Finite Group Databasebeta

← #27140369

Subgroups — #27140369

Order 12416.

238 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 59 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#20261284C211yesmaximal, characteristic
2#20260790C212yesmaximal, complemented
3#20260475C211yesmaximal, characteristic, complemented
4#20260474C212yesmaximal, complemented
5#329×C97C211yesmaximal, characteristic, complemented
6C2×#13095512C411yescharacteristic
7C2×#13095510C411yescharacteristic
8#15690808C411yescharacteristic
9#15690806C411yescharacteristic
10#15690798C2211yescharacteristic
11#15690783—22no—
12#15690770—22no—
13C42×D97C2211yescharacteristic
14#15690739—22no—
15#15690722—22no—
16#15690721C2212yes—
17#124×C97C2211yescharacteristic
18#97×C97C2212yes—
19#13095512—42no—
20#13095510—42no—
21C2×Dic194C2×C411yescharacteristic
22C2×D388—21no—
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25C2×C22⋊D97—22no—
26C22×Dic97—21no—
27C2×C4×D97—21no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D412yes—
30C2×#11715895—22no—
31#13095490—22no—
32#13095488—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D412yes—
35C2×Q8×C97C2×C411yescharacteristic
36C2×D4×C97C2×C411yescharacteristic
37C2×C8×C97—22no—
38C22⋊C4×C97—22no—
39C42×C97C2311yescharacteristic
40C23×C97—21no—
41C22×D97—21no—
42C22×D97C22⋊C411yescharacteristic
43Q8×C97—41no—
44D4×C97—41no—
45C2×C4×C97—21no—
46C2×C4×C97C22×C411yescharacteristic
47C2×C4×C97C2×D412yes—
48C22⋊D97—44no—
49C22⋊D97—42no—
50C2×Dic97—21no—
51C2×Dic97—42no—
52C2×Dic97C22⋊C412yes—
53C2×Dic97C22⋊C411yescharacteristic
54D388—41no—
55D388—42no—
56C4×D97—22no—
57C4×D97—42no—
58C4×D97—42no—
59Dic194—41no—
60C8×C97—42no—
61#11715895—42no—
62C22×C97—41no—
63C22×C97—42no—
64C22×C97C2×C22⋊C411yescharacteristic
65C2×D97—21no—
66C2×D97—22no—
67C2×D97—41no—
68C2×D97—42no—
69C4×C97—21no—
70C4×C97—22no—
71C4×C97—41no—
72Dic97—22no—
73Dic97—41no—
74Dic97#10012yes—
75C2×C97—81no—
76C2×C97C2×#10011yescharacteristic
77C2×C97#42112yes—
78D97—41no—
79D97—81no—
80#1536—971nomaximal, Sylow
81C97#153611yescharacteristic, complemented, Sylow
82#532—971no—
83#432—972no—
84#329—971no—
85#329—972no—
86#329D9711yescharacteristic, complemented
87C2×#41—971no—
88C2×#41—971no—
89C4⋊Q8—971no—
90#127—971no—
91#124—971no—
92#124C2×D9711yescharacteristic
93#121—1942no—
94C4×D4—1942no—
95C2×C42—971no—
96C4⋊C8—1942no—
97C22⋊C8—1942no—
98#97—972no—
99#97C2×D9712yes—
100#41—3882no—
101#41—3882no—
102C2×Q8—971no—
103C2×Q8C4×D9711yescharacteristic
104C2×D4—971no—
105C2×D4—1941no—
106C2×D4—1942no—
107C2×D4—1941no—
108C2×D4C4×D9711yescharacteristic
109C22×C4—972no—
110C22×C4—971no—
111C22×C4—1941no—
112C22×C4—1941no—
113C2×C8—22no—
114C2×C8—1942no—
115C4⋊C4—1942no—
116C22⋊C4—22no—
117C22⋊C4—1942no—
118C42—971no—
119C42—972no—
120C42C22×D9711yescharacteristic
121C23—21no—
122C23—971no—
123C23—1941no—
124Q8—41no—
125Q8—3881no—
126D4—41no—
127D4—3881no—
128D4—3882no—
129D4—3882no—
130D4—3884no—
131C2×C4—21no—
132C2×C4—971no—
133C2×C4—972no—
134C2×C4—3882no—
135C2×C4—3882no—
136C2×C4—3882no—
137C2×C4—1942no—
138C2×C4—1941no—
139C2×C4C2×C4×D9711yescharacteristic
140C2×C4D4×D9712yes—
141C8—42no—
142C8—3882no—
143C22—41no—
144C22—42no—
145C22—3881no—
146C22—3882no—
147C22—1941no—
148C22—1942no—
149C22C22⋊C4×D9711yescharacteristic
150C4—21no—
151C4—22no—
152C4—41no—
153C4—972no—
154C4—3881no—
155C4—1942no—
156C2—81no—
157C2—7761no—
158C2—3881no—
159C2#100×D9711yescharacteristic
160C2#2026075012yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#329×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261284C2
cosocleC42×C97C23
hypercenter#329D97
hypocenterC97#1536
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20260790C2semidirect
3. #20260475C2semidirect
4. #20260474C2semidirect
5. #329×C97C2semidirect
81. C97#1536semidirect
86. #329D97semidirect

Maximal subgroups

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

subgroupindex
#202612842
#202607902
#202604752
#202604742
#329×C972
#153697