Finite Group Databasebeta

← #27153454

Subgroups — #27153454

Order 12416.

484 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#15690830C212yesmaximal, complemented
2#20261390C211yesmaximal, characteristic, complemented
3#20261114C211yesmaximal, characteristic, complemented
4#20260974C212yesmaximal, complemented
5#20260940C211yesmaximal, characteristic, complemented
6#20260937C211yesmaximal, characteristic, complemented
7#20260897C212yesmaximal, complemented
8#20260895C212yesmaximal, complemented
9C4×#13095496C211yesmaximal, characteristic, complemented
10D4×#11715895C211yesmaximal, characteristic, complemented
11#547×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2212yes—
13#15690830—28no—
14C2×#13095501C2212yes—
15C2×#13095501C2212yes—
16C2×#13095499C2212yes—
17C2×#13095498C2212yes—
18C2×#13095496C2211yescharacteristic
19C22×#11715895C2212yes—
20#15690810C2211yescharacteristic
21#15690792—22no—
22#15690789C2212yes—
23#15690781—21no—
24C4×D388C2211yescharacteristic
25C4⋊C4×D97—21no—
26#15690743C2211yescharacteristic
27C4⋊#11715895C2211yescharacteristic
28#15690734C2212yes—
29#15690733C2212yes—
30#15690731C2211yescharacteristic
31#15690730C2211yescharacteristic
32C22⋊#11715895C2212yes—
33C4×#11715895C2211yescharacteristic
34C2×#41×C97C2212yes—
35C4⋊Q8×C97C2211yescharacteristic
36#123×C97—22no—
37C22⋊Q8×C97C2212yes—
38#121×C97—21no—
39C4×D4×C97C2211yescharacteristic
40C4×D4×C97C2211yescharacteristic
41C2×C4⋊C4×C97—21no—
42#13095511—24no—
43#13095511—42no—
44C2×Dic194C2312yes—
45C2×D388C2311yescharacteristic
46C2×C22⋊D97—22no—
47C2×C4×D97—21no—
48C2×C4×D97—22no—
49#13095501—24no—
50#13095501—42no—
51#13095499—24no—
52#13095498—24no—
53#13095496D414yescomplemented
54C2×#11715895—24no—
55C2×#11715895C2312yes—
56C2×#11715895C2311yescharacteristic
57C2×#11715895C2311yescharacteristic
58C4⋊Dic97—21no—
59C4⋊Dic97—21no—
60C4⋊Dic97C2311yescharacteristic
61#13095490—22no—
62C22⋊Dic97—22no—
63#13095488—21no—
64#41×C97—24no—
65#41×C97—42no—
66C2×Q8×C97C2312yes—
67C2×D4×C97—22no—
68C2×D4×C97D411yescharacteristic
69C2×D4×C97C2311yescharacteristic
70C22×C4×C97—21no—
71C22×C4×C97—22no—
72C22×C4×C97C2312yes—
73C4⋊C4×C97—21no—
74C4⋊C4×C97—22no—
75C4⋊C4×C97—21no—
76C4⋊C4×C97D411yescharacteristic
77C4⋊C4×C97C2311yescharacteristic
78C22⋊C4×C97—22no—
79C22⋊C4×C97—21no—
80C22⋊C4×C97D412yes—
81C42×C97C2311yescharacteristic
82C23×C97—21no—
83C23×C97C2×D412yes—
84C22×D97—21no—
85Q8×C97—22no—
86Q8×C97—22no—
87D4×C97—22no—
88D4×C97—21no—
89D4×C97—44no—
90D4×C97C2×D412yes—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—44no—
96C2×C4×C97—42no—
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97C2×D412yes—
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—44no—
102C2×Dic97—21no—
103C2×Dic97—22no—
104D388—21no—
105D388C2×D412yes—
106C4×D97—44no—
107C4×D97—42no—
108Dic194—22no—
109Dic194—22no—
110#11715895—22no—
111#11715895—21no—
112#11715895C2×D412yes—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97—22no—
116C22×C97—22no—
117C22×C97—41no—
118C22×C97—41no—
119C22×C97C22×D411yescharacteristic
120C2×D97—22no—
121C2×D97—41no—
122C2×D97—41no—
123C4×C97—22no—
124C4×C97—21no—
125C4×C97—42no—
126C4×C97—41no—
127C4×C97#13512yescomplemented
128C4×C97C22×D411yescharacteristic
129C4×C97H(5,𝔽2)11yescharacteristic
130Dic97—42no—
131Dic97—41no—
132C2×C97—42no—
133C2×C97—42no—
134C2×C97D4211yescharacteristic
135C2×C97C2×#13511yescharacteristic
136C2×C97#57811yescharacteristic
137D97—42no—
138#3283—971nomaximal, Sylow
139C97#328311yescharacteristic, complemented, Sylow
140C2×#135—972no—
141#547—971no—
142#547D9711yescharacteristic, complemented
143#494—971no—
144#467—972no—
145#461—971no—
146#461—971no—
147#452—972no—
148#452—972no—
149C4×D8—971no—
150C8×D4—971no—
151C2×#41—972no—
152C2×#41C2×D9712yes—
153#135—1948no—
154C2×Q16—972no—
155C2×Q16—972no—
156C2×SD16—972no—
157C2×SD16—972no—
158C2×D8—971no—
159C22×C8—972no—
160C4⋊Q8—971no—
161C4⋊Q8C2×D9711yescharacteristic
162#123—22no—
163#123—1942no—
164C22⋊Q8—972no—
165C22⋊Q8C2×D9712yes—
166#121—21no—
167#121—1941no—
168C4×D4—971no—
169C4×D4C2×D9711yescharacteristic
170C4×D4C2×D9711yescharacteristic
171C2×C4⋊C4—21no—
172C2×C4⋊C4—1941no—
173#107—971no—
174C4⋊C8—971no—
175#103—972no—
176#103—972no—
177#102—971no—
178#102—971no—
179C22⋊C8—972no—
180C4×C8—971no—
181#41—24no—
182#41—42no—
183#41—3882no—
184#41—1944no—
185C2×Q8—972no—
186C2×Q8C22×D9712yes—
187C2×D4—22no—
188C2×D4—971no—
189C2×D4—1942no—
190C2×D4C22⋊D9711yescharacteristic
191C2×D4C22×D9711yescharacteristic
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—1941no—
195C22×C4—1942no—
196C22×C4C22×D9712yes—
197Q16—3882no—
198Q16—1944no—
199SD16—1944no—
200SD16—1944no—
201D8—974no—
202C2×C8—971no—
203C2×C8—971no—
204C2×C8—972no—
205C2×C8—1944no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—971no—
210C4⋊C4—1942no—
211C4⋊C4—1941no—
212C4⋊C4—1941no—
213C4⋊C4C22⋊D9711yescharacteristic
214C4⋊C4C22×D9711yescharacteristic
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—1942no—
218C22⋊C4—1941no—
219C22⋊C4C22⋊D9712yes—
220C42C22×D9711yescharacteristic
221C23—21no—
222C23—1941no—
223C23C2×C22⋊D9712yes—
224Q8—22no—
225Q8—22no—
226Q8—1942no—
227Q8—1942no—
228D4—22no—
229D4—21no—
230D4—44no—
231D4—972no—
232D4—3884no—
233D4—1941no—
234D4D4×D9712yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—3884no—
242C2×C4—3882no—
243C2×C4—1941no—
244C2×C4—1942no—
245C2×C4C2×C22⋊D9711yescharacteristic
246C2×C4C2×C22⋊D9712yes—
247C2×C4C2×C22⋊D9711yescharacteristic
248C2×C4C23×D9711yescharacteristic
249C8—972no—
250C8—1942no—
251C8—1941no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—3881no—
259C22—3881no—
260C22—1942no—
261C22C22×C22⋊D9711yescharacteristic
262C4—21no—
263C4—22no—
264C4—41no—
265C4—42no—
266C4—3881no—
267C4—3882no—
268C4#1569083012yescomplemented
269C4C2×D4×D9711yescharacteristic
270C4#1569085511yescharacteristic
271C2—42no—
272C2—42no—
273C2—3882no—
274C2D4×C22⋊D9711yescharacteristic
275C2C2×#1569083011yescharacteristic
276C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#547×C97C2
socleC22×C97C22×D4
hypercenter#547D97
hypocenterC97#3283
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#15690830C2semidirect
2. #20261390C2semidirect
3. #20261114C2semidirect
4. #20260974C2semidirect
5. #20260940C2semidirect
6. #20260937C2semidirect
7. #20260897C2semidirect
8. #20260895C2semidirect
9. C4×#13095496C2semidirect
10. D4×#11715895C2semidirect
11. #547×C97C2semidirect
53. #13095496D4semidirect
127. C4×C97#135semidirect
139. C97#3283semidirect
142. #547D97semidirect
268. C4#15690830semidirect

Maximal subgroups

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

subgroupindex
C2×#156908302
#202613902
#202611142
#202609742
#202609402
#202609372
#202608972
#202608952
C4×#130954962
D4×#117158952
#547×C972
#328397