Finite Group Databasebeta

← #44558399

Subgroups — #44558399

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×#22548693C211yesmaximal, characteristic
2C2×#22548691C211yesmaximal, characteristic
3#30631628C212yesmaximal, complemented
4#30631627C211yesmaximal, characteristic, complemented
5#30631626C211yesmaximal, characteristic, complemented
6#30631616C212yesmaximal
7#30631613C212yesmaximal
8#30631607C212yesmaximal, complemented
9#30631606C211yesmaximal, characteristic, complemented
10#30631549C211yesmaximal, characteristic
11#395×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14#22548693—24no—
15#22548691—24no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2212yes—
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2211yescharacteristic
22C2×(C2×D251)⋊C4C2212yes—
23C2×(C2×D251)⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2212yes—
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2212yes—
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×Q8×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C4⋊C4×C251C2212yes—
35#95×C251C2211yescharacteristic
36#95×C251C2212yes—
37C23×D251C2311yescharacteristic
38C2×Dic502—22no—
39C2×Dic502—42no—
40C22×Dic251C2311yescharacteristic
41C22×Dic251C2312yes—
42C22×Dic251C2311yescharacteristic
43C22×Dic251C2312yes—
44C22×Dic251C2311yescharacteristic
45C2×C4×D251—42no—
46C2×C4×D251D414yes—
47C4⋊Dic251—22no—
48Dic251⋊C4—24no—
49Dic251⋊C4—22no—
50Dic251⋊C4—22no—
51Dic251⋊C4—42no—
52(C2×D251)⋊C4—24no—
53(C2×D251)⋊C4—22no—
54(C2×D251)⋊C4—22no—
55(C2×D251)⋊C4—42no—
56C2×Q8×C251—22no—
57C2×Q8×C251—42no—
58C22×C4×C251C2311yescharacteristic
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2312yes—
62C22×C4×C251C2311yescharacteristic
63C4⋊C4×C251—22no—
64C4⋊C4×C251—42no—
65C22×D251—22no—
66C22×D251—21no—
67C22×D251—21no—
68C22×D251—22no—
69C22×D251C2×D412yes—
70Dic502—44no—
71C2×Dic251—22no—
72C2×Dic251—21no—
73C2×Dic251—22no—
74C2×Dic251—21no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—21no—
78C2×Dic251—21no—
79C2×Dic251—24no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251C2×D412yes—
83C4×D251—44no—
84C23×C251C2411yescharacteristic
85Q8×C251—44no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—21no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—21no—
95C2×C4×C251—22no—
96C2×C4×C251—21no—
97C2×C4×C251—22no—
98C2×C4×C251C2×D412yes—
99C22×C251C22×D411yescharacteristic
100C22×C251H(5,𝔽2)12yes—
101C22×C2512−1+411yescharacteristic
102C22×C2512−1+411yescharacteristic
103C22×C2512−1+412yes—
104C2×D251—22no—
105C2×D251—22no—
106C2×D251—22no—
107C2×D251—42no—
108C2×D251—42no—
109C4×C251—22no—
110C4×C251—42no—
111C4×C251—41no—
112C4×C251—41no—
113C4×C251—42no—
114Dic251—22no—
115Dic251—41no—
116Dic251—41no—
117Dic251—42no—
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—2511no—
131#400—2512no—
132#397—2512no—
133#397—2512no—
134#395—2511no—
135#395—2512no—
136#395D25111yescharacteristic, complemented
137#381—2511no—
138C22×Q8—2511no—
139C22×Q8C2×D25111yescharacteristic
140C23×C4—2511no—
141C22⋊Q8—5024no—
142C22⋊Q8—5024no—
143C2×C4⋊C4—2512no—
144C2×C4⋊C4—2512no—
145C2×C4⋊C4—2511no—
146C2×C4⋊C4—2511no—
147C2×C4⋊C4—2511no—
148C2×C4⋊C4C2×D25111yescharacteristic
149C2×C4⋊C4C2×D25112yes—
150C2×C22⋊C4—2512no—
151C2×C22⋊C4—2511no—
152C2×C22⋊C4—2512no—
153C2×C22⋊C4—2511no—
154#95—2511no—
155#95—2512no—
156#95—2512no—
157#95—2512no—
158#95—2512no—
159#95—2512no—
160#95—2512no—
161#95C2×D25111yescharacteristic
162#95C2×D25112yes—
163C24—2511no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—10042no—
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×D25111yescharacteristic
177C22×C4C22×D25111yescharacteristic
178C22×C4C22×D25112yes—
179C22×C4C22×D25111yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—10042no—
183C4⋊C4—5022no—
184C4⋊C4—5024no—
185C4⋊C4—5022no—
186C4⋊C4—5022no—
187C22⋊C4—10042no—
188C22⋊C4—5022no—
189C22⋊C4—5022no—
190C22⋊C4—5024no—
191C23—5022no—
192C23—5021no—
193C23—5021no—
194C23—5022no—
195C23—2512no—
196C23C23×D25111yescharacteristic
197Q8—44no—
198Q8—10044no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—10044no—
212C2×C4—5021no—
213C2×C4—5022no—
214C2×C4—5021no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5021no—
218C2×C4—5021no—
219C2×C4—5024no—
220C2×C4—5022no—
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#2254876012yes—
232C22#2254876211yescharacteristic
233C22#2254876311yescharacteristic
234C22#2254876312yes—
235C4—22no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—42no—
240C4—10041no—
241C4—10041no—
242C4—10042no—
243C4—10042no—
244C4—5022no—
245C2—10042no—
246C2#3063224711yescharacteristic
247C2#3063226011yescharacteristic
248C2#3063226211yescharacteristic
249C2#3063243612yes—
250C2#3063243812yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#395×C251C2
prime radicalC23×D251C23
hypercenter#395D251
hypocenterC251#2817
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #30631628C2semidirect
4. #30631627C2semidirect
5. #30631626C2semidirect
8. #30631607C2semidirect
9. #30631606C2semidirect
11. #395×C251C2semidirect
125. C251#2817semidirect
136. #395D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486932
C2×#225486912
#306316282
#306316272
#306316262
#306316162
#306316132
#306316072
#306316062
#306315492
#395×C2512
#2817251