Finite Group Databasebeta

← #29571867

Subgroups — #29571867

Order 15616.

768 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445629C211yesmaximal, characteristic, complemented
2#21444426C211yesmaximal, characteristic, complemented
3#21444327C211yesmaximal, characteristic, complemented
4#21444305C211yesmaximal, characteristic, complemented
5#21444253C211yesmaximal, characteristic, complemented
6#21444236C211yesmaximal, characteristic, complemented
7C61×#3514C211yesmaximal, characteristic
8#16839819C2211yescharacteristic, complemented
9#16839815C2211yescharacteristic, complemented
10C2×#14519704—21no—
11C4×#11758194C2211yescharacteristic, complemented
12#16839608C2211yescharacteristic, complemented
13#16839521—21no—
14#16839504—21no—
15#16839502—21no—
16#16839499—21no—
17#16839496—21no—
18#16839496—22no—
19Q8⋊#92418—21no—
20D4⋊#92418—22no—
21D4⋊#92418—21no—
22C61×#559C411yescharacteristic, complemented
23C61×#556C411yescharacteristic, complemented
24C61×#555—21no—
25C61×#551—22no—
26D4×Q8×C61C411yescharacteristic, complemented
27C61×#548—21no—
28C61×#546—21no—
29C61×#546—21no—
30D42×C61C411yescharacteristic, complemented
31C61×#542—21no—
32C61×#538—22no—
33C61×#532C2211yescharacteristic
34C61×#529—21no—
35C61×#529C2211yescharacteristic
36C2×#124×C61—21no—
37C61×#518—21no—
38C4×#41×C61C2211yescharacteristic
39C2×#11758194—21no—
40C2×C4×Dic61—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43#14519704—24no—
44C22⋊#92418—21no—
45C22⋊#92418—21no—
46#14519688—21no—
47#14519688—21no—
48C4×#92418—21no—
49C4×#92418—21no—
50#14519684—21no—
51#14519684—21no—
52C2×#41×C61—21no—
53C2×#41×C61D411yescharacteristic
54C22×Q8×C61—21no—
55C22×D4×C61—21no—
56C22×D4×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61C2×C411yescharacteristic
59#127×C61C2×C411yescharacteristic
60#126×C61—22no—
61#126×C61—22no—
62#126×C61—21no—
63#125×C61D411yescharacteristic
64#124×C61—22no—
65#124×C61—22no—
66#124×C61—22no—
67#124×C61—21no—
68#124×C61—21no—
69#124×C61—21no—
70#124×C61—21no—
71#124×C61D411yescharacteristic
72#123×C61—22no—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—21no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61D411yescharacteristic
81#121×C61—22no—
82#121×C61—22no—
83#121×C61—21no—
84#121×C61—21no—
85#121×C61—21no—
86#121×C61D412yes—
87#121×C61D411yescharacteristic
88#120×C61—22no—
89#120×C61—22no—
90C4×Q8×C61—21no—
91C4×Q8×C61C2×C411yescharacteristic
92C4×Q8×C61C2×C411yescharacteristic
93C4×Q8×C61D411yescharacteristic
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61C2×C411yescharacteristic
104C4×D4×C61C2×C411yescharacteristic
105C4×D4×C61D412yes—
106C4×D4×C61D411yescharacteristic
107#117×C61—22no—
108#117×C61—21no—
109#117×C61—21no—
110#117×C61—21no—
111#117×C61D411yescharacteristic
112C2×C22⋊C4×C61—22no—
113C2×C22⋊C4×C61—22no—
114C2×C22⋊C4×C61—22no—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C24×C61—22no—
118#41×C61—22no—
119#41×C61—41no—
120#41×C61—82no—
121C2×Q8×C61—22no—
122C2×Q8×C61—21no—
123C2×Q8×C61—82no—
124C2×Q8×C61C22⋊C411yescharacteristic
125C2×Q8×C61C22⋊C411yescharacteristic
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61—82no—
136C2×D4×C61C22⋊C411yescharacteristic
137C2×D4×C61C22⋊C411yescharacteristic
138C2×D4×C61C22⋊C412yes—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×Dic61—41no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61C22⋊C411yescharacteristic
160C4⋊C4×C61C22⋊C411yescharacteristic
161C4⋊C4×C61C22⋊C411yescharacteristic
162C4⋊C4×C61C22⋊C411yescharacteristic
163C4⋊C4×C61C22⋊C412yes—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C4×Dic61—22no—
191C4×Dic61—41no—
192#11758194—22no—
193#11758194—41no—
194C2×#92418—41no—
195C2×#92418—41no—
196C2×#92418—41no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—44no—
201C23×C61—82no—
202C23×C61—82no—
203C23×C61#12011yescharacteristic
204Q8×C61—21no—
205Q8×C61—21no—
206Q8×C61—42no—
207Q8×C61—41no—
208Q8×C61—82no—
209D4×C61—21no—
210D4×C61—41no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—44no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—81no—
218D4×C61—82no—
219D4×C61—82no—
220D4×C61#10412yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—81no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×Dic61—42no—
247C2×Dic61—81no—
248C2×Dic61—81no—
249#92418—42no—
250#92418—81no—
251#92418—81no—
252#8412—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—42no—
260C22×C61—41no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61—164no—
266C22×C61#37911yescharacteristic
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61C2×#10411yescharacteristic
280C4×C61#42111yescharacteristic
281Dic61—82no—
282#3514D6111yescharacteristic
283#1759—611no—
284#1529—611no—
285#1505—611no—
286#1500—611no—
287#1488—611no—
288#1484—611no—
289C2×C61—41no—
290C2×C61—42no—
291C2×C61—82no—
292C2×C61—81no—
293C2×C61#178911yescharacteristic
294C2×C61#179411yescharacteristic
295C2×C61#179511yescharacteristic
296#559Dic6111yescharacteristic, complemented
297#556Dic6111yescharacteristic, complemented
298#555—21no—
299#551—22no—
300D4×Q8Dic6111yescharacteristic, complemented
301#548—21no—
302#546—21no—
303#546—21no—
304D42Dic6111yescharacteristic, complemented
305#542—21no—
306#538—22no—
307#532C2×D6111yescharacteristic
308#529—21no—
309#529C2×D6111yescharacteristic
310C2×#124—21no—
311#518—21no—
312C4×#41C2×D6111yescharacteristic
313#432—611no—
314#431—611no—
315C2×#104—1221no—
316C4×M16—611no—
317#376—611no—
318#339—1221no—
319#330—1221no—
320#329—1221no—
321#328—1221no—
322#327—1221no—
323#327—1222no—
324Q8⋊C8—1221no—
325D4⋊C8—1222no—
326D4⋊C8—1221no—
327C61#841211yescharacteristic, complemented, Sylow
328C2×#41—21no—
329C2×#41C22⋊D6111yescharacteristic
330C22×Q8—21no—
331C22×D4—21no—
332C22×D4—21no—
333C2×M16—1221no—
334C4⋊Q8—21no—
335C4⋊Q8C2×Dic6111yescharacteristic
336#127C2×Dic6111yescharacteristic
337#126—22no—
338#126—22no—
339#126—21no—
340#125C22⋊D6111yescharacteristic
341#124—22no—
342#124—22no—
343#124—22no—
344#124—21no—
345#124—21no—
346#124—21no—
347#124—21no—
348#124C22⋊D6111yescharacteristic
349#123—22no—
350#123—22no—
351#123—22no—
352#123—21no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8C22⋊D6111yescharacteristic
358#121—22no—
359#121—22no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121C22⋊D6111yescharacteristic
364#121C22⋊D6112yes—
365#120—22no—
366#120—22no—
367C4×Q8—21no—
368C4×Q8C2×Dic6111yescharacteristic
369C4×Q8C2×Dic6111yescharacteristic
370C4×Q8C22⋊D6111yescharacteristic
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C2×Dic6111yescharacteristic
381C4×D4C2×Dic6111yescharacteristic
382C4×D4C22⋊D6111yescharacteristic
383C4×D4C22⋊D6112yes—
384#117—22no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117C22⋊D6111yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C42—21no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397#104—1224no—
398C22⋊C8—1221no—
399C22⋊C8—1221no—
400#97—1221no—
401#97—1221no—
402C4×C8—1221no—
403C4×C8—1221no—
404#95—1221no—
405#95—1221no—
406C24—22no—
407#41—22no—
408#41—41no—
409#41—82no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—82no—
413C2×Q8C22⋊Dic6111yescharacteristic
414C2×Q8C22⋊Dic6111yescharacteristic
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—82no—
425C2×D4C22⋊Dic6111yescharacteristic
426C2×D4C22⋊Dic6111yescharacteristic
427C2×D4C22⋊Dic6112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—2441no—
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439M16—1222no—
440M16—2441no—
441C2×C8—2441no—
442C2×C8—2441no—
443C2×C8—2441no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4C22⋊Dic6111yescharacteristic
454C4⋊C4C22⋊Dic6111yescharacteristic
455C4⋊C4C22⋊Dic6111yescharacteristic
456C4⋊C4C22⋊Dic6111yescharacteristic
457C4⋊C4C22⋊Dic6112yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—1222no—
481C42—2441no—
482C42C22×Dic6111yescharacteristic
483C42C2×C22⋊D6111yescharacteristic
484C42C2×C22⋊D6111yescharacteristic
485C42C2×C22⋊D6111yescharacteristic
486C23—21no—
487C23—21no—
488C23—22no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23#1451974611yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—41no—
496Q8—42no—
497Q8—82no—
498D4—21no—
499D4—41no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—41no—
505D4—41no—
506D4—81no—
507D4—82no—
508D4—82no—
509D4#1451970412yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—4881no—
530C2×C4—4881no—
531C2×C4—2442no—
532C2×C4C2×C22⋊Dic6111yescharacteristic
533C2×C4C2×C22⋊Dic6111yescharacteristic
534C2×C4C2×C22⋊Dic6111yescharacteristic
535C2×C4#1451974611yescharacteristic
536C2×C4#1451974611yescharacteristic
537C2×C4#1451974611yescharacteristic
538C8—4881no—
539C8—4881no—
540C8—2442no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—21no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—164no—
554C22#1683961811yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—4882no—
568C4C2×#1451970411yescharacteristic
569C4#1683977911yescharacteristic
570C2—41no—
571C2—42no—
572C2—81no—
573C2—82no—
574C2#2144576311yescharacteristic
575C2#2144578311yescharacteristic
576C2#2144578911yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3514C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3514D61
hypocenterC61#8412
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3514

Complements

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

NH ≅ G/Ntype
1. #21445629C2semidirect
2. #21444426C2semidirect
3. #21444327C2semidirect
4. #21444305C2semidirect
5. #21444253C2semidirect
6. #21444236C2semidirect
8. #16839819C22semidirect
9. #16839815C22semidirect
11. C4×#11758194C22semidirect
12. #16839608C22semidirect
22. C61×#559C4semidirect
23. C61×#556C4semidirect
26. D4×Q8×C61C4semidirect
30. D42×C61C4semidirect
296. #559Dic61semidirect
297. #556Dic61semidirect
300. D4×Q8Dic61semidirect
304. D42Dic61semidirect
327. C61#8412semidirect

Maximal subgroups

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

subgroupindex
#214456292
#214444262
#214443272
#214443052
#214442532
#214442362
C61×#35142
#841261