Finite Group Databasebeta

← #27140227

Subgroups — #27140227

Order 12416.

304 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 85 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×C22⋊#11715895C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
3#20260472C212yesmaximal, complemented
4#20260469C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C22×#11715895—21no—
7C2×C22⋊Dic97—21no—
8C2×C4×Dic97—21no—
9C22⋊#11715895—21no—
10C22⋊#11715895C2212yes—
11#15690717C2212yes—
12C2×#41×C97C411yescharacteristic, complemented
13C23×C4×C97C2211yescharacteristic
14#123×C97—21no—
15C22⋊Q8×C97C411yescharacteristic, complemented
16C22⋊Q8×C97C2211yescharacteristic
17#121×C97C411yescharacteristic, complemented
18#121×C97C2211yescharacteristic
19#120×C97—21no—
20C4×D4×C97—21no—
21C4×D4×C97—21no—
22#117×C97C411yescharacteristic, complemented
23C22×Dic97—22no—
24C2×#11715895—22no—
25C2×#11715895—42no—
26C22⋊Dic97—24no—
27C4×Dic97—42no—
28C24×C97D411yescharacteristic
29#41×C97—42no—
30C2×Q8×C97C2×C411yescharacteristic
31C2×D4×C97—21no—
32C2×D4×C97—21no—
33C2×D4×C97—21no—
34C2×D4×C97C2×C411yescharacteristic
35C22×C4×C97—21no—
36C22×C4×C97—42no—
37C22×C4×C97C2×C411yescharacteristic
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C22×C4×C97C2311yescharacteristic
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97C2×C411yescharacteristic
46C4⋊C4×C97C2×C411yescharacteristic
47C22⋊C4×C97—21no—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C42×C97—21no—
53C23×C97—22no—
54C23×C97—22no—
55C23×C97—21no—
56C23×C97—41no—
57C23×C97C22⋊C411yescharacteristic
58C23×C97C22⋊C411yescharacteristic
59C23×C97C2×D411yescharacteristic
60Q8×C97—41no—
61D4×C97—41no—
62D4×C97—42no—
63D4×C97—42no—
64D4×C97—42no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97—42no—
75C2×C4×C97C22⋊C411yescharacteristic
76C2×C4×C97C22⋊C411yescharacteristic
77C2×C4×C97C22×C411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×Dic97—24no—
80C2×Dic97—42no—
81C2×Dic97—42no—
82#11715895—42no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—44no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97#10412yes—
95C22×C97C2×C22⋊C411yescharacteristic
96C22×C97#13512yes—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97—41no—
101C4×C97—41no—
102C4×C97—41no—
103C4×C97#9912yes—
104Dic97—44no—
105C2×C97—22no—
106C2×C97—22no—
107C2×C97—41no—
108C2×C97—81no—
109C2×C97C2×#9911yescharacteristic
110C2×C97#41611yescharacteristic
111C2×C97C2×#10411yescharacteristic
112#1501—971nomaximal, Sylow
113C97#150111yescharacteristic, complemented, Sylow
114#525D9711yescharacteristic
115C2×C22⋊C8—971no—
116C4×C22⋊C4—971no—
117#328—972no—
118#327—972no—
119C2×#41Dic9711yescharacteristic, complemented
120C23×C4C2×D9711yescharacteristic
121C22×C8—1941no—
122#123—21no—
123C22⋊Q8Dic9711yescharacteristic, complemented
124C22⋊Q8C2×D9711yescharacteristic
125#121Dic9711yescharacteristic, complemented
126#121C2×D9711yescharacteristic
127#120—21no—
128C4×D4—21no—
129C4×D4—21no—
130#117Dic9711yescharacteristic, complemented
131C2×C22⋊C4—1941no—
132C2×C42—1941no—
133C22⋊C8—972no—
134C22⋊C8—1941no—
135#95—972no—
136C24C22⋊D9711yescharacteristic
137#41—42no—
138C2×Q8C2×Dic9711yescharacteristic
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4C2×Dic9711yescharacteristic
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—1942no—
146C22×C4C2×Dic9711yescharacteristic
147C22×C4C2×Dic9711yescharacteristic
148C22×C4C22⋊D9711yescharacteristic
149C22×C4C22⋊D9711yescharacteristic
150C22×C4C22⋊D9711yescharacteristic
151C22×C4C22×D9711yescharacteristic
152C2×C8—3882no—
153C2×C8—1942no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4C2×Dic9711yescharacteristic
157C4⋊C4C2×Dic9711yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—1944no—
164C42—21no—
165C42—3882no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—41no—
170C23C22⋊Dic9711yescharacteristic
171C23C22⋊Dic9711yescharacteristic
172C23C2×C22⋊D9711yescharacteristic
173Q8—41no—
174D4—41no—
175D4—42no—
176D4—42no—
177D4—42no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—3882no—
189C2×C4—3882no—
190C2×C4—1944no—
191C2×C4C22⋊Dic9711yescharacteristic
192C2×C4C22⋊Dic9711yescharacteristic
193C2×C4C22×Dic9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C8—3882no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22#1569073712yes—
208C22C2×C22⋊Dic9711yescharacteristic
209C22#1569083012yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—3884no—
217C4#1569072512yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—81no—
222C2C2×#1569072511yescharacteristic
223C2#2026073111yescharacteristic
224C2C2×#1569073711yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#525×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#525D97
hypocenterC97#1501
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#11715895C2semidirect
2. C4×C22⋊Dic97C2semidirect
3. #20260472C2semidirect
4. #20260469C2semidirect
12. C2×#41×C97C4semidirect
15. C22⋊Q8×C97C4semidirect
17. #121×C97C4semidirect
22. #117×C97C4semidirect
113. C97#1501semidirect
119. C2×#41Dic97semidirect
123. C22⋊Q8Dic97semidirect
125. #121Dic97semidirect
130. #117Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#117158952
C4×C22⋊Dic972
#202604722
#202604692
#525×C972
#150197