Finite Group Databasebeta

← #27157145

Subgroups — #27157145

Order 12416.

274 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 58 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#20261706C211yesmaximal, characteristic
2#20261678C211yesmaximal, characteristic
3#20261589C212yesmaximal, complemented
4#20261588C212yesmaximal, complemented
5C22⋊Q8×D97C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×C22⋊#87604—21no—
8#15690867C2212yes—
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12C2×Q8×D97C411yescharacteristic, complemented
13C22×C4×D97C2211yescharacteristic
14#15690790C2211yescharacteristic
15#15690789C411yescharacteristic, complemented
16#15690788—21no—
17#15690787—21no—
18#15690786C2211yescharacteristic
19C4⋊C4×D97—21no—
20C4⋊C4×D97C411yescharacteristic, complemented
21C22⋊C4×D97—21no—
22C22⋊Q8×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095521—42no—
25C2×#11715897—22no—
26C22⋊#87604—42no—
27C23×D97D411yescharacteristic
28C2×Dic194—21no—
29C2×Dic194C2×C411yescharacteristic
30Q8×D97—42no—
31C22×Dic97D411yescharacteristic
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—42no—
35C2×C4×D97C2×C411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97C2311yescharacteristic
39C4⋊Dic97—21no—
40C4⋊Dic97C2×C411yescharacteristic
41#13095490—21no—
42#13095490—21no—
43#13095490—21no—
44C22⋊Dic97—21no—
45#13095488—21no—
46#13095488—21no—
47C2×Q8×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C4⋊C4×C97—21no—
50C4⋊C4×C97C2×C411yescharacteristic
51C22⋊C4×C97—21no—
52C23×C97C22⋊C411yescharacteristic
53C22×D97—22no—
54C22×D97—22no—
55C22×D97—41no—
56C22×D97C22⋊C411yescharacteristic
57C22×D97C2×D411yescharacteristic
58C2×#87604—42no—
59C2×#87604—42no—
60C2×#87604—42no—
61Q8×C97—41no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—41no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97—41no—
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72C4×D97—22no—
73C4×D97—22no—
74C4×D97—41no—
75C4×D97—41no—
76C4×D97—42no—
77C4×D97—42no—
78Dic194—42no—
79Dic194—41no—
80#11715897—42no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—21no—
86C2×D97—21no—
87C2×D97—21no—
88C2×D97—21no—
89C2×D97—44no—
90C2×D97#9912yes—
91C2×D97#13712yes—
92#87604—82no—
93C4×C97—21no—
94C4×C97—21no—
95C4×C97—41no—
96C4×C97—41no—
97Dic97—21no—
98Dic97—21no—
99Dic97—41no—
100Dic97—41no—
101C2×C97—41no—
102C2×C97C2×#9911yescharacteristic
103C2×C97#41911yescharacteristic
104C2×C97#42111yescharacteristic
105D97—22no—
106D97—22no—
107#1511—971nomaximal, Sylow
108C97#151111yescharacteristic, complemented, Sylow
109C2×C22⋊Q8—971no—
110#407—971no—
111#381—971no—
112#328—972no—
113#328—972no—
114C22×Q8—971no—
115C23×C4—971no—
116C2×M16—1941no—
117C22⋊Q8—971no—
118C22⋊Q8—971no—
119C22⋊Q8—971no—
120C22⋊Q8—1941no—
121C22⋊Q8—1941no—
122C22⋊Q8#8760411yescharacteristic, complemented
123C2×C4⋊C4—971no—
124C2×C4⋊C4—1941no—
125C2×C22⋊C4—1941no—
126C2×C22⋊C4—1941no—
127C22⋊C8—972no—
128C22⋊C8—1941no—
129#95—972no—
130#95—1941no—
131C24—971no—
132C2×Q8—971no—
133C2×Q8—3882no—
134C2×Q8—1941no—
135C2×Q8C2×#8760411yescharacteristic
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—3882no—
142C22×C4—1941no—
143C22×C4—1941no—
144C22×C4—1942no—
145C22×C4C2×#8760411yescharacteristic
146M16—3882no—
147C2×C8—1942no—
148C4⋊C4—21no—
149C4⋊C4—971no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C4⋊C4C2×#8760411yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3882no—
157C22⋊C4—1941no—
158C22⋊C4—1941no—
159C22⋊C4—1941no—
160C23—971no—
161C23—971no—
162C23—3881no—
163C23—1942no—
164C23—1942no—
165C23C22⋊#8760411yescharacteristic
166Q8—41no—
167Q8—3881no—
168Q8—3882no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—3882no—
177C2×C4—3882no—
178C2×C4—3882no—
179C2×C4—3882no—
180C2×C4—3882no—
181C2×C4—3881no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4C22⋊#8760411yescharacteristic
187C2×C4C22×#8760411yescharacteristic
188C8—3882no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—972no—
193C22—972no—
194C22—3884no—
195C22—1941no—
196C22—1941no—
197C22—1941no—
198C22—1941no—
199C22C2×C22⋊#8760411yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—7762no—
205C4—3881no—
206C4—3881no—
207C4—1941no—
208C4—1941no—
209C2—41no—
210C2—1942no—
211C2—1942no—
212C2D97⋊#9911yescharacteristic
213C2#2026172811yescharacteristic
214C2#2026173211yescharacteristic
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#1511
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20261589C2semidirect
4. #20261588C2semidirect
12. C2×Q8×D97C4semidirect
15. #15690789C4semidirect
20. C4⋊C4×D97C4semidirect
22. C22⋊Q8×C97C4semidirect
108. C97#1511semidirect
122. C22⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202616782
#202615892
#202615882
C22⋊Q8×D972
#151197