Finite Group Databasebeta

← #27154343

Subgroups — #27154343

Order 12416.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#20261435C212yesmaximal, complemented
2#20261137C211yesmaximal, characteristic, complemented
3#20261057C212yesmaximal, complemented
4#20261047C212yesmaximal, complemented
5#20261035C212yesmaximal, complemented
6#20260985C212yesmaximal, complemented
7#20260814C211yesmaximal, characteristic, complemented
8#20260776C212yesmaximal, complemented
9C4×D8×C97C211yesmaximal, characteristic, complemented
10C2×#13095494C2212yes—
11#15690803—22no—
12#15690802C2212yes—
13#15690793—22no—
14#15690783—22no—
15#15690781C2212yes—
16#15690781C2212yes—
17C4×D388C2211yescharacteristic
18C4×C22⋊D97—22no—
19#15690768C2212yes—
20C4⋊C4×D97C2212yes—
21#15690744C2211yescharacteristic
22#15690744C2211yescharacteristic
23#15690743C2212yes—
24#15690742C2212yes—
25#15690741C2212yes—
26C4⋊#11715895C2211yescharacteristic
27#15690731C2212yes—
28#15690731C2212yes—
29#15690722C2212yes—
30C2×D8×C97C2211yescharacteristic
31C4×D4×C97C2212yes—
32#107×C97C2211yescharacteristic
33#102×C97C2212yes—
34C4×C8×C97C2211yescharacteristic
35C2×D388D411yescharacteristic
36C2×C22⋊D97—22no—
37C2×C22⋊D97—22no—
38C2×C4×D97—22no—
39C2×C4×D97C2312yes—
40#13095494—24no—
41C2×#11715895C2312yes—
42C4⋊Dic97D411yescharacteristic
43C4⋊Dic97C2312yes—
44C4⋊Dic97C2312yes—
45#13095490—22no—
46#13095490—22no—
47#13095490—22no—
48C22⋊Dic97—22no—
49C22⋊Dic97—22no—
50#13095488—22no—
51#13095488D412yes—
52C4×Dic97—22no—
53C2×D4×C97C2312yes—
54C22×C4×C97—22no—
55D8×C97—22no—
56C2×C8×C97C2311yescharacteristic
57C2×C8×C97C2311yescharacteristic
58C4⋊C4×C97C2312yes—
59C22⋊C4×C97—22no—
60C42×C97C2311yescharacteristic
61C23×C97—22no—
62C22×D97C2×D412yes—
63D4×C97—22no—
64D4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—82no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2411yescharacteristic
70C22⋊D97—44no—
71C22⋊D97—82no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97C2×D412yes—
75D388—41no—
76C4×D97—22no—
77C4×D97—22no—
78C4×D97—82no—
79C8×C97—21no—
80C8×C97#4112yes—
81#11715895—22no—
82C22×C97—42no—
83C22×C97—42no—
84C22×C97—42no—
85C22×C97C22×D411yescharacteristic
86C2×D97—22no—
87C2×D97—22no—
88C2×D97—22no—
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—42no—
92C4×C97C2×#4111yescharacteristic
93C4×C972−1+411yescharacteristic
94Dic97—22no—
95Dic97—42no—
96Dic97—42no—
97C2×C97—82no—
98C2×C97#54711yescharacteristic
99C2×C97#57511yescharacteristic
100C2×C97#57611yescharacteristic
101D97—42no—
102#3355—971nomaximal, Sylow
103C97#335511yescharacteristic, complemented, Sylow
104#553—972no—
105#499—971no—
106#482—972no—
107#480—972no—
108#478—972no—
109#469—972no—
110C4×D8D9711yescharacteristic, complemented
111#436—971no—
112#428—972no—
113C2×D8C2×D9711yescharacteristic
114C2×M16—972no—
115#126—1942no—
116#125—972no—
117#123—1942no—
118#121—972no—
119#121—972no—
120#121—1942no—
121C4×D4—971no—
122C4×D4—1942no—
123C4×D4C2×D9712yes—
124#117—972no—
125C2×C4⋊C4—972no—
126#107—971no—
127#107—971no—
128#107—972no—
129#107C2×D9711yescharacteristic
130#106—972no—
131#106—972no—
132C4⋊C8—971no—
133#102—972no—
134#102—972no—
135#102C2×D9712yes—
136C22⋊C8—972no—
137C4×C8C2×D9711yescharacteristic
138C2×D4—971no—
139C2×D4—1942no—
140C2×D4—1942no—
141C2×D4C22×D9712yes—
142C22×C4—22no—
143C22×C4—972no—
144C22×C4—1942no—
145D8—22no—
146M16—1944no—
147C2×C8—972no—
148C2×C8C22×D9711yescharacteristic
149C2×C8C22×D9711yescharacteristic
150C4⋊C4—971no—
151C4⋊C4—972no—
152C4⋊C4—972no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4—1942no—
156C4⋊C4C22×D9712yes—
157C22⋊C4—22no—
158C22⋊C4—972no—
159C22⋊C4—1942no—
160C22⋊C4—1942no—
161C22⋊C4—1942no—
162C42—1942no—
163C42C22×D9711yescharacteristic
164C23—22no—
165C23—972no—
166D4—22no—
167D4—22no—
168D4—7762no—
169D4—3881no—
170D4—3884no—
171C2×C4—22no—
172C2×C4—82no—
173C2×C4—7762no—
174C2×C4—972no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4—1942no—
179C2×C4D4×D9711yescharacteristic
180C2×C4D4×D9711yescharacteristic
181C2×C4C23×D9711yescharacteristic
182C8—21no—
183C8—1942no—
184C8#1309551112yes—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—1942no—
189C22—1942no—
190C22—1942no—
191C22C2×D4×D9711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—3882no—
196C4—3882no—
197C4—1942no—
198C4C2×#1309551111yescharacteristic
199C4#1569085711yescharacteristic
200C2—82no—
201C2—3882no—
202C2#2026139011yescharacteristic
203C2#2026153211yescharacteristic
204C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
hypercenterC4×D8D97
hypocenterC97#3355
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20261435C2semidirect
2. #20261137C2semidirect
3. #20261057C2semidirect
4. #20261047C2semidirect
5. #20261035C2semidirect
6. #20260985C2semidirect
7. #20260814C2semidirect
8. #20260776C2semidirect
9. C4×D8×C97C2semidirect
103. C97#3355semidirect
110. C4×D8D97semidirect

Maximal subgroups

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

subgroupindex
#202614352
#202611372
#202610572
#202610472
#202610352
#202609852
#202608142
#202607762
C4×D8×C972
#335597