Finite Group Databasebeta

← #27140279

Subgroups — #27140279

Order 12416.

286 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 81 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#20260699C211yesmaximal, characteristic, complemented
2#20260595C211yesmaximal, characteristic, complemented
3#20260472C212yesmaximal, complemented
4#20260469C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C2×#13095495—21no—
7C2×C4⋊Dic97—21no—
8C2×C22⋊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—
24#13095495—42no—
25C2×#11715895—22no—
26C4⋊Dic97—42no—
27C22⋊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—21no—
54C23×C97—21no—
55C23×C97—21no—
56C23×C97—41no—
57C23×C97—41no—
58C23×C97C22⋊C411yescharacteristic
59C23×C97C22⋊C411yescharacteristic
60C23×C97C2×D411yescharacteristic
61Q8×C97—41no—
62D4×C97—41no—
63D4×C97—42no—
64D4×C97—42no—
65D4×C97—41no—
66D4×C97—41no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97—42no—
77C2×C4×C97C22⋊C411yescharacteristic
78C2×C4×C97C22⋊C411yescharacteristic
79C2×C4×C97C22×C411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×Dic97—42no—
82C2×Dic97—42no—
83C2×Dic97—42no—
84#11715895—42no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97C2×C22⋊C411yescharacteristic
99C22×C97AGL(1,ℤ/8)11yescharacteristic
100C22×C97#13711yescharacteristic
101C4×C97—21no—
102C4×C97—21no—
103C4×C97—21no—
104C4×C97—41no—
105C4×C97—41no—
106C4×C97—41no—
107C4×C97#9912yes—
108Dic97—82no—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97—41no—
112C2×C97—41no—
113C2×C97—81no—
114C2×C97C2×#9911yescharacteristic
115C2×C97#41711yescharacteristic
116C2×C97#42111yescharacteristic
117#1513—971nomaximal, Sylow
118C97#151311yescharacteristic, complemented, Sylow
119#525D9711yescharacteristic
120#407—971no—
121#380—971no—
122#328—972no—
123#327—972no—
124C2×#41Dic9711yescharacteristic, complemented
125C23×C4C2×D9711yescharacteristic
126C2×M16—1941no—
127#123—21no—
128C22⋊Q8Dic9711yescharacteristic, complemented
129C22⋊Q8C2×D9711yescharacteristic
130#121Dic9711yescharacteristic, complemented
131#121C2×D9711yescharacteristic
132#120—21no—
133C4×D4—21no—
134C4×D4—21no—
135#117Dic9711yescharacteristic, complemented
136C2×C4⋊C4—1941no—
137C2×C22⋊C4—1941no—
138C22⋊C8—972no—
139C22⋊C8—1941no—
140#95—972no—
141C24C22⋊D9711yescharacteristic
142#41—42no—
143C2×Q8C2×Dic9711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C2×Dic9711yescharacteristic
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—1942no—
151C22×C4C2×Dic9711yescharacteristic
152C22×C4C2×Dic9711yescharacteristic
153C22×C4C22⋊D9711yescharacteristic
154C22×C4C22⋊D9711yescharacteristic
155C22×C4C22⋊D9711yescharacteristic
156C22×C4C22×D9711yescharacteristic
157M16—3882no—
158C2×C8—1942no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—3882no—
162C4⋊C4C2×Dic9711yescharacteristic
163C4⋊C4C2×Dic9711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—3882no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23C22⋊Dic9711yescharacteristic
177C23C22⋊Dic9711yescharacteristic
178C23C2×C22⋊D9711yescharacteristic
179Q8—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183D4—41no—
184D4—41no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—3882no—
196C2×C4—3882no—
197C2×C4—3882no—
198C2×C4C22⋊Dic9711yescharacteristic
199C2×C4C22⋊Dic9711yescharacteristic
200C2×C4C22×Dic9711yescharacteristic
201C2×C4C2×C22⋊D9711yescharacteristic
202C8—3882no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22C2×C22⋊Dic9711yescharacteristic
217C22#1569083511yescharacteristic
218C22#1569084011yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—7762no—
226C4#1569072512yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—81no—
232C2C2×#1569072511yescharacteristic
233C2#2026073611yescharacteristic
234C2#2026075211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#525×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#525D97
hypocenterC97#1513
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20260699C2semidirect
2. #20260595C2semidirect
3. #20260472C2semidirect
4. #20260469C2semidirect
12. C2×#41×C97C4semidirect
15. C22⋊Q8×C97C4semidirect
17. #121×C97C4semidirect
22. #117×C97C4semidirect
118. C97#1513semidirect
124. C2×#41Dic97semidirect
128. C22⋊Q8Dic97semidirect
130. #121Dic97semidirect
135. #117Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606992
#202605952
#202604722
#202604692
#525×C972
#151397