Finite Group Databasebeta

← #28348036

Subgroups — #28348036

Order 13696.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812142C211yesmaximal, characteristic, complemented
2C22×Dic428C211yesmaximal, characteristic, complemented
3D4×Dic214C211yesmaximal, characteristic, complemented
4#20293286C211yesmaximal, characteristic
5#20293021C211yesmaximal, characteristic, complemented
6#20292885C211yesmaximal, characteristic, complemented
7#20292856C211yesmaximal, characteristic, complemented
8#20292850C211yesmaximal, characteristic, complemented
9#20292838C211yesmaximal, characteristic, complemented
10#20292791C211yesmaximal, characteristic, complemented
11#20292785C211yesmaximal, characteristic, complemented
12#20292785C211yesmaximal, characteristic, complemented
13#20292768C211yesmaximal, characteristic, complemented
14#20292735C211yesmaximal, characteristic, complemented
15#435×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C22×Dic214C2211yescharacteristic
19C22×Dic214C2211yescharacteristic
20#16812142—24no—
21C2×Dic428—24no—
22C2×Dic428C2211yescharacteristic
23C2×Dic428C2211yescharacteristic
24C2×Dic428C2211yescharacteristic
25C2×Dic428C2211yescharacteristic
26C2×Dic428C2211yescharacteristic
27C2×Dic428C2211yescharacteristic
28C2×#13133807C2211yescharacteristic
29C2×#13133807C2211yescharacteristic
30C2×#13133807C2211yescharacteristic
31#16812109C2211yescharacteristic
32#16812108—21no—
33#16812097C2211yescharacteristic
34#16812095—21no—
35#16812088C2211yescharacteristic
36#16812088C2211yescharacteristic
37#16812088C2211yescharacteristic
38#16812087—21no—
39#16812086—21no—
40#16812086—21no—
41#16812083C2211yescharacteristic
42#16812081—21no—
43C4×Dic214C2211yescharacteristic
44C4×Dic214C2211yescharacteristic
45D4×Dic107—21no—
46(C22⋊D107)⋊C4—21no—
47#16812041C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812034C2211yescharacteristic
50#16812034C2211yescharacteristic
51#16812034C2211yescharacteristic
52#16812034C2211yescharacteristic
53#16812031C2211yescharacteristic
54C2×M16×C107C2211yescharacteristic
55C22×C8×C107C2211yescharacteristic
56C4×D4×C107C2211yescharacteristic
57C4⋊C8×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59C22⋊C8×C107C2211yescharacteristic
60#97×C107C2211yescharacteristic
61#13133818—22no—
62#13133818—41no—
63C2×Dic214—22no—
64C2×Dic214—22no—
65C2×Dic214—22no—
66C2×Dic214—21no—
67C2×Dic214—21no—
68C2×Dic214—21no—
69C2×Dic214—41no—
70C2×Dic214—41no—
71C2×Dic214—41no—
72C2×Dic214C2311yescharacteristic
73C2×Dic214C2311yescharacteristic
74C2×Dic214C2311yescharacteristic
75C2×Dic214C2311yescharacteristic
76C2×Dic214C2311yescharacteristic
77C2×D428C2311yescharacteristic
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C22×Dic107—21no—
81C22×Dic107—21no—
82C22×Dic107—21no—
83C2×C4×D107—21no—
84Dic428—22no—
85Dic428—22no—
86Dic428—22no—
87Dic428—22no—
88Dic428—41no—
89Dic428D414yescomplemented
90#13133807—22no—
91#13133807—22no—
92#13133807—41no—
93C4⋊Dic107—21no—
94C4⋊Dic107—21no—
95C4⋊Dic107C2311yescharacteristic
96C4⋊Dic107C2311yescharacteristic
97Dic107⋊C4—21no—
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—
108C4×Dic107—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—
126D4×C107—22no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2×D411yescharacteristic
134C2×C4×C107C2×D411yescharacteristic
135C2×C4×C107C2411yescharacteristic
136C22⋊D107—42no—
137C22⋊D107—42no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—42no—
146C2×Dic107—42no—
147C2×Dic107—42no—
148D428—21no—
149D428—21no—
150C4×D107—42no—
151Dic214—21no—
152Dic214—21no—
153Dic214—21no—
154Dic214—21no—
155Dic214—21no—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—42no—
160Dic214—42no—
161Dic214—42no—
162Dic214C2×D412yes—
163Dic214C2×D412yes—
164C8×C107—21no—
165C8×C107—21no—
166C8×C107—21no—
167C8×C107C2×D412yes—
168C22×C107—21no—
169C22×C107—21no—
170C22×C107—21no—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107#13712yescomplemented
174C22×C107C22×D411yescharacteristic
175C2×D107—41no—
176C2×D107—41no—
177C2×D107—41no—
178C4×C107—21no—
179C4×C107—21no—
180C4×C107—21no—
181C4×C107—21no—
182C4×C107C22×D411yescharacteristic
183C4×C107H(5,𝔽2)11yescharacteristic
184Dic107—22no—
185Dic107—22no—
186Dic107—41no—
187Dic107—41no—
188Dic107—41no—
189Dic107—41no—
190Dic107—41no—
191C2×C107—22no—
192C2×C107—41no—
193C2×C107D4211yescharacteristic
194C2×C107C2×#13711yescharacteristic
195C2×C107#57811yescharacteristic
196D107—81no—
197#3270—1071nomaximal, Sylow
198C107#327011yescharacteristic, complemented, Sylow
199C2×#137—1071no—
200C22×Q16—1071no—
201D4×Q8—1071no—
202#548—1071no—
203#497—1071no—
204#470—1071no—
205#465—1071no—
206#464—1071no—
207#462—1071no—
208#452—1071no—
209#451—1071no—
210#451—1071no—
211#448—1071no—
212#441—1071no—
213#435D10711yescharacteristic, complemented
214C2×#41—1071no—
215C22×Q8—1071no—
216C22×Q8—1071no—
217C22×Q8—1071no—
218#137—2144no—
219C2×Q16—1071no—
220C2×Q16—1071no—
221C2×Q16—1071no—
222C2×Q16—1071no—
223C2×Q16—1071no—
224C2×Q16—1071no—
225C2×Q16—2144no—
226C2×SD16—1071no—
227C2×SD16—1071no—
228C2×SD16—1071no—
229C2×M16C2×D10711yescharacteristic
230C22×C8C2×D10711yescharacteristic
231C4⋊Q8—1071no—
232C4⋊Q8—2141no—
233#124—1071no—
234#124—2141no—
235C22⋊Q8—1071no—
236C22⋊Q8—1071no—
237C22⋊Q8—1071no—
238C22⋊Q8—2141no—
239C22⋊Q8—2141no—
240C22⋊Q8—2141no—
241#121—1071no—
242#121—2141no—
243C4×Q8—1071no—
244C4×Q8—1071no—
245C4×D4—2141no—
246C4×D4—2141no—
247C4×D4C2×D10711yescharacteristic
248#106—1071no—
249C4⋊C8C2×D10711yescharacteristic
250#103—1071no—
251#103—1071no—
252#103—1071no—
253#103—1071no—
254#103—1071no—
255#102—1071no—
256C22⋊C8C2×D10711yescharacteristic
257C22⋊C8C2×D10711yescharacteristic
258#97C2×D10711yescharacteristic
259#41—4281no—
260#41—2142no—
261C2×Q8—1071no—
262C2×Q8—1071no—
263C2×Q8—1071no—
264C2×Q8—1071no—
265C2×Q8—1071no—
266C2×Q8—4281no—
267C2×Q8—4281no—
268C2×Q8—4281no—
269C2×Q8—2142no—
270C2×Q8—2142no—
271C2×Q8—2142no—
272C2×Q8—2141no—
273C2×Q8—2141no—
274C2×Q8—2141no—
275C2×D4—1071no—
276C2×D4—2141no—
277C2×D4—2141no—
278C2×D4D42811yescharacteristic
279C22×C4—2141no—
280C22×C4—2141no—
281C22×C4—2141no—
282C22×C4—2141no—
283C22×C4C22×D10711yescharacteristic
284C22×C4C22×D10711yescharacteristic
285Q16—1074no—
286Q16—4281no—
287Q16—2142no—
288Q16—2142no—
289Q16—2142no—
290Q16—2142no—
291SD16—4281no—
292SD16—2142no—
293SD16—2142no—
294M16—22no—
295C2×C8—22no—
296C2×C8C22×D10711yescharacteristic
297C2×C8C22×D10711yescharacteristic
298C2×C8C22×D10711yescharacteristic
299C2×C8C22×D10711yescharacteristic
300C4⋊C4—1071no—
301C4⋊C4—1071no—
302C4⋊C4—2141no—
303C4⋊C4—2141no—
304C4⋊C4—2141no—
305C4⋊C4—2141no—
306C4⋊C4—2141no—
307C4⋊C4—2141no—
308C4⋊C4—2141no—
309C4⋊C4D42811yescharacteristic
310C22⋊C4—2141no—
311C22⋊C4—2141no—
312C22⋊C4—2141no—
313C22⋊C4—2141no—
314C22⋊C4—2141no—
315C22⋊C4—2141no—
316C22⋊C4D42811yescharacteristic
317C22⋊C4D42811yescharacteristic
318C42—2141no—
319C42—2141no—
320C42C22×D10711yescharacteristic
321C23—2141no—
322C23C2×D42811yescharacteristic
323C23C2×D42811yescharacteristic
324Q8—1072no—
325Q8—1072no—
326Q8—4282no—
327Q8—4282no—
328Q8—4282no—
329Q8—2141no—
330Q8—2141no—
331Q8—2141no—
332Q8—2141no—
333Q8—2141no—
334Q8—2141no—
335Q8—2141no—
336Q8—2141no—
337D4—22no—
338D4—4282no—
339D4—4282no—
340D4—2141no—
341D4—2141no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—4282no—
347C2×C4—4282no—
348C2×C4—4282no—
349C2×C4—4282no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4C2×D42811yescharacteristic
358C2×C4C2×D42811yescharacteristic
359C2×C4C2×D42811yescharacteristic
360C2×C4C2×D42811yescharacteristic
361C2×C4C23×D10711yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8D4×D10712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—4281no—
372C22—4281no—
373C22—4281no—
374C22#1681214212yescomplemented
375C22C22×D42811yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—4281no—
381C4—4281no—
382C4—4281no—
383C4—4281no—
384C4—4281no—
385C4—2142no—
386C4—2142no—
387C4C2×D4×D10711yescharacteristic
388C4#1681215511yescharacteristic
389C2—22no—
390C2—41no—
391C2—8561no—
392C2D4×D42811yescharacteristic
393C2C2×#1681214211yescharacteristic
394C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#435×C107C2
socleC22×C107C22×D4
prime radical#20293286C2
hypercenter#435D107
hypocenterC107#3270
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×Dic428C2semidirect
3. D4×Dic214C2semidirect
5. #20293021C2semidirect
6. #20292885C2semidirect
7. #20292856C2semidirect
8. #20292850C2semidirect
9. #20292838C2semidirect
10. #20292791C2semidirect
11. #20292785C2semidirect
12. #20292785C2semidirect
13. #20292768C2semidirect
14. #20292735C2semidirect
15. #435×C107C2semidirect
89. Dic428D4semidirect
173. C22×C107#137semidirect
198. C107#3270semidirect
213. #435D107semidirect
374. 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×Dic4282
D4×Dic2142
#202932862
#202930212
#202928852
#202928562
#202928502
#202928382
#202927912
#202927852
#202927682
#202927352
#435×C1072
#3270107