Finite Group Databasebeta

← #27155353

Subgroups — #27155353

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#20261457C213yesmaximal, complemented
3#20261426C213yesmaximal, complemented
4#20261420C213yesmaximal, complemented
5Q8×C22⋊D97C211yesmaximal, characteristic, complemented
6#20261397C211yesmaximal, characteristic, complemented
7#20261396C211yesmaximal, characteristic, complemented
8#20261386C213yesmaximal, complemented
9#20261289C213yesmaximal, complemented
10#20261272C213yesmaximal, complemented
11#20261269C213yesmaximal, complemented
12C4×#13095512C211yesmaximal, characteristic, complemented
13C4×#13095511C213yesmaximal, complemented
14C4×Q8×D97C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095511C2213yes—
18C2×Q8×D97C2211yescharacteristic
19#15690809C2213yes—
20#15690808C2213yes—
21#15690805C2216yes—
22#15690804C2216yes—
23#15690802C2213yes—
24#15690801C2213yes—
25#15690800C2216yes—
26#15690799C2213yes—
27#15690798C2213yes—
28#15690795C2216yes—
29#15690793C2216yes—
30#15690790C2213yes—
31#15690789C2213yes—
32#15690788C2216yes—
33#15690786C2213yes—
34#15690786C2213yes—
35#15690785C2213yes—
36#15690784C2213yes—
37Q8×Dic97C2211yescharacteristic
38Q8×Dic97C2211yescharacteristic
39C4×Dic194C2213yes—
40C4×Dic194C2213yes—
41C4×Dic194C2213yes—
42#15690775C2213yes—
43C4×D388C2213yes—
44C4×D388C2213yes—
45C4×D388C2213yes—
46#15690772C2213yes—
47C4×C22⋊D97C2213yes—
48C4×C22⋊D97C2213yes—
49C4×C22⋊D97C2211yescharacteristic
50C4×C22⋊D97C2213yes—
51#15690769C2213yes—
52#15690768C2213yes—
53#15690767C2213yes—
54#15690767C2213yes—
55#15690767C2216yes—
56C4⋊C4×D97C2213yes—
57C42×D97C2213yes—
58C42×D97C2213yes—
59C22×Q8×C97C2211yescharacteristic
60C4×Q8×C97C2211yescharacteristic
61C4×Q8×C97C2213yes—
62C4×Q8×C97C2211yescharacteristic
63C4×Q8×C97C2213yes—
64C2×C4⋊C4×C97C2213yes—
65C2×C42×C97C2213yes—
66#13095512—24no—
67#13095511—212no—
68C2×Dic194C2313yes—
69C2×Dic194C2313yes—
70Q8×D97—24no—
71C2×D388C2313yes—
72C2×D388C2313yes—
73C2×C22⋊D97C2313yes—
74C2×C22⋊D97C2311yescharacteristic
75C2×C4×D97C2311yescharacteristic
76C2×C4×D97C2313yes—
77C2×C4×D97C2313yes—
78C2×C4×D97C2313yes—
79C2×C4×D97C2313yes—
80C2×C4×D97C2313yes—
81C4⋊Dic97C2313yes—
82C4⋊Dic97C2313yes—
83C4⋊Dic97C2313yes—
84C4⋊Dic97C2313yes—
85#13095490C2316yes—
86#13095490C2311yescharacteristic
87#13095490C2313yes—
88#13095490C2313yes—
89#13095490C2313yes—
90#13095490C2316yes—
91#13095490C2316yes—
92C22⋊Dic97C2311yescharacteristic
93C22⋊Dic97C2313yes—
94#13095488C2311yescharacteristic
95#13095488C2313yes—
96#13095488C2313yes—
97#13095488C2313yes—
98#13095488C2316yes—
99#13095488C2316yes—
100#13095488C2316yes—
101C4×Dic97C2311yescharacteristic
102C4×Dic97C2313yes—
103C4×Dic97C2313yes—
104C4×Dic97C2313yes—
105C4×Dic97C2313yes—
106C4×Dic97C2313yes—
107C2×Q8×C97—24no—
108C2×Q8×C97C2313yes—
109C2×Q8×C97C2311yescharacteristic
110C22×C4×C97C2313yes—
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2313yes—
113C4⋊C4×C97C2313yes—
114C4⋊C4×C97C2313yes—
115C4⋊C4×C97C2316yes—
116C42×C97C2313yes—
117C42×C97C2313yes—
118C42×C97C2313yes—
119C42×C97C2313yes—
120C23×C97C2411yescharacteristic
121C22×D97C2413yes—
122C22×D97C2411yescharacteristic
123Q8×C97—26no—
124Q8×C97#4114yescomplemented
125C2×C4×C97—26no—
126C2×C4×C97—26no—
127C2×C4×C97—22no—
128C2×C4×C97C2411yescharacteristic
129C2×C4×C97C2413yes—
130C2×C4×C97C2413yes—
131C2×C4×C97C2413yes—
132C2×C4×C97C2413yes—
133C2×C4×C97C2411yescharacteristic
134C22⋊D97—26no—
135C22⋊D97#4114yescomplemented
136C2×Dic97C2411yescharacteristic
137C2×Dic97C2411yescharacteristic
138C2×Dic97C2413yes—
139C2×Dic97C2413yes—
140C2×Dic97C2413yes—
141C2×Dic97C2411yescharacteristic
142D388—26no—
143D388—26no—
144C4×D97—26no—
145C4×D97—26no—
146C4×D97—26no—
147C4×D97—22no—
148C4×D97—26no—
149C4×D97—26no—
150Dic194—26no—
151Dic194—26no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97C2×#4112yes—
155C22×C97C2511yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—23no—
159C2×D97—23no—
160C2×D97—23no—
161C2×D97C2×#4112yes—
162C4×C97—23no—
163C4×C97—23no—
164C4×C97—23no—
165C4×C97—21no—
166C4×C97C2×#4116yes—
167C4×C972−1+412yes—
168Dic97—23no—
169Dic97—23no—
170Dic97—21no—
171Dic97—21no—
172Dic97—23no—
173Dic97C2×#4112yes—
174C2×C97—22no—
175C2×C97C22×#4111yescharacteristic
176C2×C97C22×#4111yescharacteristic
177C2×C97C2×2−1+411yescharacteristic
178D97—22no—
179D97—43no—
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#548—971no—
189#547—973no—
190#533—973no—
191#529—973no—
192#529—973no—
193C4×#41—971no—
194C4×#41—973no—
195C2×C4×Q8—971no—
196C2×C4×Q8D9711yescharacteristic, complemented
197C2×#41—971no—
198C2×#41—973no—
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#124—973no—
210#123—976no—
211#123—976no—
212C22⋊Q8—973no—
213C22⋊Q8—973no—
214C22⋊Q8—973no—
215C22⋊Q8—976no—
216C22⋊Q8—973no—
217#121—973no—
218#121—973no—
219C4×Q8—971no—
220C4×Q8—971no—
221C4×Q8—973no—
222C4×Q8—973no—
223C4×Q8—973no—
224C4×Q8—973no—
225C4×Q8C2×D9711yescharacteristic
226C4×Q8C2×D9713yes—
227C4×Q8C2×D9711yescharacteristic
228C4×Q8C2×D9713yes—
229C4×D4—971no—
230C4×D4—973no—
231C4×D4—973no—
232C4×D4—973no—
233C4×D4—973no—
234C4×D4—973no—
235C4×D4—973no—
236C4×D4—973no—
237#117—973no—
238#117—973no—
239#117—973no—
240#117—973no—
241#117—976no—
242C2×C4⋊C4—973no—
243C2×C4⋊C4C2×D9713yes—
244C2×C42—973no—
245C2×C42—973no—
246C2×C42C2×D9713yes—
247#41—19412no—
248#41—1944no—
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×D4—973no—
258C2×D4—973no—
259C22×C4—973no—
260C22×C4—973no—
261C22×C4—973no—
262C22×C4—973no—
263C22×C4—971no—
264C22×C4—973no—
265C22×C4C22×D9713yes—
266C22×C4C22×D9713yes—
267C22×C4C22×D9711yescharacteristic
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—973no—
278C4⋊C4—973no—
279C4⋊C4C22×D9713yes—
280C4⋊C4C22×D9713yes—
281C4⋊C4C22×D9716yes—
282C22⋊C4—971no—
283C22⋊C4—973no—
284C22⋊C4—971no—
285C22⋊C4—973no—
286C22⋊C4—976no—
287C22⋊C4—973no—
288C22⋊C4—973no—
289C22⋊C4—976no—
290C22⋊C4—976no—
291C42—973no—
292C42—973no—
293C42—973no—
294C42—973no—
295C42—971no—
296C42—973no—
297C42C22×D9713yes—
298C42C22×D9713yes—
299C42C22×D9713yes—
300C42C22×D9713yes—
301C23—971no—
302C23—973no—
303C23C23×D9711yescharacteristic
304Q8—26no—
305Q8—1946no—
306Q8—1946no—
307Q8#1309551114yescomplemented
308D4—974no—
309D4—1946no—
310D4—1946no—
311D4—1946no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—971no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—973no—
319C2×C4—973no—
320C2×C4—973no—
321C2×C4—1946no—
322C2×C4—1946no—
323C2×C4—1946no—
324C2×C4—1946no—
325C2×C4—1946no—
326C2×C4—1942no—
327C2×C4C23×D9713yes—
328C2×C4C23×D9711yescharacteristic
329C2×C4C23×D9713yes—
330C2×C4C23×D9713yes—
331C2×C4C23×D9713yes—
332C2×C4C23×D9711yescharacteristic
333C22—21no—
334C22—21no—
335C22—972no—
336C22—1943no—
337C22—1943no—
338C22—1943no—
339C22—1941no—
340C22—1941no—
341C22#41×D9712yes—
342C22C24×D9711yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—972no—
348C4—1943no—
349C4—1941no—
350C4—1943no—
351C4—1941no—
352C4—1943no—
353C4C2×#1309551116yes—
354C4#1569085812yes—
355C2—22no—
356C2—3883no—
357C2—1942no—
358C2C2×#41×D9711yescharacteristic
359C2C22×#1309551111yescharacteristic
360C2C2×#1569085811yescharacteristic
subgroupG/H
centerC2×C4C23×D97
commutator, socle, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×C4×Q8×C97C2
hypercenterC2×C4×Q8D97
hypocenterC97#3482
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20261461C2semidirect
2. #20261457C2semidirect
3. #20261426C2semidirect
4. #20261420C2semidirect
5. Q8×C22⋊D97C2semidirect
6. #20261397C2semidirect
7. #20261396C2semidirect
8. #20261386C2semidirect
9. #20261289C2semidirect
10. #20261272C2semidirect
11. #20261269C2semidirect
12. C4×#13095512C2semidirect
13. C4×#13095511C2semidirect
14. C4×Q8×D97C2semidirect
15. C2×C4×Q8×C97C2semidirect
124. Q8×C97#41semidirect
135. C22⋊D97#41semidirect
181. C97#3482semidirect
196. C2×C4×Q8D97semidirect
307. Q8#13095511semidirect

Maximal subgroups

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

subgroupindex
#202614612
#202614572
#202614262
#202614202
Q8×C22⋊D972
#202613972
#202613962
#202613862
#202612892
#202612722
#202612692
C4×#130955122
C4×#130955112
C4×Q8×D972
C2×C4×Q8×C972
#348297