Finite Group Databasebeta

← #44558714

Subgroups — #44558714

Order 32128.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#22548713C211yesmaximal, characteristic, complemented
2C2×#22548695C211yesmaximal, characteristic
3C2×#22548691C211yesmaximal, characteristic
4C2×C42×D251C211yesmaximal, characteristic
5#30631593C211yesmaximal, characteristic, complemented
6#30631584C214yesmaximal, complemented
7#30631556C211yesmaximal, characteristic, complemented
8#30631546C214yesmaximal
9#391×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2211yescharacteristic
11C22×C4×D251C2211yescharacteristic
12C22×C4×D251C2212yes—
13#22548713—24no—
14#22548695—24no—
15#22548691—24no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2212yes—
18C2×Dic251⋊C4C2214yes—
19C2×Dic251⋊C4C2214yes—
20C2×Dic251⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×C4×Dic251C2212yes—
24C2×C4×Dic251C2211yescharacteristic
25C42×D251—24no—
26C251⋊#95C2214yes—
27C22×Q8×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C42×C251C2211yescharacteristic
30#95×C251C2214yes—
31C23×D251C2311yescharacteristic
32C2×Dic502—22no—
33C2×Dic502—22no—
34C2×Dic502—22no—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2311yescharacteristic
38C22×Dic251C2312yes—
39C2×C4×D251—24no—
40C2×C4×D251—24no—
41C2×C4×D251—22no—
42C2×C4×D251—22no—
43C2×C4×D251D414yes—
44C2×C4×D251Q818yescomplemented
45C4⋊Dic251—24no—
46C4⋊Dic251—22no—
47Dic251⋊C4—24no—
48Dic251⋊C4—28no—
49Dic251⋊C4—28no—
50(C2×D251)⋊C4—24no—
51(C2×D251)⋊C4—24no—
52C4×Dic251—24no—
53C4×Dic251D414yes—
54C2×Q8×C251—22no—
55C2×Q8×C251—22no—
56C2×Q8×C251—22no—
57C22×C4×C251C2312yes—
58C22×C4×C251C2312yes—
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C4⋊C4×C251—22no—
62C42×C251—22no—
63C22×D251—21no—
64C22×D251—22no—
65C22×D251—21no—
66C22×D251C2×D412yes—
67C22×D251C2×Q814yes—
68Dic502—44no—
69C2×Dic251—21no—
70C2×Dic251—21no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—24no—
75C2×Dic251—22no—
76C2×Dic251C2×D412yes—
77C2×Dic251C2×D414yes—
78C2×Dic251C2×Q814yes—
79C2×Dic251#4114yes—
80C4×D251—28no—
81C4×D251—216no—
82C23×C251C2411yescharacteristic
83Q8×C251—44no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—24no—
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×Q814yes—
94C2×C4×C251#4112yes—
95C22×C251C22×D411yescharacteristic
96C22×C251C22×D411yescharacteristic
97C22×C251C22×Q812yes—
98C22×C251C2×#4112yes—
99C22×C251C2×#4111yescharacteristic
100C2×D251—22no—
101C2×D251—24no—
102C2×D251—22no—
103C2×D251—24no—
104C2×D251C4⋊Q814yescomplemented
105C4×C251—24no—
106C4×C251—42no—
107C4×C251—42no—
108C4×C251#12114yes—
109Dic251—22no—
110Dic251—42no—
111Dic251—42no—
112Dic251C22⋊Q818yescomplemented
113C2×C251C2×#12111yescharacteristic
114C2×C251C2×C22⋊Q812yes—
115C2×C251C2×C4⋊Q811yescharacteristic
116C2×C251#53211yescharacteristic
117C2×C251#53312yes—
118D251—24no—
119C251#285711yescharacteristic, complemented, Sylow
120#2857—2511nomaximal, Sylow
121C2×C4⋊Q8—2511no—
122C2×C22⋊Q8—2511no—
123C2×C22⋊Q8—2511no—
124C22×C42—2511no—
125#391—2511no—
126#391D25111yescharacteristic, complemented
127#389—2514no—
128#384—2511no—
129#380—2514no—
130C22×Q8—2511no—
131C22×Q8C2×D25111yescharacteristic
132C23×C4—2511no—
133C23×C4—2512no—
134C4⋊Q8—5024no—
135C22⋊Q8—5024no—
136C22⋊Q8—5024no—
137C2×C4⋊C4—2511no—
138C2×C4⋊C4—2512no—
139C2×C4⋊C4—2512no—
140C2×C4⋊C4—2514no—
141C2×C4⋊C4—2514no—
142C2×C4⋊C4C2×D25111yescharacteristic
143C2×C22⋊C4—2512no—
144C2×C22⋊C4—2512no—
145C2×C42—5024no—
146C2×C42—2512no—
147C2×C42—2511no—
148C2×C42C2×D25111yescharacteristic
149#95—2514no—
150#95C2×D25114yes—
151C24—2511no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—5022no—
156C2×Q8—5022no—
157C2×Q8—5022no—
158C22×C4—5024no—
159C22×C4—5024no—
160C22×C4—5022no—
161C22×C4—5022no—
162C22×C4—2512no—
163C22×C4—2512no—
164C22×C4—2511no—
165C22×C4—2514no—
166C22×C4—2518no—
167C22×C4—2512no—
168C22×C4C22×D25112yes—
169C22×C4C22×D25112yes—
170C22×C4C22×D25112yes—
171C22×C4C22×D25111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—5028no—
174C4⋊C4—5024no—
175C4⋊C4—5028no—
176C4⋊C4—5024no—
177C4⋊C4—5022no—
178C22⋊C4—5024no—
179C22⋊C4—5024no—
180C42—22no—
181C42—5024no—
182C42—2514no—
183C23—5021no—
184C23—5022no—
185C23—5021no—
186C23—2512no—
187C23—2514no—
188C23C23×D25111yescharacteristic
189Q8—44no—
190Q8—10044no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—50216no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4—5024no—
203C2×C4—5022no—
204C2×C4—5021no—
205C2×C4—5021no—
206C2×C4—5028no—
207C2×C4—2512no—
208C2×C4—2514no—
209C2×C4—2514no—
210C2×C4—2514no—
211C2×C4D4×D25112yes—
212C2×C4D4×D25112yes—
213C2×C4Q8×D25114yes—
214C2×C4#1686081012yes—
215C22—5022no—
216C22—5024no—
217C22—5022no—
218C22—5024no—
219C22—2514no—
220C22C2×D4×D25111yescharacteristic
221C22C2×D4×D25111yescharacteristic
222C22C2×Q8×D25112yes—
223C22#41×D25112yes—
224C22C2×#1686081011yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—10042no—
229C4—10042no—
230C4—5022no—
231C4—2518no—
232C4#2254868814yes—
233C2—5024no—
234C2C2×#2254868811yescharacteristic
235C2C22⋊Q8×D25112yes—
236C2C4⋊Q8×D25111yescharacteristic
237C2#3063223611yescharacteristic
238C2#3063224112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#391×C251C2
prime radicalC23×D251C23
hypercenter#391D251
hypocenterC251#2857
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#22548713C2semidirect
5. #30631593C2semidirect
6. #30631584C2semidirect
7. #30631556C2semidirect
9. #391×C251C2semidirect
44. C2×C4×D251Q8semidirect
104. C2×D251C4⋊Q8semidirect
112. Dic251C22⋊Q8semidirect
119. C251#2857semidirect
126. #391D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487132
C2×#225486952
C2×#225486912
C2×C42×D2512
#306315932
#306315842
#306315562
#306315462
#391×C2512
#2857251