Finite Group Databasebeta

← #27158468

Subgroups — #27158468

Order 12416.

502 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690897C211yesmaximal, characteristic, complemented
2C22×#13095520C211yesmaximal, characteristic, complemented
3#20261770C211yesmaximal, characteristic, complemented
4#20261768C211yesmaximal, characteristic, complemented
5#20261767C211yesmaximal, characteristic, complemented
6#20261766C211yesmaximal, characteristic, complemented
7#20261766C211yesmaximal, characteristic, complemented
8#20261764C211yesmaximal, characteristic, complemented
9#20261762C211yesmaximal, characteristic, complemented
10D4×#11715897C211yesmaximal, characteristic, complemented
11#20261741C211yesmaximal, characteristic, complemented
12#20261708C211yesmaximal, characteristic, complemented
13C2×C22⋊#11715897C211yesmaximal, characteristic, complemented
14#20261699C211yesmaximal, characteristic, complemented
15#20261241C211yesmaximal, characteristic
16#15690897—24no—
17C2×#13095521C2211yescharacteristic, complemented
18C2×#13095520—24no—
19C2×#13095520C2211yescharacteristic
20C2×#13095520C2211yescharacteristic
21C2×#13095520C2211yescharacteristic
22C2×#13095520C2211yescharacteristic
23C2×#13095520C2211yescharacteristic, complemented
24C2×#13095519C2211yescharacteristic, complemented
25C22×#11715897C2211yescharacteristic
26C22×#11715897C2211yescharacteristic
27C22×#11715897C2211yescharacteristic
28C22×#11715897C2211yescharacteristic
29C22×#11715897C2211yescharacteristic, complemented
30#15690877C2211yescharacteristic, complemented
31#15690877C2211yescharacteristic, complemented
32C4⋊#11715897C2211yescharacteristic, complemented
33C4⋊#11715897C2211yescharacteristic, complemented
34#15690867C2211yescharacteristic, complemented
35#15690867C2211yescharacteristic, complemented
36C22⋊#11715897C2211yescharacteristic
37C22⋊#11715897C2211yescharacteristic
38C22⋊#11715897C2211yescharacteristic
39C22⋊#11715897C2211yescharacteristic
40C22⋊#11715897C2211yescharacteristic, complemented
41C22⋊#11715897C2211yescharacteristic, complemented
42#15690865C2211yescharacteristic, complemented
43#15690865C2211yescharacteristic, complemented
44C4×#11715897C2211yescharacteristic, complemented
45C4×#11715897C2211yescharacteristic, complemented
46C2×#13095510C2211yescharacteristic
47C23×Dic97C2211yescharacteristic
48#15690794C411yescharacteristic
49#15690791C411yescharacteristic
50#15690787C411yescharacteristic
51#15690787C411yescharacteristic
52#15690782C411yescharacteristic
53#15690782C411yescharacteristic
54#15690779C411yescharacteristic
55D4×Dic97C2211yescharacteristic
56D4×Dic97C2211yescharacteristic
57#15690770C2211yescharacteristic
58#15690770C2211yescharacteristic
59#15690766C2211yescharacteristic
60#120×C97C411yescharacteristic
61#13095521—22no—
62#13095520—22no—
63#13095520—22no—
64#13095520—22no—
65#13095520—22no—
66#13095520D414yescomplemented
67#13095519—22no—
68C2×#11715897—22no—
69C2×#11715897—22no—
70C2×#11715897—22no—
71C2×#11715897—22no—
72C2×#11715897—22no—
73C2×#11715897C2311yescharacteristic
74C2×#11715897C2311yescharacteristic
75C2×#11715897C2311yescharacteristic
76C2×#11715897C2311yescharacteristic
77C2×#11715897C2311yescharacteristic
78C2×#11715897C2311yescharacteristic
79C2×#11715897C2311yescharacteristic
80C2×#11715897C2311yescharacteristic
81#13095510—22no—
82#13095510—22no—
83C2×Dic194C2×C411yescharacteristic
84C2×C22⋊D97C2×C411yescharacteristic
85C2×C22⋊D97C2×C411yescharacteristic
86C2×C22⋊D97C2×C411yescharacteristic
87C2×C22⋊D97C2×C411yescharacteristic
88C22×Dic97—22no—
89C22×Dic97—22no—
90C22×Dic97C2×C411yescharacteristic
91C22×Dic97C2×C411yescharacteristic
92C22×Dic97C2×C411yescharacteristic
93C22×Dic97C2311yescharacteristic
94C22×Dic97C2311yescharacteristic
95C22×Dic97C2311yescharacteristic
96C22×Dic97C2311yescharacteristic
97C2×C4×D97C2311yescharacteristic
98C4⋊Dic97C2×C411yescharacteristic
99C4⋊Dic97C2×C411yescharacteristic
100#13095490C2×C411yescharacteristic
101#13095490C2×C411yescharacteristic
102#13095490C2×C411yescharacteristic
103#13095490C2×C411yescharacteristic
104C22⋊Dic97C2×C411yescharacteristic
105C22⋊Dic97C2×C411yescharacteristic
106C22⋊Dic97C2×C411yescharacteristic
107C22⋊Dic97C2×C411yescharacteristic
108C22⋊Dic97C2×C411yescharacteristic
109#13095488C2×C411yescharacteristic
110#13095488C2×C411yescharacteristic
111C4×Dic97C2311yescharacteristic
112C4×Dic97C2311yescharacteristic
113C24×C97C2×C411yescharacteristic
114C2×D4×C97C2×C411yescharacteristic
115C2×D4×C97C2×C411yescharacteristic
116C2×D4×C97C2×C411yescharacteristic
117C22⋊C4×C97C2×C411yescharacteristic
118C22⋊C4×C97C2×C411yescharacteristic
119C22⋊C4×C97C2×C411yescharacteristic
120C23×C97—22no—
121C23×C97—22no—
122C23×C97—41no—
123C23×C97C22×C411yescharacteristic
124C23×C97C22×C411yescharacteristic
125C23×C97C22×C411yescharacteristic
126C23×C97C22×C411yescharacteristic
127C22×D97C22×C411yescharacteristic
128D4×C97—22no—
129D4×C97—22no—
130D4×C97—41no—
131C2×C4×C97C22×C411yescharacteristic
132C2×C4×C97C22×C411yescharacteristic
133C2×C4×C97C22×C411yescharacteristic
134C22⋊D97—22no—
135C22⋊D97—22no—
136C22⋊D97—22no—
137C22⋊D97—41no—
138C2×Dic97—22no—
139C2×Dic97—22no—
140C2×Dic97—22no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97C22×C411yescharacteristic
148C2×Dic97C22×C411yescharacteristic
149C2×Dic97C22×C411yescharacteristic
150C2×Dic97C22×C411yescharacteristic
151C2×Dic97C22×C411yescharacteristic
152C2×Dic97C22×C411yescharacteristic
153C2×Dic97C2×D412yes—
154C2×Dic97#4112yes—
155C2×Dic97C2411yescharacteristic
156C4×D97—21no—
157C4×D97—21no—
158Dic194—22no—
159#11715897—21no—
160#11715897—21no—
161#11715897—21no—
162#11715897—21no—
163#11715897—21no—
164#11715897—21no—
165#11715897C2×D412yes—
166#11715897C2×D412yes—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—44no—
178C22×C97C4×D412yes—
179C22×C97#13112yescomplemented
180C22×C97C23×C411yescharacteristic
181C2×D97—21no—
182C2×D97—21no—
183C2×D97—21no—
184C4×C97—21no—
185C4×C97—21no—
186C4×C97—21no—
187Dic97—21no—
188Dic97—21no—
189Dic97—21no—
190Dic97—21no—
191Dic97—21no—
192Dic97C4×D412yes—
193Dic97C22×D411yescharacteristic
194Dic97C2×#4111yescharacteristic
195C2×C97—22no—
196C2×C97—22no—
197C2×C97—41no—
198C2×C97—41no—
199C2×C97C2×C4×D411yescharacteristic
200C2×C97C2×#13111yescharacteristic
201C2×C97#56811yescharacteristic
202D97—41no—
203#2933—971nomaximal, Sylow
204C97#293311yescharacteristic, complemented, Sylow
205C2×#131—971no—
206C22×M16—971no—
207#525—971no—
208#436—971no—
209#436—971no—
210#435—971no—
211#435—971no—
212#435—971no—
213#435—971no—
214C8×D4—971no—
215C8×D4—971no—
216#424—971no—
217#408—971no—
218C2×C22⋊C8—971no—
219#405—971no—
220C2×#41—971no—
221C23×C4—971no—
222#131—1944no—
223C2×M16—971no—
224C2×M16—971no—
225C2×M16—971no—
226C2×M16—971no—
227C2×M16—971no—
228C2×M16—971no—
229C2×M16—1944no—
230C22×C8—971no—
231C22×C8—971no—
232C22×C8—971no—
233C22×C8—971no—
234C22×C8—971no—
235C22×C8—971no—
236#123—971no—
237#123—971no—
238C22⋊Q8—971no—
239C22⋊Q8—971no—
240#121—971no—
241#121—971no—
242#120—971no—
243#120#8760411yescharacteristic
244C4×D4—971no—
245C4×D4—971no—
246C4×D4—971no—
247C4×D4—971no—
248#117—971no—
249C4⋊C8—971no—
250C4⋊C8—971no—
251C4⋊C8—971no—
252C4⋊C8—971no—
253C22⋊C8—971no—
254C22⋊C8—971no—
255C22⋊C8—971no—
256C22⋊C8—971no—
257C22⋊C8—971no—
258C22⋊C8—971no—
259C22⋊C8—971no—
260C22⋊C8—971no—
261#97—971no—
262#97—971no—
263C4×C8—971no—
264C4×C8—971no—
265C24C2×#8760411yescharacteristic
266#41—1942no—
267#41—1942no—
268C2×Q8—971no—
269C2×D4—971no—
270C2×D4—971no—
271C2×D4—971no—
272C2×D4—971no—
273C2×D4C2×#8760411yescharacteristic
274C2×D4C2×#8760411yescharacteristic
275C2×D4C2×#8760411yescharacteristic
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—971no—
283C22×C4—971no—
284C22×C4—1942no—
285C22×C4—1942no—
286M16—974no—
287M16—1942no—
288M16—1942no—
289M16—1942no—
290M16—1942no—
291M16—1942no—
292C2×C8—971no—
293C2×C8—971no—
294C2×C8—971no—
295C2×C8—971no—
296C2×C8—971no—
297C2×C8—971no—
298C2×C8—971no—
299C2×C8—971no—
300C2×C8—1942no—
301C2×C8—1942no—
302C2×C8—1942no—
303C2×C8—1942no—
304C2×C8—1942no—
305C2×C8—1942no—
306C4⋊C4—971no—
307C4⋊C4—971no—
308C4⋊C4—971no—
309C4⋊C4—971no—
310C4⋊C4—971no—
311C4⋊C4—971no—
312C22⋊C4—971no—
313C22⋊C4—971no—
314C22⋊C4—971no—
315C22⋊C4—971no—
316C22⋊C4—971no—
317C22⋊C4—971no—
318C22⋊C4—971no—
319C22⋊C4C2×#8760411yescharacteristic
320C22⋊C4C2×#8760411yescharacteristic
321C22⋊C4C2×#8760411yescharacteristic
322C42—971no—
323C42—971no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23—971no—
328C23C22×#8760411yescharacteristic
329C23C22×#8760411yescharacteristic
330C23C22×#8760411yescharacteristic
331C23C22×#8760411yescharacteristic
332Q8—1942no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—3881no—
337D4—1942no—
338D4—1942no—
339D4—1942no—
340C2×C4—972no—
341C2×C4—972no—
342C2×C4—971no—
343C2×C4—971no—
344C2×C4—971no—
345C2×C4—971no—
346C2×C4—971no—
347C2×C4—971no—
348C2×C4—971no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1942no—
357C2×C4—1942no—
358C2×C4—1941no—
359C2×C4—1942no—
360C2×C4C22×#8760411yescharacteristic
361C2×C4C22×#8760411yescharacteristic
362C2×C4C22×#8760411yescharacteristic
363C8—972no—
364C8—972no—
365C8—1941no—
366C8—1941no—
367C8—1941no—
368C8—1941no—
369C8—1941no—
370C8—1941no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—1941no—
383C22—1941no—
384C22—1941no—
385C22D4×#8760412yes—
386C22#1569089712yescomplemented
387C22C23×#8760411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—971no—
392C4—971no—
393C4—972no—
394C4—1941no—
395C4—1941no—
396C4—1941no—
397C4—1941no—
398C4—1941no—
399C2—22no—
400C2—22no—
401C2—41no—
402C2—41no—
403C2—3881no—
404C2C2×D4×#8760411yescharacteristic
405C2C2×#1569089711yescharacteristic
406C2#2026181311yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#120×C97C4
prime radical#20261241C2
cosocleC2×Dic97C24
hypercenter#120#87604
hypocenterC97#2933
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#15690897C2semidirect
2. C22×#13095520C2semidirect
3. #20261770C2semidirect
4. #20261768C2semidirect
5. #20261767C2semidirect
6. #20261766C2semidirect
7. #20261766C2semidirect
8. #20261764C2semidirect
9. #20261762C2semidirect
10. D4×#11715897C2semidirect
11. #20261741C2semidirect
12. #20261708C2semidirect
13. C2×C22⋊#11715897C2semidirect
14. #20261699C2semidirect
17. C2×#13095521C22semidirect
23. C2×#13095520C22semidirect
24. C2×#13095519C22semidirect
29. C22×#11715897C22semidirect
30. #15690877C22semidirect
31. #15690877C22semidirect
32. C4⋊#11715897C22semidirect
33. C4⋊#11715897C22semidirect
34. #15690867C22semidirect
35. #15690867C22semidirect
40. C22⋊#11715897C22semidirect
41. C22⋊#11715897C22semidirect
42. #15690865C22semidirect
43. #15690865C22semidirect
44. C4×#11715897C22semidirect
45. C4×#11715897C22semidirect
66. #13095520D4semidirect
179. C22×C97#131semidirect
204. C97#2933semidirect
386. C22#15690897semidirect

Maximal subgroups

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

subgroupindex
C2×#156908972
C22×#130955202
#202617702
#202617682
#202617672
#202617662
#202617642
#202617622
D4×#117158972
#202617412
#202617082
C2×C22⋊#117158972
#202616992
#202612412
#293397