Finite Group Databasebeta

← #44560198

Subgroups — #44560198

Order 32128.

398 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 141 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#30632162C211yesmaximal, characteristic
3#30631989C212yesmaximal, complemented
4#30631985C212yesmaximal, complemented
5#30631975C212yesmaximal, complemented
6#30631965C212yesmaximal, complemented
7#30631729C211yesmaximal, characteristic, complemented
8#30631723C211yesmaximal, characteristic, complemented
9#30631715C211yesmaximal, characteristic, complemented
10C2×#22548636C211yesmaximal, characteristic
11#418×C251C211yesmaximal, characteristic
12C22×C251⋊C8C2211yescharacteristic
13#22548715—21no—
14#22548714C2212yescomplemented
15#22548707—21no—
16#22548706C2212yescomplemented
17#22548692—22no—
18D4×Dic251—21no—
19D4×Dic251—21no—
20D4×Dic251C2212yes—
21D4×Dic251C2212yes—
22#22548674C2211yescharacteristic
23#22548674C2211yescharacteristic
24C2×C4⋊Dic251C2211yescharacteristic
25C2×C22⋊Dic251—22no—
26#22548649C2212yescomplemented
27#22548648C2211yescharacteristic, complemented
28#22548648C2211yescharacteristic, complemented
29#22548647C2212yescomplemented
30#22548646C2211yescharacteristic, complemented
31#22548646C2211yescharacteristic, complemented
32Dic251⋊C8C2212yescomplemented
33Dic251⋊C8C2212yescomplemented
34#22548636C2212yes—
35#22548636C2212yes—
36C22×D4×C251C2211yescharacteristic
37C2×M16×C251C2211yescharacteristic
38#117×C251C2211yescharacteristic
39#102×C251C2212yes—
40#102×C251C2212yes—
41C2×Dic502—22no—
42C22×Dic251—22no—
43C22×Dic251—22no—
44C22×Dic251—21no—
45C2×C251⋊C8—22no—
46C2×C251⋊C8C2311yescharacteristic
47C2×C251⋊C8C2311yescharacteristic
48C4⋊Dic251—22no—
49C4⋊Dic251—21no—
50C4⋊Dic251D411yescharacteristic
51C4⋊Dic251D411yescharacteristic
52C4⋊Dic251C2312yes—
53C4⋊Dic251C2311yescharacteristic
54C4⋊Dic251C2311yescharacteristic
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57C22⋊Dic251—22no—
58C22⋊Dic251—22no—
59C22⋊Dic251—21no—
60C22⋊Dic251D412yes—
61C4×Dic251—21no—
62C4×Dic251C2312yes—
63C24×C251—21no—
64C2×D4×C251—21no—
65C2×D4×C251—22no—
66C2×D4×C251Q814yescomplemented
67C2×D4×C251C2312yes—
68C22×C4×C251C2311yescharacteristic
69M16×C251—22no—
70C2×C8×C251C2312yes—
71C4⋊C4×C251C2311yescharacteristic
72C4⋊C4×C251C2311yescharacteristic
73C22⋊C4×C251—21no—
74C42×C251—21no—
75Dic502—82no—
76C2×Dic251—21no—
77C2×Dic251—21no—
78C2×Dic251—22no—
79C2×Dic251—82no—
80C2×Dic251—82no—
81C2×Dic251—81no—
82C2×Dic251C2×D412yes—
83C2×Dic251C2×D412yes—
84C251⋊C8—21no—
85C251⋊C8—21no—
86C23×C251—22no—
87C23×C251—24no—
88C23×C251—81no—
89C23×C251—81no—
90C23×C251C2×D411yescharacteristic
91D4×C251—24no—
92D4×C251—22no—
93D4×C251C2×Q814yes—
94C2×C4×C251—21no—
95C2×C4×C251—21no—
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×Q812yes—
98C2×C4×C251#4112yes—
99C2×C4×C251C2411yescharacteristic
100C8×C251—22no—
101C22×C251—21no—
102C22×C251—21no—
103C22×C251—24no—
104C22×C251—42no—
105C22×C251—42no—
106C22×C251—84no—
107C22×C251C22⋊Q812yes—
108C22×C251C22×D411yescharacteristic
109C4×C251—41no—
110C4×C251—41no—
111C4×C251C22⋊Q812yes—
112C4×C251C22×Q811yescharacteristic
113C4×C251C2×#4111yescharacteristic
114Dic251—22no—
115Dic251—22no—
116Dic251—41no—
117Dic251—42no—
118Dic251—41no—
119C2×C251—22no—
120C2×C251—44no—
121C2×C251C2×C22⋊Q811yescharacteristic
122C2×C251#57611yescharacteristic
123C2×C251#57711yescharacteristic
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#427—2511no—
134#424—2511no—
135#418D25111yescharacteristic
136C2×#102—2511no—
137C22×D4C2×D25111yescharacteristic
138C2×M16C2×D25111yescharacteristic
139C22×C8—2511no—
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#117—2511no—
151#117C2×D25111yescharacteristic
152C2×C4⋊C4—2511no—
153C2×C22⋊C4—5022no—
154#107—2512no—
155#107—2511no—
156#107—2511no—
157#106—2512no—
158#106—2511no—
159#106—2511no—
160C4⋊C8—2512no—
161C4⋊C8—2512no—
162#102—2512no—
163#102—2512no—
164#102C2×D25112yes—
165#102C2×D25112yes—
166C24—21no—
167C2×Q8—5022no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic50214yescomplemented
171C2×D4C22×D25112yes—
172C22×C4—5021no—
173C22×C4—5022no—
174C22×C4—5022no—
175C22×C4C22×D25111yescharacteristic
176M16—22no—
177C2×C8—5022no—
178C2×C8—2511no—
179C2×C8—2511no—
180C2×C8C22×D25112yes—
181C4⋊C4—5022no—
182C4⋊C4—5022no—
183C4⋊C4—5022no—
184C4⋊C4—5021no—
185C4⋊C4—2512no—
186C4⋊C4—2511no—
187C4⋊C4—2511no—
188C4⋊C4—2511no—
189C4⋊C4—2511no—
190C4⋊C4C22×D25111yescharacteristic
191C4⋊C4C22×D25111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—5021no—
194C22⋊C4—5022no—
195C22⋊C4—5022no—
196C22⋊C4—2512no—
197C42—21no—
198C42—5021no—
199C42—2512no—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23D4×D25111yescharacteristic
205Q8—20082no—
206D4—22no—
207D4—24no—
208D4C2×Dic50214yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—20081no—
212C2×C4—20082no—
213C2×C4—20082no—
214C2×C4—5021no—
215C2×C4—5021no—
216C2×C4—5022no—
217C2×C4—2512no—
218C2×C4—2512no—
219C2×C4D4×D25111yescharacteristic
220C2×C4C2×Dic50212yes—
221C2×C4#1686080912yes—
222C2×C4C23×D25111yescharacteristic
223C8—22no—
224C8—5021no—
225C8—5021no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#2254869212yes—
233C22C2×D4×D25111yescharacteristic
234C4—41no—
235C4—41no—
236C4—10041no—
237C4—10042no—
238C4—10041no—
239C4—5022no—
240C4—5022no—
241C4#2254869212yes—
242C4C22×Dic50211yescharacteristic
243C4C2×#1686080911yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#2254869211yescharacteristic
247C2#3063249111yescharacteristic
248C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×C251C22
hypercenter#418D251
hypocenterC251#3081
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #30632315C2semidirect
3. #30631989C2semidirect
4. #30631985C2semidirect
5. #30631975C2semidirect
6. #30631965C2semidirect
7. #30631729C2semidirect
8. #30631723C2semidirect
9. #30631715C2semidirect
14. #22548714C22semidirect
16. #22548706C22semidirect
26. #22548649C22semidirect
27. #22548648C22semidirect
28. #22548648C22semidirect
29. #22548647C22semidirect
30. #22548646C22semidirect
31. #22548646C22semidirect
32. Dic251⋊C8C22semidirect
33. Dic251⋊C8C22semidirect
66. C2×D4×C251Q8semidirect
124. C251#3081semidirect
170. C2×D4Dic502semidirect

Maximal subgroups

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

subgroupindex
#306323152
#306321622
#306319892
#306319852
#306319752
#306319652
#306317292
#306317232
#306317152
C2×#225486362
#418×C2512
#3081251