Finite Group Databasebeta

← #27157898

Subgroups — #27157898

Order 12416.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D97⋊#108C211yesmaximal, characteristic
2#20261728C212yesmaximal
3D97⋊#101C212yesmaximal
4#20261698C211yesmaximal, characteristic
5C2×Q16×D97C211yesmaximal, characteristic
6D97⋊M16C2212yes—
7D97⋊M16C2211yescharacteristic
8#15690889C2211yescharacteristic
9#15690881—21no—
10#15690880—21no—
11#15690873—24no—
12#15690871—24no—
13#15690864—21no—
14C8×#87604—21no—
15C2×Q8×D97C2212yes—
16C2×Dic388C411yescharacteristic, complemented
17C2×#13095501C412yescomplemented
18Q16×D97—21no—
19Q16×D97—21no—
20Q16×D97—21no—
21Q16×D97—21no—
22C2×C8×D97C2211yescharacteristic
23C2×Q16×C97C411yescharacteristic, complemented
24#13095521—21no—
25#13095521—22no—
26#13095520—22no—
27#13095520D412yes—
28#13095519—21no—
29#13095519—22no—
30C4⋊#87604D412yes—
31C22⋊#87604—21no—
32C4×#87604—21no—
33C2×Dic194—22no—
34C2×Dic194C2×C412yes—
35Q8×D97—22no—
36Q8×D97—22no—
37Q8×D97—22no—
38Q8×D97—22no—
39C2×C4×D97—22no—
40C2×C4×D97C2311yescharacteristic
41Dic388—21no—
42Dic388—21no—
43#13095501—22no—
44#13095501—22no—
45C2×#11715895C2×C411yescharacteristic
46C8×D97D411yescharacteristic
47C8×D97D411yescharacteristic
48C8×D97D411yescharacteristic
49C8×D97D411yescharacteristic
50C2×Q8×C97C2×C412yes—
51Q16×C97—21no—
52Q16×C97—21no—
53C2×C8×C97C2×C411yescharacteristic
54C22×D97#4111yescharacteristic
55C2×#87604—22no—
56Q8×C97—22no—
57Q8×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97C22×C411yescharacteristic
60C2×Dic97—22no—
61C2×Dic97C2×D411yescharacteristic
62C4×D97—42no—
63C4×D97—42no—
64C4×D97C2×D411yescharacteristic
65C4×D97C2×D411yescharacteristic
66C4×D97C2×D411yescharacteristic
67C4×D97#4111yescharacteristic
68Dic194—22no—
69Dic194—22no—
70Dic194—42no—
71Dic194—42no—
72#11715897—22no—
73#11715897—42no—
74C8×C97C22⋊C411yescharacteristic
75C8×C97C22⋊C411yescharacteristic
76#11715895C22⋊C411yescharacteristic
77#11715895C22⋊C411yescharacteristic
78C22×C97C4×D411yescharacteristic
79C2×D97—22no—
80C2×D97#12111yescharacteristic
81C2×D97#12411yescharacteristic
82#87604—42no—
83C4×C97—42no—
84C4×C97C2×C22⋊C411yescharacteristic
85C4×C97C4×D411yescharacteristic
86Dic97—42no—
87Dic97#12111yescharacteristic
88Dic97#12711yescharacteristic
89C2×C97—21no—
90C2×C97#39011yescharacteristic
91D97—21no—
92D97#47312yes—
93#1979—971nomaximal, Sylow
94C97#197911yescharacteristic, complemented, Sylow
95C22×Q16—971no—
96C2×#108—971no—
97#419—972no—
98C2×#101—972no—
99#405—971no—
100C22×Q8—972no—
101C2×Q16—972no—
102C2×Q16—971no—
103C2×Q16—1941no—
104C2×Q16—1941no—
105C2×Q16—1941no—
106C2×Q16—1941no—
107C2×Q16#8760411yescharacteristic, complemented
108C2×M16—971no—
109C2×M16—972no—
110C22×C8—971no—
111#117—971no—
112#108—1941no—
113#108—1941no—
114#103—1944no—
115#101—1944no—
116#97—1941no—
117C4×C8—1941no—
118C2×Q8—972no—
119C2×Q8—1942no—
120C2×Q8—1942no—
121C2×Q8—1942no—
122C2×Q8—1942no—
123C2×Q8—1942no—
124C2×Q8C2×#8760412yes—
125C22×C4—971no—
126C22×C4—1942no—
127Q16—21no—
128Q16—21no—
129Q16—1942no—
130Q16—1942no—
131Q16—1941no—
132Q16—1941no—
133M16—972no—
134M16—1942no—
135M16—1942no—
136M16—1941no—
137C2×C8—971no—
138C2×C8—971no—
139C2×C8—971no—
140C2×C8—971no—
141C2×C8—971no—
142C2×C8—1942no—
143C2×C8—1941no—
144C2×C8C2×#8760411yescharacteristic
145C4⋊C4—972no—
146C22⋊C4—1941no—
147C42—1941no—
148C23—971no—
149Q8—22no—
150Q8—22no—
151Q8—3882no—
152Q8—3882no—
153Q8—1942no—
154Q8—1942no—
155C2×C4—22no—
156C2×C4—971no—
157C2×C4—971no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—3882no—
162C2×C4—3882no—
163C2×C4—1942no—
164C2×C4—1942no—
165C2×C4C22×#8760411yescharacteristic
166C8—971no—
167C8—971no—
168C8—3882no—
169C8—1942no—
170C8C22⋊#8760411yescharacteristic
171C8C22⋊#8760411yescharacteristic
172C22—971no—
173C22—971no—
174C22—1942no—
175C22D4×#8760411yescharacteristic
176C4—42no—
177C4—971no—
178C4—971no—
179C4—3882no—
180C4—3882no—
181C4C2×C22⋊#8760411yescharacteristic
182C4D4×#8760411yescharacteristic
183C2—21no—
184C2—972no—
185C2—1941no—
186C2#2026168711yescharacteristic
subgroupG/H
centerC2#20261687
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×Q16×C97C4
socleC2×C97#390
prime radicalC22×D97#41
cosocleC2×C4×D97C23
hypercenterC2×Q16#87604
hypocenterC97#1979
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
16. C2×Dic388C4semidirect
17. C2×#13095501C4semidirect
23. C2×Q16×C97C4semidirect
94. C97#1979semidirect
107. C2×Q16#87604semidirect

Maximal subgroups

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

subgroupindex
D97⋊#1082
#202617282
D97⋊#1012
#202616982
C2×Q16×D972
#197997