4×C61C22⋊C412yes—
194C4⋊C4×C61C22⋊C411yescharacteristic
195C4⋊C4×C61C22⋊C412yes—
196C4⋊C4×C61C22⋊C411yescharacteristic
197C4⋊C4×C61C22⋊C412yes—
198C4⋊C4×C61C22⋊C412yes—
199C4⋊C4×C61C22⋊C412yes—
200C42×C61C22×C411yescharacteristic
201C42×C61C22×C412yes—
202C42×C61C22×C412yes—
203C42×C61C22×C411yescharacteristic
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D412yes—
206C42×C61C2×D412yes—
207C42×C61C2×D411yescharacteristic
208C22⋊Dic61—21no—
209#11758198—22no—
210#11758198—21no—
211#11758198—21no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—22no—
218#11758198—22no—
219C4⋊Dic61—22no—
220C4⋊Dic61—21no—
221C4⋊Dic61—22no—
222C4⋊Dic61—21no—
223C4⋊Dic61—44no—
224C4⋊Dic61—42no—
225C4⋊Dic61C22×C411yescharacteristic
226C4⋊Dic61C22×C411yescharacteristic
227#11758196—21no—
228#11758196—22no—
229#11758196—22no—
230#11758196—21no—
231C4×Dic61—21no—
232C4×Dic61—22no—
233C4×Dic61—22no—
234C4×Dic61—21no—
235C4×Dic61—21no—
236C4×Dic61—21no—
237C4×Dic61—21no—
238C4×Dic61—22no—
239C4×Dic61—21no—
240C4×Dic61—22no—
241C4×Dic61—42no—
242C4×Dic61—42no—
243C4×Dic61#4112yescomplemented
244C23×C61C22×D411yescharacteristic
245C22×D61—22no—
246C22×D61—21no—
247Q8×C61—22no—
248Q8×C61—22no—
249Q8×C61—42no—
250C2×C4×C61—22no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C412yes—
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C412yes—
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C22⋊D61—42no—
269C22⋊D61—41no—
270C22⋊D61—42no—
271C22⋊D61—41no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285D244—22no—
286D244—21no—
287D244—22no—
288D244—21no—
289D244—44no—
290D244—42no—
291D244—81no—
292D244—82no—
293C4×D61—44no—
294C4×D61—42no—
295C4×D61—41no—
296C4×D61—42no—
297C4×D61—41no—
298C4×D61—42no—
299C4×D61—81no—
300C4×D61—82no—
301Dic122—22no—
302Dic122—21no—
303Dic122—22no—
304Dic122—21no—
305C8×C61—22no—
306C8×C61—41no—
307C8×C61—42no—
308#22551—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61C22×C22⋊C411yescharacteristic
312C22×C61#54811yescharacteristic
313C22×C61#55011yescharacteristic
314C2×D61—42no—
315C2×D61—41no—
316C2×D61—42no—
317C2×D61—41no—
318C2×D61—42no—
319C2×D61—41no—
320C4×C61—22no—
321C4×C61—21no—
322C4×C61—22no—
323C4×C61—21no—
324C4×C61—22no—
325C4×C61—21no—
326C4×C61—22no—
327C4×C61#42112yes—
328C4×C61C4×#4111yescharacteristic
329C4×C61#52011yescharacteristic
330C4×C61#54811yescharacteristic
331C4×C61#55011yescharacteristic
332Dic61—42no—
333Dic61—42no—
334Dic61—42no—
335Dic61—41no—
336Dic61—41no—
337Dic61—41no—
338Dic61—82no—
339#3480—611no—
340Q8×M16D6111yescharacteristic, complemented
341#2295—611no—
342#1980—611no—
343#1980—612no—
344#1963—612no—
345#1962—611no—
346#1806—611no—
347#1806—612no—
348#1759—612no—
349C4×#104—611no—
350C2×C61—21no—
351C2×C61—21no—
352C2×C61#234111yescharacteristic
353C2×C61C2×#42111yescharacteristic
354C2×C61#290111yescharacteristic
355D61—82no—
356D61—81no—
357#559—612no—
358#559—611no—
359#557—611no—
360#556—611no—
361#556—612no—
362#553—1222no—
363#553—1221no—
364#550—611no—
365#548—1221no—
366#532—611no—
367#532—612no—
368#529—1221no—
369#529—1222no—
370C4×#41—611no—
371C4×#41—612no—
372C4×#41—1221no—
373C2×C4×Q8C2×D6111yescharacteristic
374#446—21no—
375#446—22no—
376C8×Q8—21no—
377#423C2×D6111yescharacteristic
378#423C2×D6112yes—
379C2×#104—612no—
380C2×#104—611no—
381C2×#104—611no—
382C2×#104—612no—
383C2×#104—612no—
384C2×#104—612no—
385C2×#104—612no—
386C4×M16C2×D6111yescharacteristic
387C4×M16C2×D6112yes—
388#391—1221no—
389#389—1221no—
390#387—1222no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—612no—
393#376—612no—
394C43—611no—
395#339—611no—
396#339—612no—
397#339—611no—
398C61#2255111yescharacteristic, complemented, Sylow
399C2×#41—611no—
400C2×#41—612no—
401C2×#41—1221no—
402C22×Q8D24411yescharacteristic
403C2×M16C22×D6111yescharacteristic
404C2×M16C22×D6112yes—
405C2×M16C22×D6111yescharacteristic
406C4⋊Q8—611no—
407C4⋊Q8—612no—
408#127—611no—
409#127—612no—
410#126—1222no—
411#126—1222no—
412#126—1222no—
413#125—1222no—
414#125—1221no—
415#124—612no—
416#124—612no—
417#124—612no—
418#124—1221no—
419#124—1222no—
420#123—1222no—
421#123—1222no—
422#123—1222no—
423C22⋊Q8—1221no—
424C22⋊Q8—1222no—
425#121—1221no—
426#121—1222no—
427#121—1221no—
428#121—1222no—
429#121—1222no—
430#121—1221no—
431C4×Q8—611no—
432C4×Q8—612no—
433C4×Q8—611no—
434C4×Q8—612no—
435C4×Q8—1221no—
436C4×Q8C4×D6111yescharacteristic
437C4×Q8C4×D6111yescharacteristic
438C4×Q8C4×D6112yes—
439C4×Q8C22⋊D6111yescharacteristic
440C4×Q8C22⋊D6111yescharacteristic
441C4×Q8C22⋊D6112yes—
442C4×D4—611no—
443C4×D4—612no—
444C4×D4—611no—
445C4×D4—612no—
446C4×D4—1221no—
447C4×D4—1222no—
448C4×D4—1221no—
449C4×D4—1222no—
450C4×D4—1222no—
451C4×D4—1221no—
452C4×D4—1221no—
453C4×D4—1222no—
454C4×D4—1222no—
455C4×D4—1221no—
456#117—611no—
457#117—1222no—
458#117—1221no—
459#117—1222no—
460#117—1221no—
461C2×C4⋊C4—1222no—
462C2×C4⋊C4—1221no—
463C2×C4⋊C4D24412yes—
464C2×C4⋊C4D24411yescharacteristic
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—612no—
468C2×C42—1221no—
469C2×C42—1222no—
470C2×C42—1221no—
471C2×C42—1221no—
472C2×C42C22×D6111yescharacteristic
473C2×C42C22×D6112yes—
474C4⋊C8—21no—
475C4⋊C8—22no—
476C4⋊C8—22no—
477C4⋊C8—21no—
478#104—1224no—
479#104—1222no—
480#104—1222no—
481#104—1224no—
482#104—2442no—
483#104—2442no—
484#104—2442no—
485#97—22no—
486#97—21no—
487C4×C8—21no—
488C4×C8—22no—
489#95—1222no—
490#95—1222no—
491#95—1221no—
492#95—1222no—
493#41—1222no—
494#41—1221no—
495#41—1221no—
496#41—1222no—
497#41—1222no—
498#41—1221no—
499#41—1221no—
500#41—1222no—
501#41—2444no—
502C2×Q8—42no—
503C2×Q8—611no—
504C2×Q8—612no—
505C2×Q8#1175819811yescharacteristic
506C2×Q8#1175819811yescharacteristic
507C2×Q8#1175819812yes—
508C2×D4—611no—
509C2×D4—612no—
510C2×D4—1221no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1221no—
514C2×D4—1222no—
515C2×D4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1222no—
520C22×C4—1221no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1221no—
524C22×C4—1222no—
525C22×C4C2×D24411yescharacteristic
526C22×C4C2×D24412yes—
527C22×C4C2×D24411yescharacteristic
528C22×C4C2×D24412yes—
529C22×C4C23×D6111yescharacteristic
530M16—21no—
531M16—22no—
532M16—21no—
533M16—22no—
534M16—21no—
535M16#1175822612yescomplemented
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—22no—
539C4⋊C4—611no—
540C4⋊C4—611no—
541C4⋊C4—1221no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1221no—
545C4⋊C4—1222no—
546C4⋊C4—1221no—
547C4⋊C4—1222no—
548C4⋊C4—1221no—
549C4⋊C4—2442no—
550C4⋊C4—2444no—
551C4⋊C4#1175819812yes—
552C4⋊C4#1175819811yescharacteristic
553C4⋊C4#1175819812yes—
554C4⋊C4#1175819811yescharacteristic
555C4⋊C4#1175819812yes—
556C4⋊C4#1175819812yes—
557C4⋊C4#1175819812yes—
558C22⋊C4—1221no—
559C22⋊C4—1222no—
560C22⋊C4—1221no—
561C22⋊C4—1221no—
562C22⋊C4—1222no—
563C22⋊C4—1222no—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C42—612no—
570C42—1221no—
571C42—1221no—
572C42—1221no—
573C42—1222no—
574C42—1222no—
575C42—1221no—
576C42—1222no—
577C42—1221no—
578C42—1222no—
579C42—1221no—
580C42—2442no—
581C42—2442no—
582C42C2×C4×D6111yescharacteristic
583C42C2×C4×D6112yes—
584C42C2×C4×D6112yes—
585C42C2×C4×D6111yescharacteristic
586C42C2×C22⋊D6111yescharacteristic
587C42C2×C22⋊D6112yes—
588C42C2×C22⋊D6112yes—
589C42C2×C22⋊D6111yescharacteristic
590C23—1222no—
591C23—1221no—
592C23C22×D24411yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—42no—
596Q8—1221no—
597Q8—1222no—
598Q8—1221no—
599Q8—1222no—
600D4—1221no—
601D4—1222no—
602D4—1221no—
603D4—1222no—
604D4—4881no—
605D4—4882no—
606D4—2444no—
607D4—2441no—
608D4—2442no—
609D4—2441no—
610D4—2442no—
611D4—2442no—
612C2×C4—22no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1222no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—4881no—
623C2×C4—4882no—
624C2×C4—2444no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2442no—
633C2×C4—2441no—
634C2×C4—2442no—
635C2×C4—2441no—
636C2×C4—2442no—
637C2×C4C2×#1175819811yescharacteristic
638C2×C4C2×#1175819812yes—
639C2×C4C2×#1175819812yes—
640C2×C4C2×#1175819811yescharacteristic
641C2×C4C2×#1175819812yes—
642C2×C4C2×#1175819812yes—
643C2×C4C2×#1175819811yescharacteristic
644C2×C4C2×#1175819811yescharacteristic
645C2×C4C22×C4×D6111yescharacteristic
646C2×C4C22×C22⋊D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4C2×#1175822611yescharacteristic
649C2×C4#1451982311yescharacteristic
650C2×C4#1451982511yescharacteristic
651C8—22no—
652C8—41no—
653C8—42no—
654C22—21no—
655C22—21no—
656C22—2442no—
657C22—2441no—
658C22—2442no—
659C22—2441no—
660C22—2442no—
661C22—2441no—
662C22C22×#1175819811yescharacteristic
663C22#1684042411yescharacteristic
664C22#1684043411yescharacteristic
665C4—21no—
666C4—22no—
667C4—22no—
668C4—21no—
669C4—22no—
670C4—21no—
671C4—22no—
672C4—4882no—
673C4—2442no—
674C4—2442no—
675C4—2441no—
676C4—2441no—
677C4—2442no—
678C4—2441no—
679C4#1683977812yes—
680C4C4×#1175822611yescharacteristic
681C4#1684024411yescharacteristic
682C4#1684042311yescharacteristic
683C4#1684043111yescharacteristic
684C2—21no—
685C2—21no—
686C2—4882no—
687C2—4881no—
688C2#2144865711yescharacteristic
689C2C2×#1683977811yescharacteristic
690C2#2145436411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingQ8×M16×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459414C2
cosocleC22×C4×C61C24
hypercenterQ8×M16D61
hypocenterC61#22551
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #21448297C2semidirect
3. #21446738C2semidirect
4. #21446738C2semidirect
5. #21446650C2semidirect
6. #21446646C2semidirect
7. #21445836C2semidirect
8. #21445836C2semidirect
9. #21445632C2semidirect
10. C4×#14519703C2semidirect
11. Q8×M16×C61C2semidirect
12. #16840497C4semidirect
13. #16840497C4semidirect
14. #16840491C4semidirect
15. #16840485C4semidirect
16. #16840485C4semidirect
19. #16840434C4semidirect
189. M16×C61#41semidirect
243. C4×Dic61#41semidirect
340. Q8×M16D61semidirect
398. C61#22551semidirect
535. M16#11758226semidirect

Maximal subgroups

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

subgroupindex
#214594142
#214482972
#214467382
#214466502
#214466462
#214458362
#214456322
C4×#145197032
Q8×M16×C612
#2255161