Finite Group Databasebeta

← #27142544

Subgroups — #27142544

Order 12416.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#15690799C211yesmaximal, characteristic
2C2×#15690734C211yesmaximal, characteristic, complemented
3C2×#15690732C211yesmaximal, characteristic
4C2×#15690730C211yesmaximal, characteristic
5C2×C8×Dic97C211yesmaximal, characteristic, complemented
6#20260600C211yesmaximal, characteristic, complemented
7C2×#103×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690799—22no—
11#15690799C414yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095488—22no—
14C2×C4×Dic97C2211yescharacteristic
15#15690734—22no—
16#15690732—22no—
17#15690730—22no—
18C8×Dic97—22no—
19#15690717—22no—
20C22×Q8×C97C2211yescharacteristic
21C22×C8×C97C2211yescharacteristic
22C2×C4⋊C4×C97C2211yescharacteristic
23#103×C97—22no—
24C23×D97—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388C2×C412yes—
31C22×Dic97—21no—
32C22×Dic97D412yes—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895D412yes—
36C4⋊Dic97—21no—
37C4⋊Dic97—21no—
38#13095488—28no—
39C4×Dic97C2×C412yes—
40C4×Dic97D412yes—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97C2×C412yes—
47C22×C4×C97—21no—
48C22×C4×C97—21no—
49C22×C4×C97C2311yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C2×C8×C97D412yes—
53C4⋊C4×C97—21no—
54C4⋊C4×C97—21no—
55C23×C97C2×D411yescharacteristic
56C22×D97—22no—
57C22×D97—41no—
58C22×D97—41no—
59C22×D97—42no—
60C22×D97—42no—
61Q8×C97—22no—
62Q8×C97—22no—
63Q8×C97—41no—
64Q8×C97—41no—
65C2×C4×C97—22no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—42no—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×C4×C97#4111yescharacteristic
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—41no—
79C2×Dic97—42no—
80C2×Dic97C22⋊C414yes—
81D388—22no—
82D388—22no—
83D388—41no—
84D388—41no—
85C8×C97—22no—
86#11715895—22no—
87C22×C97C2×C22⋊C411yescharacteristic
88C22×C97#12411yescharacteristic
89C22×C97#12711yescharacteristic
90C22×C97#13512yes—
91C22×C97#13512yes—
92C2×D97—42no—
93C2×D97—44no—
94C2×D97—82no—
95C2×D97—81no—
96C2×D97—81no—
97C4×C97—42no—
98C4×C97—81no—
99C4×C97C4×D411yescharacteristic
100C4×C97C4×D411yescharacteristic
101C4×C97#12111yescharacteristic
102C4×C97#12111yescharacteristic
103Dic97—24no—
104Dic97—81no—
105C2×C97#39011yescharacteristic
106C2×C97#41612yes—
107C2×C97#48812yes—
108C2×C97#49511yescharacteristic
109C2×C97#49511yescharacteristic
110D97—82no—
111#1958—971nomaximal, Sylow
112C97#195811yescharacteristic, complemented, Sylow
113C2×#124—971no—
114C2×#103—971no—
115C2×#103D9711yescharacteristic, complemented
116C2×#102—971no—
117C2×#102—971no—
118C2×C4×C8—971no—
119#382—971no—
120C22×Q8C2×D9711yescharacteristic
121C22×D4—971no—
122C22×C8—971no—
123C22×C8C2×D9711yescharacteristic
124#124—974no—
125#124—1942no—
126C2×C4⋊C4—971no—
127C2×C4⋊C4C2×D9711yescharacteristic
128C2×C22⋊C4—1942no—
129C2×C42—971no—
130#103—22no—
131#103—1942no—
132#102—1942no—
133#102—1942no—
134C4×C8—1942no—
135#95—1942no—
136C24—1941no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C4×D9712yes—
143C2×D4—972no—
144C2×D4—1941no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4—1941no—
148C2×D4—1941no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—972no—
152C22×C4—1941no—
153C22×C4C22×D9711yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—972no—
157C2×C8—1941no—
158C2×C8—1941no—
159C2×C8C22⋊D9712yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—1941no—
163C4⋊C4—1941no—
164C22⋊C4—1948no—
165C42—972no—
166C42—972no—
167C23—3881no—
168C23—3881no—
169C23—3882no—
170C23—3882no—
171C23—1942no—
172C23D4×D9711yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177D4—3881no—
178D4—3881no—
179D4—1942no—
180D4—1942no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—974no—
187C2×C4—3881no—
188C2×C4—3882no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4D4×D9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C2×C4#1309551011yescharacteristic
196C2×C4#1309551111yescharacteristic
197C8—22no—
198C8—1942no—
199C22—7761no—
200C22—7762no—
201C22—7761no—
202C22—3882no—
203C22—3884no—
204C22C22⋊C4×D9711yescharacteristic
205C22#1569079611yescharacteristic
206C22#1569080611yescharacteristic
207C22#1569083112yes—
208C22#1569083312yes—
209C4—42no—
210C4—81no—
211C4—7761no—
212C4—1944no—
213C4#1569077011yescharacteristic
214C4C4×C22⋊D9711yescharacteristic
215C4#1569078211yescharacteristic
216C4#1569078311yescharacteristic
217C2—7762no—
218C2#2026063511yescharacteristic
219C2#2026073012yes—
220C2#2026109112yes—
221C2#2026111611yescharacteristic
222C2#2026112011yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenterC2×#103D97
hypocenterC97#1958
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#15690734C2semidirect
5. C2×C8×Dic97C2semidirect
6. #20260600C2semidirect
7. C2×#103×C97C2semidirect
11. #15690799C4semidirect
112. C97#1958semidirect
115. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
C2×#156907342
C2×#156907322
C2×#156907302
C2×C8×Dic972
#202606002
C2×#103×C972
#195897