Finite Group Databasebeta

← #29670260

Subgroups — #29670260

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458822C211yesmaximal, characteristic, complemented
2#21458738C211yesmaximal, characteristic
3#21457689C211yesmaximal, characteristic
4#21445493C211yesmaximal, characteristic, complemented
5#21445322C211yesmaximal, characteristic, complemented
6#21445279C211yesmaximal, characteristic, complemented
7#21445213C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic
9#21445067C211yesmaximal, characteristic
10#21445061C211yesmaximal, characteristic, complemented
11#21445033C211yesmaximal, characteristic
12#21444976C211yesmaximal, characteristic, complemented
13#21444669C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic
15C61×#3381C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840141—21no—
22#16840141C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840097—21no—
26#16840060—21no—
27#16840055—21no—
28#16840054C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840001—21no—
32#16839972—21no—
33#16839972C2211yescharacteristic
34#16839972C2211yescharacteristic
35#16839969—21no—
36#16839969C2211yescharacteristic
37#16839917—21no—
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839863—21no—
41C4×#11758202—21no—
42C4×#11758202C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519700—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#557C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#499—21no—
59C61×#498C2211yescharacteristic
60C61×#479—21no—
61C61×#478—21no—
62C61×#477C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202D411yescharacteristic
74C22×#92418—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519768—21no—
80#14519768—41no—
81#14519756—21no—
82#14519754—41no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86D4×Dic61—21no—
87#14519711—21no—
88#14519711—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—41no—
92#14519710D411yescharacteristic
93#14519708—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—41no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700D411yescharacteristic
109#14519698—21no—
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×SD16×C61D411yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#125×C61—21no—
120#125×C61—21no—
121#125×C61—41no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—41no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C2×C4⋊C4×C61—21no—
131#107×C61—21no—
132#107×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—41no—
151C22×Dic61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—42no—
155C2×Dic122—41no—
156C2×Dic122—81no—
157C2×Dic122C2×D411yescharacteristic
158SD16×C61Q1614yescomplemented
159C2×C8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183#11758202—41no—
184#11758202—42no—
185#11758202—81no—
186#11758202—81no—
187#11758202—81no—
188#11758202—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195C4⋊Dic61C2×D411yescharacteristic
196#11758196—21no—
197#11758196—41no—
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—81no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C23×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215Q8×C61C2×Q1612yes—
216D4×C61—21no—
217D4×C61C2×Q1612yes—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—161no—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—82no—
236Dic122—21no—
237Dic122—21no—
238Dic122—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—41no—
242Dic122—82no—
243C8×C61—21no—
244C8×C61—41no—
245C8×C61C2×Q1612yes—
246#92418—21no—
247#92418—21no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#19815—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61D4211yescharacteristic
257C4×C61—22no—
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C22×Q1611yescharacteristic
266C4×C61#57511yescharacteristic
267C4×C61#57611yescharacteristic
268C4×C61#57811yescharacteristic
269Dic61—41no—
270Dic61—41no—
271Dic61—81no—
272Dic61—81no—
273#3395—611no—
274#3386—611no—
275#3381D6111yescharacteristic
276#3299—611no—
277#1723—611no—
278#1692—611no—
279#1684—611no—
280#1674—611no—
281#1669—611no—
282#1649—611no—
283#1648—611no—
284#1644—611no—
285#1636—611no—
286#1584—611no—
287SD16⋊C8—611no—
288C2×C61—42no—
289C2×C61#326811yescharacteristic
290C2×C61D4×Q1611yescharacteristic
291C2×C61#328911yescharacteristic
292#557—611no—
293#557C2×D6111yescharacteristic
294#554C2×D6111yescharacteristic
295D4×Q8—611no—
296#501—1221no—
297#500—611no—
298#500—611no—
299#499—21no—
300#498C2×D6111yescharacteristic
301#494—611no—
302#494—1221no—
303#487—611no—
304#487—611no—
305#484—1221no—
306#479—21no—
307#478—21no—
308#478—1221no—
309#477—611no—
310#477—1221no—
311#477C2×D6111yescharacteristic
312#475—611no—
313#474—611no—
314#474C2×D6111yescharacteristic
315#467—1221no—
316#462—611no—
317#462—611no—
318#462—611no—
319#462—1221no—
320#462—1221no—
321#451—1221no—
322#446—611no—
323C8×Q8—611no—
324C8×Q8C2×D6111yescharacteristic
325#440—1221no—
326C4×Q16—611no—
327C4×Q16—1221no—
328C4×SD16—21no—
329C4×SD16C2×D6111yescharacteristic
330C8×D4—611no—
331C2×#103—1221no—
332#335—611no—
333#335—611no—
334#333—611no—
335#332—611no—
336#332—611no—
337#332—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341D4⋊C8—611no—
342#322—611no—
343C61#1981511yescharacteristic, complemented, Sylow
344C22×Q8—1221no—
345C2×Q16—611no—
346C2×Q16—1221no—
347C2×Q16—1221no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×Q16—1221no—
351C2×SD16C22⋊D6111yescharacteristic
352C22×C8—1221no—
353C4⋊Q8—21no—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#125—21no—
361#125—21no—
362#125—41no—
363#125—1221no—
364#125—2441no—
365C22⋊Q8—21no—
366C22⋊Q8—41no—
367C22⋊Q8—1221no—
368C22⋊Q8—2441no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—611no—
372C4×Q8—1221no—
373C4×Q8—1221no—
374C4×Q8C22×D6111yescharacteristic
375C4×Q8C22×D6111yescharacteristic
376C4×D4—21no—
377C4×D4—1221no—
378C4×D4C22×D6111yescharacteristic
379C2×C4⋊C4—21no—
380#107—21no—
381#107—21no—
382#107—611no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—1221no—
387#107—2441no—
388#106—21no—
389#106—21no—
390#106—1221no—
391#106C22⋊D6111yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8C22×D6111yescharacteristic
399#103—21no—
400#103—611no—
401#103—611no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—2441no—
410#103C22⋊D6111yescharacteristic
411#102—21no—
412#102—1221no—
413#102C22⋊D6111yescharacteristic
414C22⋊C8—1221no—
415#97—1221no—
416C4×C8—21no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—1221no—
420C4×C8C22×D6111yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—611no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—4881no—
427C2×Q8—2441no—
428C2×Q8—2442no—
429C2×Q8D4×D6111yescharacteristic
430C2×Q8C2×C22⋊D6111yescharacteristic
431C2×D4C2×C22⋊D6111yescharacteristic
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—2441no—
435Q16—4881no—
436Q16—4881no—
437Q16—4881no—
438Q16—4881no—
439Q16—2441no—
440Q16—2442no—
441SD16#1175820214yescomplemented
442C2×C8—21no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—4881no—
450C2×C8C2×C22⋊D6111yescharacteristic
451C2×C8C2×C22⋊D6111yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
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⋊C4D4×D6111yescharacteristic
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6111yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—41no—
478C22⋊C4—2441no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—1221no—
483C42—2441no—
484C42C23×D6111yescharacteristic
485C23—21no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—41no—
490Q8—41no—
491Q8—81no—
492Q8—1221no—
493Q8—1221no—
494Q8—4882no—
495Q8—2441no—
496Q8—2441no—
497Q8—2441no—
498Q8—2441no—
499Q8C2×#1175820212yes—
500D4—21no—
501D4C2×#1175820212yes—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—161no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—4882no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4#1451982211yescharacteristic
520C8—21no—
521C8—41no—
522C8—1221no—
523C8—1221no—
524C8—2441no—
525C8—2441no—
526C8—2441no—
527C8—2441no—
528C8C2×#1175820212yes—
529C22—22no—
530C22—41no—
531C22—41no—
532C22D4×C22⋊D6111yescharacteristic
533C4—22no—
534C4—21no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—81no—
540C4—81no—
541C4—4881no—
542C4—4881no—
543C4—2441no—
544C4—2441no—
545C4C22×#1175820211yescharacteristic
546C4#1684055911yescharacteristic
547C4#1684056411yescharacteristic
548C4#1684057711yescharacteristic
549C2—42no—
550C2#2145735911yescharacteristic
551C2D4×#1175820211yescharacteristic
552C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3381C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3381D61
hypocenterC61#19815
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3381

Complements

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

NH ≅ G/Ntype
1. #21458822C2semidirect
4. #21445493C2semidirect
5. #21445322C2semidirect
6. #21445279C2semidirect
7. #21445213C2semidirect
10. #21445061C2semidirect
12. #21444976C2semidirect
13. #21444669C2semidirect
158. SD16×C61Q16semidirect
343. C61#19815semidirect
441. SD16#11758202semidirect

Maximal subgroups

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

subgroupindex
#214588222
#214587382
#214576892
#214454932
#214453222
#214452792
#214452132
#214451822
#214450672
#214450612
#214450332
#214449762
#214446692
SD16⋊#924182
C61×#33812
#1981561