Finite Group Databasebeta

← #27142914

Subgroups — #27142914

Order 12416.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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#20261361C211yesmaximal, characteristic, complemented
2C2×#15690724C211yesmaximal, characteristic
3C2×#15690724C211yesmaximal, characteristic
4C2×#15690724C211yesmaximal, characteristic
5#20260502C211yesmaximal, characteristic, complemented
6#20260501C211yesmaximal, characteristic, complemented
7#342×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2211yescharacteristic
9#15690789—22no—
10#15690787—22no—
11#15690787—22no—
12C2×C22⋊Dic97C2211yescharacteristic
13C2×C22⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97C2211yescharacteristic
15#15690724—22no—
16#15690724—22no—
17#15690724—22no—
18C2×C22⋊C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97C2211yescharacteristic
20C2×C22⋊C4×C97C2211yescharacteristic
21C23×D97Q811yescharacteristic, complemented
22C2×Dic194—22no—
23C2×C22⋊D97—21no—
24C2×C22⋊D97—21no—
25C2×C22⋊D97—21no—
26C2×C22⋊D97—21no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C22×Dic97—21no—
30C22×Dic97—21no—
31C22×Dic97—21no—
32C22×Dic97Q811yescharacteristic, complemented
33C4⋊Dic97—22no—
34#13095490—24no—
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97D412yes—
40C22⋊Dic97D412yes—
41C24×C97C2311yescharacteristic
42C22×C4×C97—21no—
43C22×C4×C97—21no—
44C22×C4×C97—21no—
45C22⋊C4×C97—21no—
46C22⋊C4×C97—21no—
47C22⋊C4×C97—21no—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97D412yes—
51C23×C97—81no—
52C23×C97C2×D411yescharacteristic
53C23×C97C2×D411yescharacteristic
54C23×C97C2×D411yescharacteristic
55C23×C97C2×Q811yescharacteristic
56C23×C97#4111yescharacteristic
57C23×C97#4111yescharacteristic
58C23×C97#4111yescharacteristic
59C22×D97—21no—
60C22×D97—21no—
61C22×D97—21no—
62C22×D97—81no—
63C2×C4×C97—22no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—81no—
67C2×C4×C97—81no—
68C2×C4×C97—81no—
69C22⋊D97—41no—
70C22⋊D97—41no—
71C22⋊D97—41no—
72C22⋊D97—41no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—41no—
79C2×Dic97—81no—
80C2×Dic97—81no—
81C2×Dic97—81no—
82Dic194—82no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—41no—
89C22×C97—41no—
90C22×C97#12011yescharacteristic
91C22×C97C22⋊Q811yescharacteristic
92C22×C97C22⋊Q811yescharacteristic
93C22×C97C22⋊Q811yescharacteristic
94C22×C97#12411yescharacteristic
95C22×C97#12411yescharacteristic
96C22×C97#12411yescharacteristic
97C2×D97—41no—
98C2×D97—81no—
99C2×D97—81no—
100C2×D97—81no—
101C4×C97—42no—
102C4×C97—81no—
103C4×C97—81no—
104Dic97—42no—
105Dic97—42no—
106Dic97—41no—
107Dic97—81no—
108C2×C97—21no—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97—21no—
112C2×C97—21no—
113C2×C97—21no—
114C2×C97—21no—
115C2×C97#39311yescharacteristic
116D97—81no—
117#2032—971nomaximal, Sylow
118C97#203211yescharacteristic, complemented, Sylow
119#543—971no—
120C2×#99—971no—
121C2×#99—971no—
122C2×#99—971no—
123#342—971no—
124#342—971no—
125#342D9711yescharacteristic, complemented
126C22×D4—971no—
127C22⋊Q8—1942no—
128C22⋊Q8—1942no—
129C22⋊Q8—1942no—
130C2×C22⋊C4—971no—
131C2×C22⋊C4—971no—
132C2×C22⋊C4—971no—
133C2×C22⋊C4C2×D9711yescharacteristic
134C2×C22⋊C4C2×D9711yescharacteristic
135C2×C22⋊C4C2×D9711yescharacteristic
136#99—1942no—
137#99—1942no—
138#99—1942no—
139C24—971no—
140C24C22×D9711yescharacteristic
141C2×Q8—1942no—
142C2×D4—1941no—
143C2×D4—1941no—
144C2×D4—1941no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4—1941no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—971no—
152C22×C4—1941no—
153C22×C4—1941no—
154C22×C4—1941no—
155C4⋊C4—1944no—
156C4⋊C4—1942no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—972no—
163C22⋊C4—972no—
164C22⋊C4—1941no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C22⋊C4D38812yes—
169C23—81no—
170C23—7761no—
171C23—1941no—
172C23—1941no—
173C23—1941no—
174C23D4×D9711yescharacteristic
175C23D4×D9711yescharacteristic
176C23C2×D38811yescharacteristic
177C23Q8×D9711yescharacteristic
178C23#1309551011yescharacteristic
179C23#1309551111yescharacteristic
180C23#1309551111yescharacteristic
181Q8—7762no—
182D4—3881no—
183D4—3881no—
184D4—3881no—
185D4—3881no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—7761no—
193C2×C4—7761no—
194C2×C4—7761no—
195C2×C4—3881no—
196C2×C4—1942no—
197C2×C4—1942no—
198C2×C4—1941no—
199C2×C4—1941no—
200C2×C4—1941no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—7761no—
209C22—7761no—
210C22—7761no—
211C22—3881no—
212C22#1569078011yescharacteristic
213C22#1569078811yescharacteristic
214C22#1569078811yescharacteristic
215C22#1569079011yescharacteristic
216C22#1569079611yescharacteristic
217C22#1569079611yescharacteristic
218C22#1569079811yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—7761no—
223C4—3882no—
224C4—3882no—
225C4—3881no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—7761no—
234C2#2026064811yescharacteristic
subgroupG/H
centerC2#20260648
commutator, cosocleC24×C97C23
FrattiniC24C22×D97
Fitting#342×C97C2
socleC2×C97#393
prime radicalC22×C22⋊D97C22
hypercenter#342D97
hypocenterC97#2032
LCSG ▷ C24×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #20261361C2semidirect
5. #20260502C2semidirect
6. #20260501C2semidirect
7. #342×C97C2semidirect
21. C23×D97Q8semidirect
32. C22×Dic97Q8semidirect
118. C97#2032semidirect
125. #342D97semidirect

Maximal subgroups

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

subgroupindex
#202613612
C2×#156907242
#202605022
#202605012
#342×C972
#203297