Finite Group Databasebeta

← #27153478

Subgroups — #27153478

Order 12416.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×Q16×D97C211yesmaximal, characteristic, complemented
3#20261397C212yesmaximal, complemented
4#20261119C211yesmaximal, characteristic, complemented
5#20260977C211yesmaximal, characteristic, complemented
6#20260972C211yesmaximal, characteristic, complemented
7#20260957C212yesmaximal, complemented
8#20260900C212yesmaximal, complemented
9#20260875C212yesmaximal, complemented
10C8×D388C211yesmaximal, characteristic, complemented
11C4×Q16×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×Q8×D97C2212yes—
14#15690833—24no—
15C2×Dic388C2211yescharacteristic
16C2×#13095501C2212yes—
17Q16×D97—24no—
18C2×#13095500C2212yes—
19C2×#13095498C2212yes—
20C2×D776C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690798—22no—
24#15690798C2212yes—
25#15690790—22no—
26#15690790C2212yes—
27#15690785—22no—
28#15690785C2212yes—
29C4×D388—22no—
30C4×D388C2211yescharacteristic
31C4⋊#11715895C2211yescharacteristic
32#15690733C2212yes—
33#15690730C2212yes—
34#15690722C2211yescharacteristic
35#15690722C2211yescharacteristic
36C2×Q16×C97C2211yescharacteristic
37C4×Q8×C97C2212yes—
38#107×C97C2211yescharacteristic
39#103×C97C2212yes—
40C4×C8×C97C2211yescharacteristic
41#13095512—24no—
42#13095512—42no—
43C2×Dic194—22no—
44C2×Dic194C2312yes—
45Q8×D97—24no—
46Q8×D97—42no—
47C2×D388—22no—
48C2×D388—22no—
49C2×D388D411yescharacteristic
50C2×D388C2312yes—
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97C2311yescharacteristic
54C2×C4×D97C2311yescharacteristic
55Dic388—22no—
56#13095501—24no—
57#13095500—42no—
58#13095498—24no—
59D776—22no—
60C2×#11715895C2311yescharacteristic
61C2×#11715895C2311yescharacteristic
62C8×D97—22no—
63C8×D97—22no—
64C4⋊Dic97—22no—
65C4⋊Dic97D411yescharacteristic
66#13095488—22no—
67#13095488—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488D411yescharacteristic
71#13095488D411yescharacteristic
72C2×Q8×C97C2312yes—
73Q16×C97D414yescomplemented
74C2×C8×C97C2311yescharacteristic
75C2×C8×C97C2311yescharacteristic
76C4⋊C4×C97—22no—
77C4⋊C4×C97C2312yes—
78C42×C97—22no—
79C42×C97C2311yescharacteristic
80C22×D97—22no—
81C22×D97C2×D411yescharacteristic
82C22×D97C2×D411yescharacteristic
83Q8×C97—22no—
84Q8×C97C2×D414yes—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2411yescharacteristic
90C2×Dic97—22no—
91C2×Dic97C2×D411yescharacteristic
92C2×Dic97C2×D411yescharacteristic
93D388—22no—
94D388—22no—
95D388—22no—
96D388—44no—
97D388—44no—
98C4×D97—21no—
99C4×D97—21no—
100C4×D97—21no—
101C4×D97—21no—
102C4×D97—44no—
103C4×D97—44no—
104Dic194—22no—
105Dic194—22no—
106Dic194—44no—
107C8×C97—21no—
108C8×C97C2×D412yes—
109#11715895—21no—
110#11715895—21no—
111C22×C97C22×D411yescharacteristic
112C2×D97—21no—
113C2×D97—21no—
114C2×D97—21no—
115C2×D97—21no—
116C2×D97—21no—
117C2×D97—42no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97#13512yescomplemented
121C4×C97—24no—
122C4×C97—21no—
123C4×C97—21no—
124C4×C97—42no—
125C4×C97C22×D411yescharacteristic
126C4×C97H(5,𝔽2)11yescharacteristic
127Dic97—21no—
128Dic97—21no—
129Dic97—42no—
130C2×C97D4211yescharacteristic
131C2×C97C2×#13511yescharacteristic
132C2×C97#57811yescharacteristic
133D97—22no—
134D97—41no—
135D97—82no—
136#3285—971nomaximal, Sylow
137C97#328511yescharacteristic, complemented, Sylow
138C2×#135—971no—
139C22×Q16—971no—
140#548—972no—
141#495—971no—
142#467—971no—
143#466—971no—
144#464—972no—
145#452—972no—
146#448—972no—
147C4×Q16D9711yescharacteristic, complemented
148C8×D4—971no—
149C2×#41—972no—
150C22×Q8—972no—
151#135—1944no—
152C2×Q16—972no—
153C2×Q16—971no—
154C2×Q16—1944no—
155C2×Q16C2×D9711yescharacteristic
156C2×SD16—972no—
157C2×SD16—972no—
158C2×D8—971no—
159C22×C8—971no—
160C22×C8—971no—
161#124—972no—
162#124—1942no—
163C22⋊Q8—972no—
164C22⋊Q8—1942no—
165#121—972no—
166#121—1942no—
167C4×Q8C2×D9712yes—
168C4×D4—971no—
169C4×D4—1942no—
170#107C2×D9711yescharacteristic
171C4⋊C8—971no—
172#103—972no—
173#103C2×D9712yes—
174#102—972no—
175C22⋊C8—971no—
176C22⋊C8—971no—
177C4×C8C2×D9711yescharacteristic
178#41—3882no—
179#41—1944no—
180C2×Q8—972no—
181C2×Q8—3882no—
182C2×Q8—1942no—
183C2×Q8—1944no—
184C2×Q8C22×D9712yes—
185C2×D4—971no—
186C2×D4—972no—
187C2×D4—1942no—
188C2×D4—1942no—
189C22×C4—971no—
190C22×C4—971no—
191C22×C4—1942no—
192C22×C4—1942no—
193Q16—1944no—
194Q16—1942no—
195Q16D38814yescomplemented
196SD16—3882no—
197SD16—1944no—
198D8—1942no—
199C2×C8—971no—
200C2×C8—971no—
201C2×C8—1942no—
202C2×C8—1942no—
203C2×C8C22×D9711yescharacteristic
204C2×C8C22×D9711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—971no—
207C4⋊C4—1942no—
208C4⋊C4C22×D9712yes—
209C22⋊C4—971no—
210C22⋊C4—971no—
211C22⋊C4—1942no—
212C22⋊C4—1942no—
213C22⋊C4—1942no—
214C22⋊C4—1942no—
215C42—22no—
216C42C22×D9711yescharacteristic
217C23—971no—
218C23—971no—
219C23—1942no—
220Q8—22no—
221Q8—3884no—
222Q8—1942no—
223Q8—1942no—
224Q8C2×D38814yes—
225D4—3884no—
226D4—3884no—
227D4—1942no—
228D4—1942no—
229D4—1942no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—971no—
233C2×C4—971no—
234C2×C4—3884no—
235C2×C4—3884no—
236C2×C4—1942no—
237C2×C4—1941no—
238C2×C4—1941no—
239C2×C4—1941no—
240C2×C4—1941no—
241C2×C4D4×D9711yescharacteristic
242C2×C4D4×D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—21no—
245C8—1941no—
246C8—1941no—
247C8C2×D38812yes—
248C22—972no—
249C22—3882no—
250C22—3882no—
251C22—3882no—
252C22—1941no—
253C22—1941no—
254C22—1941no—
255C22—1941no—
256C22—1941no—
257C22C2×D4×D9711yescharacteristic
258C4—21no—
259C4—21no—
260C4—24no—
261C4—42no—
262C4—3882no—
263C4—1941no—
264C4—1941no—
265C4C22×D38811yescharacteristic
266C4#1569085611yescharacteristic
267C2—7762no—
268C2—3881no—
269C2—1942no—
270C2D4×D38811yescharacteristic
271C2#135×D9711yescharacteristic
272C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×Q16×C97C2
socleC22×C97C22×D4
hypercenterC4×Q16D97
hypocenterC97#3285
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. C2×Q16×D97C2semidirect
3. #20261397C2semidirect
4. #20261119C2semidirect
5. #20260977C2semidirect
6. #20260972C2semidirect
7. #20260957C2semidirect
8. #20260900C2semidirect
9. #20260875C2semidirect
10. C8×D388C2semidirect
11. C4×Q16×C97C2semidirect
73. Q16×C97D4semidirect
120. C2×D97#135semidirect
137. C97#3285semidirect
147. C4×Q16D97semidirect
195. Q16D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
C2×Q16×D972
#202613972
#202611192
#202609772
#202609722
#202609572
#202609002
#202608752
C8×D3882
C4×Q16×C972
#328597