Finite Group Databasebeta

← D97⋊#328 #27157105

Subgroups — D97⋊#328 #27157105

Order 12416.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261703C211yesmaximal, characteristic
2C2×(C2×#87604)⋊C4C211yesmaximal, characteristic
3#20261589C212yesmaximal, complemented
4#20261588C212yesmaximal, complemented
5C22⋊Q8×D97C211yesmaximal, characteristic
6C23×#87604—21no—
7C2×#13095519—21no—
8#15690867C2212yes—
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12C2×Q8×D97C411yescharacteristic, complemented
13C22×C4×D97C2211yescharacteristic
14#15690790C2211yescharacteristic
15#15690789C411yescharacteristic, complemented
16#15690788—21no—
17#15690787—21no—
18#15690786C2211yescharacteristic
19C4⋊C4×D97—21no—
20C4⋊C4×D97C411yescharacteristic, complemented
21C22⋊C4×D97—21no—
22C22⋊Q8×C97C411yescharacteristic, complemented
23C22×#87604—24no—
24C22×#87604—22no—
25C22×#87604—42no—
26C22×#87604—42no—
27#13095519—42no—
28C2×#11715897—22no—
29C23×D97D411yescharacteristic
30C2×Dic194—21no—
31C2×Dic194C2×C411yescharacteristic, complemented
32Q8×D97—42no—
33C22×Dic97D411yescharacteristic
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—42no—
37C2×C4×D97C2×C411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97C2311yescharacteristic
41C4⋊Dic97—21no—
42C4⋊Dic97C2×C411yescharacteristic, complemented
43#13095490—21no—
44#13095490—21no—
45#13095490—21no—
46C22⋊Dic97—21no—
47#13095488—21no—
48#13095488—21no—
49C2×Q8×C97C2×C411yescharacteristic, complemented
50C22×C4×C97C2×C411yescharacteristic
51C4⋊C4×C97—21no—
52C4⋊C4×C97C2×C411yescharacteristic, complemented
53C22⋊C4×C97—21no—
54C23×C97C22⋊C411yescharacteristic
55C22×D97—22no—
56C22×D97—22no—
57C22×D97C22⋊C411yescharacteristic
58C22×D97SD1612yes—
59C22×D97Q1612yes—
60C22×D97C2×D411yescharacteristic
61C2×#87604—24no—
62C2×#87604—42no—
63C2×#87604—48no—
64C2×#87604—42no—
65Q8×C97—41no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—41no—
69C2×C4×C97C22⋊C411yescharacteristic
70C2×C4×C97C22×C411yescharacteristic
71C2×Dic97—21no—
72C2×Dic97—21no—
73C2×Dic97—41no—
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C2×D411yescharacteristic
76C4×D97—22no—
77C4×D97—22no—
78C4×D97—41no—
79C4×D97—41no—
80C4×D97—42no—
81C4×D97—42no—
82Dic194—42no—
83Dic194—41no—
84#11715897—42no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97#10312yes—
88C22×C97C2×C22⋊C411yescharacteristic
89C2×D97—21no—
90C2×D97—21no—
91C2×D97—24no—
92C2×D97—24no—
93C2×D97#9912yes—
94C2×D97#10312yes—
95C2×D97#10412yes—
96C2×D97C2×SD1611yescharacteristic
97C2×D97C2×Q1611yescharacteristic
98#87604—44no—
99C4×C97—21no—
100C4×C97—21no—
101C4×C97—41no—
102C4×C97—41no—
103Dic97—21no—
104Dic97—21no—
105Dic97—41no—
106Dic97—41no—
107C2×C97—22no—
108C2×C97C2×#9911yescharacteristic
109C2×C97C2×#10311yescharacteristic
110C2×C97C2×#10411yescharacteristic
111D97—21no—
112D97—21no—
113D97#32814yescomplemented
114C2×#328—971nomaximal, Sylow
115C97C2×#32811yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—971no—
117C2×C22⋊C8—971no—
118C2×#95—971no—
119#328—972no—
120#328—972no—
121C22×Q8—971no—
122C23×C4—971no—
123C23×C4—1941no—
124C22×C8—1941no—
125C22⋊Q8—971no—
126C22⋊Q8—971no—
127C22⋊Q8—971no—
128C22⋊Q8—1941no—
129C22⋊Q8—1941no—
130C22⋊Q8#8760411yescharacteristic, complemented
131C2×C4⋊C4—971no—
132C2×C4⋊C4—1941no—
133C2×C22⋊C4—1941no—
134C22⋊C8—972no—
135C22⋊C8—1941no—
136#95—972no—
137#95—1941no—
138C24—971no—
139C2×Q8—971no—
140C2×Q8—3882no—
141C2×Q8—1941no—
142C2×Q8C2×#8760411yescharacteristic, complemented
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—3882no—
149C22×C4—3882no—
150C22×C4—3882no—
151C22×C4—1941no—
152C22×C4—1941no—
153C22×C4—1942no—
154C22×C4—1944no—
155C22×C4C2×#8760411yescharacteristic
156C2×C8—3882no—
157C2×C8—1942no—
158C4⋊C4—21no—
159C4⋊C4—971no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C4⋊C4—1941no—
163C4⋊C4—1941no—
164C4⋊C4C2×#8760411yescharacteristic, complemented
165C22⋊C4—21no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C22⋊C4—1941no—
169C23—971no—
170C23—972no—
171C23—972no—
172C23—971no—
173C23—1942no—
174C23—1942no—
175C23C22⋊#8760411yescharacteristic
176Q8—41no—
177Q8—3881no—
178Q8—3882no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—971no—
183C2×C4—971no—
184C2×C4—3881no—
185C2×C4—3881no—
186C2×C4—3882no—
187C2×C4—3882no—
188C2×C4—3882no—
189C2×C4—3888no—
190C2×C4—3882no—
191C2×C4—3881no—
192C2×C4—1942no—
193C2×C4—1942no—
194C2×C4—1944no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4C22⋊#8760411yescharacteristic
198C2×C4C22×#8760411yescharacteristic
199C8—3882no—
200C22—21no—
201C22—21no—
202C22—972no—
203C22—972no—
204C22—972no—
205C22—971no—
206C22—971no—
207C22—1944no—
208C22—1944no—
209C22—1941no—
210C22—1941no—
211C22#1569087312yes—
212C22C2×C22⋊#8760411yescharacteristic
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—3884no—
218C4—3881no—
219C4—3881no—
220C4—1941no—
221C4—1941no—
222C2—22no—
223C2—974no—
224C2—1941no—
225C2—1941no—
226C2D97⋊#9911yescharacteristic
227C2C2×#1569087311yescharacteristic
228C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC22⋊Q8#87604
hypocenterC97C2×#328
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20261589C2semidirect
4. #20261588C2semidirect
12. C2×Q8×D97C4semidirect
15. #15690789C4semidirect
20. C4⋊C4×D97C4semidirect
22. C22⋊Q8×C97C4semidirect
31. C2×Dic194C2×C4semidirect
42. C4⋊Dic97C2×C4semidirect
49. C2×Q8×C97C2×C4semidirect
52. C4⋊C4×C97C2×C4semidirect
113. D97#328semidirect
115. C97C2×#328semidirect
130. C22⋊Q8#87604semidirect
142. C2×Q8C2×#87604semidirect
164. C4⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
C2×(C2×#87604)⋊C42
#202615892
#202615882
C22⋊Q8×D972
C2×#32897