Finite Group Databasebeta

← #44558740

Subgroups — #44558740

Order 32128.

504 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×#22548714C211yesmaximal, characteristic
2C2×#22548713C211yesmaximal, characteristic
3#30631594C211yesmaximal, characteristic
4#30631593C211yesmaximal, characteristic
5#30631592C211yesmaximal, characteristic
6#30631591C214yesmaximal
7#30631551C214yesmaximal
8C42×Dic251C211yesmaximal, characteristic
9#391×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C22×Dic502C2212yes—
12#22548714—24no—
13#22548713—24no—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×C4×Dic251C2212yes—
19C2×C4×Dic251C2211yescharacteristic
20C2×C4×Dic251C2211yescharacteristic
21C2×C4×Dic251C2212yes—
22C251⋊#95C2214yes—
23C251⋊#95C2214yes—
24C251⋊#95C2214yes—
25C22×Q8×C251C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C42×C251C2211yescharacteristic
28#95×C251C2214yes—
29C2×Dic502—24no—
30C2×Dic502—28no—
31C2×Dic502—22no—
32C2×Dic502—22no—
33C2×Dic502—22no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2311yescharacteristic
38C22×Dic251C2311yescharacteristic
39C4⋊Dic251—22no—
40Dic251⋊C4—24no—
41Dic251⋊C4—24no—
42Dic251⋊C4—24no—
43C4×Dic251—24no—
44C4×Dic251—22no—
45C4×Dic251D414yes—
46C4×Dic251Q818yescomplemented
47C2×Q8×C251—22no—
48C2×Q8×C251—22no—
49C2×Q8×C251—22no—
50C22×C4×C251C2312yes—
51C22×C4×C251C2312yes—
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C4⋊C4×C251—22no—
55C42×C251—22no—
56Dic502—44no—
57Dic502—48no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—24no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251C2×D412yes—
66C2×Dic251C2×D412yes—
67C2×Dic251C2×Q814yes—
68C2×Dic251C2×Q814yes—
69C2×Dic251#4114yes—
70C23×C251C2411yescharacteristic
71Q8×C251—44no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—24no—
79C2×C4×C251C2×D412yes—
80C2×C4×C251C2×Q814yes—
81C2×C4×C251#4112yes—
82C2×C4×C251#4112yes—
83C22×C251C22×D411yescharacteristic
84C22×C251C22×Q812yes—
85C22×C251C2×#4112yes—
86C22×C251C2×#4111yescharacteristic
87C22×C251C2×#4111yescharacteristic
88C4×C251—24no—
89C4×C251—42no—
90C4×C251—42no—
91C4×C251#12414yes—
92Dic251—24no—
93Dic251—42no—
94Dic251—42no—
95Dic251#12414yes—
96Dic251C4⋊Q814yes—
97C2×C251C2×#12411yescharacteristic
98C2×C251C2×#12411yescharacteristic
99C2×C251C2×C4⋊Q811yescharacteristic
100C2×C251#53312yes—
101C2×C251#53312yes—
102C251#286211yescharacteristic, complemented, Sylow
103#2862—2511nomaximal, Sylow
104C2×C4⋊Q8—2511no—
105C2×C4⋊Q8—2511no—
106#391—2511no—
107#391—2511no—
108#391—2511no—
109#391—2514no—
110#391D25111yescharacteristic
111#382—2514no—
112C43—2511no—
113C22×Q8—2511no—
114C22×Q8—2512no—
115C22×Q8C2×D25111yescharacteristic
116C4⋊Q8—5024no—
117C4⋊Q8—5024no—
118C2×C4⋊C4—2512no—
119C2×C4⋊C4—2512no—
120C2×C4⋊C4—2512no—
121C2×C4⋊C4—2511no—
122C2×C4⋊C4C2×D25111yescharacteristic
123C2×C42—2512no—
124C2×C42—2511no—
125C2×C42—2511no—
126C2×C42—2512no—
127C2×C42C2×D25111yescharacteristic
128#95—2514no—
129#95—2514no—
130#95—2514no—
131#95C2×D25114yes—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—5024no—
136C2×Q8—5028no—
137C2×Q8—5022no—
138C2×Q8—5022no—
139C2×Q8—5022no—
140C22×C4—2512no—
141C22×C4—2512no—
142C22×C4—2512no—
143C22×C4—2511no—
144C22×C4—2511no—
145C22×C4C22×D25112yes—
146C22×C4C22×D25112yes—
147C22×C4C22×D25112yes—
148C22×C4C22×D25111yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—5024no—
151C4⋊C4—5024no—
152C4⋊C4—5024no—
153C4⋊C4—5022no—
154C42—22no—
155C42—5024no—
156C42—5022no—
157C42—2514no—
158C42—2518no—
159C23C23×D25111yescharacteristic
160Q8—44no—
161Q8—10048no—
162Q8—10044no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—5022no—
171C2×C4—5022no—
172C2×C4—5024no—
173C2×C4—5022no—
174C2×C4—5022no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—2514no—
178C2×C4—2512no—
179C2×C4—2512no—
180C2×C4—2514no—
181C2×C4—2514no—
182C2×C4D4×D25112yes—
183C2×C4Q8×D25114yes—
184C2×C4#1686080912yes—
185C2×C4#1686081012yes—
186C22C2×D4×D25111yescharacteristic
187C22C2×Q8×D25112yes—
188C22#41×D25112yes—
189C22C2×#1686080911yescharacteristic
190C22C2×#1686081011yescharacteristic
191C4—24no—
192C4—42no—
193C4—42no—
194C4—10042no—
195C4—10042no—
196C4—5024no—
197C4—2514no—
198C4—2514no—
199C4#2254870114yes—
200C2#124×D25111yescharacteristic
201C2C2×#2254870111yescharacteristic
202C2C4⋊Q8×D25111yescharacteristic
203C2#3063223912yes—
204C2#3063224112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#391×C251C2
hypercenter#391D251
hypocenterC251#2862
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
46. C4×Dic251Q8semidirect
102. C251#2862semidirect

Maximal subgroups

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

subgroupindex
C2×#225487142
C2×#225487132
#306315942
#306315932
#306315922
#306315912
#306315512
C42×Dic2512
#391×C2512
#2862251