Finite Group Databasebeta

← #44548680

Subgroups — #44548680

Order 32128.

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

normal subgroups only

This group has 332 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
1C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic
2C22⋊C8×D251C212yesmaximal, complemented
3#30631412C212yesmaximal, complemented
4#30631411C211yesmaximal, characteristic, complemented
5#323×C251C211yesmaximal, characteristic, complemented
6C22×C22⋊D251C411yescharacteristic
7C23×Dic251C411yescharacteristic
8C22×C4×D251C2211yescharacteristic
9C2×C8×D251—22no—
10(C22⋊D251)⋊C4—22no—
11(C22⋊D251)⋊C4—22no—
12C2×Dic251⋊C4C2211yescharacteristic
13C2×(C2×D251)⋊C4C411yescharacteristic
14C2×C4×Dic251C411yescharacteristic
15C22⋊(C251⋊C8)C2212yes—
16(C2×D251)⋊C8—22no—
17C2×C22⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yes—
19C23×D251C2×C411yescharacteristic
20C2×C22⋊D251—22no—
21C2×C22⋊D251—22no—
22C2×C22⋊D251C814yescomplemented
23C22×Dic251—21no—
24C22×Dic251—21no—
25C22×Dic251—42no—
26C22×Dic251C2×C411yescharacteristic
27C22×Dic251C2×C411yescharacteristic
28C22×Dic251C2×C411yescharacteristic
29C2×C4×D251—42no—
30C2×C4×D251D412yes—
31C2×C4×D251D412yes—
32C2×C251⋊C8—22no—
33C8×D251—44no—
34Dic251⋊C4—22no—
35(C2×D251)⋊C4—22no—
36C4×Dic251—21no—
37C4×Dic251—21no—
38C24×C251C2×C411yescharacteristic
39C22×C4×C251C2×C411yescharacteristic
40C22×C4×C251C2311yescharacteristic
41C2×C8×C251—22no—
42C22⋊C4×C251—22no—
43C22×D251—41no—
44C22×D251—41no—
45C22×D251C22⋊C412yes—
46C22×D251C2×C812yes—
47C22×D251M1612yes—
48C22⋊D251—28no—
49C2×Dic251—21no—
50C2×Dic251—21no—
51C2×Dic251—22no—
52C2×Dic251—21no—
53C2×Dic251—42no—
54C2×Dic251—42no—
55C2×Dic251—41no—
56C2×Dic251C22⋊C411yescharacteristic
57C2×Dic251C22⋊C411yescharacteristic
58C2×Dic251C2×C812yescomplemented
59C4×D251—24no—
60C4×D251—42no—
61C251⋊C8—42no—
62C23×C251—21no—
63C23×C251—21no—
64C23×C251—41no—
65C23×C251—41no—
66C23×C251C2×C812yescomplemented
67C23×C251C22×C411yescharacteristic
68C2×C4×C251—21no—
69C2×C4×C251—41no—
70C2×C4×C251—41no—
71C2×C4×C251C2×D412yes—
72C8×C251—42no—
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—22no—
76C22×C251—42no—
77C22×C251—42no—
78C22×C251—41no—
79C22×C251—41no—
80C22×C251C2×C22⋊C411yescharacteristic
81C22×C251C22×C811yescharacteristic
82C22×C251C2×M1611yescharacteristic
83C2×D251—22no—
84C2×D251—22no—
85C2×D251—42no—
86C2×D251—42no—
87C2×D251C22⋊C814yes—
88C4×C251—22no—
89C4×C251—41no—
90Dic251—22no—
91Dic251—21no—
92Dic251—41no—
93Dic251#9912yes—
94C2×C251—21no—
95C2×C251—21no—
96C2×C251—42no—
97C2×C251C2×C22⋊C811yescharacteristic
98C2×C251C2×#9911yescharacteristic
99C2×C251#41311yescharacteristic
100D251—24no—
101C251#145911yescharacteristic, complemented, Sylow
102#1459—2511nomaximal, Sylow
103C2×C4×D4—2511no—
104C2×C22⋊C8—2512no—
105#324—2512no—
106#323—2511no—
107#323D25111yescharacteristic, complemented
108C22×D4—2511no—
109C23×C4—2511no—
110C23×C4—2511no—
111C22×C8—5022no—
112C4×D4—5022no—
113C4×D4—5022no—
114C2×C4⋊C4—2511no—
115C2×C22⋊C4—2511no—
116C2×C22⋊C4C2×D25111yescharacteristic
117C2×C42—2511no—
118C22⋊C8—5022no—
119C22⋊C8—2512no—
120C22⋊C8C2×D25112yes—
121C24—2511no—
122C24C4×D25111yescharacteristic
123C2×D4—5022no—
124C2×D4—5022no—
125C2×D4—2514no—
126C22×C4—10042no—
127C22×C4—10042no—
128C22×C4—5021no—
129C22×C4—5021no—
130C22×C4—2511no—
131C22×C4—2512no—
132C22×C4—2511no—
133C22×C4—2511no—
134C22×C4—2512no—
135C22×C4C4×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C2×C8—22no—
138C2×C8—10044no—
139C2×C8—5022no—
140C4⋊C4—5022no—
141C22⋊C4—22no—
142C22⋊C4—5022no—
143C42—5021no—
144C42—5021no—
145C23—21no—
146C23—21no—
147C23—41no—
148C23—41no—
149C23—10041no—
150C23—10041no—
151C23—2512no—
152C23—2512no—
153C23—2512no—
154C23C8×D25112yescomplemented
155C23C2×C4×D25111yescharacteristic
156D4—5028no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—10042no—
161C2×C4—10042no—
162C2×C4—10042no—
163C2×C4—10041no—
164C2×C4—5024no—
165C2×C4—5021no—
166C2×C4—5021no—
167C2×C4—5021no—
168C2×C4—5022no—
169C2×C4—2511no—
170C2×C4—2512no—
171C2×C4—2511no—
172C2×C4D4×D25112yes—
173C8—42no—
174C8—10042no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—10042no—
183C22—10042no—
184C22—5022no—
185C22—5022no—
186C22—2514no—
187C22C22⋊C4×D25111yescharacteristic
188C22C2×C8×D25111yescharacteristic
189C22M16×D25111yescharacteristic
190C4—22no—
191C4—41no—
192C4—10041no—
193C4—5022no—
194C4—5021no—
195C4—2512no—
196C2—21no—
197C2—21no—
198C2—42no—
199C2—5024no—
200C2C22⋊C8×D25111yescharacteristic
201C2#99×D25111yescharacteristic
202C2#3063166811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC22×C251C22×C8
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×C22⋊D251C4
cosocleC22×C4×C251C23
hypercenter#323D251
hypocenterC251#1459
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D251C2semidirect
3. #30631412C2semidirect
4. #30631411C2semidirect
5. #323×C251C2semidirect
22. C2×C22⋊D251C8semidirect
58. C2×Dic251C2×C8semidirect
66. C23×C251C2×C8semidirect
101. C251#1459semidirect
107. #323D251semidirect
154. C23C8×D251semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D251)⋊C42
C22⋊C8×D2512
#306314122
#306314112
#323×C2512
#1459251