Finite Group Databasebeta

← #342×D251 #44550093

Subgroups — #342×D251 #44550093

Order 32128.

680 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 145 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⋊C4×D251C212yesmaximal
2C2×C22⋊C4×D251C211yesmaximal, characteristic
3#30631454C211yesmaximal, characteristic, complemented
4#30631453C212yesmaximal, complemented
5#342×C251C211yesmaximal, characteristic, complemented
6C24×D251C2211yescharacteristic
7C22×C4×D251—22no—
8C22×C4×D251C412yescomplemented
9C2×C22⋊Dic251C2211yescharacteristic
10C2×C22⋊Dic251C2212yes—
11C2×(C2×D251)⋊C4—22no—
12C2×(C2×D251)⋊C4C412yescomplemented
13C22⋊C4×D251—22no—
14C22⋊C4×D251—22no—
15C22⋊C4×D251—24no—
16C22⋊C4×D251C418yescomplemented
17C2×C22⋊C4×C251C2211yescharacteristic
18C2×C22⋊C4×C251C2212yes—
19C23×D251—44no—
20C23×D251C2×C411yescharacteristic
21C23×D251C2×C414yes—
22C23×D251C2×C411yescharacteristic
23C23×D251D412yes—
24C23×D251D412yes—
25C23×D251D412yes—
26C23×D251Q812yes—
27C22×Dic251—22no—
28C22×Dic251C2×C412yescomplemented
29C2×C4×D251—28no—
30C2×C4×D251—22no—
31C2×C4×D251—22no—
32C2×C4×D251—48no—
33C2×C4×D251—44no—
34C22⋊Dic251—22no—
35C22⋊Dic251—22no—
36C22⋊Dic251C2×C414yescomplemented
37(C2×D251)⋊C4—24no—
38(C2×D251)⋊C4—28no—
39C24×C251C2311yescharacteristic
40C22×C4×C251—22no—
41C22×C4×C251C2×C412yescomplemented
42C22⋊C4×C251—22no—
43C22⋊C4×C251—22no—
44C22⋊C4×C251C2×C414yescomplemented
45C22×D251—24no—
46C22×D251—28no—
47C22×D251—48no—
48C22×D251—44no—
49C22×D251—44no—
50C22×D251—42no—
51C22×D251—44no—
52C22×D251C4212yes—
53C22×D251C4212yes—
54C22×D251C22⋊C412yes—
55C22×D251C22⋊C414yes—
56C22×D251C22⋊C411yescharacteristic
57C22×D251C22⋊C414yes—
58C22×D251C22⋊C411yescharacteristic
59C22×D251C4⋊C412yes—
60C22×D251C4⋊C414yes—
61C22×D251C4⋊C414yes—
62C22×D251C4⋊C412yes—
63C2×Dic251—22no—
64C2×Dic251—24no—
65C2×Dic251—44no—
66C2×Dic251—42no—
67C4×D251—416no—
68C4×D251—44no—
69C4×D251—44no—
70C23×C251—42no—
71C23×C251C22×C411yescharacteristic
72C23×C251C22×C412yes—
73C23×C251C2×D412yes—
74C23×C251C2×D411yescharacteristic
75C23×C251C2×Q811yescharacteristic
76C2×C4×C251—22no—
77C2×C4×C251—24no—
78C2×C4×C251—44no—
79C2×C4×C251—42no—
80C22×C251—22no—
81C22×C251—24no—
82C22×C251—42no—
83C22×C251—41no—
84C22×C251—41no—
85C22×C251C2×C4211yescharacteristic
86C22×C251C2×C22⋊C412yes—
87C22×C251C2×C22⋊C411yescharacteristic
88C22×C251C2×C4⋊C411yescharacteristic
89C22×C251C2×C4⋊C412yes—
90C2×D251—22no—
91C2×D251—24no—
92C2×D251—28no—
93C2×D251—24no—
94C2×D251—24no—
95C2×D251—28no—
96C2×D251—42no—
97C2×D251—44no—
98C2×D251—44no—
99C2×D251—42no—
100C2×D251#9512yes—
101C2×D251#9514yes—
102C2×D251#9512yes—
103C2×D251#9914yes—
104C4×C251—44no—
105C4×C251—42no—
106Dic251—44no—
107Dic251—42no—
108C2×C251—22no—
109C2×C251—21no—
110C2×C251—21no—
111C2×C251—22no—
112C2×C251C2×#9511yescharacteristic
113C2×C251C2×#9912yes—
114D251—22no—
115D251—24no—
116D251#34214yesdirect product factor
117C251C2×#34211yescharacteristic, complemented, Sylow
118C2×#342—2511nomaximal, Sylow
119C22×C22⋊C4—2511no—
120C22×C22⋊C4—2512no—
121#342—2512no—
122#342—2511no—
123#342D25111yescharacteristic, direct product factor
124C25—2511no—
125C23×C4—5022no—
126C23×C4—2512no—
127C2×C22⋊C4—5022no—
128C2×C22⋊C4—5022no—
129C2×C22⋊C4—5024no—
130C2×C22⋊C4—5022no—
131C2×C22⋊C4—2511no—
132C2×C22⋊C4—2512no—
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2518no—
135C2×C22⋊C4C2×D25111yescharacteristic
136C2×C22⋊C4C2×D25112yes—
137C24—10044no—
138C24—2511no—
139C24—2514no—
140C24—2512no—
141C24—2512no—
142C24—2512no—
143C24—2511no—
144C24—2512no—
145C24C22×D25111yescharacteristic
146C22×C4—22no—
147C22×C4—10048no—
148C22×C4—10044no—
149C22×C4—5022no—
150C22×C4—5022no—
151C22×C4—5028no—
152C22×C4—5022no—
153C22×C4—2512no—
154C22×C4C4×D25112yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—5028no—
158C22⋊C4—5024no—
159C22⋊C4—5022no—
160C22⋊C4—5022no—
161C22⋊C4—2514no—
162C22⋊C4C4×D25114yescomplemented
163C23—42no—
164C23—10042no—
165C23—10044no—
166C23—10048no—
167C23—10044no—
168C23—10044no—
169C23—5024no—
170C23—5028no—
171C23—2512no—
172C23—2514no—
173C23—2512no—
174C23—2514no—
175C23—2512no—
176C23—2514no—
177C23—2512no—
178C23—2511no—
179C23—2514no—
180C23—2512no—
181C23—2511no—
182C23C2×C4×D25111yescharacteristic
183C23C2×C4×D25112yes—
184C23D4×D25112yes—
185C23D4×D25111yescharacteristic
186C23Q8×D25111yescharacteristic
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—100416no—
192C2×C4—10044no—
193C2×C4—10044no—
194C2×C4—10044no—
195C2×C4—10042no—
196C2×C4—5024no—
197C2×C4—5022no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—10044no—
204C22—10042no—
205C22—10042no—
206C22—10044no—
207C22—5028no—
208C22—5024no—
209C22—5024no—
210C22—5028no—
211C22—5022no—
212C22—5024no—
213C22—2514no—
214C22—2512no—
215C22—2512no—
216C22—2514no—
217C22C42×D25111yescharacteristic
218C22C22⋊C4×D25112yes—
219C22C22⋊C4×D25111yescharacteristic
220C22C4⋊C4×D25111yescharacteristic
221C22C4⋊C4×D25112yes—
222C4—44no—
223C4—42no—
224C4—10044no—
225C4—10042no—
226C2—22no—
227C2—21no—
228C2—21no—
229C2—22no—
230C2—5024no—
231C2—5022no—
232C2—2514no—
233C2#95×D25111yescharacteristic
234C2#99×D25112yes—
subgroupG/H
centerC22C22⋊C4×D251
commutatorC22×C251C2×C42
FrattiniC24C22×D251
Fitting#342×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC24×D251C22
cosocleC24×C251C23
hypercenter#342D251
hypocenterC251C2×#342
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #30631454C2semidirect
4. #30631453C2semidirect
5. #342×C251C2semidirect
8. C22×C4×D251C4semidirect
12. C2×(C2×D251)⋊C4C4semidirect
16. C22⋊C4×D251C4semidirect
28. C22×Dic251C2×C4semidirect
36. C22⋊Dic251C2×C4semidirect
41. C22×C4×C251C2×C4semidirect
44. C22⋊C4×C251C2×C4semidirect
116. D251#342direct (N × H)
117. C251C2×#342semidirect
123. #342D251direct (N × H)
154. C22×C4C4×D251semidirect
162. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D2512
#306314542
#306314532
#342×C2512
C2×#342251