Finite Group Databasebeta

← #27145072

Subgroups — #27145072

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×#13095512C211yesmaximal, characteristic
2C2×#15690795C212yesmaximal, complemented
3C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
4#20260654C211yesmaximal, characteristic, complemented
5#20260653C211yesmaximal, characteristic, complemented
6#20260652C211yesmaximal, characteristic, complemented
7#20260613C212yesmaximal, complemented
8#20260609C212yesmaximal, complemented
9#20260606C212yesmaximal, complemented
10#20260597C211yesmaximal, characteristic, complemented
11#395×C97C211yesmaximal, characteristic, complemented
12C2×#13095512—28no—
13C2×#13095512—24no—
14C22×D388C2211yescharacteristic
15C22×D388C2212yes—
16C22×C4×D97C2212yes—
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690795—28no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490C2212yes—
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C4⋊C4×D97—24no—
26C2×#13095488C2212yes—
27C2×#13095488C2212yes—
28C2×#13095488C2212yes—
29C2×#13095488C2212yes—
30C2×#13095488C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33#15690717C2211yescharacteristic
34C22×Q8×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C4⋊C4×C97C2212yes—
37#95×C97C2212yes—
38#95×C97C2211yescharacteristic
39C23×D97C2312yes—
40C23×D97C2311yescharacteristic
41#13095512—232no—
42C2×D388—24no—
43C2×D388—24no—
44C2×D388—24no—
45C2×D388—22no—
46C2×D388D418yes—
47C22×Dic97C2312yes—
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2311yescharacteristic
51C2×C4×D97—22no—
52C2×C4×D97—24no—
53C2×C4×D97—24no—
54C2×C4×D97—24no—
55C2×C4×D97—22no—
56C2×C4×D97—24no—
57C2×C4×D97—22no—
58C2×C4×D97—22no—
59C2×C4×D97—24no—
60C2×C4×D97D414yes—
61C4⋊Dic97—22no—
62#13095490—22no—
63#13095490—24no—
64#13095490—22no—
65#13095488—24no—
66#13095488—24no—
67#13095488—24no—
68#13095488—24no—
69#13095488—24no—
70C2×Q8×C97—24no—
71C2×Q8×C97—22no—
72C22×C4×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2312yes—
75C22×C4×C97C2311yescharacteristic
76C22×C4×C97C2311yescharacteristic
77C4⋊C4×C97—22no—
78C4⋊C4×C97—24no—
79C23×C97C2411yescharacteristic
80C22×D97—21no—
81C22×D97—21no—
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—22no—
89C22×D97C2×D414yes—
90C22×D97C2×D414yes—
91C22×D97C2×D412yes—
92Q8×C97—28no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—21no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—21no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97C2×D414yes—
104C2×C4×C97C2×D412yes—
105C2×C4×C97#4112yescomplemented
106C2×Dic97—21no—
107C2×Dic97—22no—
108C2×Dic97—21no—
109C2×Dic97—22no—
110C2×Dic97—22no—
111C2×Dic97—21no—
112C2×Dic97—22no—
113C2×Dic97—22no—
114C2×Dic97C2×D412yes—
115C2×Dic97#4112yescomplemented
116D388—28no—
117D388—216no—
118C4×D97—28no—
119C4×D97—216no—
120C4×D97—28no—
121C4×D97—44no—
122C22×C97C22×D411yescharacteristic
123C22×C97C22×D412yes—
124C22×C97C2×#4111yescharacteristic
125C22×C97C2×#4111yescharacteristic
126C22×C972−1+412yes—
127C2×D97—22no—
128C2×D97—22no—
129C2×D97—24no—
130C2×D97—24no—
131C2×D97—24no—
132C2×D97—24no—
133C2×D97—24no—
134C2×D97—42no—
135C2×D97—42no—
136C2×D97—42no—
137C2×D97—42no—
138C2×D97#12014yes—
139C4×C97—24no—
140C4×C97—22no—
141C4×C97—22no—
142C4×C97—41no—
143C4×C97—42no—
144Dic97—22no—
145Dic97—22no—
146Dic97—41no—
147Dic97—42no—
148C2×C97C2×#12011yescharacteristic
149C2×C97#52511yescharacteristic
150C2×C97#53611yescharacteristic
151C2×C97#54712yes—
152C2×C97#54712yes—
153D97—24no—
154D97—44no—
155#2406—971nomaximal, Sylow
156C97#240611yescharacteristic, complemented, Sylow
157C22×#41—971no—
158C2×#123—972no—
159C22×C4⋊C4—971no—
160#395—971no—
161#395D9711yescharacteristic, complemented
162#394—971no—
163#394—971no—
164#386—972no—
165#385—972no—
166#385—972no—
167#381—971no—
168C2×#41—1944no—
169C2×#41—1948no—
170C22×Q8C2×D9711yescharacteristic
171C22×D4—972no—
172C22×D4—971no—
173C23×C4—971no—
174C23×C4—971no—
175C23×C4—971no—
176C23×C4—972no—
177#123—1948no—
178C2×C4⋊C4—971no—
179C2×C4⋊C4—971no—
180C2×C4⋊C4—971no—
181C2×C4⋊C4—972no—
182C2×C4⋊C4—1944no—
183C2×C4⋊C4C2×D9711yescharacteristic
184C2×C4⋊C4C2×D9712yes—
185C2×C22⋊C4—972no—
186C2×C22⋊C4—972no—
187C2×C22⋊C4—972no—
188C2×C22⋊C4—972no—
189C2×C22⋊C4—972no—
190#95—972no—
191#95—971no—
192#95—972no—
193#95C2×D9711yescharacteristic
194#95C2×D9712yes—
195C24—971no—
196C24—972no—
197#41—19432no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—978no—
201C2×D4—1944no—
202C2×D4—1944no—
203C2×D4—1944no—
204C2×D4—1942no—
205C22×C4—971no—
206C22×C4—971no—
207C22×C4—974no—
208C22×C4—972no—
209C22×C4—971no—
210C22×C4—1942no—
211C22×C4—1944no—
212C22×C4—1942no—
213C22×C4—1944no—
214C22×C4—1942no—
215C22×C4—1944no—
216C22×C4—1944no—
217C22×C4—1944no—
218C22×C4—1942no—
219C22×C4C22×D9712yes—
220C22×C4C22×D9711yescharacteristic
221C22×C4C22×D9711yescharacteristic
222C22×C4C22×D9712yes—
223C22×C4C22×D9711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—1942no—
227C4⋊C4—1942no—
228C4⋊C4—1942no—
229C4⋊C4—1944no—
230C22⋊C4—1944no—
231C22⋊C4—1944no—
232C22⋊C4—1944no—
233C22⋊C4—1944no—
234C22⋊C4—1944no—
235C23—974no—
236C23—972no—
237C23—974no—
238C23—1942no—
239C23—1942no—
240C23—1942no—
241C23—1942no—
242C23—1942no—
243C23—1942no—
244C23—1941no—
245C23—1941no—
246C23—1942no—
247C23C23×D9711yescharacteristic
248Q8—28no—
249D4—19416no—
250D4—1948no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—972no—
262C2×C4—972no—
263C2×C4—3884no—
264C2×C4—1948no—
265C2×C4—19416no—
266C2×C4—1948no—
267C2×C4—1942no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1942no—
271C2×C4—1942no—
272C2×C4—1941no—
273C2×C4—1942no—
274C2×C4—1941no—
275C2×C4D4×D9712yes—
276C2×C4D4×D9714yes—
277C2×C4#1309551212yescomplemented
278C22—974no—
279C22—3882no—
280C22—3882no—
281C22—3882no—
282C22—3882no—
283C22—1944no—
284C22—1944no—
285C22—1944no—
286C22—1944no—
287C22—1944no—
288C22—1942no—
289C22—1942no—
290C22C2×D4×D9712yes—
291C22C2×D4×D9711yescharacteristic
292C22#41×D9711yescharacteristic
293C22C2×#1309551211yescharacteristic
294C22#1569085812yes—
295C4—22no—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—3882no—
301C4—3881no—
302C4—1942no—
303C4—1942no—
304C2—3884no—
305C2—1944no—
306C2#120×D9711yescharacteristic
307C2#2026124811yescharacteristic
308C2#2026131011yescharacteristic
309C2#2026138712yes—
310C2#2026138812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#395×C97C2
prime radicalC22×#13095512C2
hypercenter#395D97
hypocenterC97#2406
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#15690795C2semidirect
3. C2×C4⋊C4×D97C2semidirect
4. #20260654C2semidirect
5. #20260653C2semidirect
6. #20260652C2semidirect
7. #20260613C2semidirect
8. #20260609C2semidirect
9. #20260606C2semidirect
10. #20260597C2semidirect
11. #395×C97C2semidirect
105. C2×C4×C97#41semidirect
115. C2×Dic97#41semidirect
156. C97#2406semidirect
161. #395D97semidirect
277. C2×C4#13095512semidirect

Maximal subgroups

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

subgroupindex
C22×#130955122
C2×#156907952
C2×C4⋊C4×D972
#202606542
#202606532
#202606522
#202606132
#202606092
#202606062
#202605972
#395×C972
#240697