Finite Group Databasebeta

← #27143409

Subgroups — #27143409

Order 12416.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261305C211yesmaximal, characteristic
2#20260707C212yesmaximal
3#20260530C211yesmaximal, characteristic, complemented
4#20260529C212yesmaximal, complemented
5#354×C97C211yesmaximal, characteristic, complemented
6C2×#13095510C2211yescharacteristic
7C2×Q8×D97C411yescharacteristic, complemented
8#15690810C411yescharacteristic, complemented
9#15690808C411yescharacteristic, complemented
10#15690797C2211yescharacteristic
11#15690792—22no—
12#15690787—22no—
13#15690767C411yescharacteristic, complemented
14#15690766—22no—
15#15690725C2212yes—
16#15690724—22no—
17#124×C97C2211yescharacteristic
18#99×C97C2212yes—
19#13095510—42no—
20C2×Dic194—21no—
21C2×Dic194C2×C411yescharacteristic
22C2×Dic194C2×C411yescharacteristic
23Q8×D97—42no—
24C2×C22⋊D97D412yes—
25C22×Dic97D412yes—
26C2×C4×D97—21no—
27C2×C4×D97C2×C411yescharacteristic
28C4⋊Dic97—22no—
29#13095490—21no—
30#13095490—22no—
31#13095490—42no—
32C22⋊Dic97—22no—
33C22⋊Dic97—22no—
34#13095488—21no—
35C4×Dic97—42no—
36C4×Dic97C2×C411yescharacteristic
37C2×Q8×C97C2×C411yescharacteristic
38C2×D4×C97C2311yescharacteristic
39C22⋊C4×C97—22no—
40C22⋊C4×C97—22no—
41C42×C97C2×C411yescharacteristic
42C23×C97C2×D412yes—
43C22×D97C22⋊C411yescharacteristic
44Q8×C97—41no—
45D4×C97—41no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97—42no—
49C2×C4×C97C22×C411yescharacteristic
50C22⋊D97—42no—
51C2×Dic97—22no—
52C2×Dic97—22no—
53C2×Dic97—21no—
54C2×Dic97—21no—
55C2×Dic97—42no—
56C2×Dic97C22⋊C412yes—
57C2×Dic97C22⋊C411yescharacteristic
58C4×D97—22no—
59C4×D97—42no—
60Dic194—41no—
61Dic194—42no—
62Dic194—41no—
63C22×C97—22no—
64C22×C97—42no—
65C22×C97C2×C22⋊C411yescharacteristic
66C2×D97—21no—
67C2×D97—41no—
68C4×C97—21no—
69C4×C97—41no—
70C4×C97—41no—
71C4×C97—82no—
72Dic97—22no—
73Dic97—41no—
74Dic97—41no—
75Dic97—82no—
76Dic97#9912yes—
77C2×C97—21no—
78C2×C97—42no—
79C2×C97C2×#9911yescharacteristic
80D97—41no—
81#2129—971nomaximal, Sylow
82C97#212911yescharacteristic, complemented, Sylow
83#536—971no—
84#410—972no—
85#354—971no—
86#354—972no—
87#354D9711yescharacteristic, complemented
88C2×#41—971no—
89C22×Q8—971no—
90C4⋊Q8—971no—
91C4⋊Q8—971no—
92#124—971no—
93#124C2×D9711yescharacteristic
94#123—1942no—
95C22⋊Q8—1942no—
96#117—971no—
97#117—1942no—
98#99—972no—
99#99—1942no—
100#99C2×D9712yes—
101#41—3882no—
102C2×Q8—971no—
103C2×Q8—971no—
104C2×Q8—3882no—
105C2×Q8—1941no—
106C2×Q8C4×D9711yescharacteristic
107C2×D4—972no—
108C2×D4C22×D9711yescharacteristic
109C22×C4—972no—
110C22×C4—971no—
111C22×C4—1941no—
112C4⋊C4—3882no—
113C4⋊C4—1941no—
114C4⋊C4—1942no—
115C4⋊C4—1942no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—1941no—
119C22⋊C4—1942no—
120C22⋊C4—1942no—
121C42—971no—
122C42—3882no—
123C42C4×D9711yescharacteristic
124C23—971no—
125C23D4×D9712yes—
126Q8—41no—
127Q8—3881no—
128Q8—3881no—
129Q8—3882no—
130D4—41no—
131D4—3882no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—971no—
136C2×C4—972no—
137C2×C4—3882no—
138C2×C4—3882no—
139C2×C4—1942no—
140C2×C4—1942no—
141C2×C4—1942no—
142C2×C4—1941no—
143C2×C4—1941no—
144C2×C4C2×C4×D9711yescharacteristic
145C22—22no—
146C22—42no—
147C22—3881no—
148C22—1941no—
149C22C22⋊C4×D9711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—7762no—
155C4—972no—
156C4—3881no—
157C4—3881no—
158C4—1942no—
159C2—21no—
160C2—42no—
161C2—3881no—
162C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#354×C97C2
socleC2×C97C2×#99
prime radicalC2×#13095510C22
cosocleC2×D4×C97C23
hypercenter#354D97
hypocenterC97#2129
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #20260530C2semidirect
4. #20260529C2semidirect
5. #354×C97C2semidirect
7. C2×Q8×D97C4semidirect
8. #15690810C4semidirect
9. #15690808C4semidirect
13. #15690767C4semidirect
82. C97#2129semidirect
87. #354D97semidirect

Maximal subgroups

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

subgroupindex
#202613052
#202607072
#202605302
#202605292
#354×C972
#212997