Finite Group Databasebeta

← #28347976

Subgroups — #28347976

Order 13696.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812142C211yesmaximal, characteristic, complemented
2C22×#13133807C211yesmaximal, characteristic, complemented
3D4×Dic214C211yesmaximal, characteristic, complemented
4#20293273C211yesmaximal, characteristic
5#20293021C211yesmaximal, characteristic, complemented
6#20292873C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292850C211yesmaximal, characteristic, complemented
9#20292833C211yesmaximal, characteristic, complemented
10#20292791C211yesmaximal, characteristic, complemented
11#20292785C211yesmaximal, characteristic, complemented
12#20292780C211yesmaximal, characteristic, complemented
13#20292768C211yesmaximal, characteristic, complemented
14#20292729C211yesmaximal, characteristic, complemented
15#435×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C22×Dic214C2211yescharacteristic, complemented
19C22×D428C2211yescharacteristic
20#16812142—24no—
21C2×Dic428C2211yescharacteristic, complemented
22C2×Dic428C2211yescharacteristic, complemented
23C2×Dic428C2211yescharacteristic, complemented
24C2×#13133807—24no—
25C2×#13133807C2211yescharacteristic
26C2×#13133807C2211yescharacteristic
27C2×#13133807C2211yescharacteristic
28C2×#13133807C2211yescharacteristic
29C2×#13133807C2211yescharacteristic
30C2×#13133807C2211yescharacteristic
31#16812109C2211yescharacteristic, complemented
32#16812108—21no—
33#16812097C2211yescharacteristic
34#16812094—21no—
35#16812088C2211yescharacteristic
36#16812088C2211yescharacteristic
37#16812088C2211yescharacteristic, complemented
38#16812086—21no—
39#16812086—21no—
40#16812083C2211yescharacteristic
41#16812082—21no—
42#16812077—21no—
43C4×Dic214C2211yescharacteristic, complemented
44C4×D428C2211yescharacteristic
45D4×Dic107—21no—
46C22⋊C4×D107—21no—
47#16812043C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812034C2211yescharacteristic, complemented
50#16812034C2211yescharacteristic, complemented
51#16812034C2211yescharacteristic, complemented
52#16812031C2211yescharacteristic
53#16812031C2211yescharacteristic
54C2×M16×C107C2211yescharacteristic, complemented
55C22×C8×C107C2211yescharacteristic
56C4×D4×C107C2211yescharacteristic
57C4⋊C8×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C22⋊C8×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133818—22no—
63#13133818—41no—
64C2×Dic214—22no—
65C2×Dic214—22no—
66C2×Dic214—21no—
67C2×Dic214—21no—
68C2×Dic214—41no—
69C2×Dic214—41no—
70C2×Dic214C2311yescharacteristic
71C2×Dic214C2311yescharacteristic
72C2×Dic214C2311yescharacteristic
73C2×Dic214C2311yescharacteristic
74C2×D428—22no—
75C2×D428—21no—
76C2×D428—41no—
77C2×D428C2311yescharacteristic
78C2×D428C2311yescharacteristic
79C2×C22⋊D107—21no—
80C2×C22⋊D107—21no—
81C22×Dic107—21no—
82C22×Dic107—21no—
83C2×C4×D107—21no—
84C2×C4×D107—21no—
85Dic428—22no—
86Dic428—22no—
87Dic428—41no—
88#13133807—22no—
89#13133807—22no—
90#13133807—22no—
91#13133807—22no—
92#13133807—41no—
93#13133807D414yescomplemented
94C4⋊Dic107—21no—
95C4⋊Dic107—21no—
96C4⋊Dic107C2311yescharacteristic
97C4⋊Dic107C2311yescharacteristic
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100Dic107⋊C4—21no—
101Dic107⋊C4—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107—21no—
104C22⋊Dic107—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109C4×Dic107—21no—
110C2×D4×C107D411yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2311yescharacteristic
113M16×C107—22no—
114C2×C8×C107—22no—
115C2×C8×C107C2311yescharacteristic
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107D411yescharacteristic
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C42×C107C2311yescharacteristic
123C23×C107C2×D411yescharacteristic
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130D4×C107—22no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107C2×D411yescharacteristic
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2411yescharacteristic
140C22⋊D107—42no—
141C22⋊D107—81no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—42no—
149C2×Dic107—42no—
150D428—21no—
151D428—21no—
152D428—21no—
153D428—42no—
154D428C2×D412yes—
155C4×D107—42no—
156C4×D107—81no—
157Dic214—21no—
158Dic214—21no—
159Dic214—21no—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214—21no—
164Dic214—42no—
165Dic214—42no—
166Dic214C2×D412yes—
167C8×C107—21no—
168C8×C107—21no—
169C8×C107—21no—
170C8×C107C2×D412yes—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107#13712yescomplemented
177C22×C107C22×D411yescharacteristic
178C2×D107—22no—
179C2×D107—41no—
180C2×D107—41no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—41no—
184C2×D107—84no—
185C4×C107—21no—
186C4×C107—21no—
187C4×C107—21no—
188C4×C107—21no—
189C4×C107C22×D411yescharacteristic
190C4×C107H(5,𝔽2)11yescharacteristic
191Dic107—22no—
192Dic107—41no—
193Dic107—41no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—22no—
198C2×C107—41no—
199C2×C107D4211yescharacteristic
200C2×C107C2×#13711yescharacteristic
201C2×C107#57711yescharacteristic
202D107—42no—
203D107—81no—
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16—1071no—
208D4×Q8—1071no—
209#546—1071no—
210#497—1071no—
211#468—1071no—
212#467—1071no—
213#464—1071no—
214#461—1071no—
215#452—1071no—
216#451—1071no—
217#450—1071no—
218#448—1071no—
219#440—1071no—
220#435D10711yescharacteristic, complemented
221C2×#41—1071no—
222C22×Q8—1071no—
223C22×Q8—1071no—
224C22×D4—1071no—
225#137—2144no—
226C2×Q16—1071no—
227C2×Q16—1071no—
228C2×Q16—1071no—
229C2×SD16—1071no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—2144no—
236C2×M16C2×D10711yescharacteristic, complemented
237C22×C8C2×D10711yescharacteristic
238C4⋊Q8—1071no—
239C4⋊Q8—2141no—
240#124—1071no—
241#123—2141no—
242C22⋊Q8—1071no—
243C22⋊Q8—1071no—
244C22⋊Q8—1071no—
245C22⋊Q8—2141no—
246C22⋊Q8—2141no—
247#121—1071no—
248#121—2141no—
249#120—2141no—
250C4×Q8—1071no—
251C4×D4—1071no—
252C4×D4—2141no—
253C4×D4C2×D10711yescharacteristic
254C2×C22⋊C4—2141no—
255#107—1071no—
256C4⋊C8C2×D10711yescharacteristic, complemented
257#103—1071no—
258#103—1071no—
259#103—1071no—
260#103—1071no—
261#102—1071no—
262#102—1071no—
263C22⋊C8C2×D10711yescharacteristic
264C22⋊C8C2×D10711yescharacteristic, complemented
265#97C2×D10711yescharacteristic, complemented
266C24—2141no—
267#41—4281no—
268#41—2142no—
269C2×Q8—1071no—
270C2×Q8—1071no—
271C2×Q8—1071no—
272C2×Q8—1071no—
273C2×Q8—4281no—
274C2×Q8—4281no—
275C2×Q8—2142no—
276C2×Q8—2142no—
277C2×Q8—2141no—
278C2×Q8—2141no—
279C2×D4—1071no—
280C2×D4—1071no—
281C2×D4—4281no—
282C2×D4—2142no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4D42811yescharacteristic
287C22×C4—2141no—
288C22×C4—2141no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4C22×D10711yescharacteristic
292C22×C4C22×D10711yescharacteristic
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—1074no—
297SD16—4281no—
298SD16—2142no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D10711yescharacteristic
305C2×C8C22×D10711yescharacteristic
306C2×C8C22×D10711yescharacteristic
307C2×C8C22×D10711yescharacteristic
308C4⋊C4—1071no—
309C4⋊C4—1071no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4—2141no—
316C4⋊C4D42811yescharacteristic
317C22⋊C4—2141no—
318C22⋊C4—2141no—
319C22⋊C4—2141no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4D42811yescharacteristic
325C22⋊C4D42811yescharacteristic
326C42—2141no—
327C42C22×D10711yescharacteristic
328C23—8561no—
329C23—4282no—
330C23—4282no—
331C23—2141no—
332C23—2141no—
333C23C2×D42811yescharacteristic
334C23C2×D42811yescharacteristic
335Q8—1072no—
336Q8—4282no—
337Q8—4282no—
338Q8—2141no—
339Q8—2141no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8—2141no—
345D4—22no—
346D4—8561no—
347D4—1072no—
348D4—4282no—
349D4—4282no—
350D4—2141no—
351D4—2141no—
352D4—2141no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—8561no—
358C2×C4—4282no—
359C2×C4—4282no—
360C2×C4—4282no—
361C2×C4—2141no—
362C2×C4—2141no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4C2×D42811yescharacteristic
368C2×C4C2×D42811yescharacteristic
369C2×C4C2×D42811yescharacteristic
370C2×C4C2×D42811yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D10712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—8564no—
382C22—4281no—
383C22—4281no—
384C22—4281no—
385C22—4281no—
386C22—4281no—
387C22—2142no—
388C22#1681214212yescomplemented
389C22C22×D42811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—4281no—
395C4—4281no—
396C4—4281no—
397C4—4281no—
398C4—4281no—
399C4—2142no—
400C4C2×D4×D10711yescharacteristic
401C4#1681215511yescharacteristic
402C2—22no—
403C2—41no—
404C2—8561no—
405C2—4282no—
406C2D4×D42811yescharacteristic
407C2C2×#1681214211yescharacteristic
408C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#435×C107C2
socleC22×C107C22×D4
prime radical#20293273C2
hypercenter#435D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C22×#13133807C2semidirect
3. D4×Dic214C2semidirect
5. #20293021C2semidirect
6. #20292873C2semidirect
7. #20292868C2semidirect
8. #20292850C2semidirect
9. #20292833C2semidirect
10. #20292791C2semidirect
11. #20292785C2semidirect
12. #20292780C2semidirect
13. #20292768C2semidirect
14. #20292729C2semidirect
15. #435×C107C2semidirect
18. C22×Dic214C22semidirect
21. C2×Dic428C22semidirect
22. C2×Dic428C22semidirect
23. C2×Dic428C22semidirect
31. #16812109C22semidirect
37. #16812088C22semidirect
43. C4×Dic214C22semidirect
47. #16812043C22semidirect
48. #16812034C22semidirect
49. #16812034C22semidirect
50. #16812034C22semidirect
51. #16812034C22semidirect
54. C2×M16×C107C22semidirect
57. C4⋊C8×C107C22semidirect
59. C22⋊C8×C107C22semidirect
60. #97×C107C22semidirect
93. #13133807D4semidirect
176. C22×C107#137semidirect
205. C107#3266semidirect
220. #435D107semidirect
236. C2×M16C2×D107semidirect
256. C4⋊C8C2×D107semidirect
264. C22⋊C8C2×D107semidirect
265. #97C2×D107semidirect
388. C22#16812142semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C22×#131338072
D4×Dic2142
#202932732
#202930212
#202928732
#202928682
#202928502
#202928332
#202927912
#202927852
#202927802
#202927682
#202927292
#435×C1072
#3266107