Finite Group Databasebeta

← #27142034

Subgroups — #27142034

Order 12416.

408 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22×#13095495C211yesmaximal, characteristic, complemented
2C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
3C2×C22⋊#11715895C212yesmaximal, complemented
4#20260490C212yesmaximal, complemented
5C2×C4×D4×C97C211yesmaximal, characteristic
6C2×#13095495—24no—
7C2×#13095495—22no—
8C22×#11715895C2212yes—
9C22×#11715895C2212yes—
10C22⋊#11715895—24no—
11C22⋊#11715895—22no—
12C22×D4×C97C411yescharacteristic
13C23×C4×C97C2211yescharacteristic
14C23×C4×C97C2211yescharacteristic
15C4×D4×C97—22no—
16C4×D4×C97—22no—
17C2×C4⋊C4×C97C411yescharacteristic
18C2×C22⋊C4×C97C411yescharacteristic
19C2×C22⋊C4×C97C411yescharacteristic
20C2×C42×C97C2211yescharacteristic
21#13095495—28no—
22C2×#11715895—22no—
23C2×#11715895—22no—
24C2×#11715895—22no—
25C2×#11715895—22no—
26C2×#11715895—22no—
27C2×#11715895D414yescomplemented
28C24×C97C2×C411yescharacteristic
29C24×C97C2×C411yescharacteristic
30C2×D4×C97—22no—
31C2×D4×C97—22no—
32C2×D4×C97—22no—
33C22×C4×C97—22no—
34C22×C4×C97—22no—
35C22×C4×C97—21no—
36C22×C4×C97—21no—
37C22×C4×C97—22no—
38C22×C4×C97—22no—
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97C2×C411yescharacteristic
43C22×C4×C97D412yes—
44C22×C4×C97D412yes—
45C22×C4×C97C2311yescharacteristic
46C4⋊C4×C97—22no—
47C22⋊C4×C97—22no—
48C22⋊C4×C97—22no—
49C42×C97—21no—
50C42×C97—21no—
51C23×C97—21no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—21no—
55C23×C97—21no—
56C23×C97—21no—
57C23×C97—22no—
58C23×C97—22no—
59C23×C97C22⋊C412yes—
60C23×C97M1612yescomplemented
61C23×C97C22×C411yescharacteristic
62D4×C97—28no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—22no—
67C2×C4×C97—24no—
68C2×C4×C97—22no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—41no—
74C2×C4×C97—42no—
75C2×C4×C97—41no—
76C2×C4×C97C22⋊C412yes—
77C2×C4×C97M1612yescomplemented
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97#4112yes—
82#11715895—24no—
83#11715895—42no—
84C22×C97—22no—
85C22×C97—22no—
86C22×C97—22no—
87C22×C97—22no—
88C22×C97—24no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—42no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97C2×C22⋊C411yescharacteristic
95C22×C97C4×D412yes—
96C22×C97C2×M1611yescharacteristic
97C22×C97C2×M1611yescharacteristic
98C22×C97#13112yes—
99C4×C97—22no—
100C4×C97—22no—
101C4×C97—41no—
102C4×C97—41no—
103C4×C97#12011yescharacteristic
104C4×C97#12112yes—
105C4×C97#12311yescharacteristic
106C2×C97—42no—
107C2×C97—42no—
108C2×C97#38611yescharacteristic
109C2×C97#40711yescharacteristic
110C2×C97#40811yescharacteristic
111C2×C97#43512yes—
112C2×C97#43612yes—
113#1870—971nomaximal, Sylow
114C97#187011yescharacteristic, complemented, Sylow
115C22×M16—971no—
116C2×C4×D4D9711yescharacteristic
117C2×C22⋊C8—972no—
118C2×C22⋊C8—971no—
119#336—972no—
120C22×D4Dic9711yescharacteristic
121C23×C4C2×D9711yescharacteristic
122C23×C4C2×D9711yescharacteristic
123C2×M16—1944no—
124C2×M16—1942no—
125C22×C8—972no—
126C22×C8—972no—
127C4×D4—22no—
128C4×D4—22no—
129C2×C4⋊C4Dic9711yescharacteristic
130C2×C22⋊C4Dic9711yescharacteristic
131C2×C22⋊C4Dic9711yescharacteristic
132C2×C42C2×D9711yescharacteristic
133C22⋊C8—1944no—
134C22⋊C8—1942no—
135C24C2×Dic9711yescharacteristic
136C24C2×Dic9711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4C2×Dic9711yescharacteristic
147C22×C4C2×Dic9711yescharacteristic
148C22×C4C2×Dic9711yescharacteristic
149C22×C4C2×Dic9711yescharacteristic
150C22×C4C22⋊D9712yes—
151C22×C4C22⋊D9712yes—
152C22×C4C22×D9711yescharacteristic
153M16—1948no—
154C2×C8—974no—
155C2×C8—1942no—
156C2×C8—1942no—
157C2×C8—1942no—
158C2×C8—1942no—
159C2×C8—1942no—
160C4⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C42—21no—
164C42—21no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23C22⋊Dic9712yes—
174C23#1309549512yescomplemented
175C23C22×Dic9711yescharacteristic
176D4—28no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4C22⋊Dic9712yes—
191C2×C4#1309549512yescomplemented
192C2×C4D4×D9712yes—
193C2×C4C2×C22⋊D9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C2×C4#1309551012yes—
196C8—3882no—
197C8—1944no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—24no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22C2×C22⋊Dic9711yescharacteristic
209C22D4×Dic9712yes—
210C22C2×#1309549511yescharacteristic
211C22C2×#1309549511yescharacteristic
212C22#1569081712yes—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—41no—
217C4#1569077811yescharacteristic
218C4#1569078212yes—
219C4#1569079111yescharacteristic
220C2—42no—
221C2—42no—
222C2#2026061511yescharacteristic
223C2#2026069911yescharacteristic
224C2#2026070211yescharacteristic
225C2#2026080712yes—
226C2#2026081312yes—
subgroupG/H
center, FrattiniC22×C4C22×D97
commutator, socleC23×C97C22×C4
FittingC2×C4×D4×C97C2
prime radicalC22×D4×C97C4
cosocleC22×C4×C97C23
hypercenterC2×C4×D4D97
hypocenterC97#1870
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#13095495C2semidirect
2. C2×C22⋊#11715895C2semidirect
3. C2×C22⋊#11715895C2semidirect
4. #20260490C2semidirect
27. C2×#11715895D4semidirect
60. C23×C97M16semidirect
77. C2×C4×C97M16semidirect
114. C97#1870semidirect
174. C23#13095495semidirect
191. C2×C4#13095495semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
C2×C22⋊#117158952
#202604902
C2×C4×D4×C972
#187097