Finite Group Databasebeta

← #356×D97 #27143415

Subgroups — #356×D97 #27143415

Order 12416.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4⋊Q8×D97C211yesmaximal, characteristic
2#101×D97C212yesmaximal
3#20260534C211yesmaximal, characteristic, complemented
4#20260533C212yesmaximal, complemented
5#356×C97C211yesmaximal, characteristic, complemented
6C2×Q8×D97C411yescharacteristic
7C2×Q8×D97C2211yescharacteristic
8M16×D97—22no—
9#15690810C411yescharacteristic
10#15690809—22no—
11#15690808C411yescharacteristic
12#15690808C2211yescharacteristic
13C4⋊C4×D97—22no—
14C42×D97C411yescharacteristic
15#15690729C2212yes—
16#15690728—22no—
17C4⋊Q8×C97C2211yescharacteristic
18#101×C97C2212yes—
19C2×Dic194—21no—
20C2×Dic194C2×C411yescharacteristic
21C2×Dic194C2×C411yescharacteristic
22C2×Dic194D412yes—
23Q8×D97—41no—
24Q8×D97—41no—
25Q8×D97—41no—
26Q8×D97—41no—
27C2×C4×D97—21no—
28C2×C4×D97—21no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D412yes—
31#13095495—22no—
32#13095494—42no—
33C8×D97—42no—
34C4⋊Dic97—22no—
35#13095490—22no—
36#13095490—22no—
37C4×Dic97—21no—
38C4×Dic97C2×C411yescharacteristic
39C2×Q8×C97C2×C411yescharacteristic
40C2×Q8×C97C2311yescharacteristic
41M16×C97—22no—
42C4⋊C4×C97—22no—
43C42×C97C2×C411yescharacteristic
44C22×D97C22⋊C411yescharacteristic
45Q8×C97—41no—
46Q8×C97—41no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97C22×C411yescharacteristic
50C2×C4×C97C2×D412yes—
51C2×Dic97—21no—
52C2×Dic97—21no—
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C22⋊C412yes—
55C4×D97—24no—
56C4×D97—22no—
57C4×D97—22no—
58C4×D97—21no—
59C4×D97—21no—
60C4×D97—41no—
61C4×D97—41no—
62Dic194—41no—
63Dic194—42no—
64Dic194—42no—
65Dic194—41no—
66C8×C97—42no—
67#11715895—42no—
68C22×C97C2×C22⋊C411yescharacteristic
69C2×D97—22no—
70C2×D97#9911yescharacteristic
71C2×D97#9911yescharacteristic
72C4×C97—22no—
73C4×C97—22no—
74C4×C97—21no—
75C4×C97—41no—
76Dic97—22no—
77Dic97—21no—
78Dic97—22no—
79Dic97—41no—
80C2×C97—21no—
81C2×C97C2×#9911yescharacteristic
82D97—21no—
83D97#35612yesdirect product factor
84C2×#356—971nomaximal, Sylow
85C97C2×#35611yescharacteristic, complemented, Sylow
86C2×C4⋊Q8—971no—
87C2×#101—972no—
88#356—971no—
89#356—972no—
90#356D9711yescharacteristic, direct product factor
91C22×Q8—971no—
92C22×Q8—971no—
93C2×M16—1942no—
94C4⋊Q8—971no—
95C4⋊Q8—971no—
96C4⋊Q8—971no—
97C4⋊Q8—1942no—
98C4⋊Q8C2×D9711yescharacteristic
99C2×C4⋊C4—1942no—
100C2×C42—971no—
101#101—972no—
102#101—1942no—
103#101C2×D9712yes—
104C2×Q8—971no—
105C2×Q8—972no—
106C2×Q8—971no—
107C2×Q8—3881no—
108C2×Q8—3881no—
109C2×Q8—3881no—
110C2×Q8—3881no—
111C2×Q8—1941no—
112C2×Q8C4×D9711yescharacteristic
113C2×Q8C22×D9711yescharacteristic
114C22×C4—971no—
115C22×C4—972no—
116C22×C4—1941no—
117C22×C4—1941no—
118M16—22no—
119M16—3882no—
120M16—1942no—
121C2×C8—3882no—
122C4⋊C4—22no—
123C4⋊C4—1942no—
124C4⋊C4—1942no—
125C4⋊C4—1942no—
126C42—971no—
127C42—1941no—
128C42C4×D9711yescharacteristic
129C23—971no—
130Q8—41no—
131Q8—41no—
132Q8—3881no—
133Q8—3882no—
134Q8—3881no—
135Q8—3882no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—971no—
139C2×C4—972no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—1942no—
143C2×C4—1942no—
144C2×C4—1941no—
145C2×C4—1941no—
146C2×C4—1944no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4C2×C4×D9711yescharacteristic
150C2×C4D4×D9712yes—
151C8—42no—
152C8—3882no—
153C22—971no—
154C22—971no—
155C22—1942no—
156C22C22⋊C4×D9711yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—3881no—
162C4—1942no—
163C4—1941no—
164C4—1942no—
165C2—21no—
166C2—972no—
167C2—1941no—
168C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#356×C97C2
socleC2×C97C2×#99
prime radicalC22×D97C22⋊C4
cosocleC2×Q8×C97C23
hypercenter#356D97
hypocenterC97C2×#356
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
3. #20260534C2semidirect
4. #20260533C2semidirect
5. #356×C97C2semidirect
83. D97#356direct (N × H)
85. C97C2×#356semidirect
90. #356D97direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D972
#101×D972
#202605342
#202605332
#356×C972
C2×#35697