Finite Group Databasebeta

← #44562233

Subgroups — #44562233

Order 32128.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#30632352C211yesmaximal, characteristic, complemented
2#30632342C211yesmaximal, characteristic
3#30632037C211yesmaximal, characteristic, complemented
4#30632016C212yesmaximal, complemented
5#30631973C211yesmaximal, characteristic, complemented
6#30631913C212yesmaximal, complemented
7#30631824C212yesmaximal, complemented
8C4×#16860798C211yesmaximal, characteristic, complemented
9#30631752C211yesmaximal, characteristic, complemented
10#103×D251C212yesmaximal, complemented
11#461×C251C211yesmaximal, characteristic, complemented
12C2×#16860810—21no—
13C2×Q8×D251C2212yes—
14C2×#16860798C2211yescharacteristic
15C2×C8×D251C2212yes—
16#22548715C2211yescharacteristic
17#22548714—21no—
18#22548697—22no—
19#22548694—21no—
20#22548693—22no—
21#22548691C2212yes—
22#22548686C2212yes—
23C4×Dic502C2211yescharacteristic
24C4×D1004C2211yescharacteristic
25#22548677—21no—
26C4⋊C4×D251C2212yes—
27#22548647C2212yes—
28#22548646C2211yescharacteristic
29#22548640C2212yes—
30#22548639C2212yes—
31#22548639C2212yes—
32#22548638C2211yescharacteristic
33#22548636C2211yescharacteristic
34(C2×D251)⋊C8C2212yes—
35C4×C251⋊C8C2211yescharacteristic
36C2×SD16×C251C2212yes—
37C4⋊Q8×C251C2211yescharacteristic
38C4×D4×C251C2211yescharacteristic
39C4⋊C8×C251C2211yescharacteristic
40#103×C251C2212yes—
41#16860810—44no—
42C2×Dic502—22no—
43C2×Dic502—21no—
44C2×Dic502C2311yescharacteristic
45Q8×D251—24no—
46Q8×D251—42no—
47C2×D1004D411yescharacteristic
48C2×C22⋊D251—22no—
49C2×C4×D251—22no—
50C2×C4×D251—22no—
51C2×C4×D251C2312yes—
52#16860798—22no—
53C2×C251⋊C8C2311yescharacteristic
54C2×C251⋊C8C2311yescharacteristic
55C8×D251—42no—
56C4⋊Dic251—21no—
57C4⋊Dic251D411yescharacteristic
58C4⋊Dic251C2311yescharacteristic
59C4⋊Dic251C2312yes—
60Dic251⋊C4—22no—
61Dic251⋊C4—22no—
62Dic251⋊C4—21no—
63C22⋊Dic251—22no—
64(C2×D251)⋊C4—22no—
65(C2×D251)⋊C4D412yes—
66C4×Dic251—21no—
67C2×Q8×C251C2312yes—
68C2×D4×C251C2311yescharacteristic
69C22×C4×C251—21no—
70SD16×C251—42no—
71C2×C8×C251C2312yes—
72C4⋊C4×C251—21no—
73C4⋊C4×C251—21no—
74C4⋊C4×C251C2311yescharacteristic
75C22⋊C4×C251—21no—
76C42×C251C2311yescharacteristic
77C22×D251C2×D412yes—
78Dic502—21no—
79Dic502—44no—
80Dic502—42no—
81Dic502#4112yescomplemented
82D1004SD1614yescomplemented
83C22⋊D251—44no—
84C2×Dic251—21no—
85C2×Dic251—21no—
86C2×Dic251—22no—
87C2×Dic251C2×D412yes—
88C4×D251—22no—
89C4×D251—22no—
90C4×D251—44no—
91C4×D251—44no—
92C251⋊C8—21no—
93C251⋊C8—21no—
94C23×C251—21no—
95Q8×C251—22no—
96Q8×C251—22no—
97D4×C251—21no—
98D4×C251—21no—
99C2×C4×C251—21no—
100C2×C4×C251—22no—
101C2×C4×C251—42no—
102C2×C4×C251C2×D411yescharacteristic
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251C2411yescharacteristic
105C8×C251—22no—
106C22×C251—41no—
107C22×C251—41no—
108C22×C251—41no—
109C22×C251C22×D411yescharacteristic
110C2×D251—22no—
111C2×D251—22no—
112C2×D251C2×SD1614yes—
113C4×C251—21no—
114C4×C251—42no—
115C4×C251—41no—
116C4×C251C2×SD1612yes—
117C4×C251C2×#4111yescharacteristic
118C4×C251H(5,𝔽2)11yescharacteristic
119Dic251—22no—
120Dic251—22no—
121Dic251—42no—
122Dic251—41no—
123C2×C251—81no—
124C2×C251#54611yescharacteristic
125C2×C251C22×SD1611yescharacteristic
126C2×C251#57811yescharacteristic
127D251—24no—
128C251#329811yescharacteristic, complemented, Sylow
129#3298—2511nomaximal, Sylow
130D4×Q8—2511no—
131#547—2511no—
132#487—2511no—
133#483—2512no—
134#475—2511no—
135#465—2512no—
136#461D25111yescharacteristic, complemented
137#448—2512no—
138C4×SD16—2511no—
139C8×D4—2511no—
140C2×#103—2512no—
141C2×#41—5021no—
142C22×Q8—2512no—
143C2×SD16—2511no—
144C2×SD16C2×D25112yes—
145C22×C8—2512no—
146C4⋊Q8—5021no—
147C4⋊Q8—2511no—
148C4⋊Q8C2×D25111yescharacteristic
149#123—5022no—
150C22⋊Q8—5021no—
151C22⋊Q8—5022no—
152C22⋊Q8—2512no—
153#121—2512no—
154C4×Q8—2511no—
155C4×D4—5021no—
156C4×D4—2511no—
157C4×D4C2×D25111yescharacteristic
158C2×C4⋊C4—2512no—
159#106—2512no—
160#106—2511no—
161C4⋊C8C2×D25111yescharacteristic
162#103—2512no—
163#103—2512no—
164#103—2512no—
165#103—2511no—
166#103C2×D25112yes—
167#102—2511no—
168C22⋊C8—2512no—
169C4×C8—2511no—
170#41—10044no—
171C2×Q8—10042no—
172C2×Q8—5022no—
173C2×Q8—5024no—
174C2×Q8—5021no—
175C2×Q8—2511no—
176C2×Q8C22×D25112yes—
177C2×D4—5022no—
178C2×D4—2511no—
179C2×D4C22×D25111yescharacteristic
180C22×C4—21no—
181C22×C4—5022no—
182C22×C4—5022no—
183C22×C4—2512no—
184SD16—42no—
185SD16—5022no—
186C2×C8—10042no—
187C2×C8—2511no—
188C2×C8—2511no—
189C2×C8C22×D25112yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—5022no—
193C4⋊C4—5022no—
194C4⋊C4—5021no—
195C4⋊C4—5021no—
196C4⋊C4—2511no—
197C4⋊C4—2512no—
198C4⋊C4—2511no—
199C4⋊C4C22×D25111yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—5022no—
202C22⋊C4—5022no—
203C22⋊C4—2512no—
204C42—5021no—
205C42C22×D25111yescharacteristic
206C23—21no—
207C23—2512no—
208Q8—22no—
209Q8—22no—
210Q8—10044no—
211Q8—10042no—
212Q8—5021no—
213Q8—2512no—
214D4—21no—
215D4—21no—
216D4—10044no—
217D4—2514no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—10044no—
222C2×C4—10044no—
223C2×C4—5021no—
224C2×C4—5022no—
225C2×C4—5021no—
226C2×C4—5022no—
227C2×C4—5022no—
228C2×C4—2512no—
229C2×C4D4×D25111yescharacteristic
230C2×C4D4×D25111yescharacteristic
231C2×C4C23×D25111yescharacteristic
232C8—22no—
233C8—5021no—
234C8—5021no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—5022no—
239C22—5022no—
240C22—2514no—
241C22C2×D4×D25111yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—10042no—
246C4—10041no—
247C4—5022no—
248C4—5022no—
249C4SD16×D25112yes—
250C4#41×D25111yescharacteristic
251C4#2254876011yescharacteristic
252C2—81no—
253C2—5024no—
254C2#3063233411yescharacteristic
255C2C2×SD16×D25111yescharacteristic
256C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#461×C251C2
socleC22×C251C22×D4
prime radical#30632342C2
hypercenter#461D251
hypocenterC251#3298
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. #30632352C2semidirect
3. #30632037C2semidirect
4. #30632016C2semidirect
5. #30631973C2semidirect
6. #30631913C2semidirect
7. #30631824C2semidirect
8. C4×#16860798C2semidirect
9. #30631752C2semidirect
10. #103×D251C2semidirect
11. #461×C251C2semidirect
81. Dic502#41semidirect
82. D1004SD16semidirect
128. C251#3298semidirect
136. #461D251semidirect

Maximal subgroups

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

subgroupindex
#306323522
#306323422
#306320372
#306320162
#306319732
#306319132
#306318242
C4×#168607982
#306317522
#103×D2512
#461×C2512
#3298251