Finite Group Databasebeta

← #27141964

Subgroups — #27141964

Order 12416.

302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 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#20261218C211yesmaximal, characteristic
2#20260777C211yesmaximal, characteristic, complemented
3#20260736C211yesmaximal, characteristic
4#20260605C211yesmaximal, characteristic, complemented
5#20260503C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9Q8×Dic97—21no—
10D4×Dic97—22no—
11D4×Dic97—21no—
12#15690769—21no—
13#15690769C2211yescharacteristic
14C2×C4⋊Dic97—21no—
15C2×C4⋊Dic97—22no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic
18#15690744—21no—
19#15690742—21no—
20#15690734—21no—
21#15690731—21no—
22C2×#41×C97C2211yescharacteristic
23C2×M16×C97C2211yescharacteristic
24C2×C42×C97C2211yescharacteristic
25#104×C97C414yescomplemented
26C22×Dic97—22no—
27C22×Dic97—21no—
28C22×Dic97D412yes—
29#13095495—21no—
30#13095495—21no—
31C2×#11715895—21no—
32C4⋊Dic97—22no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97—44no—
38C4⋊Dic97D411yescharacteristic
39C4⋊Dic97D411yescharacteristic
40C22⋊Dic97—22no—
41C22⋊Dic97—21no—
42C4×Dic97—22no—
43C4×Dic97—21no—
44#41×C97—21no—
45#41×C97—21no—
46#41×C97C2×C412yes—
47#41×C97Q812yes—
48C2×Q8×C97D411yescharacteristic
49C2×D4×C97—21no—
50C2×D4×C97D411yescharacteristic
51C22×C4×C97—21no—
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97C2×C412yescomplemented
56C2×C8×C97—21no—
57C42×C97—21no—
58C42×C97C2×C412yescomplemented
59C23×C97—21no—
60C23×C97C2×D411yescharacteristic
61Q8×C97—21no—
62Q8×C97C4⋊C412yescomplemented
63D4×C97—22no—
64D4×C97—22no—
65D4×C97—21no—
66D4×C97C4⋊C412yescomplemented
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—21no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97C22×C411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×Q811yescharacteristic
77C2×C4×C97#4111yescharacteristic
78C2×C4×C97#4111yescharacteristic
79C2×Dic97—21no—
80C2×Dic97—22no—
81C2×Dic97—21no—
82C2×Dic97—42no—
83C2×Dic97—42no—
84C2×Dic97—41no—
85C2×Dic97—82no—
86C8×C97—22no—
87#11715895—41no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—22no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97C4×D411yescharacteristic
94C22×C97#12011yescharacteristic
95C22×C97C22⋊Q811yescharacteristic
96C4×C97—24no—
97C4×C97—22no—
98C4×C97C2×C4⋊C411yescharacteristic
99C4×C97C4×D411yescharacteristic
100C4×C97C22⋊Q811yescharacteristic
101C4×C97#12311yescharacteristic
102Dic97—42no—
103Dic97—41no—
104Dic97—41no—
105Dic97—81no—
106C2×C97—21no—
107C2×C97—21no—
108C2×C97—42no—
109C2×C97#38511yescharacteristic
110C2×C97#45311yescharacteristic
111C2×C97#45611yescharacteristic
112#1859—971nomaximal, Sylow
113C97#185911yescharacteristic, complemented, Sylow
114#520—971no—
115#428—971no—
116C2×#104D9711yescharacteristic
117#417—971no—
118#384—971no—
119#343—971no—
120#339—971no—
121C2×#41C2×D9711yescharacteristic
122C2×M16—971no—
123C2×M16C2×D9711yescharacteristic
124C4×Q8—1941no—
125C4×D4—1942no—
126C4×D4—1941no—
127#117—971no—
128#117—1941no—
129C2×C4⋊C4—971no—
130C2×C4⋊C4—971no—
131C2×C4⋊C4—1941no—
132C2×C4⋊C4—1942no—
133C2×C42C2×D9711yescharacteristic
134#107—1941no—
135#106—1941no—
136#104Dic9714yescomplemented
137#103—1941no—
138#102—1941no—
139#41—21no—
140#41—21no—
141#41Dic19412yes—
142#41C2×Dic9712yes—
143C2×Q8D38811yescharacteristic
144C2×D4—21no—
145C2×D4D38811yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—972no—
149C22×C4—1942no—
150C22×C4—1941no—
151C22×C4C22×D9711yescharacteristic
152M16—21no—
153M16—1941no—
154M16—1941no—
155M16C2×Dic9712yescomplemented
156C2×C8—21no—
157C2×C8—1941no—
158C4⋊C4—971no—
159C4⋊C4—971no—
160C4⋊C4—3884no—
161C4⋊C4—1942no—
162C4⋊C4—1941no—
163C4⋊C4—1941no—
164C4⋊C4—1941no—
165C4⋊C4—1941no—
166C22⋊C4—1942no—
167C22⋊C4—1941no—
168C42—21no—
169C42—1942no—
170C42—1941no—
171C42C2×Dic9712yescomplemented
172C23—21no—
173C23D4×D9711yescharacteristic
174Q8—21no—
175Q8C4⋊Dic9712yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4C4⋊Dic9712yescomplemented
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—7762no—
187C2×C4—3882no—
188C2×C4—3881no—
189C2×C4—3882no—
190C2×C4—1941no—
191C2×C4—1941no—
192C2×C4—1942no—
193C2×C4C22×Dic9711yescharacteristic
194C2×C4D4×D9711yescharacteristic
195C2×C4C2×D38811yescharacteristic
196C2×C4C2×Dic19411yescharacteristic
197C2×C4#1309551011yescharacteristic
198C2×C4#1309551011yescharacteristic
199C8—22no—
200C8—3881no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22D4×Dic9711yescharacteristic
207C22#1569078011yescharacteristic
208C22#1569078711yescharacteristic
209C4—24no—
210C4—22no—
211C4—7761no—
212C4—3881no—
213C4—3882no—
214C4—3881no—
215C4C2×C4⋊Dic9711yescharacteristic
216C4D4×Dic9711yescharacteristic
217C4#1569078711yescharacteristic
218C4#1569079411yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#2026061011yescharacteristic
223C2#2026090411yescharacteristic
224C2#2026091911yescharacteristic
subgroupG/H
centerC22#15690780
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC22×C97#120
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1859
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #20260777C2semidirect
4. #20260605C2semidirect
5. #20260503C2semidirect
6. #20260495C2semidirect
25. #104×C97C4semidirect
55. M16×C97C2×C4semidirect
58. C42×C97C2×C4semidirect
62. Q8×C97C4⋊C4semidirect
66. D4×C97C4⋊C4semidirect
113. C97#1859semidirect
136. #104Dic97semidirect
155. M16C2×Dic97semidirect
171. C42C2×Dic97semidirect
175. Q8C4⋊Dic97semidirect
179. D4C4⋊Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612182
#202607772
#202607362
#202606052
#202605032
#202604952
C2×#104×C972
#185997