Finite Group Databasebeta

← #29662731

Subgroups — #29662731

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456401C211yesmaximal, characteristic, complemented
2#21455609C211yesmaximal, characteristic
3#21455601C211yesmaximal, characteristic
4#21454780C211yesmaximal, characteristic
5#21445404C211yesmaximal, characteristic, complemented
6#21445400C211yesmaximal, characteristic, complemented
7#21445397C211yesmaximal, characteristic, complemented
8#21445393C211yesmaximal, characteristic, complemented
9#21445346C211yesmaximal, characteristic, complemented
10#21445341C211yesmaximal, characteristic, complemented
11#21445312C211yesmaximal, characteristic, complemented
12#21445307C211yesmaximal, characteristic, complemented
13C2×#16839512C211yesmaximal, characteristic, complemented
14#21444474C211yesmaximal, characteristic, complemented
15C61×#1542C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840141C2211yescharacteristic
20#16840114C2211yescharacteristic
21#16840112C2211yescharacteristic
22#16840070—21no—
23#16840066—21no—
24#16840063—21no—
25#16840059—21no—
26#16840058C2211yescharacteristic
27#16840054C2211yescharacteristic
28#16840053C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840048C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16840043C2211yescharacteristic
33#16840039C2211yescharacteristic
34#16840001—21no—
35#16840001—21no—
36#16839889C2211yescharacteristic
37#16839889C2211yescharacteristic
38C4×#11758202—21no—
39C4×#11758202—21no—
40#16839839C2211yescharacteristic
41#16839839C2211yescharacteristic
42#16839830—21no—
43#16839830—21no—
44#16839813—21no—
45#16839800—21no—
46#16839798—21no—
47C2×C4⋊#92418C2211yescharacteristic
48#16839759—21no—
49#16839756—21no—
50C4×#11758189—21no—
51C2×C4×#92418C2211yescharacteristic
52#16839512C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839509C2211yescharacteristic
59#16839509C2211yescharacteristic
60C61×#533C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C2×C4⋊C8×C61C2211yescharacteristic
63C61×#333C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758222D411yescharacteristic
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758189—21no—
71C2×#11758189—21no—
72C22×#92418—21no—
73C22×#92418D411yescharacteristic
74C22×#92418D411yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519769—21no—
78#14519756—21no—
79#14519756—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122D411yescharacteristic
83C4×Dic122C2311yescharacteristic
84C4×D244D411yescharacteristic
85C4×D244C2311yescharacteristic
86C4×C22⋊D61—21no—
87C4×C22⋊D61—21no—
88#14519736—21no—
89#14519736D411yescharacteristic
90#14519734—21no—
91C42×D61—21no—
92#14519711—21no—
93#14519711—21no—
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519709—21no—
99#14519709—21no—
100#14519708—21no—
101#14519708—21no—
102C4⋊#92418—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706—21no—
107#14519702—21no—
108#14519702—21no—
109#14519701—21no—
110#14519701—21no—
111#14519700—21no—
112#14519700—21no—
113#14519700—21no—
114#14519700—21no—
115#14519699—21no—
116#14519699—21no—
117#14519697—21no—
118#14519697—21no—
119#14519689—21no—
120#14519689—21no—
121#14519687—21no—
122#14519687—21no—
123C4×#92418D411yescharacteristic
124C4×#92418D411yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C22×C8×C61—21no—
128C4⋊Q8×C61C2311yescharacteristic
129C4⋊Q8×C61C2311yescharacteristic
130#125×C61—21no—
131C22⋊Q8×C61—21no—
132C22⋊Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×Q8×C61—21no—
135#117×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4111yescharacteristic
149C2×C22⋊D61—21no—
150#11758222—22no—
151#11758222—22no—
152C2×D244C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122C2×D411yescharacteristic
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—42no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61—41no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61#4111yescharacteristic
175C42×C61C2411yescharacteristic
176C22⋊Dic61—21no—
177C22⋊Dic61—41no—
178#11758202—42no—
179#11758202—42no—
180#11758198—21no—
181#11758198—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C2×D411yescharacteristic
187C4⋊Dic61C2×D411yescharacteristic
188#11758196—21no—
189#11758196—21no—
190#11758196—41no—
191#11758196—41no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—41no—
196C2×#92418—22no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—41no—
200C2×#92418—42no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205#11758189—42no—
206#11758189—42no—
207C23×C61#12111yescharacteristic
208C22×D61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—81no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#13712yes—
221C2×C4×C61#13712yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C22⋊D61—42no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—21no—
232D244—21no—
233C4×D61—42no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—42no—
237Dic122—21no—
238Dic122—21no—
239Dic122—41no—
240Dic122—41no—
241Dic122—81no—
242C8×C61—41no—
243C8×C61—41no—
244C8×C61—41no—
245C8×C61—41no—
246#92418—21no—
247#92418—21no—
248#92418—21no—
249#92418—41no—
250#92418—41no—
251#92418#13512yes—
252#19158—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61#47012yes—
256C22×C61C2×#12111yescharacteristic
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—41no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61#47012yes—
266C4×C61C2×#13511yescharacteristic
267C4×C61C2×#13711yescharacteristic
268C4×C61C2×#13711yescharacteristic
269C4×C61#57511yescharacteristic
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273Dic61—81no—
274Dic61—81no—
275#3170—611no—
276#3076—611no—
277#3076—611no—
278#2965—611no—
279#1704—611no—
280#1704—611no—
281#1703—611no—
282#1703—611no—
283#1696—611no—
284#1695—611no—
285#1690—611no—
286#1689—611no—
287C2×#334—611no—
288#1542—611no—
289#1542D6111yescharacteristic, complemented
290C2×C61—22no—
291C2×C61#303511yescharacteristic
292C2×C61C2×#47011yescharacteristic
293C2×C61#306711yescharacteristic
294D61—81no—
295#533—611no—
296#533C2×D6111yescharacteristic
297C4×#41—611no—
298#500—611no—
299#494—611no—
300#487—611no—
301#487—611no—
302#479—1221no—
303#479—1221no—
304#478—1221no—
305#478—1221no—
306#477—611no—
307#477—611no—
308#476—611no—
309#476—611no—
310#475—611no—
311#475—611no—
312#474—611no—
313#474—611no—
314#467—1221no—
315#467—1221no—
316#446—611no—
317#446—611no—
318C4×Q16—1221no—
319C4×Q16—1221no—
320#436—611no—
321#436—611no—
322#435—1221no—
323#435—1221no—
324#431—1221no—
325#431C2×D6111yescharacteristic
326#427—1221no—
327#427—1221no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8C2×D6111yescharacteristic
330#416—1221no—
331#416—1221no—
332C4×M16—1221no—
333C2×C4×C8—611no—
334#334—611no—
335#334—611no—
336#334—611no—
337#334—611no—
338#333—611no—
339#333—611no—
340#333C2×D6111yescharacteristic
341#333C2×D6111yescharacteristic
342#332—611no—
343#332—611no—
344#332C2×D6111yescharacteristic
345#332C2×D6111yescharacteristic
346C61#1915811yescharacteristic, complemented, Sylow
347C2×#41—611no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×M16—1221no—
351C2×M16—1221no—
352C22×C8—21no—
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—1221no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#125—21no—
361#125—1221no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—1221no—
365C22⋊Q8—1221no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—611no—
369C4×Q8—611no—
370C4×Q8—1221no—
371C4×Q8—1221no—
372C4×D4—611no—
373C4×D4—611no—
374C4×D4—1221no—
375C4×D4—1221no—
376#117—21no—
377#117—611no—
378#117—1221no—
379#117—1221no—
380C2×C42—1221no—
381C2×C42C22×D6111yescharacteristic
382#107—1221no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—1221no—
387#107—1221no—
388#106—1221no—
389#106—1221no—
390#106—1221no—
391#106—1221no—
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6111yescharacteristic
399C4⋊C8C22×D6111yescharacteristic
400C4⋊C8C22×D6111yescharacteristic
401C4⋊C8C22×D6111yescharacteristic
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#102—1221no—
411#102—1221no—
412#102—1221no—
413#102—1221no—
414C22⋊C8—21no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417#97—1221no—
418#97—1221no—
419C4×C8—21no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—611no—
424#41—1222no—
425#41—1222no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—611no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×D4—611no—
432C2×D4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4#1175822211yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439Q16—2442no—
440Q16—2442no—
441M16—2442no—
442M16—2442no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—1222no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—2441no—
456C2×C8—2442no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—611no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C22⋊C4—41no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—2441no—
478C42—41no—
479C42—1221no—
480C42—1221no—
481C42—1221no—
482C42—2441no—
483C42#1175822211yescharacteristic
484C42D4×D6111yescharacteristic
485C42D4×D6111yescharacteristic
486C42C23×D6111yescharacteristic
487C23—1221no—
488C23#1451975011yescharacteristic
489Q8—41no—
490Q8—41no—
491Q8—81no—
492Q8—1221no—
493Q8—1221no—
494Q8—4881no—
495Q8—2441no—
496Q8—2441no—
497D4—1221no—
498D4—1221no—
499D4—2442no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4#1451975011yescharacteristic
515C2×C4#1451975011yescharacteristic
516C2×C4#1451975011yescharacteristic
517C2×C4#1451980612yes—
518C2×C4#1451980912yes—
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C2×#1175822211yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—612no—
527C8—1221no—
528C8—1221no—
529C8—1221no—
530C8—2441no—
531C8—2441no—
532C22—21no—
533C22—21no—
534C22—2441no—
535C22—2441no—
536C22—2441no—
537C22#1684001912yes—
538C22C2×#1451975011yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—81no—
543C4—81no—
544C4—4881no—
545C4—4881no—
546C4—2441no—
547C4—2441no—
548C4—2441no—
549C4#1684001912yes—
550C4#135×D6111yescharacteristic
551C4C2×#1451980611yescharacteristic
552C4C2×#1451980911yescharacteristic
553C4#1684055911yescharacteristic
554C2—22no—
555C2—4881no—
556C2#2145527211yescharacteristic
557C2C2×#1684001911yescharacteristic
558C2#2145553811yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1542D61
hypocenterC61#19158
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
5. #21445404C2semidirect
6. #21445400C2semidirect
7. #21445397C2semidirect
8. #21445393C2semidirect
9. #21445346C2semidirect
10. #21445341C2semidirect
11. #21445312C2semidirect
12. #21445307C2semidirect
13. C2×#16839512C2semidirect
14. #21444474C2semidirect
15. C61×#1542C2semidirect
289. #1542D61semidirect
346. C61#19158semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214556092
#214556012
#214547802
#214454042
#214454002
#214453972
#214453932
#214453462
#214453412
#214453122
#214453072
C2×#168395122
#214444742
C61×#15422
#1915861