Finite Group Databasebeta

← #44558409

Subgroups — #44558409

Order 32128.

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

normal subgroups only

This group has 580 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×#22548701C212yesmaximal, complemented
2#30631604C211yesmaximal, characteristic, complemented
3#30631603C212yesmaximal, complemented
4#30631600C212yesmaximal, complemented
5#30631599C212yesmaximal, complemented
6#30631596C212yesmaximal, complemented
7#30631595C212yesmaximal, complemented
8#30631552C211yesmaximal, characteristic, complemented
9#394×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2212yes—
11C22×D1004C2211yescharacteristic
12C22×C22⋊D251C2212yes—
13C22×C22⋊D251C2212yes—
14#22548701—28no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×C4×Dic251C2211yescharacteristic
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2212yes—
27C22×D4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2212yes—
30C2×C22⋊C4×C251C2212yes—
31#95×C251C2211yescharacteristic
32C23×D251C2312yes—
33C2×Dic502—24no—
34C2×Dic502—42no—
35C2×Dic502—42no—
36C2×D1004—42no—
37C2×D1004—41no—
38C2×C22⋊D251—24no—
39C2×C22⋊D251—42no—
40C2×C22⋊D251—42no—
41C2×C22⋊D251—42no—
42C2×C22⋊D251—42no—
43C2×C22⋊D251—42no—
44C22×Dic251C2312yes—
45C22×Dic251C2312yes—
46C22×Dic251C2312yes—
47C4⋊Dic251—41no—
48C22⋊Dic251—24no—
49C22⋊Dic251—42no—
50(C2×D251)⋊C4—24no—
51(C2×D251)⋊C4—24no—
52(C2×D251)⋊C4—42no—
53(C2×D251)⋊C4—42no—
54C4×Dic251D414yescomplemented
55C24×C251C2312yes—
56C2×D4×C251—41no—
57C2×D4×C251—42no—
58C22×C4×C251C2311yescharacteristic
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2311yescharacteristic
62C4⋊C4×C251—41no—
63C22⋊C4×C251—24no—
64C22⋊C4×C251—42no—
65C22×D251—22no—
66C22×D251—22no—
67C22×D251—22no—
68C22×D251—22no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—22no—
72Dic502—48no—
73D1004—44no—
74C22⋊D251—48no—
75C22⋊D251—48no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251C2×D414yes—
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—21no—
95C2×C4×C251—21no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—21no—
100C2×C4×C251—21no—
101C2×C4×C251—22no—
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×C251H(5,𝔽2)12yes—
115C22×C2512−1+411yescharacteristic
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—42no—
119C2×D251—42no—
120C2×D251—42no—
121C2×D251—42no—
122C2×D251—42no—
123C4×C251—22no—
124C4×C251—42no—
125C4×C251—41no—
126C4×C251—41no—
127Dic251—24no—
128Dic251—42no—
129Dic251—42no—
130C2×C251—82no—
131C2×C251#53512yes—
132C2×C251#53711yescharacteristic
133C2×C251H(3,𝔽4)11yescharacteristic
134C2×C251H(3,𝔽4)11yescharacteristic
135C2×C251#56212yes—
136D251—82no—
137C251#281911yescharacteristic, complemented, Sylow
138#2819—2511nomaximal, Sylow
139C2×#124—2512no—
140#394—2511no—
141#394—2512no—
142#394D25111yescharacteristic, complemented
143#393—2512no—
144#393—2512no—
145#392—2512no—
146#392—2512no—
147#382—2511no—
148C22×Q8—2512no—
149C22×D4—2512no—
150C22×D4—2512no—
151C22×D4—2511no—
152C22×D4C2×D25111yescharacteristic
153#124—5028no—
154C2×C4⋊C4—2511no—
155C2×C4⋊C4C2×D25111yescharacteristic
156C2×C22⋊C4—2512no—
157C2×C22⋊C4—2512no—
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—2512no—
166#95—2512no—
167#95—2512no—
168#95—2511no—
169#95C2×D25111yescharacteristic
170C24—2512no—
171C24C22×D25112yes—
172C2×Q8—10042no—
173C2×Q8—10042no—
174C2×Q8—5024no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—10042no—
178C2×D4—10041no—
179C2×D4—10042no—
180C2×D4—10042no—
181C2×D4—10042no—
182C2×D4—10042no—
183C2×D4—10042no—
184C2×D4—5024no—
185C22×C4—2512no—
186C22×C4—2512no—
187C22×C4—2512no—
188C22×C4C22×D25111yescharacteristic
189C22×C4C22×D25111yescharacteristic
190C22×C4C22×D25112yes—
191C22×C4C22×D25111yescharacteristic
192C4⋊C4—41no—
193C4⋊C4—10041no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—10042no—
197C22⋊C4—10042no—
198C22⋊C4—10042no—
199C22⋊C4—5024no—
200C22⋊C4—5024no—
201C22⋊C4—5024no—
202C42—2514no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—5022no—
211C23—5022no—
212C23—5022no—
213C23—5022no—
214C23—5022no—
215C23—5022no—
216C23—5022no—
217C23C23×D25111yescharacteristic
218Q8—10048no—
219D4—44no—
220D4—10044no—
221D4—10048no—
222D4—10048no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4—5022no—
236C2×C4—5022no—
237C2×C4—5022no—
238C2×C4—5022no—
239C2×C4—5022no—
240C2×C4—2514no—
241C2×C4D4×D25112yes—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—10042no—
250C22—10042no—
251C22—10042no—
252C22—10042no—
253C22—10042no—
254C22—10042no—
255C22—10042no—
256C22C2×D4×D25111yescharacteristic
257C22#2254876012yes—
258C22#2254876011yescharacteristic
259C22#2254876112yes—
260C22#2254876211yescharacteristic
261C4—22no—
262C4—42no—
263C4—41no—
264C4—41no—
265C4—10042no—
266C4—10042no—
267C4—5024no—
268C2—82no—
269C2—20082no—
270C2#3063225312yes—
271C2#3063226411yescharacteristic
272C2#3063242911yescharacteristic
273C2#3063243011yescharacteristic
274C2#3063243412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
hypercenter#394D251
hypocenterC251#2819
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548701C2semidirect
2. #30631604C2semidirect
3. #30631603C2semidirect
4. #30631600C2semidirect
5. #30631599C2semidirect
6. #30631596C2semidirect
7. #30631595C2semidirect
8. #30631552C2semidirect
9. #394×C251C2semidirect
54. C4×Dic251D4semidirect
137. C251#2819semidirect
142. #394D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487012
#306316042
#306316032
#306316002
#306315992
#306315962
#306315952
#306315522
#394×C2512
#2819251