Finite Group Databasebeta

← #44563010

Subgroups — #44563010

Order 32128.

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

normal subgroups only

This group has 350 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#30632397C211yesmaximal, characteristic, complemented
2#30632041C212yesmaximal, complemented
3#30632037C211yesmaximal, characteristic, complemented
4#30631994C212yesmaximal, complemented
5#30631977C211yesmaximal, characteristic, complemented
6#30631899C212yesmaximal
7#30631889C211yesmaximal, characteristic, complemented
8Q8×C251⋊C8C211yesmaximal, characteristic, complemented
9C4×#16860797C212yesmaximal, complemented
10C4×#16860795C211yesmaximal, characteristic
11#556×C251C211yesmaximal, characteristic, complemented
12C2×#16860797C2212yes—
13C2×#16860795C2211yescharacteristic
14#22548715C2211yescharacteristic
15#22548707C2212yes—
16#22548693—22no—
17#22548691—21no—
18C4×D1004C2211yescharacteristic
19C4×D1004C2212yes—
20C4⋊C4×D251—21no—
21#22548648C2211yescharacteristic
22#22548646C2212yes—
23C4⋊(C251⋊C8)C2211yescharacteristic
24C4⋊(C251⋊C8)C2212yes—
25#22548639C2212yes—
26#22548639C2212yes—
27#22548639C2212yes—
28#22548636C2212yes—
29#22548636C2211yescharacteristic
30#22548635C2211yescharacteristic
31C4×C251⋊C8C2211yescharacteristic
32C4×C251⋊C8C2212yes—
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—21no—
42C2×D1004C2311yescharacteristic
43C2×C4×D251—21no—
44C2×C4×D251—22no—
45#16860797—24no—
46#16860795—22no—
47C2×C251⋊C8C2311yescharacteristic
48C2×C251⋊C8C2312yes—
49C2×C251⋊C8C2311yescharacteristic
50C4⋊Dic251—21no—
51C4⋊Dic251C2312yes—
52C4⋊Dic251C2311yescharacteristic
53Dic251⋊C4—22no—
54Dic251⋊C4—22no—
55(C2×D251)⋊C4—22no—
56(C2×D251)⋊C4—21no—
57C2×Q8×C251D411yescharacteristic
58C2×Q8×C251C2312yes—
59C2×D4×C251C2311yescharacteristic
60C22×C4×C251—21no—
61C4⋊C4×C251—21no—
62C4⋊C4×C251—22no—
63C4⋊C4×C251—21no—
64C4⋊C4×C251D411yescharacteristic
65C4⋊C4×C251D412yes—
66C4⋊C4×C251C2311yescharacteristic
67C22⋊C4×C251—22no—
68C22⋊C4×C251—21no—
69C22⋊C4×C251—22no—
70C42×C251—22no—
71C42×C251C2311yescharacteristic
72C42×C251C2312yes—
73C22×D251—21no—
74Dic502—81no—
75D1004—21no—
76D1004#4112yes—
77C2×Dic251—21no—
78C2×Dic251—22no—
79C4×D251—42no—
80C4×D251—82no—
81C251⋊C8—21no—
82C251⋊C8—22no—
83C251⋊C8—21no—
84C23×C251—21no—
85Q8×C251—22no—
86Q8×C251—22no—
87Q8×C251—22no—
88D4×C251—21no—
89D4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251—22no—
92C2×C4×C251—42no—
93C2×C4×C251C2×D412yes—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2411yescharacteristic
98C22×C251—41no—
99C22×C251—41no—
100C22×C251—41no—
101C22×C251C22×D411yescharacteristic
102C2×D251—22no—
103C2×D251—41no—
104C2×D251—41no—
105C4×C251—22no—
106C4×C251—22no—
107C4×C251—21no—
108C4×C251—42no—
109C4×C251—41no—
110C4×C251#13512yes—
111C4×C251C2×#4111yescharacteristic
112C4×C2512−1+411yescharacteristic
113Dic251—41no—
114Dic251—42no—
115C2×C251—81no—
116C2×C251#54811yescharacteristic
117C2×C251C2×#13511yescharacteristic
118C2×C251#57711yescharacteristic
119D251—42no—
120C251#336011yescharacteristic, complemented, Sylow
121#3360—2511nomaximal, Sylow
122#556D25111yescharacteristic, complemented
123#554—2511no—
124#488—2512no—
125#487—2511no—
126#479—2512no—
127#476—2511no—
128#463—2512no—
129#461—2511no—
130C8×Q8—2511no—
131C4×SD16—2512no—
132C4×D8—2511no—
133C2×SD16—2512no—
134C2×D8—2511no—
135C4⋊Q8—2511no—
136C4⋊Q8C2×D25111yescharacteristic
137#126—22no—
138#125—2512no—
139#124C2×D25112yes—
140C22⋊Q8—21no—
141C22⋊Q8—5021no—
142C22⋊Q8—5022no—
143C4×Q8C2×D25112yes—
144C4×Q8C2×D25111yescharacteristic
145C4×D4—2511no—
146C4×D4—2512no—
147C4×D4C2×D25111yescharacteristic
148#117—21no—
149C2×C4⋊C4—5021no—
150#107—2511no—
151#106—2512no—
152C4⋊C8—2511no—
153C4⋊C8—2512no—
154#103—2512no—
155#103—2512no—
156#103—2512no—
157#102—2511no—
158#102—2512no—
159#102—2511no—
160C4×C8—2511no—
161C4×C8—2512no—
162C2×Q8—5021no—
163C2×Q8C22⋊D25111yescharacteristic
164C2×Q8C22×D25112yes—
165C2×D4—2511no—
166C2×D4C22×D25111yescharacteristic
167C22×C4—21no—
168C22×C4—5022no—
169C22×C4—5021no—
170SD16—5024no—
171D8—5022no—
172C2×C8—2511no—
173C2×C8—2512no—
174C2×C8—2511no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—5022no—
179C4⋊C4—5021no—
180C4⋊C4—5022no—
181C4⋊C4—2512no—
182C4⋊C4—2511no—
183C4⋊C4C22⋊D25111yescharacteristic
184C4⋊C4C22⋊D25112yes—
185C4⋊C4C22×D25111yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—5021no—
190C22⋊C4—5022no—
191C42—22no—
192C42C22×D25111yescharacteristic
193C42C22×D25112yes—
194C23—21no—
195C23—5021no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—20081no—
200D4—21no—
201D4—21no—
202D4—5021no—
203D4—2512no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—20082no—
208C2×C4—10042no—
209C2×C4—5021no—
210C2×C4—5022no—
211C2×C4C2×C22⋊D25112yes—
212C2×C4C2×C22⋊D25112yes—
213C2×C4C2×C22⋊D25111yescharacteristic
214C2×C4C2×C22⋊D25111yescharacteristic
215C2×C4C23×D25111yescharacteristic
216C8—5021no—
217C8—5022no—
218C8—5021no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—10041no—
223C22—10041no—
224C22—5022no—
225C22C22×C22⋊D25111yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—10041no—
232C4—10042no—
233C4#2254873512yes—
234C4#41×D25111yescharacteristic
235C4#2254876211yescharacteristic
236C2—81no—
237C2—10042no—
238C2#3063234411yescharacteristic
239C2C2×#2254873511yescharacteristic
240C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#556×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860795C22
hypercenter#556D251
hypocenterC251#3360
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. #30632397C2semidirect
2. #30632041C2semidirect
3. #30632037C2semidirect
4. #30631994C2semidirect
5. #30631977C2semidirect
7. #30631889C2semidirect
8. Q8×C251⋊C8C2semidirect
9. C4×#16860797C2semidirect
11. #556×C251C2semidirect
120. C251#3360semidirect
122. #556D251semidirect

Maximal subgroups

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

subgroupindex
#306323972
#306320412
#306320372
#306319942
#306319772
#306318992
#306318892
Q8×C251⋊C82
C4×#168607972
C4×#168607952
#556×C2512
#3360251