Finite Group Databasebeta

← #27155352

Subgroups — #27155352

Order 12416.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261461C211yesmaximal, characteristic, complemented
2#20261454C213yesmaximal, complemented
3#20261425C213yesmaximal, complemented
4#20261419C213yesmaximal, complemented
5Q8×C22⋊D97C211yesmaximal, characteristic, complemented
6#20261392C211yesmaximal, characteristic, complemented
7#20261385C213yesmaximal, complemented
8#20261287C213yesmaximal, complemented
9#20261266C213yesmaximal, complemented
10#20261264C213yesmaximal, complemented
11#41×Dic97C211yesmaximal, characteristic, complemented
12C4×#13095510C213yesmaximal, complemented
13C2×Q8×Dic97C211yesmaximal, characteristic, complemented
14C4×Q8×D97C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2213yes—
17C2×Q8×D97C2211yescharacteristic
18#15690809C2213yes—
19#15690808C2213yes—
20#15690803C2216yes—
21#15690803C2216yes—
22#15690802C2213yes—
23#15690801C2213yes—
24#15690800C2216yes—
25#15690797C2213yes—
26#15690792C2216yes—
27#15690791C2216yes—
28#15690789C2213yes—
29#15690789C2213yes—
30#15690787C2216yes—
31#15690786C2213yes—
32#15690781C2213yes—
33Q8×Dic97C2211yescharacteristic
34Q8×Dic97C2211yescharacteristic
35Q8×Dic97C2213yes—
36Q8×Dic97C2211yescharacteristic
37Q8×Dic97C2213yes—
38C4×Dic194C2213yes—
39C4×Dic194C2213yes—
40#15690775C2213yes—
41#15690775C2213yes—
42D4×Dic97C2213yes—
43D4×Dic97C2213yes—
44D4×Dic97C2213yes—
45C4×C22⋊D97C2213yes—
46C4×C22⋊D97C2213yes—
47#15690770C2211yescharacteristic
48#15690770C2213yes—
49#15690769C2213yes—
50#15690768C2213yes—
51#15690766C2213yes—
52#15690766C2213yes—
53#15690766C2216yes—
54C2×C4⋊Dic97C2213yes—
55C4⋊C4×D97C2213yes—
56C2×C4×Dic97C2213yes—
57C2×C4×Dic97C2213yes—
58C42×D97C2213yes—
59C2×#41×C97C2211yescharacteristic
60C22×Q8×C97C2211yescharacteristic
61#124×C97C2213yes—
62C22⋊Q8×C97C2213yes—
63#121×C97C2213yes—
64C4×Q8×C97C2211yescharacteristic
65C4×D4×C97C2213yes—
66#13095510—212no—
67C2×Dic194C2313yes—
68C2×Dic194C2313yes—
69Q8×D97—24no—
70C2×C22⋊D97C2313yes—
71C2×C22⋊D97C2311yescharacteristic
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2313yes—
74C22×Dic97C2313yes—
75C22×Dic97C2313yes—
76C2×C4×D97C2311yescharacteristic
77C2×C4×D97C2313yes—
78C2×C4×D97C2313yes—
79C4⋊Dic97C2313yes—
80C4⋊Dic97C2313yes—
81C4⋊Dic97C2316yes—
82C4⋊Dic97C2313yes—
83C4⋊Dic97C2313yes—
84#13095490C2316yes—
85#13095490C2311yescharacteristic
86#13095490C2313yes—
87#13095490C2313yes—
88#13095490C2313yes—
89#13095490C2316yes—
90#13095490C2316yes—
91C22⋊Dic97C2313yes—
92C22⋊Dic97C2313yes—
93C22⋊Dic97C2316yes—
94C22⋊Dic97C2316yes—
95#13095488C2311yescharacteristic
96#13095488C2313yes—
97C4×Dic97C2313yes—
98C4×Dic97C2313yes—
99C4×Dic97C2313yes—
100C4×Dic97C2311yescharacteristic
101C4×Dic97C2313yes—
102C4×Dic97C2313yes—
103C4×Dic97C2313yes—
104C4×Dic97C2313yes—
105C4×Dic97C2313yes—
106#41×C97—24no—
107C2×Q8×C97—24no—
108C2×Q8×C97C2313yes—
109C2×Q8×C97C2311yescharacteristic
110C2×D4×C97C2313yes—
111C2×D4×C97C2313yes—
112C22×C4×C97C2313yes—
113C22×C4×C97C2313yes—
114C4⋊C4×C97C2313yes—
115C4⋊C4×C97C2313yes—
116C22⋊C4×C97C2311yescharacteristic
117C22⋊C4×C97C2313yes—
118C22⋊C4×C97C2316yes—
119C42×C97C2313yes—
120C23×C97C2413yes—
121C23×C97C2411yescharacteristic
122C22×D97C2411yescharacteristic
123Q8×C97—26no—
124Q8×C97#4114yescomplemented
125D4×C97—26no—
126D4×C97—26no—
127C2×C4×C97—26no—
128C2×C4×C97—26no—
129C2×C4×C97C2411yescharacteristic
130C2×C4×C97C2413yes—
131C2×C4×C97C2413yes—
132C2×C4×C97C2411yescharacteristic
133C2×C4×C97C2413yes—
134C22⋊D97—26no—
135C22⋊D97#4114yescomplemented
136C2×Dic97—26no—
137C2×Dic97—26no—
138C2×Dic97—22no—
139C2×Dic97—26no—
140C2×Dic97C2411yescharacteristic
141C2×Dic97C2413yes—
142C2×Dic97C2413yes—
143C2×Dic97C2413yes—
144C2×Dic97C2413yes—
145C2×Dic97C2411yescharacteristic
146C2×Dic97C2411yescharacteristic
147C4×D97—26no—
148C4×D97—26no—
149C4×D97—22no—
150Dic194—26no—
151Dic194—26no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—23no—
155C22×C97—23no—
156C22×C97—23no—
157C22×C97C2×#4112yes—
158C22×C97C2511yescharacteristic
159C2×D97—21no—
160C2×D97—21no—
161C2×D97C2×#4112yes—
162C4×C97—23no—
163C4×C97—21no—
164C4×C97—23no—
165C4×C97—21no—
166C4×C97C2×#4116yes—
167Dic97—23no—
168Dic97—23no—
169Dic97—23no—
170Dic97—21no—
171Dic97—23no—
172Dic97C2×#4112yes—
173Dic972−1+412yes—
174C2×C97—22no—
175C2×C97—43no—
176C2×C97C22×#4111yescharacteristic
177C2×C97C22×#4111yescharacteristic
178C2×C97C2×2−1+411yescharacteristic
179D97—22no—
180#3482—971nomaximal, Sylow
181C97#348211yescharacteristic, complemented, Sylow
182#557—971no—
183#556—973no—
184#552—973no—
185#552—973no—
186D4×Q8—971no—
187#548—971no—
188#548D9711yescharacteristic, complemented
189#547—973no—
190#533—973no—
191#529—973no—
192#529—973no—
193C4×#41—971no—
194C4×#41—973no—
195C2×C4×Q8—971no—
196C2×C4×Q8—971no—
197C2×#41—973no—
198C2×#41C2×D9711yescharacteristic
199C22×Q8—971no—
200C22×Q8C2×D9711yescharacteristic
201C4⋊Q8—973no—
202C4⋊Q8—973no—
203#126—976no—
204#126—976no—
205#125—973no—
206#125—976no—
207#125—973no—
208#124—973no—
209#124C2×D9713yes—
210#123—976no—
211#123—976no—
212C22⋊Q8—973no—
213C22⋊Q8—973no—
214C22⋊Q8—973no—
215C22⋊Q8—976no—
216C22⋊Q8C2×D9713yes—
217#121—973no—
218#121C2×D9713yes—
219C4×Q8—971no—
220C4×Q8—973no—
221C4×Q8—971no—
222C4×Q8—973no—
223C4×Q8—971no—
224C4×Q8—973no—
225C4×Q8—973no—
226C4×Q8—973no—
227C4×Q8—973no—
228C4×Q8C2×D9711yescharacteristic
229C4×D4—973no—
230C4×D4—973no—
231C4×D4—973no—
232C4×D4—973no—
233C4×D4—973no—
234C4×D4—971no—
235C4×D4—973no—
236C4×D4C2×D9713yes—
237#117—973no—
238#117—973no—
239#117—973no—
240#117—973no—
241#117—976no—
242C2×C4⋊C4—973no—
243C2×C4⋊C4—973no—
244C2×C42—973no—
245C2×C42—973no—
246C2×C42—973no—
247#41—24no—
248#41—19412no—
249C2×Q8—24no—
250C2×Q8—973no—
251C2×Q8—973no—
252C2×Q8—1944no—
253C2×Q8C22×D9713yes—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—971no—
256C2×D4—973no—
257C2×D4C22×D9713yes—
258C2×D4C22×D9713yes—
259C22×C4—971no—
260C22×C4—973no—
261C22×C4—973no—
262C22×C4—973no—
263C22×C4—971no—
264C22×C4—973no—
265C22×C4—973no—
266C22×C4C22×D9713yes—
267C22×C4C22×D9713yes—
268C4⋊C4—976no—
269C4⋊C4—971no—
270C4⋊C4—973no—
271C4⋊C4—973no—
272C4⋊C4—973no—
273C4⋊C4—976no—
274C4⋊C4—976no—
275C4⋊C4—973no—
276C4⋊C4—973no—
277C4⋊C4—976no—
278C4⋊C4—973no—
279C4⋊C4—973no—
280C4⋊C4C22×D9713yes—
281C4⋊C4C22×D9713yes—
282C22⋊C4—973no—
283C22⋊C4—976no—
284C22⋊C4—973no—
285C22⋊C4—976no—
286C22⋊C4—971no—
287C22⋊C4—973no—
288C22⋊C4C22×D9711yescharacteristic
289C22⋊C4C22×D9713yes—
290C22⋊C4C22×D9716yes—
291C42—971no—
292C42—973no—
293C42—973no—
294C42—973no—
295C42—973no—
296C42—973no—
297C42—973no—
298C42—973no—
299C42—973no—
300C42C22×D9713yes—
301C23—971no—
302C23C23×D9713yes—
303C23C23×D9711yescharacteristic
304Q8—26no—
305Q8—1946no—
306Q8—1946no—
307Q8#1309551014yescomplemented
308D4—26no—
309D4—26no—
310D4—974no—
311D4—1946no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—971no—
315C2×C4—971no—
316C2×C4—971no—
317C2×C4—973no—
318C2×C4—973no—
319C2×C4—973no—
320C2×C4—973no—
321C2×C4—1942no—
322C2×C4—1946no—
323C2×C4—1946no—
324C2×C4—1942no—
325C2×C4—1946no—
326C2×C4—1946no—
327C2×C4—1946no—
328C2×C4C23×D9711yescharacteristic
329C2×C4C23×D9711yescharacteristic
330C2×C4C23×D9713yes—
331C2×C4C23×D9713yes—
332C2×C4C23×D9713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—972no—
339C22—1941no—
340C22—1941no—
341C22#41×D9712yes—
342C22C24×D9711yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—972no—
348C4—972no—
349C4—1943no—
350C4—1943no—
351C4—1943no—
352C4—1941no—
353C4—1943no—
354C4C2×#1309551016yes—
355C2—22no—
356C2—43no—
357C2—1942no—
358C2C2×#41×D9711yescharacteristic
359C2C22×#1309551011yescharacteristic
360C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#548×C97C2
hypercenter#548D97
hypocenterC97#3482
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20261461C2semidirect
2. #20261454C2semidirect
3. #20261425C2semidirect
4. #20261419C2semidirect
5. Q8×C22⋊D97C2semidirect
6. #20261392C2semidirect
7. #20261385C2semidirect
8. #20261287C2semidirect
9. #20261266C2semidirect
10. #20261264C2semidirect
11. #41×Dic97C2semidirect
12. C4×#13095510C2semidirect
13. C2×Q8×Dic97C2semidirect
14. C4×Q8×D97C2semidirect
15. #548×C97C2semidirect
124. Q8×C97#41semidirect
135. C22⋊D97#41semidirect
181. C97#3482semidirect
188. #548D97semidirect
307. Q8#13095510semidirect

Maximal subgroups

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

subgroupindex
#202614612
#202614542
#202614252
#202614192
Q8×C22⋊D972
#202613922
#202613852
#202612872
#202612662
#202612642
#41×Dic972
C4×#130955102
C2×Q8×Dic972
C4×Q8×D972
#548×C972
#348297