Finite Group Databasebeta

← #27157607

Subgroups — #27157607

Order 12416.

292 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 84 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#20261736C211yesmaximal, characteristic
2C2×#15690873C212yesmaximal
3#20261679C211yesmaximal, characteristic
4#20261608C212yesmaximal
5C4×Q8×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604C2212yes—
8#15690877—21no—
9C4⋊#11715897—21no—
10#15690873—24no—
11(C2×#87604)⋊C4—21no—
12(C2×#87604)⋊C4—21no—
13C2×Q8×D97C2211yescharacteristic
14Q8×Dic97C411yescharacteristic, complemented
15C4×Dic194—21no—
16C4×Dic194C411yescharacteristic, complemented
17#15690775—21no—
18#15690775C411yescharacteristic, complemented
19C4⋊C4×D97—21no—
20C4⋊C4×D97C2211yescharacteristic
21C42×D97—21no—
22C42×D97C2211yescharacteristic
23C4×Q8×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25#13095519—22no—
26#13095519—22no—
27C2×#11715897—22no—
28C4⋊#87604—22no—
29C4⋊#87604—22no—
30C2×Dic194—21no—
31C2×Dic194C2×C411yescharacteristic
32Q8×D97—21no—
33Q8×D97—21no—
34Q8×D97D412yes—
35Q8×D97Q812yes—
36C2×C4×D97—21no—
37C2×C4×D97—21no—
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97C2311yescharacteristic
41C4⋊Dic97—21no—
42C4⋊Dic97C2×C411yescharacteristic
43#13095490—21no—
44#13095490—21no—
45#13095490—21no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C4×Dic97—21no—
49C4×Dic97C2×C411yescharacteristic
50C4×Dic97D411yescharacteristic
51C4×Dic97D411yescharacteristic
52C2×Q8×C97C2×C411yescharacteristic
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C42×C97—21no—
56C42×C97C2×C411yescharacteristic
57C22×D97C2×D411yescharacteristic
58C2×#87604—42no—
59C2×#87604—42no—
60C2×#87604—42no—
61Q8×C97—21no—
62Q8×C97C4⋊C412yes—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22×C411yescharacteristic
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73C4×D97—22no—
74C4×D97—22no—
75C4×D97—22no—
76C4×D97—22no—
77C4×D97—42no—
78C4×D97C2×D411yescharacteristic
79C4×D97C2×Q811yescharacteristic
80C4×D97#4111yescharacteristic
81C4×D97#4111yescharacteristic
82Dic194—21no—
83Dic194—22no—
84Dic194—22no—
85Dic194C4⋊C412yes—
86#11715897—42no—
87C22×C97C2×C22⋊C411yescharacteristic
88C2×D97#12111yescharacteristic
89C2×D97C22⋊Q811yescharacteristic
90C2×D97#13512yes—
91C2×D97#13712yes—
92#87604—82no—
93C4×C97—22no—
94C4×C97—21no—
95C4×C97—21no—
96C4×C97—41no—
97C4×C97C2×C4⋊C411yescharacteristic
98C4×C97#11711yescharacteristic
99Dic97—22no—
100Dic97—21no—
101Dic97—21no—
102Dic97—41no—
103Dic97#12111yescharacteristic
104Dic97C22⋊Q811yescharacteristic
105C2×C97#38011yescharacteristic
106C2×C97#41611yescharacteristic
107C2×C97#41911yescharacteristic
108D97#46412yes—
109D97#47912yes—
110#1805—971nomaximal, Sylow
111C97#180511yescharacteristic, complemented, Sylow
112C2×C4×Q8—971no—
113C2×C4⋊C8—971no—
114C2×#103—972no—
115#382—971no—
116#340—972no—
117C22×Q8—971no—
118C22×C8—972no—
119C4×Q8—971no—
120C4×Q8—971no—
121C4×Q8—971no—
122C4×Q8—1941no—
123C4×Q8—1941no—
124C4×Q8#8760411yescharacteristic, complemented
125C2×C4⋊C4—971no—
126C2×C4⋊C4—972no—
127C2×C4⋊C4—1941no—
128C2×C42—971no—
129C2×C42—1941no—
130C4⋊C8—1941no—
131C4⋊C8—1941no—
132#103—1944no—
133#95—1941no—
134#95—1941no—
135C2×Q8—972no—
136C2×Q8—972no—
137C2×Q8—971no—
138C2×Q8—1941no—
139C2×Q8—1941no—
140C2×Q8—1941no—
141C2×Q8C2×#8760411yescharacteristic
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—1941no—
146C22×C4—1941no—
147C22×C4—1942no—
148C2×C8—1942no—
149C2×C8—1942no—
150C2×C8—1942no—
151C4⋊C4—21no—
152C4⋊C4—971no—
153C4⋊C4—1941no—
154C4⋊C4—1941no—
155C4⋊C4—1941no—
156C4⋊C4—1942no—
157C4⋊C4—1942no—
158C4⋊C4—1941no—
159C4⋊C4C2×#8760411yescharacteristic
160C42—21no—
161C42—971no—
162C42—971no—
163C42—971no—
164C42—1941no—
165C42—1941no—
166C42—1941no—
167C42C2×#8760411yescharacteristic
168C23—971no—
169Q8—21no—
170Q8—972no—
171Q8—1942no—
172Q8—1942no—
173Q8—1941no—
174Q8C4⋊#8760412yes—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—971no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—971no—
183C2×C4—971no—
184C2×C4—3882no—
185C2×C4—3882no—
186C2×C4—3882no—
187C2×C4—3882no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4—1942no—
192C2×C4—1941no—
193C2×C4—1941no—
194C2×C4C22⋊#8760411yescharacteristic
195C2×C4C22⋊#8760411yescharacteristic
196C2×C4C22×#8760411yescharacteristic
197C8—3882no—
198C22—971no—
199C22—971no—
200C22—972no—
201C22—972no—
202C22C2×C22⋊#8760411yescharacteristic
203C4—22no—
204C4—21no—
205C4—21no—
206C4—41no—
207C4—7762no—
208C4—971no—
209C4—971no—
210C4—3881no—
211C4—1942no—
212C4—1941no—
213C4—1941no—
214C4C2×C4⋊#8760411yescharacteristic
215C4#1569088911yescharacteristic
216C2—972no—
217C2—972no—
218C2#2026167711yescharacteristic
219C2#2026172511yescharacteristic
220C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4×Q8#87604
hypocenterC97#1805
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
14. Q8×Dic97C4semidirect
16. C4×Dic194C4semidirect
18. #15690775C4semidirect
23. C4×Q8×C97C4semidirect
111. C97#1805semidirect
124. C4×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617362
C2×#156908732
#202616792
#202616082
C4×Q8×D972
#180597