Finite Group Databasebeta

← #27143424

Subgroups — #27143424

Order 12416.

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

normal subgroups only

This group has 230 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#20261310C211yesmaximal, characteristic
2#101×D97C212yesmaximal, complemented
3#20260534C211yesmaximal, characteristic, complemented
4#20260531C212yesmaximal, complemented
5#356×C97C211yesmaximal, characteristic, complemented
6C2×#13095512C411yescharacteristic
7C2×Q8×D97C2211yescharacteristic
8M16×D97—22no—
9#15690808C411yescharacteristic
10#15690799C2211yescharacteristic
11#15690798C411yescharacteristic
12#15690795—22no—
13#15690788—22no—
14#15690767C411yescharacteristic
15#15690729C2212yes—
16#15690728—22no—
17C4⋊Q8×C97C2211yescharacteristic
18#101×C97C2212yes—
19#13095512—42no—
20C2×Dic194C2×C411yescharacteristic
21C2×Dic194D412yes—
22Q8×D97—42no—
23C2×D388—21no—
24C2×D388C2×C411yescharacteristic
25C2×C4×D97—21no—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D412yes—
28#13095495—22no—
29#13095494—42no—
30C8×D97—42no—
31#13095490—21no—
32#13095490—22no—
33#13095488—21no—
34#13095488—22no—
35#13095488—22no—
36C4×Dic97C2×C411yescharacteristic
37C2×Q8×C97C2×C411yescharacteristic
38C2×Q8×C97C2311yescharacteristic
39M16×C97—22no—
40C4⋊C4×C97—22no—
41C42×C97C2×C411yescharacteristic
42C22×D97—21no—
43C22×D97C22⋊C411yescharacteristic
44Q8×C97—41no—
45Q8×C97—41no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97C22×C411yescharacteristic
49C2×C4×C97C2×D412yes—
50C2×Dic97—21no—
51C2×Dic97C22⋊C412yes—
52C2×Dic97C22⋊C411yescharacteristic
53D388—41no—
54D388—42no—
55C4×D97—22no—
56C4×D97—22no—
57C4×D97—22no—
58C4×D97—42no—
59Dic194—41no—
60Dic194—42no—
61C8×C97—42no—
62#11715895—42no—
63C22×C97C2×C22⋊C411yescharacteristic
64C2×D97—41no—
65C2×D97—42no—
66C2×D97—41no—
67C2×D97#9912yes—
68C4×C97—22no—
69C4×C97—21no—
70C4×C97—41no—
71C4×C97—41no—
72Dic97—21no—
73Dic97—22no—
74Dic97—41no—
75C2×C97—21no—
76C2×C97C2×#9911yescharacteristic
77D97—22no—
78D97—81no—
79#2133—971nomaximal, Sylow
80C97#213311yescharacteristic, complemented, Sylow
81#536—971no—
82C2×#101—972no—
83#356—971no—
84#356D9711yescharacteristic, complemented
85#355—972no—
86C2×#41—971no—
87C22×Q8—971no—
88C2×M16—1942no—
89C4⋊Q8—971no—
90C4⋊Q8C2×D9711yescharacteristic
91#124—971no—
92#124—971no—
93#123—1942no—
94C22⋊Q8—1942no—
95#117—971no—
96#101—972no—
97#101—1942no—
98#101C2×D9712yes—
99#41—3882no—
100C2×Q8—971no—
101C2×Q8—972no—
102C2×Q8—3882no—
103C2×Q8C4×D9711yescharacteristic
104C2×Q8C22×D9711yescharacteristic
105C2×D4—971no—
106C2×D4—1941no—
107C22×C4—971no—
108C22×C4—972no—
109C22×C4—1941no—
110M16—22no—
111M16—3882no—
112M16—1942no—
113C2×C8—3882no—
114C4⋊C4—22no—
115C4⋊C4—1941no—
116C4⋊C4—1942no—
117C22⋊C4—1942no—
118C22⋊C4—1941no—
119C22⋊C4—1942no—
120C42—971no—
121C42C4×D9711yescharacteristic
122C23—971no—
123C23—1941no—
124Q8—41no—
125Q8—41no—
126Q8—3882no—
127Q8—3881no—
128D4—3881no—
129D4—3882no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—972no—
133C2×C4—971no—
134C2×C4—3882no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4—1942no—
138C2×C4—1941no—
139C2×C4C2×C4×D9711yescharacteristic
140C2×C4D4×D9712yes—
141C8—42no—
142C8—3882no—
143C22—972no—
144C22—3882no—
145C22—3881no—
146C22—3881no—
147C22C22⋊C4×D9711yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—3881no—
153C4—1941no—
154C4—1942no—
155C2—21no—
156C2—7761no—
157C2—1942no—
158C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#356×C97C2
socleC2×C97C2×#99
prime radicalC2×#13095512C4
cosocleC2×Q8×C97C23
hypercenter#356D97
hypocenterC97#2133
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #101×D97C2semidirect
3. #20260534C2semidirect
4. #20260531C2semidirect
5. #356×C97C2semidirect
80. C97#2133semidirect
84. #356D97semidirect

Maximal subgroups

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

subgroupindex
#202613102
#101×D972
#202605342
#202605312
#356×C972
#213397