Finite Group Databasebeta

← #44560199

Subgroups — #44560199

Order 32128.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#30632315C211yesmaximal, characteristic, complemented
2#30632161C211yesmaximal, characteristic
3#30631990C212yesmaximal, complemented
4#30631986C212yesmaximal, complemented
5#30631976C212yesmaximal, complemented
6#30631966C212yesmaximal, complemented
7#30631730C211yesmaximal, characteristic, complemented
8#30631713C211yesmaximal, characteristic, complemented
9#30631691C211yesmaximal, characteristic
10C2×#22548637C211yesmaximal, characteristic
11#427×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C2×#16860794C2211yescharacteristic
14#22548715—21no—
15#22548714C2212yescomplemented
16#22548707—21no—
17#22548706C2212yescomplemented
18#22548692—22no—
19C4×D1004—21no—
20C4×D1004—21no—
21#22548677C2212yes—
22#22548677C2212yes—
23#22548674C2211yescharacteristic
24C2×C4⋊Dic251C2211yescharacteristic
25C22⋊C4×D251—22no—
26#22548648C2212yescomplemented
27#22548646C2212yescomplemented
28Dic251⋊C8C2212yescomplemented
29Dic251⋊C8C2212yescomplemented
30#22548637C2212yes—
31#22548637C2212yes—
32#22548635C2212yes—
33#22548635C2212yes—
34C22×C8×C251C2211yescharacteristic
35#117×C251C2211yescharacteristic
36#117×C251C2211yescharacteristic
37#107×C251C2211yescharacteristic, complemented
38#107×C251C2211yescharacteristic, complemented
39#106×C251C2211yescharacteristic, complemented
40#106×C251C2211yescharacteristic, complemented
41C23×D251—21no—
42C2×Dic502—22no—
43C2×D1004—21no—
44C2×D1004—22no—
45C2×D1004Q814yescomplemented
46C2×D1004C2312yes—
47C22×Dic251—21no—
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50#16860794—22no—
51C2×C251⋊C8C2312yes—
52C4⋊Dic251—22no—
53C4⋊Dic251—21no—
54C4⋊Dic251D411yescharacteristic
55C4⋊Dic251D411yescharacteristic
56C4⋊Dic251C2312yes—
57Dic251⋊C4—22no—
58Dic251⋊C4—22no—
59C22⋊Dic251D412yes—
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4—22no—
62C4×Dic251C2312yes—
63C22×C4×C251C2311yescharacteristic
64C2×C8×C251—22no—
65C2×C8×C251C2311yescharacteristic
66C2×C8×C251C2311yescharacteristic
67C4⋊C4×C251C2311yescharacteristic
68C4⋊C4×C251C2311yescharacteristic
69C4⋊C4×C251C2311yescharacteristic
70C4⋊C4×C251C2311yescharacteristic
71C22⋊C4×C251—21no—
72C22⋊C4×C251—21no—
73C42×C251—21no—
74C42×C251—21no—
75C22×D251—22no—
76C22×D251—24no—
77C22×D251—81no—
78C22×D251—81no—
79Dic502—82no—
80D1004—24no—
81D1004—22no—
82D1004C2×Q814yes—
83C2×Dic251—22no—
84C2×Dic251—81no—
85C2×Dic251C2×D412yes—
86C2×Dic251C2×D412yes—
87C4×D251—82no—
88C4×D251—82no—
89C251⋊C8—22no—
90C23×C251C2×D411yescharacteristic
91C2×C4×C251—21no—
92C2×C4×C251—21no—
93C2×C4×C251—21no—
94C2×C4×C251—21no—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2×Q812yes—
97C2×C4×C251#4112yes—
98C2×C4×C251C2411yescharacteristic
99C8×C251—21no—
100C8×C251—21no—
101C22×C251—21no—
102C22×C251—21no—
103C22×C251C22⋊Q812yes—
104C22×C251C22×D411yescharacteristic
105C2×D251—24no—
106C2×D251—42no—
107C2×D251—42no—
108C2×D251—84no—
109C4×C251—41no—
110C4×C251—41no—
111C4×C251—41no—
112C4×C251—41no—
113C4×C251C22⋊Q812yes—
114C4×C251C22×Q811yescharacteristic
115C4×C251C2×#4111yescharacteristic
116Dic251—22no—
117Dic251—22no—
118Dic251—42no—
119C2×C251—22no—
120C2×C251C2×C22⋊Q811yescharacteristic
121C2×C251#57611yescharacteristic
122C2×C251#57711yescharacteristic
123D251—44no—
124C251#308111yescharacteristic, complemented, Sylow
125#3081—2511nomaximal, Sylow
126#544—2511no—
127#518—2511no—
128#478—2512no—
129#478—2512no—
130#476—2512no—
131#474—2512no—
132#428—2511no—
133#427D25111yescharacteristic, complemented
134#424—2511no—
135#418—2511no—
136C2×#102—2511no—
137C22×D4—2511no—
138C2×M16—2511no—
139C22×C8C2×D25111yescharacteristic
140C4⋊Q8—5021no—
141C4⋊Q8—2512no—
142#125—5021no—
143#125—2512no—
144C22⋊Q8—5022no—
145C4×D4—5021no—
146C4×D4—5021no—
147C4×D4—2512no—
148C4×D4—2512no—
149#117—2511no—
150#117C2×D25111yescharacteristic
151#117C2×D25111yescharacteristic
152C2×C4⋊C4—2511no—
153C2×C22⋊C4—5022no—
154#107—2512no—
155#107C2×D25111yescharacteristic, complemented
156#107C2×D25111yescharacteristic, complemented
157#106—2512no—
158#106C2×D25111yescharacteristic, complemented
159#106C2×D25111yescharacteristic, complemented
160C4⋊C8—2512no—
161C4⋊C8—2512no—
162#102—2512no—
163#102—2512no—
164#102—2512no—
165#102—2512no—
166C24—5021no—
167C2×Q8—5022no—
168C2×D4—5021no—
169C2×D4—5022no—
170C2×D4—2514no—
171C2×D4—2512no—
172C22×C4—5022no—
173C22×C4—5022no—
174C22×C4—5021no—
175C22×C4C22×D25111yescharacteristic
176M16—5022no—
177C2×C8—22no—
178C2×C8—2512no—
179C2×C8C22×D25111yescharacteristic
180C2×C8C22×D25111yescharacteristic
181C4⋊C4—5022no—
182C4⋊C4—5022no—
183C4⋊C4—5022no—
184C4⋊C4—5021no—
185C4⋊C4—2512no—
186C4⋊C4—2511no—
187C4⋊C4—2511no—
188C4⋊C4C22×D25111yescharacteristic
189C4⋊C4C22×D25111yescharacteristic
190C4⋊C4C22×D25111yescharacteristic
191C4⋊C4C22×D25111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—5022no—
195C22⋊C4—5022no—
196C22⋊C4—2512no—
197C42—21no—
198C42—21no—
199C42—2512no—
200C23—20081no—
201C23—20081no—
202C23—5024no—
203C23—5022no—
204C23D4×D25111yescharacteristic
205Q8—20082no—
206D4—5022no—
207D4—5024no—
208D4—2514no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—20082no—
214C2×C4—20082no—
215C2×C4—20081no—
216C2×C4—5022no—
217C2×C4—2512no—
218C2×C4—2512no—
219C2×C4D4×D25111yescharacteristic
220C2×C4Q8×D25112yes—
221C2×C4#1686081012yes—
222C2×C4C23×D25111yescharacteristic
223C8—21no—
224C8—21no—
225C8—5022no—
226C22—21no—
227C22—21no—
228C22—20084no—
229C22—10042no—
230C22—10042no—
231C22—5024no—
232C22#2254869312yes—
233C22C2×D4×D25111yescharacteristic
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—10042no—
239C4—5022no—
240C4—5022no—
241C4#2254869312yes—
242C4C2×Q8×D25111yescharacteristic
243C4C2×#1686081011yescharacteristic
244C2—22no—
245C2—10044no—
246C2C2×#2254869311yescharacteristic
247C2#3063248911yescharacteristic
248C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#427×C251C2
socleC22×C251C22×D4
prime radicalC22×D1004C22
hypercenter#427D251
hypocenterC251#3081
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #30632315C2semidirect
3. #30631990C2semidirect
4. #30631986C2semidirect
5. #30631976C2semidirect
6. #30631966C2semidirect
7. #30631730C2semidirect
8. #30631713C2semidirect
11. #427×C251C2semidirect
15. #22548714C22semidirect
17. #22548706C22semidirect
26. #22548648C22semidirect
27. #22548646C22semidirect
28. Dic251⋊C8C22semidirect
29. Dic251⋊C8C22semidirect
37. #107×C251C22semidirect
38. #107×C251C22semidirect
39. #106×C251C22semidirect
40. #106×C251C22semidirect
45. C2×D1004Q8semidirect
124. C251#3081semidirect
133. #427D251semidirect
155. #107C2×D251semidirect
156. #107C2×D251semidirect
158. #106C2×D251semidirect
159. #106C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323152
#306321612
#306319902
#306319862
#306319762
#306319662
#306317302
#306317132
#306316912
C2×#225486372
#427×C2512
#3081251