Finite Group Databasebeta

← #27153477

Subgroups — #27153477

Order 12416.

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

normal subgroups only

This group has 482 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×#15690833C211yesmaximal, characteristic, complemented
2#20261396C212yesmaximal, complemented
3#20261120C211yesmaximal, characteristic, complemented
4#20260973C211yesmaximal, characteristic, complemented
5#20260971C211yesmaximal, characteristic, complemented
6#20260959C212yesmaximal, complemented
7#20260896C212yesmaximal, complemented
8#20260874C212yesmaximal, complemented
9Q16×Dic97C211yesmaximal, characteristic, complemented
10C8×C22⋊D97C211yesmaximal, characteristic, complemented
11C22×Q16×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13#15690833—24no—
14C2×#13095498C2212yes—
15C2×#13095498C2212yes—
16C2×D776C2211yescharacteristic
17C2×C8×D97C2211yescharacteristic
18#15690799—22no—
19#15690799C2212yes—
20#15690786—22no—
21#15690786C2212yes—
22#15690784—22no—
23#15690784C2212yes—
24Q8×Dic97C2212yes—
25C4×C22⋊D97—22no—
26C4×C22⋊D97C2211yescharacteristic
27#15690744C2211yescharacteristic
28#15690739C2211yescharacteristic
29#15690734C2212yes—
30#15690734C2212yes—
31#15690732C2212yes—
32C22⋊#11715895C2211yescharacteristic
33#15690722C2211yescharacteristic
34C8×Dic97C2211yescharacteristic
35C22×Q8×C97C2212yes—
36C2×Q16×C97—24no—
37C2×Q16×C97C2211yescharacteristic
38C2×Q16×C97C2212yes—
39C2×Q16×C97C2211yescharacteristic
40C22×C8×C97C2211yescharacteristic
41#13095512—24no—
42#13095512—42no—
43C2×D388—22no—
44C2×D388C2312yes—
45C2×C22⋊D97—22no—
46C2×C22⋊D97D411yescharacteristic
47C2×C4×D97—22no—
48C2×C4×D97C2311yescharacteristic
49#13095498—24no—
50#13095498—42no—
51D776—22no—
52C2×#11715895C2311yescharacteristic
53C2×#11715895C2311yescharacteristic
54C8×D97—22no—
55C4⋊Dic97—22no—
56C4⋊Dic97C2312yes—
57#13095490—22no—
58#13095490D411yescharacteristic
59C22⋊Dic97D411yescharacteristic
60#13095488—22no—
61#13095488—22no—
62#13095488—22no—
63#13095488—22no—
64#13095488D411yescharacteristic
65C4×Dic97—22no—
66C4×Dic97C2311yescharacteristic
67C2×Q8×C97—22no—
68C2×Q8×C97—24no—
69C2×Q8×C97—42no—
70C2×Q8×C97C2312yes—
71C2×Q8×C97C2312yes—
72C22×C4×C97—22no—
73C22×C4×C97C2311yescharacteristic
74Q16×C97—24no—
75Q16×C97—22no—
76Q16×C97D414yescomplemented
77C2×C8×C97—22no—
78C2×C8×C97C2311yescharacteristic
79C2×C8×C97C2311yescharacteristic
80C23×C97C2×D411yescharacteristic
81C22×D97—22no—
82C22×D97C2×D411yescharacteristic
83Q8×C97—22no—
84Q8×C97—22no—
85Q8×C97—22no—
86Q8×C97—44no—
87Q8×C97C2×D414yes—
88C2×C4×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—44no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2411yescharacteristic
95C22⋊D97—22no—
96C22⋊D97—44no—
97C2×Dic97—22no—
98C2×Dic97C2×D411yescharacteristic
99C2×Dic97C2×D411yescharacteristic
100C2×Dic97C2×D411yescharacteristic
101D388—22no—
102D388—22no—
103D388—44no—
104C4×D97—21no—
105C4×D97—21no—
106C4×D97—44no—
107C8×C97—21no—
108C8×C97C2×D412yes—
109#11715895—21no—
110#11715895—21no—
111C22×C97—21no—
112C22×C97—21no—
113C22×C97#13512yescomplemented
114C22×C97C22×D411yescharacteristic
115C2×D97—21no—
116C2×D97—21no—
117C2×D97—21no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—42no—
121C4×C97—24no—
122C4×C97—21no—
123C4×C97—42no—
124C4×C97C22×D411yescharacteristic
125C4×C97H(5,𝔽2)11yescharacteristic
126Dic97—21no—
127Dic97—21no—
128Dic97—21no—
129Dic97—42no—
130C2×C97—22no—
131C2×C97D4211yescharacteristic
132C2×C97C2×#13511yescharacteristic
133C2×C97#57811yescharacteristic
134D97—41no—
135D97—82no—
136#3285—971nomaximal, Sylow
137C97#328511yescharacteristic, complemented, Sylow
138C2×#135—971no—
139C22×Q16D9711yescharacteristic, complemented
140#548—972no—
141#495—971no—
142#467—971no—
143#466—971no—
144#464—972no—
145#452—972no—
146#448—972no—
147C4×Q16—971no—
148C8×D4—971no—
149C2×#41—972no—
150C22×Q8C2×D9712yes—
151#135—1944no—
152C2×Q16—24no—
153C2×Q16C2×D9712yes—
154C2×Q16C2×D9711yescharacteristic
155C2×Q16C2×D9711yescharacteristic
156C2×SD16—972no—
157C2×SD16—972no—
158C2×D8—971no—
159C22×C8—971no—
160C22×C8C2×D9711yescharacteristic
161#124—972no—
162#124—1942no—
163C22⋊Q8—972no—
164C22⋊Q8—1942no—
165#121—972no—
166#121—1942no—
167C4×Q8—972no—
168C4×D4—971no—
169C4×D4—1942no—
170#107—971no—
171C4⋊C8—971no—
172#103—972no—
173#103—972no—
174#102—972no—
175C22⋊C8—971no—
176C22⋊C8—971no—
177C4×C8—971no—
178#41—3882no—
179#41—1944no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D9712yes—
184C2×Q8C22×D9712yes—
185C2×D4—971no—
186C2×D4—972no—
187C2×D4—1942no—
188C2×D4—1942no—
189C22×C4—22no—
190C22×C4—971no—
191C22×C4—1942no—
192C22×C4C22×D9711yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D9714yescomplemented
196SD16—3882no—
197SD16—1944no—
198D8—1942no—
199C2×C8—22no—
200C2×C8—971no—
201C2×C8—971no—
202C2×C8—1942no—
203C2×C8C22×D9711yescharacteristic
204C2×C8C22×D9711yescharacteristic
205C4⋊C4—971no—
206C4⋊C4—972no—
207C4⋊C4—1942no—
208C4⋊C4—1942no—
209C22⋊C4—971no—
210C22⋊C4—971no—
211C22⋊C4—1942no—
212C22⋊C4—1942no—
213C22⋊C4—1942no—
214C22⋊C4—1942no—
215C42—971no—
216C42—1942no—
217C23—971no—
218C23—1942no—
219C23D4×D9711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D9714yes—
225D4—3884no—
226D4—3884no—
227D4—1942no—
228D4—1942no—
229D4—1942no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—971no—
236C2×C4—971no—
237C2×C4—971no—
238C2×C4—3884no—
239C2×C4—1942no—
240C2×C4—1941no—
241C2×C4—1941no—
242C2×C4D4×D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—21no—
245C8—1941no—
246C8—1941no—
247C8C2×C22⋊D9712yes—
248C22—21no—
249C22—21no—
250C22—3882no—
251C22—3882no—
252C22—3882no—
253C22—1941no—
254C22—1941no—
255C22—1941no—
256C22#1569083312yescomplemented
257C22C2×D4×D9711yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—3882no—
262C4—1941no—
263C4—1941no—
264C4—1941no—
265C4C22×C22⋊D9711yescharacteristic
266C4#1569085511yescharacteristic
267C2—22no—
268C2—7762no—
269C2—3881no—
270C2D4×C22⋊D9711yescharacteristic
271C2C2×#1569083311yescharacteristic
272C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×Q16×C97C2
socleC22×C97C22×D4
hypercenterC22×Q16D97
hypocenterC97#3285
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. #20261396C2semidirect
3. #20261120C2semidirect
4. #20260973C2semidirect
5. #20260971C2semidirect
6. #20260959C2semidirect
7. #20260896C2semidirect
8. #20260874C2semidirect
9. Q16×Dic97C2semidirect
10. C8×C22⋊D97C2semidirect
11. C22×Q16×C97C2semidirect
76. Q16×C97D4semidirect
113. C22×C97#135semidirect
137. C97#3285semidirect
139. C22×Q16D97semidirect
195. Q16C22⋊D97semidirect
256. C22#15690833semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
#202613962
#202611202
#202609732
#202609712
#202609592
#202608962
#202608742
Q16×Dic972
C8×C22⋊D972
C22×Q16×C972
#328597