Finite Group Databasebeta

← #29670886

Subgroups — #29670886

Order 15616.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458872C211yesmaximal, characteristic, complemented
2#21458692C211yesmaximal, characteristic, complemented
3#21458207C211yesmaximal, characteristic, complemented
4#21445440C211yesmaximal, characteristic, complemented
5#21445345C211yesmaximal, characteristic, complemented
6#21445232C211yesmaximal, characteristic, complemented
7#21445201C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic
9#21445033C211yesmaximal, characteristic, complemented
10#21444976C211yesmaximal, characteristic, complemented
11#21444940C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic
13#21444656C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3361C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840162—21no—
20#16840160C2211yescharacteristic
21#16840157—21no—
22#16840134C2211yescharacteristic
23#16840134C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840097—21no—
26#16840095—21no—
27#16840067—21no—
28#16840064—21no—
29#16840057C2211yescharacteristic
30#16840057C2211yescharacteristic
31#16840054C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16840001—21no—
34#16839981—21no—
35#16839972C2211yescharacteristic
36#16839967—21no—
37#16839959C2211yescharacteristic
38#16839893C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40#16839862—21no—
41C4×#11758202C2211yescharacteristic
42SD16×Dic61—21no—
43D4×#92418C2211yescharacteristic
44C2×#14519710—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#559C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#488—21no—
59C61×#486C2211yescharacteristic
60C61×#478—21no—
61C61×#477C2211yescharacteristic
62C61×#463—21no—
63C61×#459C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61C2211yescharacteristic
66C4×D8×C61—21no—
67C2×#11758202—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—41no—
73C22×#92418—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519768—21no—
78#14519768—21no—
79#14519768—41no—
80#14519756—21no—
81#14519754—41no—
82Q8×Dic61—21no—
83C4×Dic122—21no—
84C4×Dic122C2311yescharacteristic
85D4×Dic61—21no—
86C2×C4⋊Dic61—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710D411yescharacteristic
94#14519709—21no—
95#14519709—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—41no—
104#14519701D411yescharacteristic
105#14519700—21no—
106#14519700—21no—
107#14519700D411yescharacteristic
108#14519698—21no—
109#14519698—21no—
110C22⋊#92418—21no—
111#14519688—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×SD16×C61D411yescharacteristic
116C2×D8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#125×C61—21no—
120#124×C61—21no—
121#124×C61—21no—
122#123×C61—41no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61C2311yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#106×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D411yescharacteristic
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—41no—
153C22×C4×C61—41no—
154C22×Dic61—41no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122C2×D411yescharacteristic
158SD16×C61—22no—
159D8×C61—42no—
160C2×C8×C61—21no—
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61C2×D411yescharacteristic
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C22⋊C4×C61—21no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61C2411yescharacteristic
180C22⋊Dic61—41no—
181#11758202—22no—
182#11758202—81no—
183#11758200—42no—
184#11758200—81no—
185#11758200—81no—
186#11758200—82no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61—41no—
193C4⋊Dic61—41no—
194C4⋊Dic61C2×D411yescharacteristic
195#11758196—21no—
196#11758196—41no—
197#11758196—41no—
198C4×Dic61—21no—
199C4×Dic61—41no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—42no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61—41no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—81no—
220D4×C61#13712yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—161no—
230C2×C4×C61—161no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—161no—
239Dic122—21no—
240Dic122—21no—
241Dic122—41no—
242Dic122—41no—
243Dic122—81no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—41no—
247#92418—21no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#92418#13512yes—
253#19857—611nomaximal, Sylow
254C22×C61—22no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61D4211yescharacteristic
261C4×C61—21no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—81no—
268C4×C61C2×#13511yescharacteristic
269C4×C61C2×#13711yescharacteristic
270C4×C61#57711yescharacteristic
271C4×C61#57811yescharacteristic
272Dic61—41no—
273Dic61—41no—
274Dic61—81no—
275Dic61—81no—
276#3400—611no—
277#3381—611no—
278#3361D6111yescharacteristic
279#3338—611no—
280#1714—611no—
281#1696—611no—
282#1678—611no—
283#1672—611no—
284#1665—611no—
285#1644—611no—
286#1636—611no—
287#1630—611no—
288#1622—611no—
289#1582—611no—
290#1579—611no—
291C2×C61—42no—
292C2×C61—161no—
293C2×C61#327011yescharacteristic
294C2×C61#328411yescharacteristic
295C2×C61#328811yescharacteristic
296#559C2×D6111yescharacteristic
297#557—611no—
298#554—611no—
299#554C2×D6111yescharacteristic
300#500—611no—
301#499—1221no—
302#498—611no—
303#497—1221no—
304#492—611no—
305#492—611no—
306#488—21no—
307#487—611no—
308#486C2×D6111yescharacteristic
309#484—1221no—
310#484—1221no—
311#479—1221no—
312#478—21no—
313#478—1221no—
314#477—611no—
315#477—611no—
316#477—611no—
317#477C2×D6111yescharacteristic
318#474—611no—
319#467—1221no—
320#464—1221no—
321#463—21no—
322#462—611no—
323#461—1221no—
324#460—611no—
325#459C2×D6111yescharacteristic
326#446—611no—
327C8×Q8—611no—
328C8×Q8C2×D6111yescharacteristic
329#440—1221no—
330C4×Q16—611no—
331C4×SD16—1221no—
332C4×SD16C2×D6111yescharacteristic
333C4×D8—21no—
334C8×D4—611no—
335C2×#107—1221no—
336#334—611no—
337#334—611no—
338#333—611no—
339#333—611no—
340#332—611no—
341#331—611no—
342Q8⋊C8—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345D4⋊C8—611no—
346#322—611no—
347C61#1985711yescharacteristic, complemented, Sylow
348C2×Q16—611no—
349C2×Q16—1221no—
350C2×SD16—1221no—
351C2×SD16—1221no—
352C2×SD16—1221no—
353C2×SD16—2441no—
354C2×SD16C22⋊D6111yescharacteristic
355C2×D8—21no—
356C22×C8—1221no—
357C4⋊Q8—611no—
358C4⋊Q8—611no—
359C4⋊Q8—1221no—
360C4⋊Q8C22×D6111yescharacteristic
361#127C22×D6111yescharacteristic
362#125—21no—
363#125—1221no—
364#125—1221no—
365#125—2441no—
366#124—21no—
367#124—21no—
368#123—41no—
369C22⋊Q8—21no—
370C22⋊Q8—41no—
371C22⋊Q8—1221no—
372C22⋊Q8—2441no—
373C4×Q8—611no—
374C4×Q8—1221no—
375C4×Q8—1221no—
376C4×Q8C22×D6111yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—1221no—
382C4×D4C22×D6111yescharacteristic
383C2×C4⋊C4—21no—
384C2×C4⋊C4—1221no—
385#107—21no—
386#107—611no—
387#107—1221no—
388#107—1221no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—1221no—
394#106—1221no—
395#106C22⋊D6111yescharacteristic
396C4⋊C8—21no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403#103—21no—
404#103—611no—
405#103—611no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—2441no—
411#103C22⋊D6111yescharacteristic
412#102—21no—
413#102—21no—
414#102—21no—
415#102—1221no—
416#102—1221no—
417#102C22⋊D6111yescharacteristic
418C22⋊C8—1221no—
419#97—1221no—
420C4×C8—21no—
421C4×C8—611no—
422C4×C8—611no—
423C4×C8—1221no—
424C4×C8C22×D6111yescharacteristic
425C2×Q8—21no—
426C2×Q8—611no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8D4×D6111yescharacteristic
430C2×Q8C2×C22⋊D6111yescharacteristic
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C22⋊D6111yescharacteristic
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—2441no—
439Q16—1222no—
440Q16—4881no—
441SD16—22no—
442SD16—4881no—
443SD16—4881no—
444SD16—4882no—
445SD16—2442no—
446D8—42no—
447C2×C8—21no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—2442no—
455C2×C8C2×C22⋊D6111yescharacteristic
456C2×C8C2×C22⋊D6111yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—611no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4D4×D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C4⋊C4C2×C22⋊D6111yescharacteristic
477C22⋊C4—21no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—2441no—
483C42—21no—
484C42—21no—
485C42—1221no—
486C42—2441no—
487C42C23×D6111yescharacteristic
488C23—21no—
489C23—41no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—1221no—
496Q8—1221no—
497Q8—4881no—
498Q8—2441no—
499Q8—2441no—
500D4—21no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—81no—
506D4#1451980712yes—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—161no—
516C2×C4—161no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—9761no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#1451982211yescharacteristic
525C8—21no—
526C8—21no—
527C8—41no—
528C8—612no—
529C8—1221no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C8—2441no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22D4×C22⋊D6111yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—81no—
548C4—4881no—
549C4—4881no—
550C4—2441no—
551C4—2441no—
552C4#135×D6111yescharacteristic
553C4C2×#1451980711yescharacteristic
554C4#1684056911yescharacteristic
555C4#1684057711yescharacteristic
556C2—42no—
557C2—161no—
558C2#2145738511yescharacteristic
559C2#2145754311yescharacteristic
560C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3361C2
socleC22×C61D42
prime radicalC61×#459C22
hypercenter#3361D61
hypocenterC61#19857
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3361

Complements

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

NH ≅ G/Ntype
1. #21458872C2semidirect
2. #21458692C2semidirect
3. #21458207C2semidirect
4. #21445440C2semidirect
5. #21445345C2semidirect
6. #21445232C2semidirect
7. #21445201C2semidirect
9. #21445033C2semidirect
10. #21444976C2semidirect
11. #21444940C2semidirect
13. #21444656C2semidirect
14. #21444636C2semidirect
347. C61#19857semidirect

Maximal subgroups

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

subgroupindex
#214588722
#214586922
#214582072
#214454402
#214453452
#214452322
#214452012
#214451582
#214450332
#214449762
#214449402
#214448892
#214446562
#214446362
C61×#33612
#1985761