Finite Group Databasebeta

← #44558308

Subgroups — #44558308

Order 32128.

480 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 137 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#30631629C212yesmaximal, complemented
2#30631599C212yesmaximal, complemented
3#30631599C212yesmaximal, complemented
4#30631591C212yesmaximal, complemented
5#30631590C211yesmaximal, characteristic
6#30631567C211yesmaximal, characteristic
7#30631560C212yesmaximal, complemented
8#30631559C212yesmaximal, complemented
9#386×C251C211yesmaximal, characteristic
10C22×Dic502C2212yes—
11C23×Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C22⋊Dic251C2212yes—
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2211yescharacteristic
19C2×C4×Dic251C2211yescharacteristic
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C22×D4×C251C2211yescharacteristic
27C23×C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2211yescharacteristic
30#95×C251C2212yes—
31C2×Dic502—24no—
32C2×Dic502—42no—
33C2×Dic502—42no—
34C22×Dic251—42no—
35C22×Dic251—41no—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C4⋊Dic251—41no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43C22⋊Dic251—24no—
44C22⋊Dic251—24no—
45C22⋊Dic251—42no—
46C22⋊Dic251—41no—
47C4×Dic251—41no—
48C24×C251C2311yescharacteristic
49C24×C251C2311yescharacteristic
50C2×D4×C251—41no—
51C2×D4×C251—41no—
52C2×D4×C251Q814yescomplemented
53C22×C4×C251—42no—
54C22×C4×C251—41no—
55C22×C4×C251C2312yes—
56C22×C4×C251C2312yes—
57C22×C4×C251C2311yescharacteristic
58C22⋊C4×C251—42no—
59C22⋊C4×C251—41no—
60Dic502—48no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—44no—
73C2×Dic251#4114yes—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251—21no—
77C23×C251—22no—
78C23×C251—21no—
79C23×C251—22no—
80C23×C251—21no—
81C23×C251—21no—
82C23×C251C2×Q812yes—
83C23×C251C2×Q812yes—
84C23×C251C2411yescharacteristic
85D4×C251—44no—
86C2×C4×C251—21no—
87C2×C4×C251—21no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—44no—
95C2×C4×C251C2×Q812yes—
96C22×C251—22no—
97C22×C251—22no—
98C22×C251—24no—
99C22×C251—24no—
100C22×C251—42no—
101C22×C251—42no—
102C22×C251—41no—
103C22×C251—41no—
104C22×C251—41no—
105C22×C251—41no—
106C22×C251C22×Q811yescharacteristic
107C22×C251C2×#4112yes—
108C22×C251H(5,𝔽2)11yescharacteristic
109C22×C251H(5,𝔽2)12yes—
110C22×C251H(5,𝔽2)11yescharacteristic
111C4×C251—22no—
112C4×C251—42no—
113C4×C251—42no—
114Dic251—24no—
115Dic251—42no—
116Dic251—42no—
117Dic251—42no—
118C2×C251—42no—
119C2×C251—42no—
120C2×C251#53812yes—
121C2×C251#54311yescharacteristic
122C2×C251#55412yes—
123C2×C251#55911yescharacteristic
124C2×C251H(3,𝔽4)11yescharacteristic
125C251#280411yescharacteristic, complemented, Sylow
126#2804—2511nomaximal, Sylow
127#400—2512no—
128#393—2512no—
129#393—2512no—
130#391—2512no—
131#390—2511no—
132#386—2511no—
133#386D25111yescharacteristic
134#385—2512no—
135#385—2512no—
136C22×Q8—2512no—
137C22×D4C2×D25111yescharacteristic
138C23×C4—2511no—
139C23×C4C2×D25111yescharacteristic
140C2×C4⋊C4—2512no—
141C2×C4⋊C4—2512no—
142C2×C4⋊C4—2511no—
143C2×C22⋊C4—2512no—
144C2×C22⋊C4—2511no—
145C2×C22⋊C4—2512no—
146C2×C22⋊C4—2512no—
147C2×C22⋊C4C2×D25112yes—
148C2×C22⋊C4C2×D25111yescharacteristic
149C2×C42—2511no—
150#95—2512no—
151#95—2512no—
152#95—2512no—
153#95—2512no—
154#95—2512no—
155#95—2512no—
156#95C2×D25112yes—
157C24C22×D25111yescharacteristic
158C24C22×D25111yescharacteristic
159C2×Q8—10042no—
160C2×Q8—10042no—
161C2×Q8—5024no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4Dic50214yescomplemented
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—10042no—
168C22×C4—10041no—
169C22×C4—2512no—
170C22×C4—2512no—
171C22×C4—2512no—
172C22×C4—2512no—
173C22×C4C22×D25112yes—
174C22×C4C22×D25112yes—
175C22×C4C22×D25111yescharacteristic
176C4⋊C4—10042no—
177C4⋊C4—10042no—
178C4⋊C4—10041no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—10042no—
182C22⋊C4—10041no—
183C22⋊C4—5024no—
184C22⋊C4—5024no—
185C42—10041no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C2×Dic50212yes—
195C23C2×Dic50212yes—
196C23C23×D25111yescharacteristic
197Q8—10048no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—10044no—
209C2×C4—5022no—
210C2×C4—5022no—
211C2×C4—5022no—
212C2×C4—5022no—
213C2×C4—5022no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5022no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—2514no—
221C2×C4C2×Dic50212yes—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22C22×Dic50211yescharacteristic
233C22#41×D25112yes—
234C22#2254876011yescharacteristic
235C22#2254876012yes—
236C22#2254876011yescharacteristic
237C4—22no—
238C4—42no—
239C4—42no—
240C4—10042no—
241C4—10042no—
242C4—10042no—
243C4—5024no—
244C2—42no—
245C2—42no—
246C2#3063226912yes—
247C2#3063231311yescharacteristic
248C2#3063239612yes—
249C2#3063242111yescharacteristic
250C2#3063243011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#386×C251C2
prime radicalC22×D4×C251C22
hypercenter#386D251
hypocenterC251#2804
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631599C2semidirect
3. #30631599C2semidirect
4. #30631591C2semidirect
7. #30631560C2semidirect
8. #30631559C2semidirect
52. C2×D4×C251Q8semidirect
125. C251#2804semidirect
164. C2×D4Dic502semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306315992
#306315912
#306315902
#306315672
#306315602
#306315592
#386×C2512
#2804251