Finite Group Databasebeta

← #44563079

Subgroups — #44563079

Order 32128.

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

normal subgroups only

This group has 372 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
1Q8×D1004C211yesmaximal, characteristic, complemented
2#30632046C212yesmaximal, complemented
3#30632037C211yesmaximal, characteristic, complemented
4#30631982C212yesmaximal, complemented
5#30631967C211yesmaximal, characteristic, complemented
6#30631899C212yesmaximal
7#30631889C211yesmaximal, characteristic, complemented
8#30631818C211yesmaximal, characteristic, complemented
9#30631785C212yesmaximal, complemented
10C4×#16860795C211yesmaximal, characteristic
11#556×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251—21no—
13C2×#16860797C2212yes—
14C2×#16860795C2211yescharacteristic
15#22548715C2211yescharacteristic
16#22548715C2212yes—
17#22548695—21no—
18#22548695—22no—
19C4×D1004C2211yescharacteristic
20C4×D1004C2212yes—
21#22548648C2212yes—
22#22548648C2211yescharacteristic
23C4⋊(C251⋊C8)C2211yescharacteristic
24C4⋊(C251⋊C8)C2212yes—
25#22548639C2212yes—
26#22548639C2212yes—
27#22548639C2212yes—
28#22548636C2212yes—
29#22548636C2211yescharacteristic
30#22548635C2211yescharacteristic
31#22548626C2212yes—
32C4×C251⋊C8C2211yescharacteristic
33C4⋊Q8×C251C2211yescharacteristic
34#126×C251—22no—
35#124×C251C2212yes—
36C22⋊Q8×C251—21no—
37C4×Q8×C251C2212yes—
38C4×Q8×C251C2211yescharacteristic
39C4×D4×C251C2211yescharacteristic
40#117×C251—21no—
41C2×Dic502—22no—
42C2×Dic502—21no—
43Q8×D251—44no—
44C2×D1004C2311yescharacteristic
45C2×C4×D251—22no—
46C2×C4×D251—21no—
47#16860797—24no—
48#16860795—22no—
49C2×C251⋊C8C2311yescharacteristic
50C2×C251⋊C8C2311yescharacteristic
51C2×C251⋊C8C2312yes—
52C4⋊Dic251—22no—
53C4⋊Dic251—21no—
54C4⋊Dic251C2312yes—
55C4⋊Dic251C2311yescharacteristic
56(C2×D251)⋊C4—22no—
57(C2×D251)⋊C4—21no—
58C2×Q8×C251D411yescharacteristic
59C2×Q8×C251C2312yes—
60C2×D4×C251C2311yescharacteristic
61C22×C4×C251—21no—
62C4⋊C4×C251—21no—
63C4⋊C4×C251—22no—
64C4⋊C4×C251—21no—
65C4⋊C4×C251D411yescharacteristic
66C4⋊C4×C251D412yes—
67C4⋊C4×C251C2311yescharacteristic
68C22⋊C4×C251—22no—
69C22⋊C4×C251—21no—
70C22⋊C4×C251—22no—
71C42×C251—22no—
72C42×C251C2311yescharacteristic
73C42×C251C2312yes—
74C22×D251—21no—
75Dic502—44no—
76Dic502—42no—
77D1004—21no—
78D1004#4112yes—
79C2×Dic251—22no—
80C2×Dic251—21no—
81C4×D251—44no—
82C4×D251—42no—
83C251⋊C8—21no—
84C251⋊C8—22no—
85C251⋊C8—21no—
86C23×C251—21no—
87Q8×C251—22no—
88Q8×C251—22no—
89Q8×C251—22no—
90D4×C251—21no—
91D4×C251—21no—
92C2×C4×C251—21no—
93C2×C4×C251—22no—
94C2×C4×C251—42no—
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×D412yes—
98C2×C4×C251C2×D411yescharacteristic
99C2×C4×C251C2411yescharacteristic
100C22×C251—41no—
101C22×C251—41no—
102C22×C251—41no—
103C22×C251C22×D411yescharacteristic
104C2×D251—22no—
105C2×D251—41no—
106C2×D251—41no—
107C4×C251—22no—
108C4×C251—22no—
109C4×C251—21no—
110C4×C251—42no—
111C4×C251—41no—
112C4×C251AGL(1,ℤ/8)12yes—
113C4×C251C2×#4111yescharacteristic
114C4×C2512−1+411yescharacteristic
115Dic251—42no—
116Dic251—41no—
117C2×C251—81no—
118C2×C251#54811yescharacteristic
119C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C251#57811yescharacteristic
121D251—42no—
122C251#336711yescharacteristic, complemented, Sylow
123#3367—2511nomaximal, Sylow
124#556D25111yescharacteristic, complemented
125D4×Q8—2511no—
126#489—2512no—
127#487—2511no—
128#477—2512no—
129#474—2511no—
130#463—2512no—
131#461—2511no—
132#446—2511no—
133#440—2512no—
134C4×D8—2511no—
135C22×Q8—5021no—
136C2×SD16—2512no—
137C2×D8—2511no—
138C4⋊Q8—2511no—
139C4⋊Q8—2512no—
140C4⋊Q8C2×D25111yescharacteristic
141#126—22no—
142#124C2×D25112yes—
143C22⋊Q8—21no—
144C22⋊Q8—5021no—
145C22⋊Q8—5022no—
146C4×Q8C2×D25112yes—
147C4×Q8C2×D25111yescharacteristic
148C4×D4—2511no—
149C4×D4—2512no—
150C4×D4C2×D25111yescharacteristic
151#117—21no—
152#107—2511no—
153#107—2512no—
154C4⋊C8—2511no—
155C4⋊C8—2512no—
156#103—2512no—
157#103—2512no—
158#103—2512no—
159#102—2511no—
160#102—2512no—
161#102—2511no—
162#97—2512no—
163C4×C8—2511no—
164C2×Q8—10044no—
165C2×Q8—5022no—
166C2×Q8—5021no—
167C2×Q8C22⋊D25111yescharacteristic
168C2×Q8C22×D25112yes—
169C2×D4—2511no—
170C2×D4C22×D25111yescharacteristic
171C22×C4—21no—
172C22×C4—5022no—
173C22×C4—5021no—
174SD16—5024no—
175D8—5022no—
176C2×C8—2511no—
177C2×C8—2511no—
178C2×C8—2512no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—5022no—
183C4⋊C4—5021no—
184C4⋊C4—2511no—
185C4⋊C4—2512no—
186C4⋊C4C22⋊D25111yescharacteristic
187C4⋊C4C22⋊D25112yes—
188C4⋊C4C22×D25111yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—5022no—
193C22⋊C4—5021no—
194C42—22no—
195C42C22×D25111yescharacteristic
196C42C22×D25112yes—
197C23—21no—
198C23—5021no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—10044no—
203Q8—10042no—
204D4—21no—
205D4—21no—
206D4—5021no—
207D4—2512no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—10044no—
212C2×C4—10042no—
213C2×C4—5022no—
214C2×C4—5021no—
215C2×C4C2×C22⋊D25112yes—
216C2×C4C2×C22⋊D25111yescharacteristic
217C2×C4C2×C22⋊D25112yes—
218C2×C4C2×C22⋊D25111yescharacteristic
219C2×C4C23×D25111yescharacteristic
220C8—5021no—
221C8—5022no—
222C8—5021no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—10041no—
227C22—10041no—
228C22—5022no—
229C22C22×C22⋊D25111yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—10042no—
236C4—10041no—
237C4#2254874012yes—
238C4#41×D25111yescharacteristic
239C4#2254876211yescharacteristic
240C2—81no—
241C2—10042no—
242C2#3063234411yescharacteristic
243C2C2×#2254874011yescharacteristic
244C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#556×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860795C22
hypercenter#556D251
hypocenterC251#3367
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D1004C2semidirect
2. #30632046C2semidirect
3. #30632037C2semidirect
4. #30631982C2semidirect
5. #30631967C2semidirect
7. #30631889C2semidirect
8. #30631818C2semidirect
9. #30631785C2semidirect
11. #556×C251C2semidirect
122. C251#3367semidirect
124. #556D251semidirect

Maximal subgroups

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

subgroupindex
Q8×D10042
#306320462
#306320372
#306319822
#306319672
#306318992
#306318892
#306318182
#306317852
C4×#168607952
#556×C2512
#3367251