Finite Group Databasebeta

← #27153455

Subgroups — #27153455

Order 12416.

484 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 123 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×#15690833C211yesmaximal, characteristic, complemented
2C2×#15690832C211yesmaximal, characteristic, complemented
3#20261387C212yesmaximal, complemented
4#20261113C211yesmaximal, characteristic, complemented
5#20260977C211yesmaximal, characteristic, complemented
6#20260939C212yesmaximal, complemented
7#20260900C212yesmaximal, complemented
8#20260895C212yesmaximal, complemented
9#20260817C211yesmaximal, characteristic, complemented
10C8×C22⋊D97C211yesmaximal, characteristic, complemented
11#467×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×#13095511C2212yes—
14#15690833—24no—
15#15690832—24no—
16C2×Dic388C2211yescharacteristic
17C2×#13095501C2212yes—
18C2×#13095500C2212yes—
19C2×#13095498C2212yes—
20C2×D776C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22#15690808C2212yes—
23#15690795—22no—
24#15690792—22no—
25#15690790C2212yes—
26#15690783—22no—
27#15690772C2212yes—
28C4×C22⋊D97C2211yescharacteristic
29C4⋊C4×D97—22no—
30#15690739C2211yescharacteristic
31#15690733C2212yes—
32#15690730C2212yes—
33C22⋊#11715895C2211yescharacteristic
34#15690722C2211yescharacteristic
35C8×Dic97C2211yescharacteristic
36C2×Q16×C97C2211yescharacteristic
37C22×C8×C97C2211yescharacteristic
38C22⋊Q8×C97C2212yes—
39#107×C97C2211yescharacteristic
40#103×C97C2212yes—
41#13095512—24no—
42#13095512—42no—
43#13095511—24no—
44#13095511—42no—
45C2×Dic194C2312yes—
46C2×D388—22no—
47C2×D388C2312yes—
48C2×C22⋊D97—22no—
49C2×C22⋊D97D411yescharacteristic
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97C2311yescharacteristic
54Dic388—22no—
55#13095501—42no—
56#13095500—24no—
57#13095498—24no—
58D776D414yescomplemented
59C2×#11715895C2311yescharacteristic
60C2×#11715895C2311yescharacteristic
61C8×D97—22no—
62C4⋊Dic97—22no—
63#13095490—22no—
64#13095490—22no—
65#13095490D411yescharacteristic
66C22⋊Dic97D411yescharacteristic
67#13095488—22no—
68#13095488—22no—
69#13095488D411yescharacteristic
70C4×Dic97C2311yescharacteristic
71C2×Q8×C97C2312yes—
72C22×C4×C97C2311yescharacteristic
73Q16×C97—22no—
74C2×C8×C97—22no—
75C2×C8×C97C2311yescharacteristic
76C2×C8×C97C2311yescharacteristic
77C4⋊C4×C97—22no—
78C4⋊C4×C97C2312yes—
79C22⋊C4×C97—22no—
80C23×C97C2×D411yescharacteristic
81C22×D97—22no—
82C22×D97C2×D411yescharacteristic
83Q8×C97—22no—
84Q8×C97—22no—
85C2×C4×C97—21no—
86C2×C4×C97—21no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2411yescharacteristic
91C22⋊D97—22no—
92C22⋊D97—44no—
93C2×Dic97—22no—
94C2×Dic97C2×D411yescharacteristic
95C2×Dic97C2×D411yescharacteristic
96C2×Dic97C2×D411yescharacteristic
97D388—22no—
98D388—44no—
99D388C2×D414yes—
100C4×D97—21no—
101C4×D97—21no—
102C4×D97—44no—
103C4×D97—44no—
104C4×D97—44no—
105Dic194—22no—
106Dic194—22no—
107C8×C97—21no—
108C8×C97C2×D412yes—
109#11715895—21no—
110#11715895—21no—
111C22×C97—21no—
112C22×C97—21no—
113C22×C97—21no—
114C22×C97C22×D411yescharacteristic
115C2×D97—21no—
116C2×D97—24no—
117C2×D97—21no—
118C2×D97—21no—
119C2×D97—42no—
120C2×D97—42no—
121C4×C97—21no—
122C4×C97—42no—
123C4×C97—42no—
124C4×C97C22×D411yescharacteristic
125C4×C97H(5,𝔽2)11yescharacteristic
126Dic97—21no—
127Dic97—21no—
128Dic97—42no—
129Dic97#13512yescomplemented
130C2×C97—41no—
131C2×C97D4211yescharacteristic
132C2×C97C2×#13511yescharacteristic
133C2×C97#57811yescharacteristic
134D97—41no—
135D97—44no—
136#3283—971nomaximal, Sylow
137C97#328311yescharacteristic, complemented, Sylow
138C2×#135—971no—
139C2×#135—971no—
140#547—972no—
141#494—971no—
142#467—971no—
143#467D9711yescharacteristic, complemented
144#461—972no—
145#452—972no—
146#452—972no—
147C4×D8—971no—
148C8×D4—971no—
149C2×#41—972no—
150C2×#41—972no—
151#135—1944no—
152#135—1944no—
153C2×Q16—972no—
154C2×Q16—971no—
155C2×Q16C2×D9711yescharacteristic
156C2×SD16—972no—
157C2×SD16—972no—
158C2×D8—971no—
159C22×C8—971no—
160C22×C8C2×D9711yescharacteristic
161C4⋊Q8—972no—
162#123—1942no—
163#123—1942no—
164C22⋊Q8—972no—
165C22⋊Q8C2×D9712yes—
166#121—1942no—
167C4×D4—971no—
168C4×D4—972no—
169C2×C4⋊C4—1942no—
170#107C2×D9711yescharacteristic
171C4⋊C8—971no—
172#103—972no—
173#103C2×D9712yes—
174#102—972no—
175C22⋊C8—971no—
176C22⋊C8—971no—
177C4×C8—971no—
178#41—3882no—
179#41—3882no—
180#41—1944no—
181#41—1944no—
182C2×Q8—972no—
183C2×Q8C22×D9712yes—
184C2×D4—971no—
185C2×D4—972no—
186C2×D4—1942no—
187C2×D4—1942no—
188C22×C4—971no—
189C22×C4—1942no—
190C22×C4—1942no—
191C22×C4—1942no—
192C22×C4C22×D9711yescharacteristic
193Q16—22no—
194Q16—3882no—
195Q16—1942no—
196SD16—1944no—
197SD16—1944no—
198D8—974no—
199C2×C8—22no—
200C2×C8—971no—
201C2×C8—971no—
202C2×C8—1942no—
203C2×C8C22×D9711yescharacteristic
204C2×C8C22×D9711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—971no—
207C4⋊C4—1942no—
208C4⋊C4—1942no—
209C4⋊C4—1942no—
210C4⋊C4C22×D9712yes—
211C22⋊C4—22no—
212C22⋊C4—971no—
213C22⋊C4—971no—
214C22⋊C4—1942no—
215C22⋊C4—1942no—
216C42—971no—
217C23—971no—
218C23—1942no—
219C23D4×D9711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—1942no—
223Q8—1942no—
224D4—974no—
225D4—3884no—
226D4—3884no—
227D4—1942no—
228D4—1942no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—971no—
234C2×C4—971no—
235C2×C4—971no—
236C2×C4—3884no—
237C2×C4—3884no—
238C2×C4—3884no—
239C2×C4—1942no—
240C2×C4—1941no—
241C2×C4—1941no—
242C2×C4D4×D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—21no—
245C8—1941no—
246C8—1941no—
247C8D4×D9712yes—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—3882no—
252C22—3882no—
253C22—1941no—
254C22—1941no—
255C22—1944no—
256C22—1941no—
257C22C2×D4×D9711yescharacteristic
258C4—21no—
259C4—42no—
260C4—42no—
261C4—972no—
262C4—3882no—
263C4—1941no—
264C4—1941no—
265C4C2×D4×D9711yescharacteristic
266C4#1569085611yescharacteristic
267C2—41no—
268C2—3884no—
269C2—3881no—
270C2#2026136911yescharacteristic
271C2#135×D9711yescharacteristic
272C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#467×C97C2
socleC22×C97C22×D4
hypercenter#467D97
hypocenterC97#3283
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. C2×#15690832C2semidirect
3. #20261387C2semidirect
4. #20261113C2semidirect
5. #20260977C2semidirect
6. #20260939C2semidirect
7. #20260900C2semidirect
8. #20260895C2semidirect
9. #20260817C2semidirect
10. C8×C22⋊D97C2semidirect
11. #467×C97C2semidirect
58. D776D4semidirect
129. Dic97#135semidirect
137. C97#3283semidirect
143. #467D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
C2×#156908322
#202613872
#202611132
#202609772
#202609392
#202609002
#202608952
#202608172
C8×C22⋊D972
#467×C972
#328397