Finite Group Databasebeta

← #44550281

Subgroups — #44550281

Order 32128.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#16860809C211yesmaximal, characteristic
2C2×#22548630C211yesmaximal, characteristic, complemented
3C2×#22548629C212yesmaximal, complemented
4#30631548C212yesmaximal, complemented
5C2×#99×C251C211yesmaximal, characteristic, complemented
6C2×#16860809—24no—
7C2×#16860809—24no—
8C2×#16860809—22no—
9C2×#16860809—22no—
10C22×Dic502C411yescharacteristic, complemented
11C22×C22⋊D251C411yescharacteristic, complemented
12C22×C22⋊D251C2211yescharacteristic
13C23×Dic251C411yescharacteristic, complemented
14C23×Dic251C2211yescharacteristic
15C22×C4×D251C411yescharacteristic, complemented
16C2×C22⋊Dic251C2212yes—
17#22548630—22no—
18#22548629—24no—
19C251⋊#95—24no—
20C22×D4×C251C2211yescharacteristic
21C2×C22⋊C4×C251C2212yes—
22#99×C251—22no—
23C23×D251C2×C411yescharacteristic
24#16860809—216no—
25#16860809—216no—
26C2×Dic502—22no—
27C2×Dic502—22no—
28C2×Dic502—22no—
29C2×C22⋊D251—22no—
30C2×C22⋊D251—22no—
31C2×C22⋊D251—22no—
32C2×C22⋊D251—22no—
33C2×C22⋊D251—21no—
34C2×C22⋊D251—21no—
35C2×C22⋊D251D414yes—
36C22×Dic251—22no—
37C22×Dic251—22no—
38C22×Dic251—22no—
39C22×Dic251—22no—
40C22×Dic251—21no—
41C22×Dic251—21no—
42C22×Dic251—22no—
43C22×Dic251C2×C411yescharacteristic
44C22×Dic251C2×C411yescharacteristic
45C22×Dic251C2×C411yescharacteristic
46C22×Dic251D414yes—
47C2×C4×D251—22no—
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50C22⋊Dic251—22no—
51C22⋊Dic251—22no—
52C24×C251C2×C411yescharacteristic
53C24×C251C2311yescharacteristic
54C2×D4×C251—22no—
55C2×D4×C251—22no—
56C2×D4×C251D412yes—
57C2×D4×C251D412yes—
58C22×C4×C251—22no—
59C22×C4×C251C2×C411yescharacteristic
60C22⋊C4×C251—22no—
61C22⋊C4×C251—22no—
62C22×D251—21no—
63C22×D251—21no—
64C22×D251—42no—
65C22×D251C22⋊C412yes—
66Dic502—28no—
67C22⋊D251—28no—
68C22⋊D251—24no—
69C22⋊D251—24no—
70C2×Dic251—28no—
71C2×Dic251—21no—
72C2×Dic251—21no—
73C2×Dic251—21no—
74C2×Dic251—21no—
75C2×Dic251—21no—
76C2×Dic251—21no—
77C2×Dic251—24no—
78C2×Dic251—24no—
79C2×Dic251—44no—
80C2×Dic251—42no—
81C2×Dic251C22⋊C412yes—
82C2×Dic251C22⋊C412yes—
83C2×Dic251C22⋊C412yes—
84C4×D251—28no—
85C23×C251—21no—
86C23×C251—21no—
87C23×C251—42no—
88C23×C251—42no—
89C23×C251C22⋊C412yes—
90C23×C251C22×C411yescharacteristic
91C23×C251C2×D412yes—
92C23×C251C2×D412yes—
93C23×C251C2×D411yescharacteristic
94C23×C251C2×D411yescharacteristic
95D4×C251—24no—
96D4×C251—24no—
97C2×C4×C251—21no—
98C2×C4×C251—21no—
99C2×C4×C251—44no—
100C2×C4×C251—42no—
101C2×C4×C251C22⋊C412yes—
102C22×C251—24no—
103C22×C251—22no—
104C22×C251—22no—
105C22×C251—42no—
106C22×C251—41no—
107C22×C251—41no—
108C22×C251—42no—
109C22×C251—42no—
110C22×C251—42no—
111C22×C251C2×C22⋊C411yescharacteristic
112C22×C251C2×C22⋊C411yescharacteristic
113C22×C251C2×C22⋊C411yescharacteristic
114C22×C251#12012yes—
115C22×C251#12012yes—
116C2×D251—22no—
117C2×D251—42no—
118C2×D251—42no—
119C2×D251—42no—
120C4×C251—22no—
121C4×C251—82no—
122Dic251—22no—
123Dic251—22no—
124Dic251—22no—
125Dic251—82no—
126C2×C251—22no—
127C2×C251—22no—
128C2×C251—21no—
129C2×C251—21no—
130C2×C251—42no—
131C2×C251#37911yescharacteristic
132C2×C251#41012yes—
133D251—42no—
134C251#178211yescharacteristic, complemented, Sylow
135#1782—2511nomaximal, Sylow
136C22×#41—2511no—
137C2×#99—2511no—
138C2×#99—2512no—
139C2×#99D25111yescharacteristic, complemented
140#381—2512no—
141C2×#41—5024no—
142C2×#41—5022no—
143C2×#41—5022no—
144C2×#41—5024no—
145C22×Q8—2511no—
146C22×D4—2511no—
147C22×D4—2511no—
148C22×D4C2×D25111yescharacteristic
149C23×C4—2511no—
150C23×C4—2511no—
151C23×C4—2511no—
152C2×C22⋊C4—2512no—
153C2×C22⋊C4C2×D25112yes—
154#99—22no—
155#99—5024no—
156#99—5022no—
157#95—5024no—
158C24—2511no—
159C24C4×D25111yescharacteristic
160C24C22×D25111yescharacteristic
161#41—50216no—
162#41—50216no—
163C2×Q8—5022no—
164C2×Q8—5022no—
165C2×Q8—5022no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—5022no—
169C2×D4—5022no—
170C2×D4—5022no—
171C2×D4—5021no—
172C2×D4—5021no—
173C2×D4—5022no—
174C2×D4—2514no—
175C2×D4C22⋊D25112yes—
176C2×D4D100412yes—
177C22×C4—22no—
178C22×C4—5022no—
179C22×C4—5022no—
180C22×C4—5022no—
181C22×C4—5022no—
182C22×C4—5022no—
183C22×C4—5021no—
184C22×C4—5021no—
185C22×C4—5022no—
186C22×C4—5022no—
187C22×C4—5022no—
188C22×C4—2511no—
189C22×C4—2511no—
190C22×C4—2511no—
191C22×C4—2514no—
192C22×C4C4×D25111yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—5022no—
196C22⋊C4—5022no—
197C23—21no—
198C23—21no—
199C23—42no—
200C23—42no—
201C23—10042no—
202C23—5021no—
203C23—5021no—
204C23—2512no—
205C23(C2×D251)⋊C412yes—
206C23C2×C4×D25111yescharacteristic
207C23D4×D25112yes—
208C23D4×D25112yes—
209C23C2×C22⋊D25111yescharacteristic
210C23C2×D100411yescharacteristic
211Q8—5028no—
212D4—24no—
213D4—24no—
214D4—5028no—
215D4—5024no—
216D4—5024no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—10044no—
222C2×C4—10042no—
223C2×C4—5028no—
224C2×C4—5028no—
225C2×C4—5021no—
226C2×C4—5021no—
227C2×C4—5021no—
228C2×C4—5021no—
229C2×C4—5021no—
230C2×C4—5021no—
231C2×C4—5024no—
232C2×C4—5024no—
233C2×C4—2512no—
234C2×C4—2512no—
235C2×C4—2512no—
236C2×C4(C2×D251)⋊C412yes—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—10042no—
247C22—10042no—
248C22—10042no—
249C22—5022no—
250C22C22⋊C4×D25111yescharacteristic
251C22C22⋊C4×D25111yescharacteristic
252C22C2×(C2×D251)⋊C411yescharacteristic
253C22#2254868312yes—
254C22#2254868512yes—
255C4—22no—
256C4—82no—
257C4—20082no—
258C4—5022no—
259C4—5022no—
260C4—5022no—
261C2—22no—
262C2—22no—
263C2—21no—
264C2—21no—
265C2—42no—
266C2—10042no—
267C2#3063154211yescharacteristic
268C2#3063165912yes—
subgroupG/H
centerC22C22⋊C4×D251
commutatorC23×C251C22×C4
FrattiniC24C22×D251
FittingC2×#99×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×#16860809C2
cosocleC24×C251C23
hypercenterC2×#99D251
hypocenterC251#1782
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#22548630C2semidirect
3. C2×#22548629C2semidirect
4. #30631548C2semidirect
5. C2×#99×C251C2semidirect
10. C22×Dic502C4semidirect
11. C22×C22⋊D251C4semidirect
13. C23×Dic251C4semidirect
15. C22×C4×D251C4semidirect
134. C251#1782semidirect
139. C2×#99D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168608092
C2×#225486302
C2×#225486292
#306315482
C2×#99×C2512
#1782251