Finite Group Databasebeta

← #27157509

Subgroups — #27157509

Order 12416.

324 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 98 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×Q8×#87604C211yesmaximal, characteristic
2C2×#15690873C211yesmaximal, characteristic
3C2×#15690864C211yesmaximal, characteristic
4C4⋊C4×#87604C211yesmaximal, characteristic
5#20261609C211yesmaximal, characteristic
6#20261608C211yesmaximal, characteristic
7#103×D97C211yesmaximal, characteristic
8C2×#13095519C2211yescharacteristic
9Q8×#87604—22no—
10Q8×#87604C414yescomplemented
11C2×C4⋊#87604—21no—
12C2×C4⋊#87604C2211yescharacteristic
13C2×C4⋊#87604C2211yescharacteristic
14C2×C4×#87604—21no—
15C2×C4×#87604—21no—
16C2×C4×#87604C2211yescharacteristic
17#15690873C414yescomplemented
18#15690864—22no—
19(C2×#87604)⋊C4—21no—
20C2×Q8×D97C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C4⋊C4×D97C2211yescharacteristic
23#15690735C411yescharacteristic, complemented
24#15690734C411yescharacteristic, complemented
25#15690733C411yescharacteristic, complemented
26#103×C97C411yescharacteristic, complemented
27C22×#87604—21no—
28C22×#87604—21no—
29C22×#87604D411yescharacteristic
30C22×#87604D411yescharacteristic
31#13095519—21no—
32#13095519C2×C412yes—
33C2×#11715897—21no—
34C4⋊#87604—24no—
35C4⋊#87604—21no—
36C4⋊#87604—21no—
37C4⋊#87604C2×C412yes—
38C4⋊#87604C2×C412yes—
39C4×#87604—24no—
40C4×#87604—42no—
41C4×#87604C2×C412yes—
42C4×#87604D412yes—
43C2×Dic194—21no—
44C2×Dic194C2×C411yescharacteristic
45Q8×D97—21no—
46Q8×D97—21no—
47Q8×D97C2×C412yes—
48Q8×D97C2×C412yes—
49C2×C4×D97—21no—
50C2×C4×D97—21no—
51C2×C4×D97C2311yescharacteristic
52C2×#11715895C2×C411yescharacteristic
53C8×D97—21no—
54C8×D97—21no—
55C4⋊Dic97C2×C411yescharacteristic
56#13095490—21no—
57C2×Q8×C97C2×C411yescharacteristic
58C2×C8×C97C2×C411yescharacteristic
59C4⋊C4×C97C2×C411yescharacteristic
60C22×D97C2×D411yescharacteristic
61C2×#87604—21no—
62C2×#87604—21no—
63C2×#87604—21no—
64C2×#87604—21no—
65C2×#87604—22no—
66C2×#87604—22no—
67C2×#87604—41no—
68C2×#87604—41no—
69C2×#87604—41no—
70C2×#87604—41no—
71C2×#87604C22⋊C412yes—
72C2×#87604C22⋊C412yes—
73Q8×C97—21no—
74Q8×C97C4212yescomplemented
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97C22×C411yescharacteristic
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97#4111yescharacteristic
81C4×D97—22no—
82C4×D97—22no—
83C4×D97—41no—
84C4×D97—41no—
85C4×D97C22×C411yescharacteristic
86C4×D97C22×C411yescharacteristic
87C4×D97C2×D411yescharacteristic
88C4×D97#4111yescharacteristic
89Dic194—21no—
90Dic194—22no—
91Dic194—22no—
92Dic194C4212yescomplemented
93#11715897—22no—
94C8×C97—21no—
95#11715895—21no—
96C22×C97C4×D411yescharacteristic
97C2×D97C2×C22⋊C411yescharacteristic
98C2×D97C4×D411yescharacteristic
99C2×D97AGL(1,ℤ/8)11yescharacteristic
100C2×D97#13711yescharacteristic
101C2×D97#13711yescharacteristic
102C2×D97#13711yescharacteristic
103#87604—22no—
104#87604—22no—
105#87604—42no—
106#87604—42no—
107C4×C97—22no—
108C4×C97—41no—
109C4×C97C2×C4211yescharacteristic
110C4×C97C4×D411yescharacteristic
111Dic97—22no—
112Dic97—41no—
113Dic97#11711yescharacteristic
114Dic97C4×D411yescharacteristic
115C2×C97C4×C22⋊C411yescharacteristic
116C2×C97#44011yescharacteristic
117C2×C97#44111yescharacteristic
118D97#41711yescharacteristic
119D97#41911yescharacteristic
120D97#44011yescharacteristic
121D97#44111yescharacteristic
122#1763—971nomaximal, Sylow
123C97#176311yescharacteristic, complemented, Sylow
124C2×C4×Q8—971no—
125C2×#103—971no—
126C2×#103—971no—
127C2×#97—971no—
128C4×C4⋊C4—971no—
129#340—971no—
130#340—971no—
131C22×Q8—971no—
132C22×C8—971no—
133C22×C8—971no—
134C4×Q8—974no—
135C4×Q8—1942no—
136C2×C4⋊C4—971no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—1941no—
140C2×C42—971no—
141C2×C42—1941no—
142C2×C42—1941no—
143#103—974no—
144#103—971no—
145#103—971no—
146#103—971no—
147#103#8760411yescharacteristic, complemented
148#97—1942no—
149#95—1941no—
150C2×Q8—972no—
151C2×Q8—972no—
152C2×Q8—971no—
153C2×Q8—1941no—
154C2×Q8—1941no—
155C2×Q8—1941no—
156C2×Q8C2×#8760411yescharacteristic
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—1941no—
161C22×C4—1941no—
162C22×C4—1941no—
163C22×C4—1941no—
164C2×C8—972no—
165C2×C8—971no—
166C2×C8—1941no—
167C2×C8—1941no—
168C2×C8—1941no—
169C2×C8—1941no—
170C2×C8C2×#8760411yescharacteristic
171C4⋊C4—972no—
172C4⋊C4—972no—
173C4⋊C4—971no—
174C4⋊C4—1941no—
175C4⋊C4—1941no—
176C4⋊C4—1941no—
177C4⋊C4—1944no—
178C4⋊C4C2×#8760411yescharacteristic
179C42—972no—
180C42—972no—
181C42—3882no—
182C42—1944no—
183C23—971no—
184Q8—21no—
185Q8—972no—
186Q8—1942no—
187Q8—1942no—
188Q8—1941no—
189Q8C4×#8760412yescomplemented
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—971no—
193C2×C4—971no—
194C2×C4—971no—
195C2×C4—971no—
196C2×C4—971no—
197C2×C4—972no—
198C2×C4—972no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—3881no—
202C2×C4—3881no—
203C2×C4—3881no—
204C2×C4—3881no—
205C2×C4—1942no—
206C2×C4—1942no—
207C2×C4—1942no—
208C2×C4—1942no—
209C2×C4—1941no—
210C2×C4—1941no—
211C2×C4—1941no—
212C2×C4—1941no—
213C2×C4—1941no—
214C2×C4—1941no—
215C2×C4C22×#8760411yescharacteristic
216C8—21no—
217C8—1942no—
218C8—1941no—
219C22—971no—
220C22—971no—
221C22—971no—
222C22—971no—
223C22—971no—
224C22—971no—
225C22D4×#8760411yescharacteristic
226C4—22no—
227C4—41no—
228C4—971no—
229C4—971no—
230C4—3882no—
231C4—3882no—
232C4—3881no—
233C4—1942no—
234C4—1942no—
235C4—1942no—
236C4C2×C4×#8760411yescharacteristic
237C4D4×#8760411yescharacteristic
238C2—971no—
239C2—971no—
240C2—971no—
241C2—971no—
242C2C22⋊C4×#8760411yescharacteristic
243C2#2026177411yescharacteristic
244C2#2026177511yescharacteristic
subgroupG/H
centerC22D4×#87604
commutatorC4×C97C2×C42
FrattiniC2×C4C22×#87604
Fitting#103×C97C4
socleC22×C97C4×D4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenter#103#87604
hypocenterC97#1763
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
10. Q8×#87604C4semidirect
17. #15690873C4semidirect
23. #15690735C4semidirect
24. #15690734C4semidirect
25. #15690733C4semidirect
26. #103×C97C4semidirect
74. Q8×C97C42semidirect
92. Dic194C42semidirect
123. C97#1763semidirect
147. #103#87604semidirect
189. Q8C4×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#876042
C2×#156908732
C2×#156908642
C4⋊C4×#876042
#202616092
#202616082
#103×D972
#176397