Finite Group Databasebeta

← #29696917

Subgroups — #29696917

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#21459412C211yesmaximal, characteristic
2M16×Dic122C211yesmaximal, characteristic, complemented
3#21448296C211yesmaximal, characteristic, complemented
4#21446738C211yesmaximal, characteristic, complemented
5#21446736C212yesmaximal, complemented
6#21446649C212yesmaximal, complemented
7#21446647C211yesmaximal, characteristic, complemented
8#21445839C211yesmaximal, characteristic, complemented
9#21445838C212yesmaximal, complemented
10#21445628C212yesmaximal, complemented
11C4×#104×C61C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic, complemented
13#16840495C412yescomplemented
14#16840490C411yescharacteristic, complemented
15#16840484C412yescomplemented
16#16840482C411yescharacteristic, complemented
17#16840461—22no—
18#16840458—21no—
19#16840433C411yescharacteristic, complemented
20#16840421—21no—
21#16840312C2211yescharacteristic
22#16840308C2212yes—
23#16840296—21no—
24#16840290—22no—
25C4×#11758222—21no—
26#16840230C2212yes—
27C2×C4×Dic122C2211yescharacteristic
28#16839892—21no—
29#16839890—22no—
30C8×Dic122—21no—
31#16839786C2211yescharacteristic
32#16839785C2212yes—
33C2×#14519705C2212yes—
34C2×#14519705C2212yes—
35C2×#14519704C2212yes—
36C2×#14519703C2212yes—
37C2×#14519703C2212yes—
38M16×Dic61C2212yes—
39#16839668—21no—
40#16839660—21no—
41#16839646—22no—
42C4×#11758196C2212yes—
43C4×C4⋊Dic61C2211yescharacteristic
44#16839608C2212yes—
45#16839521C2212yes—
46C4×#41×C61C2211yescharacteristic
47C2×#104×C61C2211yescharacteristic
48C2×#104×C61C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic
50C43×C61C2211yescharacteristic
51C61×#339C2211yescharacteristic
52C61×#339C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758222C2312yes—
55C22×Dic122D411yescharacteristic
56C2×#11758194C2312yes—
57#14519777C2×C411yescharacteristic
58#14519775C2×C412yes—
59#14519774C2×C411yescharacteristic
60#14519773C2×C412yes—
61#14519772—22no—
62#14519771—22no—
63#14519770—22no—
64#14519769—21no—
65#14519767—22no—
66#14519766C2×C412yes—
67#14519765—21no—
68#14519765C2×C412yes—
69#14519764C2×C412yes—
70#14519763—22no—
71#14519762—22no—
72#14519760—22no—
73#14519759—22no—
74#14519756—21no—
75#14519755—22no—
76#14519751—21no—
77#14519751—21no—
78#14519751—21no—
79#14519750—22no—
80#14519749—22no—
81#14519749—22no—
82C4×Dic122—21no—
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C412yes—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122D411yescharacteristic
87C4×Dic122D412yes—
88C4×Dic122D411yescharacteristic
89#14519742C2×C412yes—
90#14519742C2×C412yes—
91C4×D244—21no—
92C4×D244—21no—
93C4×D244—21no—
94#14519739C2×C412yes—
95#14519739C2×C412yes—
96C4×C22⋊D61—22no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99#14519737—22no—
100#14519737—22no—
101#14519736—21no—
102#14519734—22no—
103#14519734—22no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196D412yes—
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2312yes—
110C42×D61—22no—
111C4⋊#92418—21no—
112C4⋊#92418—21no—
113#14519706—22no—
114#14519706—22no—
115#14519705—24no—
116#14519705—42no—
117#14519704—42no—
118#14519703—24no—
119#14519703—42no—
120#14519687—22no—
121C8×Dic61—22no—
122#14519684—21no—
123#14519684—22no—
124#14519684—22no—
125#14519684—22no—
126C2×#41×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C2×M16×C61C2311yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×Q8×C61C2×C411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61C2×C411yescharacteristic
134C4×D4×C61C2×C411yescharacteristic
135#117×C61—21no—
136#117×C61C2311yescharacteristic
137C2×C42×C61—21no—
138C2×C42×C61—21no—
139C2×C42×C61—21no—
140C2×C42×C61C2311yescharacteristic
141C2×C42×C61C2311yescharacteristic
142C2×C42×C61C2311yescharacteristic
143#104×C61—22no—
144#104×C61—22no—
145#97×C61—21no—
146C4×C8×C61—21no—
147#41×C61—21no—
148#41×C61—21no—
149#41×C61—21no—
150#41×C61—21no—
151C2×Q8×C61C22×C411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61C22×C411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C22×Dic61—22no—
166C22×Dic61C2×D412yes—
167C22×Dic61C2×D412yes—
168#11758222—22no—
169#11758222—22no—
170#11758222—22no—
171#11758222—22no—
172#11758222—44no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244C22×C412yes—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×Dic122—42no—
180C2×Dic122C22⋊C411yescharacteristic
181C2×Dic122C22⋊C411yescharacteristic
182C2×Dic122C22⋊C412yes—
183C2×Dic122C22×C412yes—
184M16×C61—21no—
185M16×C61—21no—
186M16×C61—21no—
187M16×C61#4112yescomplemented
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22×C411yescharacteristic
193C4⋊C4×C61C22×C411yescharacteristic
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—42no—
204C42×C61—42no—
205C42×C61C22×C411yescharacteristic
206C42×C61C22×C411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61#4112yescomplemented
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—22no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—42no—
221C4⋊Dic61C22⋊C411yescharacteristic
222C4⋊Dic61C22⋊C411yescharacteristic
223C4⋊Dic61C22⋊C412yes—
224#11758196—22no—
225#11758196—22no—
226#11758196—22no—
227#11758196—22no—
228#11758196—44no—
229#11758196C22⋊C412yes—
230#11758196C22⋊C412yes—
231#11758196C22⋊C412yes—
232#11758196C22⋊C412yes—
233C4×Dic61—22no—
234C4×Dic61—22no—
235C4×Dic61—22no—
236C4×Dic61—22no—
237C4×Dic61C22×C412yes—
238C4×Dic61C22×C412yes—
239C4×Dic61C2×D412yes—
240C4×Dic61C2×D412yes—
241#11758194—22no—
242#11758194—22no—
243C2×#92418—22no—
244C23×C61—21no—
245C23×C61C22×D411yescharacteristic
246C22×D61—22no—
247Q8×C61—21no—
248Q8×C61—21no—
249D4×C61—21no—
250D4×C61—21no—
251D4×C61—41no—
252D4×C61—41no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—42no—
267C2×C4×C61—81no—
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C2×C22⋊C411yescharacteristic
270C2×C4×C61C2×C22⋊C411yescharacteristic
271C2×C4×C61C2×C22⋊C411yescharacteristic
272C2×C4×C61C23×C411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C2×C4×C612−1+411yescharacteristic
278C22⋊D61—44no—
279C22⋊D61—42no—
280C22⋊D61—42no—
281C22⋊D61—82no—
282C2×Dic61—22no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61C2×C22⋊C412yes—
289C2×Dic61C2×C22⋊C412yes—
290C2×Dic61C2×C22⋊C412yes—
291C2×Dic61C2×C22⋊C412yes—
292D244—22no—
293D244—22no—
294D244—42no—
295D244—81no—
296C4×D61—44no—
297C4×D61—42no—
298C4×D61—42no—
299C4×D61—82no—
300Dic122—22no—
301Dic122—22no—
302Dic122—22no—
303Dic122—22no—
304Dic122—42no—
305C8×C61—22no—
306C8×C61—41no—
307#92418—42no—
308#22551—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61C22×C22⋊C411yescharacteristic
315C22×C61#54811yescharacteristic
316C22×C61#55011yescharacteristic
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—42no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61#42112yes—
329C4×C61C4×#4111yescharacteristic
330C4×C61#52011yescharacteristic
331C4×C61#54811yescharacteristic
332C4×C61#55011yescharacteristic
333Dic61—22no—
334Dic61—22no—
335Dic61—22no—
336Dic61—22no—
337Dic61—42no—
338Dic61—82no—
339#3480—611no—
340Q8×M16—611no—
341#2295—611no—
342#1980—612no—
343#1980—611no—
344#1963—612no—
345#1962—611no—
346#1806—612no—
347#1806—611no—
348#1759—612no—
349C4×#104D6111yescharacteristic, complemented
350C2×C61—21no—
351C2×C61—21no—
352C2×C61—81no—
353C2×C61#234111yescharacteristic
354C2×C61C2×#42111yescharacteristic
355C2×C61#290111yescharacteristic
356D61—82no—
357#559—611no—
358#559—612no—
359#557—611no—
360#556—612no—
361#556—611no—
362#553—1221no—
363#553—1222no—
364#550—611no—
365#548—1221no—
366#532—612no—
367#532—611no—
368#529—1221no—
369#529—1222no—
370C4×#41—612no—
371C4×#41—1221no—
372C4×#41C2×D6111yescharacteristic
373C2×C4×Q8—611no—
374#446—1221no—
375#446—1222no—
376C8×Q8—1221no—
377#423—612no—
378#423—611no—
379C2×#104—612no—
380C2×#104—612no—
381C2×#104—612no—
382C2×#104—612no—
383C2×#104—612no—
384C2×#104C2×D6111yescharacteristic
385C2×#104C2×D6111yescharacteristic
386C4×M16—612no—
387C4×M16C2×D6111yescharacteristic
388#391—1221no—
389#389—1221no—
390#387—1222no—
391C4×C4⋊C4—612no—
392C4×C4⋊C4—611no—
393#376—612no—
394C43C2×D6111yescharacteristic
395#339—612no—
396#339C2×D6111yescharacteristic
397#339C2×D6111yescharacteristic
398C61#2255111yescharacteristic, complemented, Sylow
399C2×#41—612no—
400C2×#41—1221no—
401C2×#41C22×D6111yescharacteristic
402C22×Q8—611no—
403C2×M16—612no—
404C2×M16C22×D6111yescharacteristic
405C2×M16C22×D6111yescharacteristic
406C4⋊Q8—612no—
407C4⋊Q8—611no—
408#127—612no—
409#127—611no—
410#126—1222no—
411#126—1222no—
412#126—1222no—
413#125—1221no—
414#125—1222no—
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—1222no—
424C22⋊Q8—1221no—
425#121—1222no—
426#121—1221no—
427#121—1222no—
428#121—1222no—
429#121—1221no—
430#121—1221no—
431C4×Q8—611no—
432C4×Q8—611no—
433C4×Q8—612no—
434C4×Q8—611no—
435C4×Q8—611no—
436C4×Q8—612no—
437C4×Q8—612no—
438C4×Q8—612no—
439C4×Q8—1221no—
440C4×Q8C4×D6111yescharacteristic
441C4×Q8C4×D6111yescharacteristic
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—612no—
445C4×D4—612no—
446C4×D4—1221no—
447C4×D4—1222no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4—1222no—
451C4×D4—1222no—
452C4×D4—1221no—
453C4×D4—1221no—
454C4×D4C4×D6111yescharacteristic
455C4×D4C4×D6111yescharacteristic
456#117—21no—
457#117—1222no—
458#117—1222no—
459#117—1221no—
460#117C22×D6111yescharacteristic
461C2×C4⋊C4—612no—
462C2×C4⋊C4—611no—
463C2×C4⋊C4—1222no—
464C2×C4⋊C4—1221no—
465C2×C42—21no—
466C2×C42—21no—
467C2×C42—21no—
468C2×C42—612no—
469C2×C42—612no—
470C2×C42—1222no—
471C2×C42C22×D6111yescharacteristic
472C2×C42C22×D6111yescharacteristic
473C2×C42C22×D6111yescharacteristic
474C4⋊C8—1222no—
475C4⋊C8—1222no—
476C4⋊C8—1221no—
477C4⋊C8—1221no—
478#104—22no—
479#104—22no—
480#104—1224no—
481#104—1224no—
482#104—2442no—
483#104—2442no—
484#104—2442no—
485#97—21no—
486#97—1222no—
487C4×C8—21no—
488C4×C8—1222no—
489#95—1222no—
490#95—1222no—
491#95—1222no—
492#95—1221no—
493#41—21no—
494#41—21no—
495#41—21no—
496#41—21no—
497#41—1222no—
498#41—1222no—
499#41—1222no—
500#41—1222no—
501#41—2444no—
502C2×Q8—611no—
503C2×Q8—611no—
504C2×Q8—612no—
505C2×Q8—612no—
506C2×Q8—2442no—
507C2×Q8C2×C4×D6111yescharacteristic
508C2×D4—21no—
509C2×D4—612no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1221no—
514C2×D4—1221no—
515C2×D4C2×C4×D6111yescharacteristic
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—612no—
522C22×C4—612no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4D4×D6111yescharacteristic
528C22×C4D4×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530M16—21no—
531M16—21no—
532M16—21no—
533M16—1222no—
534M16—1222no—
535M16#1175822212yescomplemented
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—1222no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—612no—
542C4⋊C4—612no—
543C4⋊C4—612no—
544C4⋊C4—612no—
545C4⋊C4—611no—
546C4⋊C4—611no—
547C4⋊C4—612no—
548C4⋊C4—1222no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—2444no—
555C4⋊C4—2442no—
556C4⋊C4C2×C4×D6111yescharacteristic
557C4⋊C4C2×C4×D6111yescharacteristic
558C22⋊C4—21no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—1222no—
562C22⋊C4—1222no—
563C22⋊C4—1222no—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—42no—
576C42—42no—
577C42—612no—
578C42—612no—
579C42—612no—
580C42—612no—
581C42—1222no—
582C42—1222no—
583C42—1222no—
584C42—1222no—
585C42C2×C4×D6111yescharacteristic
586C42C2×C4×D6111yescharacteristic
587C42#1175822212yescomplemented
588C42D4×D6111yescharacteristic
589C42D4×D6111yescharacteristic
590C23—21no—
591C23—1222no—
592C23C2×D4×D6111yescharacteristic
593Q8—21no—
594Q8—21no—
595Q8—1222no—
596Q8—1222no—
597Q8—1222no—
598Q8—1222no—
599Q8—2442no—
600D4—21no—
601D4—21no—
602D4—41no—
603D4—41no—
604D4—1222no—
605D4—1222no—
606D4—4881no—
607D4—4882no—
608D4—2444no—
609D4—2442no—
610D4—2442no—
611D4—2442no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—81no—
627C2×C4—612no—
628C2×C4—612no—
629C2×C4—612no—
630C2×C4—612no—
631C2×C4—1222no—
632C2×C4—4882no—
633C2×C4—2444no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4C22⋊C4×D6111yescharacteristic
642C2×C4C22⋊C4×D6111yescharacteristic
643C2×C4C22⋊C4×D6111yescharacteristic
644C2×C4C22⋊C4×D6111yescharacteristic
645C2×C4C22×C4×D6111yescharacteristic
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4C2×#1175822211yescharacteristic
648C2×C4C2×#1175822211yescharacteristic
649C2×C4#1451982311yescharacteristic
650C2×C4#1451982411yescharacteristic
651C8—22no—
652C8—41no—
653C8—2442no—
654C22—21no—
655C22—21no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—2442no—
660C22—2442no—
661C22—2442no—
662C22C2×C22⋊C4×D6111yescharacteristic
663C22#1684042111yescharacteristic
664C22#1684043311yescharacteristic
665C4—21no—
666C4—21no—
667C4—21no—
668C4—42no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—1222no—
674C4—1222no—
675C4—1222no—
676C4—1222no—
677C4—4882no—
678C4—2442no—
679C4#1683977712yes—
680C4C4×#1175822211yescharacteristic
681C4#1684024211yescharacteristic
682C4#1684042111yescharacteristic
683C4#1684043311yescharacteristic
684C2—21no—
685C2—21no—
686C2—81no—
687C2—4882no—
688C2#2144865911yescharacteristic
689C2C2×#1683977711yescharacteristic
690C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#104×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459412C2
cosocleC22×C4×C61C24
hypercenterC4×#104D61
hypocenterC61#22551
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. M16×Dic122C2semidirect
3. #21448296C2semidirect
4. #21446738C2semidirect
5. #21446736C2semidirect
6. #21446649C2semidirect
7. #21446647C2semidirect
8. #21445839C2semidirect
9. #21445838C2semidirect
10. #21445628C2semidirect
11. C4×#104×C61C2semidirect
12. #16840497C4semidirect
13. #16840495C4semidirect
14. #16840490C4semidirect
15. #16840484C4semidirect
16. #16840482C4semidirect
19. #16840433C4semidirect
187. M16×C61#41semidirect
209. C42×C61#41semidirect
349. C4×#104D61semidirect
398. C61#22551semidirect
535. M16#11758222semidirect
587. C42#11758222semidirect

Maximal subgroups

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

subgroupindex
#214594122
M16×Dic1222
#214482962
#214467382
#214467362
#214466492
#214466472
#214458392
#214458382
#214456282
C4×#104×C612
#2255161