Finite Group Databasebeta

← #27157164

Subgroups — #27157164

Order 12416.

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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×#15690867C211yesmaximal, characteristic, complemented
2#20261678C211yesmaximal, characteristic, complemented
3#20261588C212yesmaximal, complemented
4#20261587C212yesmaximal, complemented
5#20261247C211yesmaximal, characteristic
6C22×#11715897—21no—
7C2×C22⋊#87604—21no—
8#15690867—21no—
9#15690867C2212yes—
10(C2×#87604)⋊C4—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—
24C2×#11715897—22no—
25C2×#11715897—42no—
26C22⋊#87604—42no—
27C23×D97D411yescharacteristic
28#13095511—42no—
29C2×Dic194C2×C411yescharacteristic
30C2×D388—21no—
31C2×D388—21no—
32C2×D388C2×C411yescharacteristic
33C2×C22⋊D97—21no—
34C22×Dic97D411yescharacteristic
35C2×C4×D97—21no—
36C2×C4×D97—42no—
37C2×C4×D97C2×C411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97C2311yescharacteristic
41C4⋊Dic97—21no—
42C4⋊Dic97—21no—
43#13095488—21no—
44#13095488—21no—
45#13095488—21no—
46#13095488—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—41no—
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
67C22⋊D97—42no—
68C2×Dic97—22no—
69C2×Dic97—21no—
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72D388—41no—
73D388—42no—
74D388—42no—
75C4×D97—41no—
76C4×D97—41no—
77C4×D97—41no—
78C4×D97—41no—
79C4×D97—42no—
80Dic194—41no—
81#11715897—42no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97#9912yes—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—21no—
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#13512yes—
96#87604—82no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—41no—
100C4×C97—41no—
101Dic97—21no—
102Dic97—21no—
103Dic97—41no—
104C2×C97—22no—
105C2×C97C2×#9911yescharacteristic
106C2×C97#41611yescharacteristic
107C2×C97#42111yescharacteristic
108D97—22no—
109D97—41no—
110D97—81no—
111#1516—971nomaximal, Sylow
112C97#151611yescharacteristic, complemented, Sylow
113#525—971no—
114C2×C22⋊C8—971no—
115#381—971no—
116#328—972no—
117#327—972no—
118C2×#41—971no—
119C23×C4—971no—
120C22×C8—1941no—
121#123—1941no—
122C22⋊Q8—971no—
123C22⋊Q8—971no—
124#121—971no—
125#121—971no—
126#120—1941no—
127C4×D4—1941no—
128C4×D4—1941no—
129#117#8760411yescharacteristic, complemented
130C2×C22⋊C4—1941no—
131C22⋊C8—972no—
132C22⋊C8—1941no—
133#95—972no—
134#95—1941no—
135C24—971no—
136#41—3882no—
137C2×Q8—971no—
138C2×D4—971no—
139C2×D4—1941no—
140C2×D4—1941no—
141C2×D4—1941no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—3882no—
148C22×C4—1942no—
149C22×C4—1941no—
150C22×C4C2×#8760411yescharacteristic
151C2×C8—3882no—
152C2×C8—1942no—
153C4⋊C4—1941no—
154C4⋊C4—1941no—
155C4⋊C4C2×#8760411yescharacteristic
156C4⋊C4C2×#8760411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3882no—
159C22⋊C4—1941no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C42—21no—
164C23—971no—
165C23—971no—
166C23—3881no—
167C23—1942no—
168C23—1941no—
169C23—1942no—
170C23C22⋊#8760411yescharacteristic
171Q8—3881no—
172D4—3881no—
173D4—3882no—
174D4—3882no—
175D4—3882no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—3882no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—3881no—
186C2×C4—3882no—
187C2×C4—3882no—
188C2×C4—3882no—
189C2×C4—1941no—
190C2×C4—1942no—
191C2×C4C22⋊#8760411yescharacteristic
192C2×C4C22×#8760411yescharacteristic
193C8—3882no—
194C22—21no—
195C22—21no—
196C22—972no—
197C22—3884no—
198C22—3881no—
199C22—3881no—
200C22—3881no—
201C22—1941no—
202C22—1941no—
203C22—1941no—
204C22—1941no—
205C22—1941no—
206C22#1569086912yes—
207C22C2×C22⋊#8760411yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—7762no—
213C4—3881no—
214C4—1941no—
215C4—1941no—
216C2—22no—
217C2—7761no—
218C2—3881no—
219C2—1942no—
220C2C2×#1569086911yescharacteristic
221C2#2026172511yescharacteristic
222C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#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#1516
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#15690867C2semidirect
2. #20261678C2semidirect
3. #20261588C2semidirect
4. #20261587C2semidirect
12. C2×#13095511C4semidirect
15. #15690790C4semidirect
17. #15690785C4semidirect
22. #117×C97C4semidirect
112. C97#1516semidirect
129. #117#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908672
#202616782
#202615882
#202615872
#202612472
#151697