Finite Group Databasebeta

← #27153600

Subgroups — #27153600

Order 12416.

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

normal subgroups only

This group has 420 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
1#20261400C211yesmaximal, characteristic, complemented
2#20261388C211yesmaximal, characteristic
3#20261064C212yesmaximal, complemented
4#20261017C211yesmaximal, characteristic, complemented
5#20260967C212yesmaximal, complemented
6#20260937C211yesmaximal, characteristic, complemented
7#20260875C212yesmaximal, complemented
8C4×#13095500C211yesmaximal, characteristic, complemented
9C8×D388C211yesmaximal, characteristic, complemented
10#103×D97C212yesmaximal, complemented
11#487×C97C211yesmaximal, characteristic, complemented
12C2×#13095512—21no—
13C2×Q8×D97C2212yes—
14C2×#13095500C2211yescharacteristic
15C2×#13095498C2212yes—
16C2×C8×D97C2212yes—
17#15690809—21no—
18#15690795—22no—
19#15690788—22no—
20#15690786—21no—
21#15690786C2212yes—
22#15690785C2212yes—
23C4×Dic194C2211yescharacteristic
24C4×D388C2211yescharacteristic
25C4×D388C2211yescharacteristic
26#15690772—21no—
27C4⋊C4×D97C2212yes—
28#15690742C2211yescharacteristic
29#15690741C2212yes—
30C4⋊#11715895C2211yescharacteristic
31#15690735C2211yescharacteristic
32#15690734C2212yes—
33#15690733C2212yes—
34#15690732C2211yescharacteristic
35#15690722C2212yes—
36C4⋊Q8×C97C2211yescharacteristic
37C4⋊Q8×C97C2211yescharacteristic
38#103×C97C2212yes—
39#103×C97C2212yes—
40C4×C8×C97C2211yescharacteristic
41#13095512—44no—
42C2×Dic194—22no—
43C2×Dic194C2311yescharacteristic
44Q8×D97—24no—
45Q8×D97—42no—
46C2×D388—22no—
47C2×D388D411yescharacteristic
48C2×D388C2311yescharacteristic
49C2×C4×D97—22no—
50C2×C4×D97—22no—
51C2×C4×D97—21no—
52C2×C4×D97C2312yes—
53#13095500—22no—
54#13095498—42no—
55C2×#11715895C2312yes—
56C8×D97—42no—
57C4⋊Dic97D411yescharacteristic
58C4⋊Dic97C2311yescharacteristic
59C4⋊Dic97C2311yescharacteristic
60#13095490—21no—
61#13095490—22no—
62#13095490—22no—
63#13095488—22no—
64#13095488—21no—
65#13095488—22no—
66#13095488D412yes—
67C4×Dic97—21no—
68C2×Q8×C97—21no—
69C2×Q8×C97C2312yes—
70C2×C8×C97C2311yescharacteristic
71C2×C8×C97C2311yescharacteristic
72C4⋊C4×C97—21no—
73C4⋊C4×C97—21no—
74C4⋊C4×C97—21no—
75C4⋊C4×C97C2312yes—
76C42×C97C2311yescharacteristic
77C22×D97—21no—
78C22×D97C2×D412yes—
79Q8×C97—22no—
80Q8×C97—22no—
81Q8×C97—42no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97C2411yescharacteristic
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97C2×D412yes—
91D388—21no—
92D388—21no—
93D388—44no—
94D388SD1614yescomplemented
95C4×D97—22no—
96C4×D97—22no—
97C4×D97—44no—
98C4×D97—44no—
99C4×D97—42no—
100Dic194—21no—
101Dic194—44no—
102Dic194#4112yescomplemented
103C8×C97—21no—
104C8×C97—21no—
105#11715895—22no—
106C22×C97C22×D411yescharacteristic
107C2×D97—22no—
108C2×D97—22no—
109C2×D97—41no—
110C2×D97—41no—
111C2×D97—41no—
112C2×D97C2×SD1614yes—
113C4×C97—21no—
114C4×C97—42no—
115C4×C97—42no—
116C4×C97C2×SD1612yes—
117C4×C97C2×#4111yescharacteristic
118C4×C97H(5,𝔽2)11yescharacteristic
119Dic97—22no—
120Dic97—22no—
121Dic97—41no—
122Dic97—41no—
123C2×C97#54611yescharacteristic
124C2×C97C22×SD1611yescharacteristic
125C2×C97#57811yescharacteristic
126D97—24no—
127D97—81no—
128#3298—971nomaximal, Sylow
129C97#329811yescharacteristic, complemented, Sylow
130D4×Q8—971no—
131#547—971no—
132#487D9711yescharacteristic, complemented
133#483—972no—
134#475—971no—
135#465—972no—
136#461—971no—
137#448—972no—
138C4×SD16—971no—
139C8×D4—971no—
140C2×#103—972no—
141C2×#41—1941no—
142C22×Q8—972no—
143C2×SD16—972no—
144C2×SD16—971no—
145C22×C8—972no—
146C4⋊Q8—1941no—
147C4⋊Q8C2×D9711yescharacteristic
148C4⋊Q8C2×D9711yescharacteristic
149#123—1942no—
150C22⋊Q8—972no—
151C22⋊Q8—1942no—
152C22⋊Q8—1941no—
153#121—972no—
154C4×Q8—971no—
155C4×D4—971no—
156C4×D4—971no—
157C4×D4—1941no—
158C2×C4⋊C4—972no—
159#106—971no—
160#106—972no—
161C4⋊C8—971no—
162#103—972no—
163#103—971no—
164#103—972no—
165#103C2×D9712yes—
166#103C2×D9712yes—
167#102—971no—
168C22⋊C8—972no—
169C4×C8C2×D9711yescharacteristic
170#41—3884no—
171C2×Q8—21no—
172C2×Q8—971no—
173C2×Q8—3882no—
174C2×Q8—1942no—
175C2×Q8—1944no—
176C2×Q8C22×D9712yes—
177C2×D4—971no—
178C2×D4—971no—
179C2×D4—1942no—
180C22×C4—972no—
181C22×C4—1942no—
182C22×C4—1942no—
183C22×C4—1941no—
184SD16—3882no—
185SD16—1942no—
186C2×C8—972no—
187C2×C8—3882no—
188C2×C8C22×D9711yescharacteristic
189C2×C8C22×D9711yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—971no—
194C4⋊C4—971no—
195C4⋊C4—971no—
196C4⋊C4—1942no—
197C4⋊C4—1942no—
198C4⋊C4—1941no—
199C4⋊C4C22×D9712yes—
200C22⋊C4—972no—
201C22⋊C4—1942no—
202C22⋊C4—1942no—
203C22⋊C4—1941no—
204C42—1941no—
205C42C22×D9711yescharacteristic
206C23—972no—
207C23—1941no—
208Q8—22no—
209Q8—22no—
210Q8—42no—
211Q8—972no—
212Q8—3884no—
213Q8—1941no—
214D4—974no—
215D4—3884no—
216D4—1941no—
217D4—1941no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—972no—
221C2×C4—3884no—
222C2×C4—3884no—
223C2×C4—3882no—
224C2×C4—1941no—
225C2×C4—1941no—
226C2×C4—1941no—
227C2×C4—1942no—
228C2×C4—1942no—
229C2×C4D4×D9711yescharacteristic
230C2×C4D4×D9711yescharacteristic
231C2×C4C23×D9711yescharacteristic
232C8—21no—
233C8—21no—
234C8—1942no—
235C22—974no—
236C22—3881no—
237C22—3881no—
238C22—3881no—
239C22—1942no—
240C22—1942no—
241C22C2×D4×D9711yescharacteristic
242C4—21no—
243C4—42no—
244C4—42no—
245C4—3881no—
246C4—3881no—
247C4—1942no—
248C4—1942no—
249C4SD16×D9712yes—
250C4#41×D9711yescharacteristic
251C4#1569085611yescharacteristic
252C2—7761no—
253C2—1944no—
254C2#2026137811yescharacteristic
255C2C2×SD16×D9711yescharacteristic
256C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#487×C97C2
socleC22×C97C22×D4
prime radical#20261388C2
hypercenter#487D97
hypocenterC97#3298
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #20261400C2semidirect
3. #20261064C2semidirect
4. #20261017C2semidirect
5. #20260967C2semidirect
6. #20260937C2semidirect
7. #20260875C2semidirect
8. C4×#13095500C2semidirect
9. C8×D388C2semidirect
10. #103×D97C2semidirect
11. #487×C97C2semidirect
94. D388SD16semidirect
102. Dic194#41semidirect
129. C97#3298semidirect
132. #487D97semidirect

Maximal subgroups

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

subgroupindex
#202614002
#202613882
#202610642
#202610172
#202609672
#202609372
#202608752
C4×#130955002
C8×D3882
#103×D972
#487×C972
#329897