Finite Group Databasebeta

← #44562296

Subgroups — #44562296

Order 32128.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#30632402C211yesmaximal, characteristic, complemented
2#30632340C211yesmaximal, characteristic, complemented
3#30632037C211yesmaximal, characteristic, complemented
4#30632026C212yesmaximal, complemented
5#30631978C211yesmaximal, characteristic, complemented
6#30631913C212yesmaximal, complemented
7#30631831C212yesmaximal, complemented
8C4×#16860795C211yesmaximal, characteristic, complemented
9#30631752C211yesmaximal, characteristic, complemented
10#30631682C212yesmaximal, complemented
11#461×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C2×#16860795C2211yescharacteristic
14C2×C8×D251C2212yes—
15#22548715C2211yescharacteristic
16#22548701—22no—
17#22548700—22no—
18#22548691C2212yes—
19#22548690—21no—
20#22548689—21no—
21#22548686C2212yes—
22C4×D1004C2211yescharacteristic
23C4×D1004C2211yescharacteristic
24#22548674—21no—
25#22548673C2212yes—
26C4⋊C4×D251—21no—
27#22548648C2211yescharacteristic
28#22548647C2212yes—
29#22548639C2212yes—
30#22548639C2212yes—
31#22548637C2212yes—
32#22548636C2211yescharacteristic
33#22548635C2211yescharacteristic
34(C2×D251)⋊C8C2212yes—
35C4×C251⋊C8C2211yescharacteristic
36C2×SD16×C251C2212yes—
37C4⋊Q8×C251C2211yescharacteristic
38C4×D4×C251C2211yescharacteristic
39C4⋊C8×C251C2211yescharacteristic
40#103×C251C2212yes—
41#16860811—24no—
42#16860811—42no—
43C2×Dic502—21no—
44C2×D1004—22no—
45C2×D1004D411yescharacteristic
46C2×D1004C2311yescharacteristic
47C2×C22⋊D251—22no—
48C2×C4×D251—22no—
49C2×C4×D251—21no—
50C2×C4×D251C2312yes—
51#16860795—22no—
52C2×C251⋊C8C2311yescharacteristic
53C2×C251⋊C8C2311yescharacteristic
54C8×D251—42no—
55C4⋊Dic251—21no—
56C4⋊Dic251D411yescharacteristic
57C4⋊Dic251C2311yescharacteristic
58C4⋊Dic251C2312yes—
59Dic251⋊C4—22no—
60C22⋊Dic251—22no—
61(C2×D251)⋊C4—22no—
62(C2×D251)⋊C4—22no—
63(C2×D251)⋊C4—21no—
64(C2×D251)⋊C4D412yes—
65C4×Dic251—22no—
66C2×Q8×C251C2312yes—
67C2×D4×C251C2311yescharacteristic
68C22×C4×C251—21no—
69SD16×C251—42no—
70C2×C8×C251C2312yes—
71C4⋊C4×C251—21no—
72C4⋊C4×C251—21no—
73C4⋊C4×C251C2311yescharacteristic
74C22⋊C4×C251—21no—
75C42×C251C2311yescharacteristic
76C22×D251—21no—
77C22×D251C2×D412yes—
78Dic502—81no—
79D1004—22no—
80D1004—21no—
81D1004—44no—
82D1004#4112yes—
83C22⋊D251—82no—
84C2×Dic251—22no—
85C2×Dic251—21no—
86C2×Dic251C2×D412yes—
87C4×D251—22no—
88C4×D251—22no—
89C4×D251—44no—
90C4×D251—42no—
91C251⋊C8—21no—
92C251⋊C8—21no—
93C23×C251—21no—
94Q8×C251—22no—
95Q8×C251—22no—
96D4×C251—21no—
97D4×C251—21no—
98C2×C4×C251—21no—
99C2×C4×C251—22no—
100C2×C4×C251—42no—
101C2×C4×C251C2×D411yescharacteristic
102C2×C4×C251C2×D411yescharacteristic
103C2×C4×C251C2411yescharacteristic
104C8×C251—22no—
105C22×C251—41no—
106C22×C251—41no—
107C22×C251—41no—
108C22×C251C22×D411yescharacteristic
109C2×D251—22no—
110C2×D251—22no—
111C2×D251—22no—
112C2×D251—22no—
113C2×D251—41no—
114C2×D251—41no—
115C4×C251—21no—
116C4×C251—42no—
117C4×C251—41no—
118C4×C251#13512yes—
119C4×C251C2×#4111yescharacteristic
120C4×C251H(5,𝔽2)11yescharacteristic
121Dic251—22no—
122Dic251—41no—
123Dic251—42no—
124C2×C251—81no—
125C2×C251#54611yescharacteristic
126C2×C251C2×#13511yescharacteristic
127C2×C251#57711yescharacteristic
128D251—42no—
129D251—42no—
130C251#330311yescharacteristic, complemented, Sylow
131#3303—2511nomaximal, Sylow
132#555—2511no—
133#547—2511no—
134#487—2511no—
135#485—2512no—
136#476—2511no—
137#465—2512no—
138#461D25111yescharacteristic, complemented
139#449—2512no—
140C4×D8—2511no—
141C8×D4—2511no—
142#416—2512no—
143C2×#41—2512no—
144C2×SD16C2×D25112yes—
145C2×D8—2511no—
146C22×C8—2512no—
147C4⋊Q8—2511no—
148C4⋊Q8C2×D25111yescharacteristic
149#124—5022no—
150#123—5022no—
151C22⋊Q8—2512no—
152#121—5021no—
153#121—5021no—
154#121—2512no—
155C4×D4—2511no—
156C4×D4—2511no—
157C4×D4C2×D25111yescharacteristic
158#117—5021no—
159#117—2512no—
160C2×C4⋊C4—5021no—
161#107—2511no—
162#106—2512no—
163C4⋊C8C2×D25111yescharacteristic
164#103—2512no—
165#103—2512no—
166#103C2×D25112yes—
167#102—2511no—
168#102—2512no—
169#102—2511no—
170C22⋊C8—2512no—
171C4×C8—2511no—
172#41—10042no—
173#41—5024no—
174C2×Q8—5021no—
175C2×Q8C22×D25112yes—
176C2×D4—5022no—
177C2×D4—5022no—
178C2×D4—2511no—
179C2×D4—2511no—
180C2×D4C22×D25111yescharacteristic
181C22×C4—21no—
182C22×C4—5022no—
183C22×C4—5021no—
184C22×C4—2512no—
185SD16—42no—
186D8—5022no—
187C2×C8—10042no—
188C2×C8—2511no—
189C2×C8—2511no—
190C2×C8C22×D25112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—5022no—
194C4⋊C4—5021no—
195C4⋊C4—2512no—
196C4⋊C4—2511no—
197C4⋊C4—2511no—
198C4⋊C4C22×D25111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—5022no—
201C22⋊C4—5022no—
202C22⋊C4—5022no—
203C22⋊C4—5021no—
204C22⋊C4—2512no—
205C42—5022no—
206C42C22×D25111yescharacteristic
207C23—21no—
208C23—5021no—
209C23—2512no—
210Q8—22no—
211Q8—22no—
212Q8—20081no—
213D4—21no—
214D4—21no—
215D4—20082no—
216D4—10044no—
217D4—5022no—
218D4—5021no—
219D4—2512no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—10044no—
224C2×C4—10042no—
225C2×C4—5022no—
226C2×C4—5021no—
227C2×C4—5022no—
228C2×C4—5022no—
229C2×C4—2512no—
230C2×C4D4×D25111yescharacteristic
231C2×C4D4×D25111yescharacteristic
232C2×C4C23×D25111yescharacteristic
233C8—22no—
234C8—5021no—
235C8—5021no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—10041no—
240C22—10041no—
241C22—5022no—
242C22—5022no—
243C22—5022no—
244C22—5022no—
245C22C2×D4×D25111yescharacteristic
246C4—21no—
247C4—41no—
248C4—42no—
249C4—10041no—
250C4—10042no—
251C4—5022no—
252C4#2254873612yes—
253C4#41×D25111yescharacteristic
254C4#2254876011yescharacteristic
255C2—81no—
256C2—10042no—
257C2—10042no—
258C2#3063233411yescharacteristic
259C2C2×#2254873611yescharacteristic
260C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#461×C251C2
socleC22×C251C22×D4
hypercenter#461D251
hypocenterC251#3303
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #30632402C2semidirect
2. #30632340C2semidirect
3. #30632037C2semidirect
4. #30632026C2semidirect
5. #30631978C2semidirect
6. #30631913C2semidirect
7. #30631831C2semidirect
8. C4×#16860795C2semidirect
9. #30631752C2semidirect
10. #30631682C2semidirect
11. #461×C251C2semidirect
130. C251#3303semidirect
138. #461D251semidirect

Maximal subgroups

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

subgroupindex
#306324022
#306323402
#306320372
#306320262
#306319782
#306319132
#306318312
C4×#168607952
#306317522
#306316822
#461×C2512
#3303251