Finite Group Databasebeta

← #44558401

Subgroups — #44558401

Order 32128.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22548695C211yesmaximal, characteristic
2C2×#22548693C211yesmaximal, characteristic
3#30631629C212yesmaximal, complemented
4#30631625C211yesmaximal, characteristic, complemented
5#30631616C212yesmaximal
6#30631614C212yesmaximal
7#30631608C211yesmaximal, characteristic, complemented
8#30631608C212yesmaximal, complemented
9#30631607C211yesmaximal, characteristic, complemented
10#30631549C211yesmaximal, characteristic
11#400×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×Dic502C2211yescharacteristic
14C22×C4×D251C2211yescharacteristic
15#22548695—24no—
16#22548693—24no—
17C2×C4⋊Dic251C2212yes—
18C2×C4⋊Dic251C2212yes—
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2211yescharacteristic
23C2×(C2×D251)⋊C4C2212yes—
24C2×(C2×D251)⋊C4C2211yescharacteristic
25C2×(C2×D251)⋊C4C2212yes—
26C251⋊#95C2211yescharacteristic
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34#95×C251C2211yescharacteristic
35#95×C251C2212yes—
36#95×C251C2212yes—
37C23×D251C2311yescharacteristic
38C2×Dic502—22no—
39C2×Dic502—22no—
40C2×Dic502—42no—
41C2×Dic502—42no—
42C22×Dic251C2311yescharacteristic
43C22×Dic251C2312yes—
44C22×Dic251C2312yes—
45C22×Dic251C2311yescharacteristic
46C22×Dic251C2311yescharacteristic
47C2×C4×D251—42no—
48C2×C4×D251D414yes—
49C4⋊Dic251—24no—
50C4⋊Dic251—42no—
51Dic251⋊C4—22no—
52Dic251⋊C4—22no—
53Dic251⋊C4—42no—
54(C2×D251)⋊C4—22no—
55(C2×D251)⋊C4—22no—
56(C2×D251)⋊C4—24no—
57(C2×D251)⋊C4—42no—
58C22×C4×C251C2311yescharacteristic
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2312yes—
62C22×C4×C251C2311yescharacteristic
63C4⋊C4×C251—22no—
64C4⋊C4×C251—22no—
65C22×D251—22no—
66C22×D251—21no—
67C22×D251—22no—
68C22×D251—21no—
69C22×D251C2×D412yes—
70Dic502—44no—
71Dic502—44no—
72C2×Dic251—22no—
73C2×Dic251—21no—
74C2×Dic251—21no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—21no—
81C2×Dic251—22no—
82C2×Dic251—21no—
83C2×Dic251—22no—
84C2×Dic251C2×D412yes—
85C4×D251—44no—
86C23×C251C2411yescharacteristic
87C2×C4×C251—22no—
88C2×C4×C251—21no—
89C2×C4×C251—22no—
90C2×C4×C251—21no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—21no—
94C2×C4×C251—24no—
95C2×C4×C251—22no—
96C2×C4×C251—22no—
97C2×C4×C251—21no—
98C2×C4×C251C2×D412yes—
99C22×C251C22×D411yescharacteristic
100C22×C251H(5,𝔽2)12yes—
101C22×C2512−1+412yes—
102C22×C2512−1+411yescharacteristic
103C22×C2512−1+411yescharacteristic
104C2×D251—22no—
105C2×D251—22no—
106C2×D251—22no—
107C2×D251—42no—
108C2×D251—42no—
109C4×C251—22no—
110C4×C251—41no—
111C4×C251—42no—
112C4×C251—41no—
113C4×C251—42no—
114Dic251—22no—
115Dic251—41no—
116Dic251—42no—
117Dic251—41no—
118Dic251—42no—
119C2×C251#53411yescharacteristic
120C2×C251#53611yescharacteristic
121C2×C251#53611yescharacteristic
122C2×C251#56312yes—
123C2×C251#56312yes—
124D251—42no—
125C251#281711yescharacteristic, complemented, Sylow
126#2817—2511nomaximal, Sylow
127C2×C22⋊Q8—2511no—
128C2×C22⋊Q8—2511no—
129#400—2511no—
130#400—2512no—
131#400D25111yescharacteristic, complemented
132#397—2512no—
133#397—2512no—
134#395—2511no—
135#395—2512no—
136#395—2511no—
137#381—2511no—
138C22×Q8—2511no—
139C22×Q8—2511no—
140C23×C4—2511no—
141C22⋊Q8—5024no—
142C22⋊Q8—5024no—
143C2×C4⋊C4—2512no—
144C2×C4⋊C4—2512no—
145C2×C4⋊C4—2511no—
146C2×C4⋊C4—2512no—
147C2×C4⋊C4—2511no—
148C2×C4⋊C4C2×D25111yescharacteristic
149C2×C4⋊C4C2×D25111yescharacteristic
150C2×C22⋊C4—2512no—
151C2×C22⋊C4—2511no—
152C2×C22⋊C4—2511no—
153C2×C22⋊C4—2512no—
154#95—2511no—
155#95—2512no—
156#95—2512no—
157#95—2512no—
158#95—2512no—
159#95—2512no—
160#95C2×D25111yescharacteristic
161#95C2×D25112yes—
162#95C2×D25112yes—
163C24—2511no—
164C2×Q8—10042no—
165C2×Q8—10042no—
166C2×Q8—5022no—
167C2×Q8—5022no—
168C22×C4—10042no—
169C22×C4—2514no—
170C22×C4—2512no—
171C22×C4—2511no—
172C22×C4—2512no—
173C22×C4—2511no—
174C22×C4—2511no—
175C22×C4C22×D25112yes—
176C22×C4C22×D25112yes—
177C22×C4C22×D25111yescharacteristic
178C22×C4C22×D25111yescharacteristic
179C22×C4C22×D25111yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—10042no—
183C4⋊C4—10042no—
184C4⋊C4—5022no—
185C4⋊C4—5024no—
186C4⋊C4—5022no—
187C22⋊C4—10042no—
188C22⋊C4—5022no—
189C22⋊C4—5024no—
190C22⋊C4—5022no—
191C23—5022no—
192C23—5021no—
193C23—5022no—
194C23—5021no—
195C23—2512no—
196C23C23×D25111yescharacteristic
197Q8—10044no—
198Q8—10044no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—10044no—
211C2×C4—5021no—
212C2×C4—5022no—
213C2×C4—5021no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5022no—
218C2×C4—5021no—
219C2×C4—5022no—
220C2×C4—5021no—
221C2×C4—5022no—
222C2×C4—5022no—
223C2×C4—2512no—
224C2×C4D4×D25112yes—
225C22—10042no—
226C22—10042no—
227C22—5022no—
228C22—5022no—
229C22—5022no—
230C22C2×D4×D25111yescharacteristic
231C22#2254876112yes—
232C22#2254876212yes—
233C22#2254876211yescharacteristic
234C22#2254876311yescharacteristic
235C4—22no—
236C4—41no—
237C4—42no—
238C4—41no—
239C4—42no—
240C4—10042no—
241C4—10041no—
242C4—10041no—
243C4—10042no—
244C4—5022no—
245C2—10042no—
246C2#3063224811yescharacteristic
247C2#3063225711yescharacteristic
248C2#3063226011yescharacteristic
249C2#3063243912yes—
250C2#3063244112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#400×C251C2
prime radicalC23×D251C23
hypercenter#400D251
hypocenterC251#2817
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #30631629C2semidirect
4. #30631625C2semidirect
7. #30631608C2semidirect
8. #30631608C2semidirect
9. #30631607C2semidirect
11. #400×C251C2semidirect
125. C251#2817semidirect
131. #400D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486952
C2×#225486932
#306316292
#306316252
#306316162
#306316142
#306316082
#306316072
#306315492
#400×C2512
#2817251