Finite Group Databasebeta

← #27151908

Subgroups — #27151908

Order 12416.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#20261271C211yesmaximal, characteristic
2C2×Q8×Dic97C211yesmaximal, characteristic, complemented
3#20261068C211yesmaximal, characteristic, complemented
4#20261054C211yesmaximal, characteristic
5#20261040C212yesmaximal, complemented
6#20260959C212yesmaximal, complemented
7#20260874C211yesmaximal, characteristic
8#20260842C212yesmaximal, complemented
9#20260838C212yesmaximal, complemented
10#20260789C211yesmaximal, characteristic, complemented
11#451×C97C211yesmaximal, characteristic, complemented
12C2×#13095498C2212yes—
13#15690805—21no—
14#15690801C2211yescharacteristic
15#15690799C2211yescharacteristic
16#15690794—21no—
17#15690784C2211yescharacteristic
18Q8×Dic97—21no—
19Q8×Dic97—21no—
20Q8×Dic97C2212yes—
21Q8×Dic97C2212yes—
22Q8×Dic97C2211yescharacteristic
23#15690772C2211yescharacteristic
24#15690770—21no—
25#15690768—21no—
26C2×C4⋊Dic97—21no—
27C2×C4⋊Dic97C2211yescharacteristic
28C2×C4×Dic97—21no—
29C2×C4×Dic97C2211yescharacteristic
30#15690743C2212yes—
31#15690742C2212yes—
32#15690739C2212yes—
33#15690734C2212yes—
34#15690734C2212yes—
35#15690732C2212yes—
36C22⋊#11715895C2211yescharacteristic
37#15690720C2212yes—
38C22×Q8×C97C2211yescharacteristic
39C2×Q16×C97C2212yes—
40C22⋊Q8×C97C2211yescharacteristic
41#103×C97C2212yes—
42C22⋊C8×C97C2211yescharacteristic
43C2×D388C2311yescharacteristic
44C2×C22⋊D97—21no—
45C22×Dic97—21no—
46C22×Dic97D411yescharacteristic
47C22×Dic97D411yescharacteristic
48C2×C4×D97—21no—
49#13095498—24no—
50C2×#11715895C2312yes—
51C4⋊Dic97—21no—
52C4⋊Dic97—22no—
53C4⋊Dic97—22no—
54C4⋊Dic97—21no—
55C4⋊Dic97—21no—
56C4⋊Dic97C2312yes—
57C4⋊Dic97C2311yescharacteristic
58#13095490—21no—
59#13095488—21no—
60#13095488—21no—
61#13095488—21no—
62#13095488—21no—
63C4×Dic97—22no—
64C4×Dic97—22no—
65C4×Dic97—21no—
66C4×Dic97D411yescharacteristic
67C4×Dic97D411yescharacteristic
68C4×Dic97C2311yescharacteristic
69C4×Dic97C2311yescharacteristic
70C2×Q8×C97—24no—
71C2×Q8×C97—21no—
72C2×Q8×C97C2311yescharacteristic
73C2×Q8×C97C2312yes—
74C22×C4×C97—21no—
75C22×C4×C97C2311yescharacteristic
76Q16×C97—24no—
77C2×C8×C97C2312yes—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2311yescharacteristic
80C22⋊C4×C97—21no—
81C23×C97C2×D411yescharacteristic
82C22×D97—21no—
83Q8×C97—24no—
84Q8×C97—22no—
85Q8×C97—22no—
86Q8×C97#4114yes—
87C2×C4×C97—24no—
88C2×C4×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—42no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—41no—
99C2×Dic97—42no—
100C2×Dic97C2×D411yescharacteristic
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C2×Dic97C2×D411yescharacteristic
104D388—22no—
105C4×D97—42no—
106C8×C97—22no—
107#11715895—22no—
108C22×C97—21no—
109C22×C97—21no—
110C22×C97#13712yes—
111C22×C97C22×D411yescharacteristic
112C2×D97—41no—
113C2×D97—41no—
114C2×D97—41no—
115C4×C97—24no—
116C4×C97—21no—
117C4×C97—41no—
118C4×C97—41no—
119C4×C97C2×#4112yes—
120Dic97—21no—
121Dic97—21no—
122Dic97—21no—
123Dic97—21no—
124Dic97—41no—
125Dic97—42no—
126C2×C97—22no—
127C2×C97#52511yescharacteristic
128C2×C97C2×#13711yescharacteristic
129C2×C97#57511yescharacteristic
130D97—81no—
131#3117—971nomaximal, Sylow
132C97#311711yescharacteristic, complemented, Sylow
133#529—971no—
134C2×C4×Q8—971no—
135#484—971no—
136#481—971no—
137#479—972no—
138#464—972no—
139#451D9711yescharacteristic, complemented
140#448—971no—
141#441—972no—
142#440—972no—
143#432—971no—
144C22×Q8C2×D9711yescharacteristic
145C2×Q16C2×D9712yes—
146C2×SD16—972no—
147#126—1941no—
148#125—971no—
149#124—971no—
150#123—1941no—
151C22⋊Q8C2×D9711yescharacteristic
152#121—971no—
153C4×Q8—972no—
154C4×Q8—972no—
155C4×Q8—971no—
156C4×Q8—1941no—
157C4×Q8—1941no—
158C4×D4—971no—
159C4×D4—1941no—
160#117—1941no—
161C2×C4⋊C4—971no—
162C2×C4⋊C4—1941no—
163C2×C42—971no—
164C2×C42—1941no—
165#107—972no—
166#106—972no—
167C4⋊C8—972no—
168#103—972no—
169#103—972no—
170#103C2×D9712yes—
171#102—972no—
172C22⋊C8—971no—
173C22⋊C8C2×D9711yescharacteristic
174#97—972no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D9711yescharacteristic
178C2×Q8C22×D9712yes—
179C2×D4—971no—
180C2×D4—1941no—
181C22×C4—21no—
182C22×C4—971no—
183C22×C4—971no—
184C22×C4—1941no—
185C22×C4—1941no—
186C22×C4C22×D9711yescharacteristic
187Q16—24no—
188SD16—1944no—
189C2×C8—972no—
190C2×C8C22×D9712yes—
191C4⋊C4—21no—
192C4⋊C4—972no—
193C4⋊C4—971no—
194C4⋊C4—1941no—
195C4⋊C4—1942no—
196C4⋊C4—1942no—
197C4⋊C4—1941no—
198C4⋊C4—1941no—
199C4⋊C4—1941no—
200C4⋊C4C22×D9711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1941no—
203C22⋊C4—1941no—
204C22⋊C4—1941no—
205C22⋊C4—1941no—
206C42—971no—
207C42—971no—
208C42—971no—
209C42—971no—
210C42—1942no—
211C42—1942no—
212C42—1941no—
213C23—1941no—
214C23D4×D9711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1309551014yes—
219D4—3882no—
220D4—1942no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—971no—
227C2×C4—971no—
228C2×C4—971no—
229C2×C4—971no—
230C2×C4—3882no—
231C2×C4—3882no—
232C2×C4—3881no—
233C2×C4—1942no—
234C2×C4—1941no—
235C2×C4—1942no—
236C2×C4D4×D9711yescharacteristic
237C2×C4C23×D9711yescharacteristic
238C8—22no—
239C8—1942no—
240C22—21no—
241C22—21no—
242C22—3881no—
243C22—3881no—
244C22—3881no—
245C22#1569084212yes—
246C22C2×D4×D9711yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—3881no—
252C4—3882no—
253C4—1941no—
254C4—1941no—
255C4—1941no—
256C4—1941no—
257C4C2×#1309551012yes—
258C2—22no—
259C2—7761no—
260C2#2026124111yescharacteristic
261C2C2×#1569084211yescharacteristic
262C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#451×C97C2
socleC22×C97C22×D4
prime radical#15690784C22
hypercenter#451D97
hypocenterC97#3117
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic97C2semidirect
3. #20261068C2semidirect
5. #20261040C2semidirect
6. #20260959C2semidirect
8. #20260842C2semidirect
9. #20260838C2semidirect
10. #20260789C2semidirect
11. #451×C97C2semidirect
132. C97#3117semidirect
139. #451D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
C2×Q8×Dic972
#202610682
#202610542
#202610402
#202609592
#202608742
#202608422
#202608382
#202607892
#451×C972
#311797