Finite Group Databasebeta

← #27157425

Subgroups — #27157425

Order 12416.

448 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 138 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×D97⋊M16C211yesmaximal, characteristic
2C22×C4⋊#87604C211yesmaximal, characteristic
3#20261608C214yesmaximal, complemented
4#117×D97C211yesmaximal, characteristic
5C23×#87604—21no—
6C2×#13095521—21no—
7D97⋊M16—22no—
8D97⋊M16C414yescomplemented
9C2×#13095520—21no—
10C2×#13095519C2212yes—
11C2×C4⋊#87604—21no—
12C2×C4⋊#87604—22no—
13C2×C4⋊#87604C414yescomplemented
14C2×C4⋊#87604C2212yes—
15C22×C4×D97C2211yescharacteristic
16#15690769C411yescharacteristic, complemented
17#15690768C412yescomplemented
18#15690767—21no—
19#15690766—21no—
20C4⋊C4×D97C2212yes—
21C22⋊C4×D97—21no—
22C42×D97—21no—
23#117×C97C411yescharacteristic, complemented
24C22×#87604—24no—
25C22×#87604—22no—
26C22×#87604—44no—
27#13095521—24no—
28#13095520—24no—
29#13095519—22no—
30#13095519C2×C414yescomplemented
31C2×#11715897—22no—
32C4⋊#87604—24no—
33C4⋊#87604—22no—
34C4⋊#87604C2×C414yes—
35C23×D97D411yescharacteristic
36C22×Dic97D411yescharacteristic
37C2×C4×D97—22no—
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97C2×C412yes—
40C2×C4×D97C2×C412yes—
41C2×C4×D97D411yescharacteristic
42C2×C4×D97D411yescharacteristic
43C2×C4×D97D412yes—
44C2×C4×D97Q812yes—
45C2×C4×D97C2311yescharacteristic
46C4⋊Dic97C2×C412yescomplemented
47#13095490—22no—
48C22⋊Dic97—21no—
49#13095488—22no—
50C4×Dic97—21no—
51C4×Dic97—22no—
52C22×C4×C97C2×C411yescharacteristic
53C4⋊C4×C97C2×C412yescomplemented
54C22⋊C4×C97—21no—
55C42×C97—21no—
56C23×C97C22⋊C411yescharacteristic
57C22×D97—22no—
58C22×D97—22no—
59C22×D97C22⋊C411yescharacteristic
60C22×D97C22⋊C412yes—
61C22×D97C4⋊C412yes—
62C22×D97C2×D411yescharacteristic
63C2×#87604—24no—
64C2×#87604—48no—
65C2×#87604—44no—
66C2×C4×C97—22no—
67C2×C4×C97C4212yes—
68C2×C4×C97C22⋊C411yescharacteristic
69C2×C4×C97C4⋊C412yes—
70C2×C4×C97C22×C411yescharacteristic
71C2×Dic97—22no—
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C22⋊C412yes—
74C2×Dic97C4⋊C412yes—
75C2×Dic97C2×D411yescharacteristic
76C4×D97—42no—
77C4×D97—42no—
78C4×D97C4212yes—
79C4×D97C22⋊C412yes—
80C4×D97C22⋊C412yes—
81C4×D97C4⋊C412yes—
82C4×D97C4⋊C412yes—
83C4×D97C4⋊C412yes—
84C4×D97C22×C411yescharacteristic
85C4×D97C22×C411yescharacteristic
86C4×D97C2×D411yescharacteristic
87C4×D97C2×Q811yescharacteristic
88#11715897—24no—
89C22×C97—22no—
90C22×C97#9512yes—
91C22×C97C2×C22⋊C411yescharacteristic
92C2×D97—22no—
93C2×D97—22no—
94C2×D97—22no—
95C2×D97—24no—
96C2×D97#9512yes—
97C2×D97C2×C22⋊C411yescharacteristic
98C2×D97C2×C4⋊C411yescharacteristic
99C2×D97AGL(1,ℤ/8)12yes—
100C2×D97#13712yes—
101#87604—44no—
102C4×C97—42no—
103C4×C97#9512yes—
104C4×C97C2×C4211yescharacteristic
105C4×C97C2×C4⋊C411yescharacteristic
106Dic97—42no—
107Dic97#9512yes—
108Dic97C2×C22⋊C411yescharacteristic
109Dic97C2×C4⋊C411yescharacteristic
110C2×C97—22no—
111C2×C97C2×#9511yescharacteristic
112C2×C97#41712yes—
113D97—22no—
114D97#41811yescharacteristic
115D97#41911yescharacteristic
116D97#42812yes—
117#1736—971nomaximal, Sylow
118C97#173611yescharacteristic, complemented, Sylow
119C22×M16—971no—
120C2×#117—971no—
121C22×C4⋊C4—971no—
122#340—974no—
123C23×C4—971no—
124C23×C4—1941no—
125C2×M16—974no—
126C2×M16—1941no—
127C2×M16—1942no—
128C2×M16—1941no—
129C22×C8—972no—
130#117—971no—
131#117—972no—
132#117—1941no—
133#117—1941no—
134#117#8760411yescharacteristic, complemented
135C2×C4⋊C4—972no—
136C2×C4⋊C4—974no—
137C2×C4⋊C4—972no—
138C2×C4⋊C4—1941no—
139C2×C4⋊C4—1942no—
140C2×C22⋊C4—1941no—
141C2×C42—1941no—
142C24—971no—
143C22×C4—971no—
144C22×C4—972no—
145C22×C4—972no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4—971no—
151C22×C4—971no—
152C22×C4—3884no—
153C22×C4—1942no—
154C22×C4—1942no—
155C22×C4—1944no—
156C22×C4C2×#8760411yescharacteristic
157M16—1944no—
158M16—1944no—
159C2×C8—974no—
160C2×C8—1942no—
161C2×C8—1942no—
162C4⋊C4—974no—
163C4⋊C4—972no—
164C4⋊C4—1942no—
165C4⋊C4—1944no—
166C4⋊C4—1942no—
167C4⋊C4C2×#8760412yescomplemented
168C22⋊C4—21no—
169C22⋊C4—1942no—
170C22⋊C4—1941no—
171C42—21no—
172C42—1942no—
173C42—1941no—
174C23—971no—
175C23—972no—
176C23—972no—
177C23—971no—
178C23—1942no—
179C23—1942no—
180C23C22⋊#8760411yescharacteristic
181C2×C4—22no—
182C2×C4—972no—
183C2×C4—972no—
184C2×C4—971no—
185C2×C4—971no—
186C2×C4—972no—
187C2×C4—972no—
188C2×C4—972no—
189C2×C4—972no—
190C2×C4—971no—
191C2×C4—971no—
192C2×C4—972no—
193C2×C4—971no—
194C2×C4—972no—
195C2×C4—971no—
196C2×C4—3882no—
197C2×C4—3882no—
198C2×C4—3884no—
199C2×C4—3888no—
200C2×C4—1944no—
201C2×C4—1942no—
202C2×C4C4×#8760412yes—
203C2×C4C22⋊#8760411yescharacteristic
204C2×C4C4⋊#8760412yes—
205C2×C4C22×#8760411yescharacteristic
206C8—1944no—
207C22—22no—
208C22—971no—
209C22—971no—
210C22—972no—
211C22—972no—
212C22—972no—
213C22—1942no—
214C22—1942no—
215C22—1944no—
216C22—1942no—
217C22(C2×#87604)⋊C412yes—
218C22C2×C22⋊#8760411yescharacteristic
219C4—42no—
220C4—971no—
221C4—972no—
222C4—971no—
223C4—3884no—
224C4—3882no—
225C4(C2×#87604)⋊C412yes—
226C4C2×C4×#8760411yescharacteristic
227C4C2×C4⋊#8760411yescharacteristic
228C2—22no—
229C2—971no—
230C2—971no—
231C2—972no—
232C2—1942no—
233C2C2×(C2×#87604)⋊C411yescharacteristic
234C2#2026172612yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C2×C42
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenter#117#87604
hypocenterC97#1736
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20261608C2semidirect
8. D97⋊M16C4semidirect
13. C2×C4⋊#87604C4semidirect
16. #15690769C4semidirect
17. #15690768C4semidirect
23. #117×C97C4semidirect
30. #13095519C2×C4semidirect
46. C4⋊Dic97C2×C4semidirect
53. C4⋊C4×C97C2×C4semidirect
118. C97#1736semidirect
134. #117#87604semidirect
167. C4⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×D97⋊M162
C22×C4⋊#876042
#202616082
#117×D972
#173697