Finite Group Databasebeta

← #29749945

Subgroups — #29749945

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839866C211yesmaximal, characteristic
2#21449676C211yesmaximal, characteristic
3#21449591C211yesmaximal, characteristic
4#21447196C211yesmaximal, characteristic
5#21447154C211yesmaximal, characteristic
6#21447140C211yesmaximal, characteristic
7#21446919C211yesmaximal, characteristic
8#21446677C211yesmaximal, characteristic
9#21446509C211yesmaximal, characteristic
10#21446506C211yesmaximal, characteristic
11#21446497C211yesmaximal, characteristic
12#21446157C211yesmaximal, characteristic
13#21446071C211yesmaximal, characteristic
14#21445687C211yesmaximal, characteristic
15C2×C61×#491C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20#16839866—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839701—21no—
31#16839700—21no—
32#16839695—21no—
33#16839692—21no—
34#16839669C2211yescharacteristic
35#16839669C2211yescharacteristic
36#16839669C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839658—21no—
42#16839658C2211yescharacteristic
43#16839643—21no—
44#16839643—21no—
45#16839643—21no—
46C4×#11758196C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C61×#491—24no—
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70Q8×Dic61—22no—
71Q8×Dic61—41no—
72#14519742—22no—
73#14519742—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519710—41no—
89#14519708—22no—
90#14519706—22no—
91#14519702—22no—
92#14519702—22no—
93#14519701—22no—
94#14519701—22no—
95#14519687—22no—
96C8×Dic61—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C4⋊Q8×C61—22no—
117C4⋊Q8×C61—41no—
118#125×C61—22no—
119#125×C61—41no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#103×C61—22no—
128#103×C61—22no—
129#103×C61—22no—
130#103×C61D414yes—
131#97×C61—22no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C2×D412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—42no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61C2×D412yes—
179C42×C61—21no—
180C42×C61—21no—
181#11758202—28no—
182#11758202—44no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—42no—
190C4⋊Dic61—42no—
191#11758196—22no—
192#11758196—41no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418#4112yes—
207C2×#92418#4112yes—
208C23×C61C22×D411yescharacteristic
209Q8×C61—22no—
210Q8×C61—22no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61#13712yes—
265Dic122—22no—
266Dic122—22no—
267Dic122—41no—
268Dic122—41no—
269C8×C61—22no—
270C8×C61—41no—
271#92418—22no—
272#92418—22no—
273#26847—611nomaximal, Sylow
274C22×C61#53211yescharacteristic
275C22×C61D4211yescharacteristic
276C22×C61#54811yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57811yescharacteristic
280C22×C61#57811yescharacteristic
281C4×C61—42no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61#52511yescharacteristic
288C4×C61#54211yescharacteristic
289C4×C61#54711yescharacteristic
290C4×C61#55111yescharacteristic
291Dic61—22no—
292Dic61—41no—
293Dic61—42no—
294Dic61—41no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299C2×#491D6111yescharacteristic
300C2×#441—611no—
301#2453—611no—
302#2446—611no—
303#2066—611no—
304#2059—611no—
305#2056—611no—
306#2016—611no—
307#1969—611no—
308#1935—611no—
309#1935—611no—
310#1933—611no—
311#1878—611no—
312#1863—611no—
313#1770—611no—
314C2×C61#246111yescharacteristic
315C2×C61#317711yescharacteristic
316C2×C61#318111yescharacteristic
317C2×C61#327611yescharacteristic
318C2×C61#327711yescharacteristic
319C2×C61#337411yescharacteristic
320C2×C61#337711yescharacteristic
321C22×Q16—611no—
322C22×Q16—611no—
323C2×C4⋊Q8C2×D6111yescharacteristic
324C2×#125C2×D6111yescharacteristic
325C2×C4×Q8—611no—
326C2×C4×Q8—611no—
327#491—24no—
328#441—1224no—
329C2×#107—611no—
330C2×#106—611no—
331C2×C4⋊C8—611no—
332C2×#103—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#103C2×D6111yescharacteristic
337C2×#103C2×D6111yescharacteristic
338C2×#103C2×D6111yescharacteristic
339C2×#103C2×D6111yescharacteristic
340C2×#97—611no—
341C2×#97C2×D6111yescharacteristic
342C2×C4×C8—611no—
343#400—1221no—
344#400—1221no—
345#398—1221no—
346#398—1221no—
347#391—611no—
348#391—611no—
349#391—611no—
350#391—611no—
351#391—611no—
352#389—611no—
353#389—611no—
354#389—611no—
355#389—1221no—
356#387—1221no—
357#387—1221no—
358#387—1221no—
359C4×C4⋊C4—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#336—611no—
365#336—611no—
366C61#2684711yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×Q8C22×D6111yescharacteristic
369C22×Q8C22×D6111yescharacteristic
370C2×Q16—1222no—
371C2×Q16—1222no—
372C2×Q16—1222no—
373C2×Q16—1222no—
374C2×Q16—1222no—
375C2×Q16—2441no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C4⋊Q8—22no—
381C4⋊Q8—41no—
382#125—22no—
383#125—41no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×Q8—2441no—
387C4×Q8—2441no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#107—2441no—
410#106—1222no—
411C4⋊C8—1222no—
412#103—22no—
413#103—22no—
414#103—22no—
415#103—1222no—
416#103—1222no—
417#103—1222no—
418#103—1222no—
419#103C22⋊D6114yes—
420#97—22no—
421#97—1222no—
422C4×C8—1222no—
423#95—611no—
424#95—611no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—612no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8C2×C22⋊D6112yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4D4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4C23×D6111yescharacteristic
471Q16—1228no—
472Q16—2444no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—612no—
479C2×C8—612no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8C2×C22⋊D6112yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—1222no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—2441no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4C2×C22⋊D6112yes—
506C42—21no—
507C42—21no—
508C42—1221no—
509C42—1221no—
510C42—1221no—
511C42—1221no—
512C42—2442no—
513C42—2442no—
514C23C2×D4×D6111yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—1222no—
524Q8—1222no—
525Q8—2441no—
526Q8—2441no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—612no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4C22×C22⋊D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4#1451982211yescharacteristic
574C2×C4#1451982411yescharacteristic
575C8—22no—
576C8—41no—
577C8—1222no—
578C8—1222no—
579C22#1684031111yescharacteristic
580C22D4×C22⋊D6111yescharacteristic
581C22#1684042011yescharacteristic
582C22#137×D6111yescharacteristic
583C22#1684056111yescharacteristic
584C22#1684057811yescharacteristic
585C22#1684057811yescharacteristic
586C4—41no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—1222no—
593C4—4881no—
594C4—4881no—
595C4—4881no—
596C4—2441no—
597C4—2442no—
598C4—2441no—
599C4—2441no—
600C4#1684026911yescharacteristic
601C4#1684037611yescharacteristic
602C4#1684041111yescharacteristic
603C4#1684044211yescharacteristic
604C2#2144978211yescharacteristic
605C2#2145647711yescharacteristic
606C2#2145652711yescharacteristic
607C2#2145745911yescharacteristic
608C2#2145747011yescharacteristic
609C2#2145861911yescharacteristic
610C2#2145865611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#491C2
socle, prime radicalC23×C61C22×D4
hypercenterC2×#491D61
hypocenterC61#26847
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
366. C61#26847semidirect

Maximal subgroups

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

subgroupindex
C2×#168398662
#214496762
#214495912
#214471962
#214471542
#214471402
#214469192
#214466772
#214465092
#214465062
#214464972
#214461572
#214460712
#214456872
C2×C61×#4912
#2684761