Finite Group Databasebeta

← #27157641

Subgroups — #27157641

Order 12416.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261703C211yesmaximal, characteristic
2#20261677C211yesmaximal, characteristic
3#20261608C212yesmaximal, complemented
4#20261608C212yesmaximal, complemented
5#117×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604C2212yes—
8C2×C22⋊#87604—21no—
9#15690867—21no—
10C22⋊#11715897—21no—
11(C2×#87604)⋊C4—21no—
12C22×C4×D97C2211yescharacteristic
13#15690769C411yescharacteristic, complemented
14#15690768C411yescharacteristic, complemented
15#15690768C411yescharacteristic, complemented
16#15690767—21no—
17#15690766—21no—
18C4⋊C4×D97C2211yescharacteristic
19C4⋊C4×D97C2211yescharacteristic
20C22⋊C4×D97—21no—
21C42×D97—21no—
22#117×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095519—24no—
25C2×#11715897—22no—
26C4⋊#87604—24no—
27C22⋊#87604—42no—
28C23×D97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C2×C4×D97—24no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97C2311yescharacteristic
37C4⋊Dic97C2×C411yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40#13095490—21no—
41C22⋊Dic97—21no—
42#13095488—21no—
43#13095488—21no—
44C4×Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C22×C4×C97C2×C411yescharacteristic
48C4⋊C4×C97C2×C411yescharacteristic
49C4⋊C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C42×C97—21no—
52C23×C97C22⋊C411yescharacteristic
53C22×D97—22no—
54C22×D97—22no—
55C22×D97—21no—
56C22×D97—21no—
57C22×D97C22⋊C411yescharacteristic
58C22×D97C2×D411yescharacteristic
59C2×#87604—42no—
60C2×#87604—42no—
61C2×#87604—42no—
62C2×C4×C97—22no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72C4×D97—22no—
73C4×D97—22no—
74C4×D97—22no—
75C4×D97—42no—
76C4×D97—41no—
77C4×D97—41no—
78C4×D97#4114yes—
79#11715897—42no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97C2×C22⋊C411yescharacteristic
84C2×D97—21no—
85C2×D97—21no—
86C2×D97—21no—
87C2×D97—42no—
88C2×D97—42no—
89C2×D97#12312yes—
90C2×D97#13512yes—
91C2×D97AGL(1,ℤ/8)11yescharacteristic
92C2×D97#13711yescharacteristic
93#87604—82no—
94C4×C97—21no—
95C4×C97—41no—
96C4×C97—41no—
97C4×C97#11712yes—
98Dic97—21no—
99Dic97—41no—
100Dic97—41no—
101Dic97#12312yes—
102C2×C97—41no—
103C2×C97#38111yescharacteristic
104C2×C97#41611yescharacteristic
105C2×C97#41711yescharacteristic
106D97—41no—
107D97#48212yes—
108D97#48512yes—
109#1825—971nomaximal, Sylow
110C97#182511yescharacteristic, complemented, Sylow
111C2×#117—971no—
112C2×C22⋊C8—971no—
113#380—971no—
114#340—972no—
115#340—972no—
116C23×C4—971no—
117C22×C8—972no—
118#117—971no—
119#117—971no—
120#117—971no—
121#117—1941no—
122#117—1941no—
123#117#8760411yescharacteristic, complemented
124C2×C4⋊C4—971no—
125C2×C4⋊C4—971no—
126C2×C4⋊C4—972no—
127C2×C22⋊C4—1941no—
128C2×C22⋊C4—1941no—
129C2×C42—1941no—
130C22⋊C8—1941no—
131C22⋊C8—1941no—
132#95—1941no—
133C24—971no—
134C22×C4—971no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—1944no—
140C22×C4—1941no—
141C22×C4—1941no—
142C22×C4—1942no—
143C22×C4C2×#8760411yescharacteristic
144C2×C8—1942no—
145C2×C8—1944no—
146C4⋊C4—971no—
147C4⋊C4—971no—
148C4⋊C4—1941no—
149C4⋊C4—1941no—
150C4⋊C4—1944no—
151C4⋊C4C2×#8760411yescharacteristic
152C4⋊C4C2×#8760411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—3882no—
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C42—21no—
159C42—1941no—
160C42—1941no—
161C42—1941no—
162C23—971no—
163C23—971no—
164C23—1941no—
165C23—1942no—
166C23—1942no—
167C23—1941no—
168C23C22⋊#8760411yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—971no—
173C2×C4—974no—
174C2×C4—971no—
175C2×C4—3882no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—3882no—
179C2×C4—3882no—
180C2×C4—3882no—
181C2×C4—1942no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4—1942no—
187C2×C4C22⋊#8760411yescharacteristic
188C2×C4C22×#8760411yescharacteristic
189C8—3882no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—972no—
194C22—972no—
195C22—971no—
196C22—971no—
197C22—3882no—
198C22—3882no—
199C22—1941no—
200C22—1941no—
201C22—1941no—
202C22C2×C22⋊#8760411yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—7762no—
207C4—972no—
208C4—3881no—
209C4—3881no—
210C4—1941no—
211C4#1569088912yes—
212C2—41no—
213C2—972no—
214C2—972no—
215C2—3881no—
216C2#2026167811yescharacteristic
217C2#2026172511yescharacteristic
218C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenter#117#87604
hypocenterC97#1825
LCSG ▷ C2×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
4. #20261608C2semidirect
13. #15690769C4semidirect
14. #15690768C4semidirect
15. #15690768C4semidirect
22. #117×C97C4semidirect
110. C97#1825semidirect
123. #117#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
#202616772
#202616082
#117×D972
#182597