Finite Group Databasebeta

← #487×D97 #27152044

Subgroups — #487×D97 #27152044

Order 12416.

424 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 145 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
2C4⋊Q8×D97C211yesmaximal, characteristic
3#20261087C211yesmaximal, characteristic, complemented
4#20261086C211yesmaximal, characteristic, complemented
5#20261085C211yesmaximal, characteristic, complemented
6#20261084C214yesmaximal, complemented
7#103×D97C214yesmaximal
8C4×C8×D97C211yesmaximal, characteristic
9#487×C97C211yesmaximal, characteristic, complemented
10C2×Q8×D97—21no—
11C2×Q8×D97C2212yes—
12C2×C8×D97C2212yes—
13#15690810C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690809—21no—
16#15690809—21no—
17#15690809—21no—
18#15690809C2212yes—
19#15690808—21no—
20#15690808C2212yes—
21C4⋊C4×D97—21no—
22C4⋊C4×D97—21no—
23C4⋊C4×D97—21no—
24C4⋊C4×D97C2212yes—
25C42×D97C2211yescharacteristic
26#15690735C2214yes—
27#15690734C2214yes—
28#15690733C2214yes—
29C4×#11715895C2211yescharacteristic
30C8×Dic97C2212yes—
31C4⋊Q8×C97C2211yescharacteristic
32C4⋊Q8×C97C2211yescharacteristic
33#103×C97C2214yes—
34C4×C8×C97C2211yescharacteristic
35C2×Dic194—21no—
36C2×Dic194—22no—
37C2×Dic194—22no—
38C2×Dic194C2312yes—
39Q8×D97—28no—
40Q8×D97—44no—
41C2×C4×D97—22no—
42C2×C4×D97—22no—
43C2×C4×D97D411yescharacteristic
44C2×C4×D97D411yescharacteristic
45C2×C4×D97C2311yescharacteristic
46C2×#11715895C2312yes—
47C8×D97—24no—
48C4⋊Dic97—21no—
49C4⋊Dic97—21no—
50C4⋊Dic97—21no—
51C4⋊Dic97C2312yes—
52#13095490—22no—
53#13095490—22no—
54#13095490—22no—
55#13095490—22no—
56C4×Dic97D411yescharacteristic
57C4×Dic97D411yescharacteristic
58C4×Dic97C2311yescharacteristic
59C2×Q8×C97—21no—
60C2×Q8×C97C2312yes—
61C2×C8×C97C2312yes—
62C4⋊C4×C97—21no—
63C4⋊C4×C97—21no—
64C4⋊C4×C97—21no—
65C4⋊C4×C97C2312yes—
66C42×C97C2311yescharacteristic
67C22×D97C2×D411yescharacteristic
68Q8×C97—24no—
69Q8×C97—42no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2411yescharacteristic
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97C2×D411yescharacteristic
78C2×Dic97C2×D411yescharacteristic
79C2×Dic97C2×D411yescharacteristic
80C4×D97—44no—
81C4×D97—44no—
82C4×D97SD1614yes—
83C4×D97Q1614yes—
84C4×D97#4114yes—
85Dic194—24no—
86Dic194—42no—
87Dic194—44no—
88Dic194—44no—
89C8×C97—22no—
90#11715895—22no—
91C22×C97C22×D411yescharacteristic
92C2×D97#12412yes—
93C2×D97C2×SD1612yes—
94C2×D97C2×Q1612yes—
95C4×C97—42no—
96C4×C97—42no—
97C4×C97C2×SD1612yes—
98C4×C97C2×Q1612yes—
99C4×C97C2×#4112yes—
100Dic97—42no—
101Dic97—42no—
102Dic97#12412yes—
103Dic97C2×SD1612yes—
104Dic97C2×Q1612yes—
105C2×C97C2×#12411yescharacteristic
106C2×C97C22×SD1611yescharacteristic
107C2×C97C22×Q1611yescharacteristic
108D97#48714yesdirect product factor
109C2×#487—971nomaximal, Sylow
110C97C2×#48711yescharacteristic, complemented, Sylow
111C2×C4⋊Q8—971no—
112C2×C4⋊Q8—971no—
113#487—971no—
114#487—971no—
115#487—974no—
116#487—971no—
117#487D9711yescharacteristic, direct product factor
118C2×#103—974no—
119C2×C4×C8—971no—
120C22×Q8—972no—
121C22×Q8—1941no—
122C22×C8—972no—
123C4⋊Q8—972no—
124C4⋊Q8—972no—
125C4⋊Q8—971no—
126C4⋊Q8—971no—
127C4⋊Q8—1941no—
128C4⋊Q8—1941no—
129C4⋊Q8—1941no—
130C4⋊Q8—1941no—
131C4⋊Q8C2×D9711yescharacteristic
132C4⋊Q8C2×D9711yescharacteristic
133C2×C4⋊C4—972no—
134C2×C4⋊C4—1941no—
135C2×C4⋊C4—1941no—
136C2×C4⋊C4—1941no—
137C2×C42—971no—
138#103—974no—
139#103—974no—
140#103—974no—
141#103C2×D9714yes—
142C4×C8—972no—
143C4×C8—971no—
144C4×C8C2×D9711yescharacteristic
145C2×Q8—21no—
146C2×Q8—972no—
147C2×Q8—3884no—
148C2×Q8—1942no—
149C2×Q8—1942no—
150C2×Q8—1948no—
151C2×Q8—1941no—
152C2×Q8C22×D9712yes—
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—1942no—
157C22×C4—1942no—
158C2×C8—972no—
159C2×C8—1944no—
160C2×C8C22×D9712yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—972no—
165C4⋊C4—1942no—
166C4⋊C4—1942no—
167C4⋊C4—1942no—
168C4⋊C4—1942no—
169C4⋊C4—1941no—
170C4⋊C4—1941no—
171C4⋊C4—1941no—
172C4⋊C4C22×D9712yes—
173C42—971no—
174C42—971no—
175C42—971no—
176C42C22×D9711yescharacteristic
177C23—971no—
178Q8—24no—
179Q8—42no—
180Q8—3884no—
181Q8—3884no—
182Q8—3882no—
183Q8—1944no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—971no—
187C2×C4—971no—
188C2×C4—971no—
189C2×C4—974no—
190C2×C4—974no—
191C2×C4—974no—
192C2×C4—3884no—
193C2×C4—3884no—
194C2×C4—1942no—
195C2×C4—1942no—
196C2×C4D4×D9711yescharacteristic
197C2×C4D4×D9711yescharacteristic
198C2×C4C23×D9711yescharacteristic
199C8—22no—
200C8—1942no—
201C22—972no—
202C22—972no—
203C22—972no—
204C22C2×D4×D9711yescharacteristic
205C4—42no—
206C4—42no—
207C4—972no—
208C4—972no—
209C4—972no—
210C4—3882no—
211C4—3882no—
212C4SD16×D9712yes—
213C4Q16×D9712yes—
214C4#41×D9712yes—
215C2—974no—
216C2#124×D9711yescharacteristic
217C2C2×SD16×D9711yescharacteristic
218C2C2×Q16×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#487×C97C2
socleC22×C97C22×D4
prime radicalC22×D97C2×D4
hypercenter#487D97
hypocenterC97C2×#487
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #20261087C2semidirect
4. #20261086C2semidirect
5. #20261085C2semidirect
6. #20261084C2semidirect
9. #487×C97C2semidirect
108. D97#487direct (N × H)
110. C97C2×#487semidirect
117. #487D97direct (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
#202610872
#202610862
#202610852
#202610842
#103×D972
C4×C8×D972
#487×C972
C2×#48797