Finite Group Databasebeta

← #27143276

Subgroups — #27143276

Order 12416.

254 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15690741C211yesmaximal, characteristic
2C2×#15690735C211yesmaximal, characteristic
3C2×#15690733C211yesmaximal, characteristic
4#20260640C211yesmaximal, characteristic
5#20260604C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic
7C2×#106×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10C2×C4⋊Dic97C2211yescharacteristic
11C2×#13095490—22no—
12C2×C4×Dic97C2211yescharacteristic
13#15690741—22no—
14#15690735—22no—
15#15690733—22no—
16#15690717—22no—
17C22×C8×C97C2211yescharacteristic
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×C4⋊C4×C97C2211yescharacteristic
20#106×C97—22no—
21C2×Dic194—21no—
22C2×Dic194—21no—
23C2×Dic194—21no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194Q812yes—
27C22×Dic97—21no—
28C22×Dic97—21no—
29C22×Dic97D412yes—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895D412yes—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35#13095490—44no—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C22×C4×C97—21no—
39C22×C4×C97—21no—
40C22×C4×C97C2311yescharacteristic
41C2×C8×C97—21no—
42C2×C8×C97—21no—
43C2×C8×C97D412yes—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C23×C97C2×D411yescharacteristic
49C2×C4×C97—41no—
50C2×C4×C97—41no—
51C2×C4×C97—42no—
52C2×C4×C97—42no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×Q811yescharacteristic
56C2×C4×C97#4111yescharacteristic
57C2×C4×C97#4111yescharacteristic
58C2×C4×C97#4111yescharacteristic
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—22no—
62C2×Dic97—41no—
63C2×Dic97—41no—
64C2×Dic97—41no—
65C2×Dic97—42no—
66C2×Dic97SD1614yes—
67Dic194—22no—
68Dic194—22no—
69Dic194—41no—
70Dic194—41no—
71C8×C97—22no—
72#11715895—22no—
73C22×C97C22⋊Q811yescharacteristic
74C22×C97#12311yescharacteristic
75C22×C97#12711yescharacteristic
76C22×C97C2×SD1612yes—
77C22×C97#13712yes—
78C4×C97—81no—
79C4×C97—81no—
80C4×C97C22⋊Q811yescharacteristic
81C4×C97C22⋊Q811yescharacteristic
82C4×C97#12311yescharacteristic
83C4×C97#12311yescharacteristic
84Dic97—24no—
85Dic97—42no—
86Dic97—81no—
87C2×C97#39911yescharacteristic
88C2×C97#47512yes—
89C2×C97#48312yes—
90C2×C97#49211yescharacteristic
91C2×C97#49711yescharacteristic
92#2096—971nomaximal, Sylow
93C97#209611yescharacteristic, complemented, Sylow
94C2×#106—971no—
95C2×#106D9711yescharacteristic
96C2×#103—971no—
97C2×#103—971no—
98#391—971no—
99#384—971no—
100#336—971no—
101C22×Q8—971no—
102C22×C8—971no—
103C22×C8C2×D9711yescharacteristic
104C2×C4⋊C4—971no—
105C2×C4⋊C4—1942no—
106C2×C4⋊C4C2×D9711yescharacteristic
107C2×C4⋊C4C2×D9711yescharacteristic
108C2×C42—971no—
109#106—22no—
110#106—1942no—
111#103—1942no—
112#103—1942no—
113#95—1942no—
114C2×Q8—972no—
115C2×Q8—1941no—
116C2×Q8—1941no—
117C2×Q8—1941no—
118C2×Q8—1941no—
119C2×Q8—1941no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—972no—
123C22×C4—1941no—
124C22×C4—1941no—
125C22×C4C22×D9711yescharacteristic
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—972no—
129C2×C8—1941no—
130C2×C8—1941no—
131C2×C8C22⋊D9712yes—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—3884no—
137C4⋊C4—1941no—
138C4⋊C4—1941no—
139C42—1941no—
140C42—1941no—
141C23D4×D9711yescharacteristic
142Q8—3881no—
143Q8—3881no—
144Q8—1942no—
145Q8—1942no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—974no—
151C2×C4—3881no—
152C2×C4—3882no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—1942no—
156C2×C4—1942no—
157C2×C4—1942no—
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C2×C4Q8×D9711yescharacteristic
161C2×C4#1309551111yescharacteristic
162C2×C4#1309551111yescharacteristic
163C2×C4#1309551211yescharacteristic
164C8—22no—
165C8—1942no—
166C22#1569078811yescharacteristic
167C22#1569079511yescharacteristic
168C22#1569080611yescharacteristic
169C22SD16×D9712yes—
170C22#1569083912yes—
171C4—81no—
172C4—81no—
173C4—7761no—
174C4—3882no—
175C4—1944no—
176C4#1569078611yescharacteristic
177C4#1569078811yescharacteristic
178C4#1569079311yescharacteristic
179C4#1569079511yescharacteristic
180C2#2026067011yescharacteristic
181C2#2026101912yes—
182C2#2026106712yes—
183C2#2026110611yescharacteristic
184C2#2026112611yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#106×C97C2
socle, prime radicalC23×C97C2×D4
hypercenterC2×#106D97
hypocenterC97#2096
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
93. C97#2096semidirect

Maximal subgroups

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

subgroupindex
C2×#156907412
C2×#156907352
C2×#156907332
#202606402
#202606042
#202604892
C2×#106×C972
#209697