Finite Group Databasebeta

← #44563018

Subgroups — #44563018

Order 32128.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632411C211yesmaximal, characteristic, complemented
2#30632394C211yesmaximal, characteristic, complemented
3#30632043C212yesmaximal, complemented
4#30632037C211yesmaximal, characteristic, complemented
5#30631992C212yesmaximal, complemented
6#30631978C211yesmaximal, characteristic, complemented
7#30631904C212yesmaximal
8#30631811C211yesmaximal, characteristic, complemented
9SD16×Dic251C212yesmaximal, complemented
10C4×#16860795C211yesmaximal, characteristic
11#461×C251C211yesmaximal, characteristic, complemented
12C2×#16860795C2211yescharacteristic
13#22548715C2211yescharacteristic
14#22548710—22no—
15#22548706C2212yes—
16#22548704C2212yes—
17#22548695—21no—
18#22548694—21no—
19#22548692—22no—
20Q8×Dic251C2212yes—
21C4×Dic502C2211yescharacteristic
22C4×D1004C2211yescharacteristic
23D4×Dic251C2212yes—
24#22548673—21no—
25C2×C4⋊Dic251—21no—
26#22548648C2211yescharacteristic
27#22548647C2212yes—
28Dic251⋊C8C2212yes—
29#22548639C2212yes—
30#22548639C2212yes—
31#22548637C2212yes—
32#22548636C2211yescharacteristic
33#22548635C2211yescharacteristic
34C4×C251⋊C8C2211yescharacteristic
35C8×Dic251C2212yes—
36C2×SD16×C251C2212yes—
37C4⋊Q8×C251C2211yescharacteristic
38C4×D4×C251C2211yescharacteristic
39C4⋊C8×C251C2211yescharacteristic
40#103×C251C2212yes—
41C2×Dic502—21no—
42C2×Dic502D411yescharacteristic
43C2×D1004C2311yescharacteristic
44C22×Dic251—22no—
45C2×C4×D251—21no—
46#16860795—22no—
47C2×C251⋊C8C2311yescharacteristic
48C2×C251⋊C8C2311yescharacteristic
49C4⋊Dic251—22no—
50C4⋊Dic251—22no—
51C4⋊Dic251—21no—
52C4⋊Dic251D411yescharacteristic
53C4⋊Dic251C2311yescharacteristic
54C4⋊Dic251C2312yes—
55Dic251⋊C4—22no—
56Dic251⋊C4D412yes—
57C22⋊Dic251—22no—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—21no—
61C4×Dic251—22no—
62C4×Dic251C2312yes—
63C2×Q8×C251C2312yes—
64C2×D4×C251C2311yescharacteristic
65C22×C4×C251—21no—
66SD16×C251—24no—
67C2×C8×C251C2312yes—
68C4⋊C4×C251—21no—
69C4⋊C4×C251—21no—
70C4⋊C4×C251C2311yescharacteristic
71C22⋊C4×C251—21no—
72C42×C251C2311yescharacteristic
73C22×D251—21no—
74Dic502—22no—
75Dic502—81no—
76D1004—21no—
77D1004—21no—
78C2×Dic251—21no—
79C2×Dic251—22no—
80C2×Dic251—82no—
81C2×Dic251C2×D412yes—
82C2×Dic251C2×D412yes—
83C4×D251—42no—
84C251⋊C8—21no—
85C251⋊C8—21no—
86C23×C251—21no—
87Q8×C251—22no—
88Q8×C251—22no—
89D4×C251—21no—
90D4×C251#4112yes—
91C2×C4×C251—21no—
92C2×C4×C251—22no—
93C2×C4×C251—42no—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2411yescharacteristic
97C8×C251—22no—
98C22×C251—22no—
99C22×C251—41no—
100C22×C251—41no—
101C22×C251C22×D411yescharacteristic
102C2×D251—41no—
103C2×D251—41no—
104C2×D251—41no—
105C4×C251—21no—
106C4×C251—42no—
107C4×C251—41no—
108C4×C251#13512yes—
109C4×C251C2×#4111yescharacteristic
110C4×C2512−1+411yescharacteristic
111Dic251—22no—
112Dic251—22no—
113Dic251—41no—
114Dic251—42no—
115C2×C251—42no—
116C2×C251#54811yescharacteristic
117C2×C251C2×#13511yescharacteristic
118C2×C251#57711yescharacteristic
119D251—81no—
120C251#336011yescharacteristic, complemented, Sylow
121#3360—2511nomaximal, Sylow
122#556—2511no—
123#554—2511no—
124#488—2512no—
125#487—2511no—
126#479—2512no—
127#476—2511no—
128#463—2512no—
129#461D25111yescharacteristic, complemented
130C8×Q8—2511no—
131C4×SD16—2512no—
132C4×D8—2511no—
133C2×SD16C2×D25112yes—
134C2×D8—2511no—
135C4⋊Q8—2511no—
136C4⋊Q8C2×D25111yescharacteristic
137#126—5022no—
138#125—2512no—
139#124—2512no—
140C22⋊Q8—5022no—
141C22⋊Q8—5021no—
142C22⋊Q8—5021no—
143C4×Q8—2512no—
144C4×Q8—2511no—
145C4×D4—2512no—
146C4×D4—2511no—
147C4×D4C2×D25111yescharacteristic
148#117—5021no—
149C2×C4⋊C4—5021no—
150#107—2511no—
151#106—2512no—
152C4⋊C8—2512no—
153C4⋊C8C2×D25111yescharacteristic
154#103—2512no—
155#103—2512no—
156#103C2×D25112yes—
157#102—2511no—
158#102—2512no—
159#102—2511no—
160C4×C8—2512no—
161C4×C8—2511no—
162C2×Q8—5021no—
163C2×Q8—2511no—
164C2×Q8C22×D25112yes—
165C2×D4—2511no—
166C2×D4C22×D25111yescharacteristic
167C22×C4—21no—
168C22×C4—5022no—
169C22×C4—5021no—
170SD16—24no—
171D8—5022no—
172C2×C8—2511no—
173C2×C8—2511no—
174C2×C8C22×D25112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—5022no—
178C4⋊C4—5022no—
179C4⋊C4—5022no—
180C4⋊C4—5021no—
181C4⋊C4—2512no—
182C4⋊C4—2511no—
183C4⋊C4—2511no—
184C4⋊C4—2512no—
185C4⋊C4C22×D25111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—5022no—
188C22⋊C4—5022no—
189C22⋊C4—5022no—
190C22⋊C4—5021no—
191C42—5022no—
192C42—2512no—
193C42C22×D25111yescharacteristic
194C23—21no—
195C23—5021no—
196Q8—22no—
197Q8—22no—
198Q8—20081no—
199Q8—5022no—
200D4—21no—
201D4—5021no—
202D4—5021no—
203D4#1686080912yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—20082no—
208C2×C4—10042no—
209C2×C4—5021no—
210C2×C4—5022no—
211C2×C4—2512no—
212C2×C4—2512no—
213C2×C4D4×D25111yescharacteristic
214C2×C4D4×D25111yescharacteristic
215C2×C4C23×D25111yescharacteristic
216C8—22no—
217C8—5021no—
218C8—5021no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—10041no—
223C22—10041no—
224C22—10041no—
225C22C2×D4×D25111yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—10041no—
230C4—10042no—
231C4—5022no—
232C4—5022no—
233C4#2254873612yes—
234C4C2×#1686080911yescharacteristic
235C4#2254876211yescharacteristic
236C2—42no—
237C2—20081no—
238C2#3063234311yescharacteristic
239C2C2×#2254873611yescharacteristic
240C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#461×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860795C22
hypercenter#461D251
hypocenterC251#3360
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. #30632411C2semidirect
2. #30632394C2semidirect
3. #30632043C2semidirect
4. #30632037C2semidirect
5. #30631992C2semidirect
6. #30631978C2semidirect
8. #30631811C2semidirect
9. SD16×Dic251C2semidirect
11. #461×C251C2semidirect
120. C251#3360semidirect
129. #461D251semidirect

Maximal subgroups

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

subgroupindex
#306324112
#306323942
#306320432
#306320372
#306319922
#306319782
#306319042
#306318112
SD16×Dic2512
C4×#168607952
#461×C2512
#3360251