Finite Group Databasebeta

← #501×D251 #44560880

Subgroups — #501×D251 #44560880

Order 32128.

384 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D251C211yesmaximal, characteristic
2#125×D251C211yesmaximal, characteristic
3#30632097C211yesmaximal, characteristic, complemented
4#30632096C211yesmaximal, characteristic, complemented
5#30632095C212yesmaximal, complemented
6#30632094C212yesmaximal, complemented
7#30632093C211yesmaximal, characteristic, complemented
8#107×D251C212yesmaximal
9#106×D251C212yesmaximal
10#97×D251C211yesmaximal, characteristic
11#501×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251—21no—
13C2×C8×D251C2212yes—
14#22548715C2211yescharacteristic
15#22548714—21no—
16#22548714C2212yes—
17#22548713—21no—
18#22548707C2211yescharacteristic
19#22548706C2211yescharacteristic
20#22548706C2211yescharacteristic
21#22548705—21no—
22#22548705—21no—
23C4⋊C4×D251—21no—
24C4⋊C4×D251—21no—
25C4⋊C4×D251—21no—
26C4⋊C4×D251C2211yescharacteristic
27C4⋊C4×D251C2211yescharacteristic
28C4⋊C4×D251C2212yes—
29C42×D251C2211yescharacteristic
30#22548649C2212yes—
31#22548648C2212yes—
32#22548648C2212yes—
33#22548647C2212yes—
34#22548646C2212yes—
35#22548646C2212yes—
36#22548626C2211yescharacteristic
37#22548625C2212yes—
38C4⋊Q8×C251C2211yescharacteristic
39#125×C251C2211yescharacteristic
40#107×C251C2212yes—
41#106×C251C2212yes—
42#97×C251C2211yescharacteristic
43C2×Dic502—22no—
44C2×Dic502—21no—
45Q8×D251—44no—
46C2×C4×D251—22no—
47C2×C4×D251—21no—
48C2×C4×D251—21no—
49C2×C4×D251D411yescharacteristic
50C2×C4×D251D411yescharacteristic
51C2×C4×D251C2311yescharacteristic
52C2×C251⋊C8C2312yes—
53C8×D251Q818yescomplemented
54C4⋊Dic251—21no—
55C4⋊Dic251—21no—
56C4⋊Dic251—21no—
57C4⋊Dic251C2311yescharacteristic
58C4⋊Dic251C2312yes—
59C4⋊Dic251C2311yescharacteristic
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—22no—
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65Dic251⋊C4—22no—
66Dic251⋊C4—21no—
67Dic251⋊C4—21no—
68C4×Dic251D411yescharacteristic
69C4×Dic251D411yescharacteristic
70C4×Dic251C2311yescharacteristic
71C2×Q8×C251—21no—
72C2×C8×C251C2312yes—
73C4⋊C4×C251—21no—
74C4⋊C4×C251—21no—
75C4⋊C4×C251—21no—
76C4⋊C4×C251C2311yescharacteristic
77C4⋊C4×C251C2312yes—
78C4⋊C4×C251C2311yescharacteristic
79C42×C251C2311yescharacteristic
80C22×D251C2×D411yescharacteristic
81Dic502—44no—
82Dic502—42no—
83C2×Dic251—21no—
84C2×Dic251—21no—
85C2×Dic251—22no—
86C2×Dic251C2×D411yescharacteristic
87C2×Dic251C2×D411yescharacteristic
88C2×Dic251C2×D411yescharacteristic
89C4×D251—22no—
90C4×D251—22no—
91C4×D251—42no—
92C4×D251—44no—
93C4×D251—42no—
94C4×D251C2×Q814yes—
95C251⋊C8C2×Q814yescomplemented
96Q8×C251—42no—
97C2×C4×C251—21no—
98C2×C4×C251—21no—
99C2×C4×C251—22no—
100C2×C4×C251C2×D411yescharacteristic
101C2×C4×C251C2×D411yescharacteristic
102C2×C4×C251C2411yescharacteristic
103C8×C251C2×Q814yescomplemented
104C22×C251C22×D411yescharacteristic
105C2×D251C4⋊Q812yes—
106C2×D251AGL(1,ℤ/8)12yes—
107C2×D251#13712yes—
108C4×C251—21no—
109C4×C251—21no—
110C4×C251—42no—
111C4×C251—41no—
112C4×C251—41no—
113C4×C251C22×Q812yes—
114Dic251—21no—
115Dic251—21no—
116Dic251—41no—
117Dic251—42no—
118Dic251—41no—
119Dic251C4⋊Q812yes—
120C2×C251C2×C4⋊Q811yescharacteristic
121C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C251C2×#13711yescharacteristic
123D251#50114yesdirect product factor
124C251C2×#50111yescharacteristic, complemented, Sylow
125C2×#501—2511nomaximal, Sylow
126C2×C4⋊Q8—2511no—
127C2×#125—2511no—
128#501—2511no—
129#501—2511no—
130#501—2512no—
131#501—2512no—
132#501—2511no—
133#501D25111yescharacteristic, direct product factor
134C2×#107—2512no—
135C2×#106—2512no—
136C2×#97—2511no—
137C22×Q8—5021no—
138C22×C8—2512no—
139C4⋊Q8—5021no—
140C4⋊Q8—5021no—
141C4⋊Q8—2512no—
142C4⋊Q8—2511no—
143C4⋊Q8C2×D25111yescharacteristic
144#125—5021no—
145#125—5021no—
146#125—2511no—
147#125—2511no—
148#125—2511no—
149#125C2×D25111yescharacteristic
150C2×C4⋊C4—5021no—
151C2×C4⋊C4—5021no—
152C2×C4⋊C4—5021no—
153C2×C4⋊C4—2511no—
154C2×C4⋊C4—2511no—
155C2×C4⋊C4—2512no—
156C2×C42—2511no—
157#107—2512no—
158#107—2512no—
159#107—2512no—
160#107C2×D25112yes—
161#106—2512no—
162#106—2512no—
163#106—2512no—
164#106C2×D25112yes—
165#97—2512no—
166#97—2511no—
167#97C2×D25111yescharacteristic
168C2×Q8—21no—
169C2×Q8—10044no—
170C2×Q8—5022no—
171C2×Q8—5021no—
172C22×C4—5021no—
173C22×C4—5021no—
174C22×C4—5022no—
175C22×C4—2511no—
176C22×C4—2511no—
177C22×C4—2511no—
178C2×C8—2512no—
179C2×C8—2518no—
180C2×C8C22×D25112yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—5021no—
185C4⋊C4—5021no—
186C4⋊C4—5022no—
187C4⋊C4—5021no—
188C4⋊C4—5021no—
189C4⋊C4—5021no—
190C4⋊C4—5021no—
191C4⋊C4—5022no—
192C4⋊C4—5021no—
193C4⋊C4—5021no—
194C4⋊C4—5021no—
195C4⋊C4—2511no—
196C4⋊C4—2512no—
197C4⋊C4—2511no—
198C4⋊C4C22×D25111yescharacteristic
199C4⋊C4C22×D25112yes—
200C4⋊C4C22×D25111yescharacteristic
201C42—2511no—
202C42—2511no—
203C42—2511no—
204C42C22×D25111yescharacteristic
205C23—2511no—
206Q8—42no—
207Q8—10042no—
208Q8—10044no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—10044no—
213C2×C4—10042no—
214C2×C4—10042no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5021no—
218C2×C4—5021no—
219C2×C4—5022no—
220C2×C4—2511no—
221C2×C4—2511no—
222C2×C4—2511no—
223C2×C4—2514no—
224C2×C4D4×D25111yescharacteristic
225C2×C4D4×D25111yescharacteristic
226C2×C4C23×D25111yescharacteristic
227C8—2514no—
228C8Q8×D25114yescomplemented
229C22—2512no—
230C22—2512no—
231C22—2512no—
232C22C2×D4×D25111yescharacteristic
233C4—21no—
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—10042no—
239C4—10041no—
240C4—10041no—
241C4—5021no—
242C4—5021no—
243C4—2512no—
244C4C2×Q8×D25112yes—
245C2—2514no—
246C2C4⋊Q8×D25111yescharacteristic
247C2AGL(1,ℤ/8)×D25111yescharacteristic
248C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#501×C251C2
socleC22×C251C22×D4
prime radicalC22×D251C2×D4
hypercenter#501D251
hypocenterC251C2×#501
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #30632097C2semidirect
4. #30632096C2semidirect
5. #30632095C2semidirect
6. #30632094C2semidirect
7. #30632093C2semidirect
11. #501×C251C2semidirect
53. C8×D251Q8semidirect
95. C251⋊C8C2×Q8semidirect
103. C8×C251C2×Q8semidirect
123. D251#501direct (N × H)
124. C251C2×#501semidirect
133. #501D251direct (N × H)
228. C8Q8×D251semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2512
#125×D2512
#306320972
#306320962
#306320952
#306320942
#306320932
#107×D2512
#106×D2512
#97×D2512
#501×C2512
C2×#501251