Finite Group Databasebeta

← #27157575

Subgroups — #27157575

Order 12416.

592 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 104 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×#15690873C214yesmaximal, complemented
2#20261703C211yesmaximal, characteristic
3#20261677C211yesmaximal, characteristic
4C22×Q8×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7C2×C22⋊#87604—21no—
8#15690873—28no—
9#15690867—21no—
10C22⋊#11715897—21no—
11(C2×#87604)⋊C4—21no—
12C22×Dic194—21no—
13C22×Dic194C411yescharacteristic, complemented
14C2×Q8×D97—28no—
15C2×Q8×D97—21no—
16C2×Q8×D97—21no—
17C2×Q8×D97C412yescomplemented
18C2×Q8×D97C2212yes—
19C22×C4×D97—21no—
20C22×C4×D97C2211yescharacteristic
21C22×Q8×C97C411yescharacteristic, complemented
22C22×#87604—22no—
23#13095519—24no—
24C2×#11715897—22no—
25C4⋊#87604—24no—
26C22⋊#87604—42no—
27C23×D97D411yescharacteristic
28C2×Dic194—28no—
29C2×Dic194—21no—
30C2×Dic194—22no—
31C2×Dic194—22no—
32C2×Dic194—24no—
33C2×Dic194C2×C412yes—
34Q8×D97—28no—
35Q8×D97—28no—
36Q8×D97—24no—
37Q8×D97—24no—
38Q8×D97—24no—
39Q8×D97D418yes—
40C22×Dic97—21no—
41C22×Dic97D411yescharacteristic
42C2×C4×D97—28no—
43C2×C4×D97—22no—
44C2×C4×D97—22no—
45C2×C4×D97—24no—
46C2×C4×D97C2×C411yescharacteristic
47C2×C4×D97D411yescharacteristic
48C2×C4×D97D411yescharacteristic
49C2×C4×D97C2311yescharacteristic
50C2×Q8×C97—24no—
51C2×Q8×C97—21no—
52C2×Q8×C97C2×C412yes—
53C22×C4×C97—21no—
54C22×C4×C97C2×C411yescharacteristic
55C23×C97C22⋊C411yescharacteristic
56C22×D97—22no—
57C22×D97—22no—
58C22×D97C22⋊C411yescharacteristic
59C22×D97SD1612yes—
60C22×D97Q1612yes—
61C22×D97C2×D411yescharacteristic
62C2×#87604—42no—
63C2×#87604—42no—
64C2×#87604—42no—
65Q8×C97—24no—
66Q8×C97—22no—
67Q8×C97C22⋊C414yescomplemented
68C2×C4×C97—24no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97C22⋊C411yescharacteristic
72C2×C4×C97C22×C411yescharacteristic
73C2×Dic97—24no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97C22⋊C411yescharacteristic
77C2×Dic97C2×D411yescharacteristic
78C4×D97—28no—
79C4×D97—28no—
80C4×D97—22no—
81C4×D97—22no—
82C4×D97—22no—
83C4×D97C2×D414yes—
84Dic194—28no—
85Dic194—24no—
86Dic194—22no—
87Dic194—28no—
88Dic194C22⋊C414yescomplemented
89#11715897—42no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97#10312yescomplemented
93C22×C97C2×C22⋊C411yescharacteristic
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—24no—
97C2×D97—24no—
98C2×D97#10312yescomplemented
99C2×D97#12012yes—
100C2×D97C2×SD1611yescharacteristic
101C2×D97C2×Q1611yescharacteristic
102C2×D97#13712yes—
103#87604—82no—
104C4×C97—24no—
105C4×C97—21no—
106C4×C97C2×C22⋊C412yes—
107Dic97—21no—
108Dic97—24no—
109Dic97#12012yes—
110C2×C97—22no—
111C2×C97#37911yescharacteristic
112C2×C97C2×#10311yescharacteristic
113C2×C97#41911yescharacteristic
114D97—22no—
115D97#44812yes—
116D97#45112yes—
117#1787—971nomaximal, Sylow
118C97#178711yescharacteristic, complemented, Sylow
119C23×Q8—971no—
120C2×#103—974no—
121C2×C22⋊C8—971no—
122#380—971no—
123C22×Q8—972no—
124C22×Q8—972no—
125C22×Q8—971no—
126C22×Q8—1941no—
127C22×Q8—1941no—
128C22×Q8—1941no—
129C22×Q8—1948no—
130C22×Q8#8760411yescharacteristic, complemented
131C23×C4—971no—
132C23×C4—1941no—
133C22×C8—972no—
134C2×C4⋊C4—972no—
135C2×C22⋊C4—1941no—
136#103—1948no—
137C22⋊C8—1941no—
138C22⋊C8—1941no—
139#95—1941no—
140C24—971no—
141C2×Q8—24no—
142C2×Q8—21no—
143C2×Q8—978no—
144C2×Q8—972no—
145C2×Q8—1942no—
146C2×Q8—1942no—
147C2×Q8—1944no—
148C2×Q8—1948no—
149C2×Q8—1944no—
150C2×Q8—1944no—
151C2×Q8—1948no—
152C2×Q8—1948no—
153C2×Q8—1944no—
154C2×Q8—1941no—
155C2×Q8C2×#8760412yes—
156C22×C4—21no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—1944no—
163C22×C4—1942no—
164C22×C4—1942no—
165C22×C4—1948no—
166C22×C4—1942no—
167C22×C4—1941no—
168C22×C4C2×#8760411yescharacteristic
169C2×C8—1942no—
170C2×C8—1944no—
171C4⋊C4—1944no—
172C22⋊C4—3882no—
173C23—971no—
174C23—972no—
175C23—972no—
176C23—971no—
177C23—1942no—
178C23—1942no—
179C23C22⋊#8760411yescharacteristic
180Q8—24no—
181Q8—22no—
182Q8—974no—
183Q8—1948no—
184Q8—1948no—
185Q8—1944no—
186Q8—1942no—
187Q8C22⋊#8760414yescomplemented
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—971no—
192C2×C4—974no—
193C2×C4—971no—
194C2×C4—3882no—
195C2×C4—3882no—
196C2×C4—3882no—
197C2×C4—1942no—
198C2×C4—1942no—
199C2×C4—1948no—
200C2×C4—1948no—
201C2×C4—1942no—
202C2×C4—1944no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4C22⋊#8760411yescharacteristic
206C2×C4C22×#8760411yescharacteristic
207C8—3882no—
208C22—21no—
209C22—21no—
210C22—972no—
211C22—971no—
212C22—971no—
213C22—972no—
214C22—972no—
215C22—1944no—
216C22—1944no—
217C22—1941no—
218C22—1941no—
219C22#1569087312yescomplemented
220C22C2×C22⋊#8760411yescharacteristic
221C4—24no—
222C4—21no—
223C4—7762no—
224C4—972no—
225C4—1941no—
226C4—1944no—
227C4C2×C22⋊#8760412yes—
228C2—22no—
229C2—972no—
230C2—972no—
231C2—1942no—
232C2#2026167611yescharacteristic
233C2C2×#1569087311yescharacteristic
234C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC22×Q8#87604
hypocenterC97#1787
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690873C2semidirect
13. C22×Dic194C4semidirect
17. C2×Q8×D97C4semidirect
21. C22×Q8×C97C4semidirect
67. Q8×C97C22⋊C4semidirect
88. Dic194C22⋊C4semidirect
92. C22×C97#103semidirect
98. C2×D97#103semidirect
118. C97#1787semidirect
130. C22×Q8#87604semidirect
187. Q8C22⋊#87604semidirect
219. C22#15690873semidirect

Maximal subgroups

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

subgroupindex
C2×#156908732
#202617032
#202616772
C22×Q8×D972
#178797