Finite Group Databasebeta

← #27140169

Subgroups — #27140169

Order 12416.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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×#13095512C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic, complemented
3#20260786C211yesmaximal, characteristic, complemented
4Q8⋊#11715895C211yesmaximal, characteristic, complemented
5#20260464C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095512C411yescharacteristic
9Q8×Dic97C411yescharacteristic
10C4×D388—21no—
11C4×D388C2211yescharacteristic
12#15690772—21no—
13#15690772C411yescharacteristic
14#15690768—21no—
15#15690768C411yescharacteristic
16C42×D97—21no—
17C42×D97C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690722—21no—
22#15690720—21no—
23C4×#11715895C2211yescharacteristic
24C4×Q8×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27#13095512—22no—
28#13095512—22no—
29C2×D388—21no—
30C2×D388C2×C411yescharacteristic
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×#11715895—21no—
38C2×#11715895—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97C2×C411yescharacteristic
41#13095488—21no—
42#13095488—21no—
43#13095488—21no—
44C4×Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×Q8×C97C2×C411yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C42×C97—21no—
55C42×C97C2311yescharacteristic
56C22×D97—21no—
57C22×D97C22⋊C411yescharacteristic
58Q8×C97—21no—
59Q8×C97M1612yes—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×Dic97—21no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69D388—24no—
70D388—21no—
71D388M1612yes—
72C4×D97—24no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—21no—
76C4×D97—21no—
77C4×D97—41no—
78C4×D97—42no—
79C4×D97—41no—
80C4×D97—41no—
81C8×C97—41no—
82C8×C97—41no—
83#11715895—41no—
84#11715895—41no—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—21no—
87C2×D97—22no—
88C2×D97—21no—
89C2×D97—21no—
90C2×D97—41no—
91C2×D97—41no—
92C4×C97—22no—
93C4×C97—21no—
94C4×C97—41no—
95C4×C97C2×M1611yescharacteristic
96C4×C97C2×M1611yescharacteristic
97C4×C97#13511yescharacteristic
98C4×C97#13511yescharacteristic
99Dic97—21no—
100Dic97—21no—
101Dic97—21no—
102Dic97—41no—
103C2×C97#40711yescharacteristic
104C2×C97#41611yescharacteristic
105C2×C97#42111yescharacteristic
106D97—41no—
107D97—42no—
108#1488—971nomaximal, Sylow
109C97#148811yescharacteristic, complemented, Sylow
110C4×#41—971no—
111#432—971no—
112#431—971no—
113Q8⋊C8—971no—
114Q8⋊C8D9711yescharacteristic, complemented
115D4⋊C8—971no—
116D4⋊C8—971no—
117C2×#41—971no—
118C4×Q8—971no—
119C4×Q8C2×D9711yescharacteristic
120C4×D4—971no—
121C4×D4—971no—
122C4×D4—1941no—
123C4×D4—1941no—
124#117—971no—
125#117—1941no—
126C2×C42—971no—
127C2×C42—1941no—
128C4⋊C8—971no—
129C4⋊C8—1941no—
130C4⋊C8C2×D9711yescharacteristic
131C22⋊C8—1941no—
132C22⋊C8—1941no—
133#97—1941no—
134C4×C8—971no—
135C4×C8C2×D9711yescharacteristic
136#41—1942no—
137#41—1942no—
138C2×Q8C4×D9711yescharacteristic
139C2×D4—971no—
140C2×D4—1941no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—1941no—
145C22×C4—1941no—
146C22×C4—1941no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—1941no—
150C2×C8—1941no—
151C4⋊C4—21no—
152C4⋊C4—971no—
153C4⋊C4—1941no—
154C4⋊C4C4×D9711yescharacteristic
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C42—21no—
159C42—971no—
160C42—971no—
161C42—971no—
162C42—1941no—
163C42—1941no—
164C42C22×D9711yescharacteristic
165C23—971no—
166C23—1941no—
167Q8—21no—
168Q8#1309549412yes—
169D4—972no—
170D4—1944no—
171D4—1941no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—971no—
175C2×C4—971no—
176C2×C4—971no—
177C2×C4—3881no—
178C2×C4—3882no—
179C2×C4—3881no—
180C2×C4—3881no—
181C2×C4—1944no—
182C2×C4—1941no—
183C2×C4—1941no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4C2×C4×D9711yescharacteristic
188C2×C4D4×D9711yescharacteristic
189C2×C4D4×D9711yescharacteristic
190C8—41no—
191C8—41no—
192C8—3881no—
193C8—3881no—
194C22—3881no—
195C22—3881no—
196C22—1942no—
197C22—1941no—
198C22—1941no—
199C22—1941no—
200C22C22⋊C4×D9711yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—3881no—
205C4—1941no—
206C4—1941no—
207C4—1941no—
208C4M16×D9711yescharacteristic
209C4C2×#1309549411yescharacteristic
210C4#1569083111yescharacteristic
211C4#1569083311yescharacteristic
212C2—3882no—
213C2—3881no—
214C2#2026069711yescharacteristic
215C2#2026073011yescharacteristic
216C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095512C4
cosocleC42×C97C23
hypercenterQ8⋊C8D97
hypocenterC97#1488
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260791C2semidirect
3. #20260786C2semidirect
4. Q8⋊#11715895C2semidirect
5. #20260464C2semidirect
6. #20260462C2semidirect
7. Q8⋊C8×C97C2semidirect
109. C97#1488semidirect
114. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
C4×#130955122
#202607912
#202607862
Q8⋊#117158952
#202604642
#202604622
Q8⋊C8×C972
#148897