Finite Group Databasebeta

← #27151541

Subgroups — #27151541

Order 12416.

376 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 153 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#20261292C211yesmaximal, characteristic
2#20261042C214yesmaximal, complemented
3#20261030C212yesmaximal, complemented
4#20261020C212yesmaximal, complemented
5#20260775C212yesmaximal
6#20260759C211yesmaximal, characteristic
7#20260744C212yesmaximal
8C2×C4×Q8×C97C211yesmaximal, characteristic
9C2×#13095495C2212yes—
10#15690810C2211yescharacteristic
11#15690810C2211yescharacteristic
12#15690802C2212yes—
13#15690789—21no—
14#15690789—21no—
15C4×Dic194—21no—
16C4×Dic194—21no—
17#15690769C2212yes—
18#15690743C2214yes—
19#15690741C2214yes—
20C4⋊#11715895C2212yes—
21C4⋊#11715895C2212yes—
22#15690734C2214yes—
23#15690734C2214yes—
24C22×Q8×C97C2211yescharacteristic
25C4×Q8×C97—21no—
26C4×Q8×C97—21no—
27C4×Q8×C97C2212yes—
28C4×Q8×C97C2212yes—
29C2×C4⋊C4×C97—21no—
30C2×C4⋊C4×C97C2211yescharacteristic
31C2×C42×C97—21no—
32C2×C42×C97C2211yescharacteristic
33C2×Dic194—21no—
34C2×Dic194—21no—
35#13095495—24no—
36C2×#11715895C2314yes—
37C4⋊Dic97—21no—
38C4⋊Dic97—21no—
39C4⋊Dic97C2312yes—
40C4⋊Dic97C2312yes—
41#13095490—22no—
42#13095490—22no—
43C22⋊Dic97—22no—
44C4×Dic97—22no—
45C2×Q8×C97—21no—
46C2×Q8×C97—22no—
47C2×Q8×C97Q814yescomplemented
48C2×Q8×C97C2312yes—
49C22×C4×C97—21no—
50C22×C4×C97—21no—
51C22×C4×C97D411yescharacteristic
52C22×C4×C97D411yescharacteristic
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97—22no—
55C4⋊C4×C97—22no—
56C4⋊C4×C97—21no—
57C4⋊C4×C97C2312yes—
58C42×C97—22no—
59C42×C97—22no—
60C42×C97D411yescharacteristic
61C42×C97D411yescharacteristic
62C42×C97C2311yescharacteristic
63C42×C97C2311yescharacteristic
64C23×C97C2×D411yescharacteristic
65Q8×C97—24no—
66Q8×C97—22no—
67Q8×C97C2×Q814yes—
68C2×C4×C97—24no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—41no—
73C2×C4×C97—42no—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×Q812yes—
80C2×C4×C97#4112yes—
81C2×C4×C97C2411yescharacteristic
82C2×Dic97—22no—
83C2×Dic97—22no—
84Dic194—42no—
85Dic194—42no—
86#11715895—24no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97C22⋊Q812yes—
90C22×C97C22×D411yescharacteristic
91C4×C97—24no—
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—21no—
95C4×C97—42no—
96C4×C97C22⋊Q812yes—
97C4×C97#13712yes—
98C4×C97C22×Q811yescharacteristic
99C4×C97C2×#4111yescharacteristic
100Dic97—42no—
101Dic97—42no—
102C2×C97—22no—
103C2×C97C2×C22⋊Q811yescharacteristic
104C2×C97C2×#13711yescharacteristic
105C2×C97#57511yescharacteristic
106#3078—971nomaximal, Sylow
107C97#307811yescharacteristic, complemented, Sylow
108#533—971no—
109C2×C4×Q8D9711yescharacteristic
110#479—974no—
111#477—972no—
112#475—972no—
113#428—972no—
114#423—971no—
115#419—972no—
116C22×Q8C2×D9711yescharacteristic
117C2×M16—972no—
118C4⋊Q8—971no—
119C4⋊Q8—971no—
120#125—972no—
121C22⋊Q8—1941no—
122C22⋊Q8—1941no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—1941no—
126C4×Q8—1941no—
127C4×Q8C2×D9712yes—
128C4×Q8C2×D9712yes—
129#117—972no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C2×D9711yescharacteristic
132C2×C42—21no—
133C2×C42C2×D9711yescharacteristic
134#107—974no—
135#106—974no—
136C4⋊C8—972no—
137C4⋊C8—972no—
138#103—974no—
139#103—974no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—1941no—
143C2×Q8—1941no—
144C2×Q8Dic19414yescomplemented
145C2×Q8C22×D9712yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22⋊D9711yescharacteristic
149C22×C4C22⋊D9711yescharacteristic
150C22×C4C22×D9711yescharacteristic
151M16—1944no—
152C2×C8—974no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—972no—
157C4⋊C4—972no—
158C4⋊C4—1942no—
159C4⋊C4—1942no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C4⋊C4C22×D9712yes—
163C22⋊C4—1942no—
164C42—22no—
165C42—22no—
166C42—1942no—
167C42C22⋊D9711yescharacteristic
168C42C22⋊D9711yescharacteristic
169C42C22×D9711yescharacteristic
170C42C22×D9711yescharacteristic
171C23C2×C22⋊D9711yescharacteristic
172Q8—24no—
173Q8—22no—
174Q8—3882no—
175Q8—3882no—
176Q8C2×Dic19414yes—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4C2×C22⋊D9711yescharacteristic
186C2×C4C2×C22⋊D9711yescharacteristic
187C2×C4C2×C22⋊D9711yescharacteristic
188C2×C4C2×C22⋊D9711yescharacteristic
189C2×C4C2×C22⋊D9711yescharacteristic
190C2×C4C2×Dic19412yes—
191C2×C4#1309551112yes—
192C2×C4C23×D9711yescharacteristic
193C8—1944no—
194C22—21no—
195C22—21no—
196C22#1569078912yes—
197C22C22×C22⋊D9711yescharacteristic
198C4—24no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—3882no—
204C4—3882no—
205C4#1569078912yes—
206C4#1569084112yes—
207C4C22×Dic19411yescharacteristic
208C4C2×#1309551111yescharacteristic
209C2—22no—
210C2C2×#1569078911yescharacteristic
211C2C2×#1569084111yescharacteristic
212C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C4×Q8×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenterC2×C4×Q8D97
hypocenterC97#3078
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #20261042C2semidirect
3. #20261030C2semidirect
4. #20261020C2semidirect
47. C2×Q8×C97Q8semidirect
107. C97#3078semidirect
144. C2×Q8Dic194semidirect

Maximal subgroups

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

subgroupindex
#202612922
#202610422
#202610302
#202610202
#202607752
#202607592
#202607442
C2×C4×Q8×C972
#307897