Finite Group Databasebeta

← #27157113

Subgroups — #27157113

Order 12416.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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, complemented
2C4×C22⋊#87604C211yesmaximal, characteristic, complemented
3#20261588C212yesmaximal, complemented
4#20261587C212yesmaximal, complemented
5#20261247C211yesmaximal, characteristic
6C2×#13095519—21no—
7C2×C22⋊#87604—21no—
8C2×C4×#87604—21no—
9#15690867C2212yes—
10C22⋊#11715897—21no—
11(C2×#87604)⋊C4C2212yes—
12C2×#13095511C411yescharacteristic, complemented
13C22×C4×D97C2211yescharacteristic
14#15690794—21no—
15#15690790C411yescharacteristic, complemented
16#15690790C2211yescharacteristic
17#15690785C411yescharacteristic, complemented
18#15690785C2211yescharacteristic
19#15690780—21no—
20C4×D388—21no—
21C4×D388—21no—
22#117×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095519—42no—
25C2×#11715897—22no—
26C22⋊#87604—24no—
27C4×#87604—42no—
28C23×D97D411yescharacteristic
29#13095511—42no—
30C2×Dic194C2×C411yescharacteristic
31C2×D388—21no—
32C2×D388—21no—
33C2×D388C2×C411yescharacteristic
34C2×C22⋊D97—21no—
35C22×Dic97D411yescharacteristic
36C2×C4×D97—21no—
37C2×C4×D97—42no—
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41C2×C4×D97C2311yescharacteristic
42C4⋊Dic97—21no—
43C4⋊Dic97—21no—
44#13095488—21no—
45#13095488—21no—
46#13095488—21no—
47#13095488—21no—
48C22×C4×C97C2×C411yescharacteristic
49C4⋊C4×C97C2×C411yescharacteristic
50C4⋊C4×C97C2×C411yescharacteristic
51C22⋊C4×C97—21no—
52C42×C97—21no—
53C23×C97C22⋊C411yescharacteristic
54C22×D97—22no—
55C22×D97—22no—
56C22×D97—21no—
57C22×D97—41no—
58C22×D97C22⋊C411yescharacteristic
59C22×D97C2×D411yescharacteristic
60C2×#87604—24no—
61C2×#87604—42no—
62C2×#87604—42no—
63C2×C4×C97—22no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22×C411yescharacteristic
68C22⋊D97—42no—
69C2×Dic97—21no—
70C2×Dic97—41no—
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73D388—41no—
74D388—42no—
75D388—42no—
76C4×D97—22no—
77C4×D97—22no—
78C4×D97—41no—
79C4×D97—41no—
80C4×D97—42no—
81Dic194—41no—
82#11715897—42no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97C2×C22⋊C411yescharacteristic
87C2×D97—21no—
88C2×D97—21no—
89C2×D97—21no—
90C2×D97—21no—
91C2×D97—44no—
92C2×D97—41no—
93C2×D97—41no—
94C2×D97—41no—
95C2×D97#10412yes—
96C2×D97#13512yes—
97#87604—44no—
98C4×C97—21no—
99C4×C97—41no—
100C4×C97—41no—
101C4×C97#9912yes—
102Dic97—21no—
103Dic97—21no—
104Dic97—41no—
105C2×C97—41no—
106C2×C97C2×#9911yescharacteristic
107C2×C97#41611yescharacteristic
108C2×C97C2×#10411yescharacteristic
109D97—22no—
110D97—22no—
111D97—81no—
112#1501—971nomaximal, Sylow
113C97#150111yescharacteristic, complemented, Sylow
114#525—971no—
115C2×C22⋊C8—971no—
116C4×C22⋊C4—971no—
117#328—972no—
118#327—972no—
119C2×#41—971no—
120C23×C4—971no—
121C22×C8—1941no—
122#123—1941no—
123C22⋊Q8—971no—
124C22⋊Q8—971no—
125#121—971no—
126#121—971no—
127#120—1941no—
128C4×D4—1941no—
129C4×D4—1941no—
130#117#8760411yescharacteristic, complemented
131C2×C22⋊C4—1941no—
132C2×C42—1941no—
133C22⋊C8—972no—
134C22⋊C8—1941no—
135#95—972no—
136C24—971no—
137#41—3882no—
138C2×Q8—971no—
139C2×D4—971no—
140C2×D4—1941no—
141C2×D4—1941no—
142C2×D4—1941no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—3882no—
149C22×C4—1942no—
150C22×C4—1941no—
151C22×C4C2×#8760411yescharacteristic
152C2×C8—3882no—
153C2×C8—1942no—
154C4⋊C4—1941no—
155C4⋊C4—1941no—
156C4⋊C4C2×#8760411yescharacteristic
157C4⋊C4C2×#8760411yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—1941no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C22⋊C4—1944no—
164C42—21no—
165C42—3882no—
166C23—971no—
167C23—971no—
168C23—3881no—
169C23—1942no—
170C23—1942no—
171C23—1941no—
172C23C22⋊#8760411yescharacteristic
173Q8—3881no—
174D4—3881no—
175D4—3882no—
176D4—3882no—
177D4—3882no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—971no—
182C2×C4—971no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—3882no—
186C2×C4—3882no—
187C2×C4—3882no—
188C2×C4—3881no—
189C2×C4—1944no—
190C2×C4—1941no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4C22⋊#8760411yescharacteristic
194C2×C4C22×#8760411yescharacteristic
195C8—3882no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—972no—
200C22—972no—
201C22—3884no—
202C22—3881no—
203C22—3881no—
204C22—3881no—
205C22—1941no—
206C22—1941no—
207C22—1941no—
208C22—1941no—
209C22C2×C22⋊#8760411yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—3884no—
214C4—3881no—
215C4—1941no—
216C4—1941no—
217C4#1569086912yes—
218C2—41no—
219C2—7761no—
220C2—1942no—
221C2—1942no—
222C2C2×#1569086911yescharacteristic
223C2#2026172511yescharacteristic
224C2D97⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261247C2
cosocleC2×C4×D97C23
hypercenter#117#87604
hypocenterC97#1501
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261703C2semidirect
2. C4×C22⋊#87604C2semidirect
3. #20261588C2semidirect
4. #20261587C2semidirect
12. C2×#13095511C4semidirect
15. #15690790C4semidirect
17. #15690785C4semidirect
22. #117×C97C4semidirect
113. C97#1501semidirect
130. #117#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
C4×C22⋊#876042
#202615882
#202615872
#202612472
#150197