Finite Group Databasebeta

← #49129704

Subgroups — #49129704

Order 36224.

764 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C22×#17985392C211yesmaximal, characteristic
2C2×M16×D283C212yesmaximal, complemented
3#31826430C218yesmaximal, complemented
4#31826429C211yesmaximal, characteristic, complemented
5#31826428C212yesmaximal, complemented
6#407×C283C211yesmaximal, characteristic, complemented
7C2×#17985392—28no—
8C2×#17985392C414yes—
9C2×#17985392C2214yes—
10C22×Dic566C411yescharacteristic
11C22×D1132C411yescharacteristic
12C22×C22⋊D283C411yescharacteristic
13C22×C22⋊D283C411yescharacteristic
14C22×C4×D283C2211yescharacteristic
15C22×C4×D283C2211yescharacteristic
16C2×#17985376C2212yes—
17C2×#17985375C2214yescomplemented
18M16×D283—28no—
19C2×C8×D283C2214yescomplemented
20C22⋊(C283⋊C8)C2214yescomplemented
21(C2×D283)⋊C8C2218yescomplemented
22C23×C4×C283C2211yescharacteristic
23C2×M16×C283C2212yes—
24C22⋊C8×C283C2214yescomplemented
25C23×D283C2×C411yescharacteristic
26C23×D283C2×C411yescharacteristic
27#17985392—216no—
28#17985392—28no—
29#17985392—28no—
30C2×Dic566—24no—
31C2×Dic566C2×C414yes—
32C2×D1132—24no—
33C2×D1132C2×C414yes—
34C2×C22⋊D283—24no—
35C2×C22⋊D283—24no—
36C2×C22⋊D283C2×C414yes—
37C2×C22⋊D283C2×C414yes—
38C22×Dic283C2×C411yescharacteristic
39C22×Dic283C2×C411yescharacteristic
40C2×C4×D283—24no—
41C2×C4×D283C2×C412yes—
42C2×C4×D283C2×C412yes—
43C2×C4×D283D418yes—
44C2×C4×D283C2312yes—
45C2×C4×D283C2312yes—
46#17985376—24no—
47#17985375—28no—
48C2×C283⋊C8C2314yes—
49C8×D283—28no—
50C24×C283C2×C411yescharacteristic
51C22×C4×C283—24no—
52C22×C4×C283C2×C411yescharacteristic
53C22×C4×C283C2×C412yes—
54C22×C4×C283C2312yes—
55C22×C4×C283C2311yescharacteristic
56M16×C283—24no—
57C2×C8×C283C2314yes—
58C22×D283—22no—
59C22×D283—42no—
60C22×D283—42no—
61C22×D283C22⋊C414yes—
62C22×D283C22×C412yes—
63C22×D283C22×C412yes—
64Dic566—28no—
65D1132—28no—
66C22⋊D283—28no—
67C22⋊D283—28no—
68C2×Dic283—22no—
69C2×Dic283C22⋊C414yes—
70C2×Dic283C22×C412yes—
71C2×Dic283C22×C412yes—
72C4×D283—24no—
73C4×D283—24no—
74C4×D283C22⋊C418yes—
75C4×D283C2×D414yes—
76C4×D283C2×D414yes—
77C283⋊C8—24no—
78C23×C283—22no—
79C23×C283—42no—
80C23×C283C22×C412yes—
81C23×C283C22×C411yescharacteristic
82C2×C4×C283—22no—
83C2×C4×C283—24no—
84C2×C4×C283—24no—
85C2×C4×C283C22×C412yes—
86C2×C4×C283C22×C411yescharacteristic
87C2×C4×C283C2×D414yes—
88C2×C4×C283C2411yescharacteristic
89C8×C283—24no—
90C22×C283—22no—
91C22×C283—22no—
92C22×C283—24no—
93C22×C283—44no—
94C22×C283C2×C22⋊C412yes—
95C22×C283C23×C411yescharacteristic
96C2×D283—24no—
97C2×D283—22no—
98C2×D283—24no—
99C2×D283—44no—
100C2×D283—44no—
101C2×D283C2×C22⋊C414yes—
102C4×C283—22no—
103C4×C283C2×C22⋊C412yes—
104C4×C283C22×D412yes—
105Dic283—22no—
106Dic283C2×C22⋊C414yes—
107C2×C283—22no—
108C2×C283—42no—
109C2×C283C22×C22⋊C411yescharacteristic
110C2×C283#56812yes—
111D283—24no—
112D283—42no—
113C283#287911yescharacteristic, complemented, Sylow
114#2879—2831nomaximal, Sylow
115C22×#41—2831no—
116C22×M16—2832no—
117#408—2838no—
118#407—2831no—
119#407—2832no—
120#407D28311yescharacteristic, complemented
121C2×#41—2834no—
122C2×#41—2834no—
123C2×#41—5668no—
124C22×Q8—2831no—
125C22×D4—2831no—
126C22×D4—2831no—
127C22×D4—2831no—
128C23×C4—2831no—
129C23×C4—2831no—
130C23×C4C2×D28311yescharacteristic
131C2×M16—2832no—
132C2×M16—2834no—
133C2×M16—5668no—
134C2×M16C2×D28312yes—
135C22×C8—2834no—
136C22⋊C8—2834no—
137C22⋊C8—2838no—
138C22⋊C8C2×D28314yescomplemented
139C24—2831no—
140C24—2831no—
141C24C4×D28311yescharacteristic
142#41—56616no—
143#41—5668no—
144#41—5668no—
145C2×Q8—2834no—
146C2×Q8—5664no—
147C2×D4—2834no—
148C2×D4—2834no—
149C2×D4—2834no—
150C2×D4—5664no—
151C2×D4—5664no—
152C2×D4—5664no—
153C22×C4—24no—
154C22×C4—2831no—
155C22×C4—2832no—
156C22×C4—2838no—
157C22×C4—2832no—
158C22×C4—2831no—
159C22×C4—2832no—
160C22×C4—2832no—
161C22×C4—5664no—
162C22×C4C4×D28311yescharacteristic
163C22×C4C4×D28312yes—
164C22×C4C22×D28312yes—
165C22×C4C22×D28311yescharacteristic
166M16—24no—
167M16—5668no—
168M16—5664no—
169C2×C8—2834no—
170C2×C8—5668no—
171C2×C8C22×D28314yes—
172C23—22no—
173C23—42no—
174C23—2832no—
175C23—2834no—
176C23—2832no—
177C23—5662no—
178C23—11322no—
179C23—11322no—
180C23C2×C4×D28312yes—
181C23C2×C4×D28311yescharacteristic
182Q8—5668no—
183D4—5668no—
184D4—5668no—
185D4—5668no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—2834no—
190C2×C4—2832no—
191C2×C4—2838no—
192C2×C4—2834no—
193C2×C4—2834no—
194C2×C4—2832no—
195C2×C4—5662no—
196C2×C4—5664no—
197C2×C4—5664no—
198C2×C4C2×C4×D28312yes—
199C2×C4C2×C4×D28311yescharacteristic
200C2×C4D4×D28314yes—
201C2×C4C23×D28311yescharacteristic
202C8—24no—
203C8—5664no—
204C22—22no—
205C22—24no—
206C22—22no—
207C22—44no—
208C22—2834no—
209C22—5664no—
210C22—5664no—
211C22—5662no—
212C22—11324no—
213C22—11324no—
214C22C22⋊C4×D28312yes—
215C22C22×C4×D28311yescharacteristic
216C4—22no—
217C4—2834no—
218C4—5662no—
219C4C22⋊C4×D28312yes—
220C4C2×D4×D28312yes—
221C2—22no—
222C2—42no—
223C2—5664no—
224C2—11322no—
225C2C2×C22⋊C4×D28311yescharacteristic
226C2#3182723112yes—
subgroupG/H
center, FrattiniC2×C4C23×D283
commutator, socleC22×C283C23×C4
Fitting#407×C283C2
prime radicalC22×#17985392C2
cosocleC2×C4×C283C24
hypercenter#407D283
hypocenterC283#2879
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C2×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×M16×D283C2semidirect
3. #31826430C2semidirect
4. #31826429C2semidirect
5. #31826428C2semidirect
6. #407×C283C2semidirect
17. C2×#17985375C22semidirect
19. C2×C8×D283C22semidirect
20. C22⋊(C283⋊C8)C22semidirect
21. (C2×D283)⋊C8C22semidirect
24. C22⋊C8×C283C22semidirect
113. C283#2879semidirect
120. #407D283semidirect
138. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C22×#179853922
C2×M16×D2832
#318264302
#318264292
#318264282
#407×C2832
#2879283