Finite Group Databasebeta

← #27145066

Subgroups — #27145066

Order 12416.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095510C211yesmaximal, characteristic
2C2×#15690792C212yesmaximal, complemented
3C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
4#20260656C211yesmaximal, characteristic, complemented
5#20260655C211yesmaximal, characteristic, complemented
6#20260649C211yesmaximal, characteristic, complemented
7#20260611C212yesmaximal, complemented
8#20260610C212yesmaximal, complemented
9#20260607C212yesmaximal, complemented
10#20260597C211yesmaximal, characteristic, complemented
11#394×C97C211yesmaximal, characteristic, complemented
12C2×#13095510—28no—
13C2×#13095510—24no—
14C22×Dic194C2211yescharacteristic
15C22×C22⋊D97C2212yes—
16C23×Dic97C2212yes—
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690792—28no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×C4⋊Dic97C2212yes—
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2212yes—
25C2×#13095490C2211yescharacteristic
26C4⋊C4×D97—24no—
27C2×C22⋊Dic97C2212yes—
28C2×C22⋊Dic97C2212yes—
29C2×#13095488C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33#15690717C2211yescharacteristic
34C22×D4×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C22⋊C4×C97C2212yes—
37C2×C22⋊C4×C97C2212yes—
38#95×C97C2211yescharacteristic
39C23×D97C2311yescharacteristic
40#13095510—232no—
41C2×Dic194—22no—
42C2×Dic194—24no—
43C2×C22⋊D97—24no—
44C2×C22⋊D97—24no—
45C2×C22⋊D97D418yes—
46C22×Dic97—24no—
47C22×Dic97—24no—
48C22×Dic97—24no—
49C22×Dic97C2312yes—
50C22×Dic97C2311yescharacteristic
51C22×Dic97C2311yescharacteristic
52C22×Dic97C2312yes—
53C22×Dic97C2311yescharacteristic
54C2×C4×D97—22no—
55C2×C4×D97—24no—
56C2×C4×D97—22no—
57C2×C4×D97—22no—
58C2×C4×D97—22no—
59C2×C4×D97—24no—
60C2×C4×D97D414yes—
61C4⋊Dic97—22no—
62C4⋊Dic97—24no—
63#13095490—22no—
64#13095490—22no—
65#13095490—24no—
66C22⋊Dic97—24no—
67C22⋊Dic97—24no—
68#13095488—24no—
69C24×C97C2312yes—
70C2×D4×C97—24no—
71C2×D4×C97—22no—
72C22×C4×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2311yescharacteristic
75C22×C4×C97C2311yescharacteristic
76C4⋊C4×C97—22no—
77C22⋊C4×C97—24no—
78C22⋊C4×C97—24no—
79C23×C97—22no—
80C23×C97—22no—
81C23×C97—22no—
82C23×C97—22no—
83C23×C97—22no—
84C23×C97—22no—
85C23×C97C2×D414yes—
86C23×C97C2411yescharacteristic
87C22×D97—21no—
88C22×D97—21no—
89C22×D97—22no—
90C22×D97C2×D414yes—
91C22×D97C2×D412yes—
92D4×C97—28no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—21no—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97—21no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97C2×D412yes—
102C2×C4×C97#4112yescomplemented
103C22⋊D97—216no—
104C2×Dic97—216no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—21no—
110C2×Dic97—22no—
111C2×Dic97—22no—
112C2×Dic97—21no—
113C2×Dic97—22no—
114C2×Dic97—22no—
115C2×Dic97C2×D412yes—
116C2×Dic97C2×D414yes—
117C2×Dic97#4112yescomplemented
118C4×D97—28no—
119C4×D97—28no—
120C4×D97—44no—
121Dic194—28no—
122C22×C97—24no—
123C22×C97—24no—
124C22×C97—24no—
125C22×C97—42no—
126C22×C97—42no—
127C22×C97—42no—
128C22×C97—42no—
129C22×C97C22×D411yescharacteristic
130C22×C97C22×D412yes—
131C22×C97C2×#4111yescharacteristic
132C22×C97C2×#4111yescharacteristic
133C22×C972−1+412yes—
134C2×D97—22no—
135C2×D97—22no—
136C2×D97—24no—
137C2×D97—24no—
138C2×D97#12014yes—
139C4×C97—22no—
140C4×C97—22no—
141C4×C97—41no—
142C4×C97—42no—
143Dic97—24no—
144Dic97—22no—
145Dic97—22no—
146Dic97—41no—
147Dic97—42no—
148C2×C97—44no—
149C2×C97C2×#12011yescharacteristic
150C2×C97#52511yescharacteristic
151C2×C97#53611yescharacteristic
152C2×C97#54712yes—
153C2×C97#54712yes—
154D97—24no—
155#2406—971nomaximal, Sylow
156C97#240611yescharacteristic, complemented, Sylow
157C22×#41—971no—
158C2×#123—972no—
159C22×C4⋊C4—971no—
160#395—971no—
161#395—971no—
162#394—971no—
163#394D9711yescharacteristic, complemented
164#386—972no—
165#385—972no—
166#385—972no—
167#381—971no—
168C2×#41—1944no—
169C2×#41—1948no—
170C22×Q8—971no—
171C22×D4—972no—
172C22×D4C2×D9711yescharacteristic
173C23×C4—971no—
174C23×C4—972no—
175C23×C4—971no—
176C23×C4—971no—
177#123—1948no—
178C2×C4⋊C4—971no—
179C2×C4⋊C4—971no—
180C2×C4⋊C4—971no—
181C2×C4⋊C4—972no—
182C2×C4⋊C4—972no—
183C2×C4⋊C4—1944no—
184C2×C4⋊C4C2×D9711yescharacteristic
185C2×C22⋊C4—972no—
186C2×C22⋊C4—972no—
187C2×C22⋊C4—972no—
188C2×C22⋊C4C2×D9712yes—
189C2×C22⋊C4C2×D9712yes—
190#95—972no—
191#95—971no—
192#95—972no—
193#95—972no—
194#95C2×D9711yescharacteristic
195C24—971no—
196C24C22×D9712yes—
197#41—19432no—
198C2×Q8—1944no—
199C2×Q8—1942no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—978no—
203C2×D4—1944no—
204C2×D4—1944no—
205C22×C4—971no—
206C22×C4—971no—
207C22×C4—972no—
208C22×C4—974no—
209C22×C4—972no—
210C22×C4—971no—
211C22×C4—1942no—
212C22×C4—1942no—
213C22×C4—1944no—
214C22×C4—1942no—
215C22×C4—1944no—
216C22×C4—1944no—
217C22×C4—1944no—
218C22×C4—1944no—
219C22×C4—1942no—
220C22×C4C22×D9712yes—
221C22×C4C22×D9711yescharacteristic
222C22×C4C22×D9711yescharacteristic
223C22×C4C22×D9711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—1942no—
226C4⋊C4—1944no—
227C4⋊C4—1942no—
228C4⋊C4—1944no—
229C4⋊C4—1942no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—1944no—
233C22⋊C4—1944no—
234C22⋊C4—1944no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—974no—
242C23—972no—
243C23—1941no—
244C23—1941no—
245C23—1942no—
246C23D4×D9714yes—
247C23C23×D9711yescharacteristic
248Q8—1948no—
249D4—28no—
250D4—19416no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—972no—
260C2×C4—974no—
261C2×C4—972no—
262C2×C4—3884no—
263C2×C4—1948no—
264C2×C4—1948no—
265C2×C4—1942no—
266C2×C4—1942no—
267C2×C4—1941no—
268C2×C4—1942no—
269C2×C4—1942no—
270C2×C4—19416no—
271C2×C4—1941no—
272C2×C4—1942no—
273C2×C4—1942no—
274C2×C4—1942no—
275C2×C4—1941no—
276C2×C4D4×D9712yes—
277C2×C4#1309551012yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—974no—
286C22—1944no—
287C22—1944no—
288C22—1942no—
289C22—1942no—
290C22C2×D4×D9712yes—
291C22C2×D4×D9711yescharacteristic
292C22#41×D9711yescharacteristic
293C22C2×#1309551011yescharacteristic
294C22#1569085712yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—3882no—
300C4—3881no—
301C4—1942no—
302C4—1944no—
303C4—1942no—
304C2—44no—
305C2—1944no—
306C2#120×D9711yescharacteristic
307C2#2026124511yescharacteristic
308C2#2026130511yescharacteristic
309C2#2026138312yes—
310C2#2026138512yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#394×C97C2
prime radicalC22×#13095510C2
hypercenter#394D97
hypocenterC97#2406
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#15690792C2semidirect
3. C2×C4⋊C4×D97C2semidirect
4. #20260656C2semidirect
5. #20260655C2semidirect
6. #20260649C2semidirect
7. #20260611C2semidirect
8. #20260610C2semidirect
9. #20260607C2semidirect
10. #20260597C2semidirect
11. #394×C97C2semidirect
102. C2×C4×C97#41semidirect
117. C2×Dic97#41semidirect
156. C97#2406semidirect
163. #394D97semidirect
277. C2×C4#13095510semidirect

Maximal subgroups

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

subgroupindex
C22×#130955102
C2×#156907922
C2×C4⋊C4×D972
#202606562
#202606552
#202606492
#202606112
#202606102
#202606072
#202605972
#394×C972
#240697