Finite Group Databasebeta

← #44558418

Subgroups — #44558418

Order 32128.

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

normal subgroups only

This group has 464 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×#22548705C211yesmaximal, characteristic, complemented
2C2×#22548702C211yesmaximal, characteristic
3#30631623C212yesmaximal, complemented
4#30631615C212yesmaximal, complemented
5#30631610C212yesmaximal, complemented
6#30631610C212yesmaximal, complemented
7#30631608C211yesmaximal, characteristic, complemented
8#30631607C211yesmaximal, characteristic, complemented
9#30631601C211yesmaximal, characteristic
10#30631551C211yesmaximal, characteristic, complemented
11#394×C251C211yesmaximal, characteristic
12C22×Dic502C2211yescharacteristic
13#22548705—24no—
14#22548702—24no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×C4⋊Dic251C2212yes—
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2212yes—
22C2×Dic251⋊C4C2212yes—
23C2×C22⋊Dic251C2212yes—
24C2×C22⋊Dic251C2212yes—
25C2×C22⋊Dic251C2212yes—
26C2×C22⋊Dic251C2212yes—
27C2×C4×Dic251C2211yescharacteristic
28C251⋊#95C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×D4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2212yes—
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C2×Dic502—22no—
38C2×Dic502—42no—
39C22×Dic251C2312yes—
40C22×Dic251C2312yes—
41C22×Dic251C2311yescharacteristic
42C22×Dic251C2311yescharacteristic
43C22×Dic251C2311yescharacteristic
44C22×Dic251C2311yescharacteristic
45C4⋊Dic251—22no—
46C4⋊Dic251—42no—
47Dic251⋊C4—22no—
48Dic251⋊C4—22no—
49Dic251⋊C4—22no—
50Dic251⋊C4—22no—
51Dic251⋊C4—42no—
52Dic251⋊C4—42no—
53C22⋊Dic251—24no—
54C22⋊Dic251—24no—
55C22⋊Dic251—42no—
56C22⋊Dic251—42no—
57C4×Dic251D414yes—
58C24×C251C2312yes—
59C2×D4×C251—22no—
60C2×D4×C251—42no—
61C22×C4×C251C2312yes—
62C22×C4×C251C2311yescharacteristic
63C22×C4×C251C2311yescharacteristic
64C22×C4×C251C2311yescharacteristic
65C4⋊C4×C251—22no—
66C22⋊C4×C251—42no—
67C22⋊C4×C251—42no—
68Dic502—44no—
69C2×Dic251—21no—
70C2×Dic251—22no—
71C2×Dic251—21no—
72C2×Dic251—22no—
73C2×Dic251—21no—
74C2×Dic251—21no—
75C2×Dic251—21no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—21no—
82C2×Dic251—22no—
83C2×Dic251C2×D412yes—
84C2×Dic251C2×D412yes—
85C23×C251—22no—
86C23×C251—22no—
87C23×C251—22no—
88C23×C251—22no—
89C23×C251—22no—
90C23×C251—22no—
91C23×C251—22no—
92C23×C251C2411yescharacteristic
93D4×C251—44no—
94C2×C4×C251—22no—
95C2×C4×C251—21no—
96C2×C4×C251—21no—
97C2×C4×C251—21no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—22no—
101C2×C4×C251—21no—
102C2×C4×C251—22no—
103C2×C4×C251C2×D412yes—
104C22×C251—42no—
105C22×C251—42no—
106C22×C251—42no—
107C22×C251—42no—
108C22×C251—42no—
109C22×C251—42no—
110C22×C251—42no—
111C22×C251C22×D411yescharacteristic
112C22×C251H(5,𝔽2)12yes—
113C22×C251H(5,𝔽2)11yescharacteristic
114C22×C2512−1+412yes—
115C22×C2512−1+411yescharacteristic
116C4×C251—22no—
117C4×C251—41no—
118C4×C251—41no—
119C4×C251—42no—
120Dic251—22no—
121Dic251—22no—
122Dic251—41no—
123Dic251—42no—
124Dic251—41no—
125Dic251—42no—
126C2×C251—82no—
127C2×C251#53511yescharacteristic
128C2×C251#53611yescharacteristic
129C2×C251#53711yescharacteristic
130C2×C251#56212yes—
131C2×C251#56312yes—
132C251#282011yescharacteristic, complemented, Sylow
133#2820—2511nomaximal, Sylow
134C2×#125—2511no—
135C2×#124—2511no—
136#399—2512no—
137#397—2512no—
138#396—2512no—
139#396—2512no—
140#395—2511no—
141#395—2511no—
142#394—2511no—
143#394D25111yescharacteristic
144#382—2511no—
145C22×Q8—2511no—
146C22×D4C2×D25111yescharacteristic
147#125—5024no—
148#124—5024no—
149C2×C4⋊C4—2512no—
150C2×C4⋊C4—2512no—
151C2×C4⋊C4—2512no—
152C2×C4⋊C4—2511no—
153C2×C4⋊C4—2511no—
154C2×C4⋊C4—2511no—
155C2×C4⋊C4—2511no—
156C2×C4⋊C4—2511no—
157C2×C4⋊C4C2×D25111yescharacteristic
158C2×C22⋊C4—2512no—
159C2×C22⋊C4—2512no—
160C2×C22⋊C4—2512no—
161C2×C22⋊C4—2512no—
162C2×C22⋊C4C2×D25112yes—
163C2×C22⋊C4C2×D25112yes—
164C2×C42—2511no—
165#95—2511no—
166#95—2512no—
167#95—2512no—
168#95—2512no—
169#95C2×D25111yescharacteristic
170C24C22×D25112yes—
171C2×Q8—10042no—
172C2×Q8—5022no—
173C2×D4—22no—
174C2×D4—42no—
175C22×C4—2512no—
176C22×C4—2511no—
177C22×C4—2512no—
178C22×C4—2511no—
179C22×C4—2511no—
180C22×C4—2511no—
181C22×C4C22×D25112yes—
182C22×C4C22×D25111yescharacteristic
183C22×C4C22×D25111yescharacteristic
184C22×C4C22×D25111yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—10042no—
187C4⋊C4—10042no—
188C4⋊C4—10042no—
189C4⋊C4—5022no—
190C4⋊C4—5022no—
191C4⋊C4—5022no—
192C4⋊C4—5022no—
193C4⋊C4—5022no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—10042no—
197C22⋊C4—10042no—
198C22⋊C4—5024no—
199C22⋊C4—5024no—
200C42—2514no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23C23×D25111yescharacteristic
209Q8—10044no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—5022no—
221C2×C4—5021no—
222C2×C4—5021no—
223C2×C4—5021no—
224C2×C4—5022no—
225C2×C4—5022no—
226C2×C4—5022no—
227C2×C4—5022no—
228C2×C4—5022no—
229C2×C4—5021no—
230C2×C4—5022no—
231C2×C4—5021no—
232C2×C4—5022no—
233C2×C4—5021no—
234C2×C4—2512no—
235C2×C4—2512no—
236C2×C4D4×D25112yes—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D25111yescharacteristic
245C22#2254876012yes—
246C22#2254876011yescharacteristic
247C22#2254876212yes—
248C22#2254876211yescharacteristic
249C4—22no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—10042no—
254C4—10041no—
255C4—10041no—
256C4—10042no—
257C4—5022no—
258C4—5022no—
259C2—82no—
260C2#3063225211yescharacteristic
261C2#3063225711yescharacteristic
262C2#3063226611yescharacteristic
263C2#3063243212yes—
264C2#3063244012yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC22×D4×C251C22
hypercenter#394D251
hypocenterC251#2820
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548705C2semidirect
3. #30631623C2semidirect
4. #30631615C2semidirect
5. #30631610C2semidirect
6. #30631610C2semidirect
7. #30631608C2semidirect
8. #30631607C2semidirect
10. #30631551C2semidirect
132. C251#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#225487052
C2×#225487022
#306316232
#306316152
#306316102
#306316082
#306316072
#306316012
#306315512
#394×C2512
#2820251