Finite Group Databasebeta

← #44559623

Subgroups — #44559623

Order 32128.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16860810C211yesmaximal, characteristic
2C2×#22548746C212yesmaximal, complemented
3#30631847C214yesmaximal, complemented
4#30631842C212yesmaximal, complemented
5#30631837C212yesmaximal, complemented
6#30631686C212yesmaximal, complemented
7#30631651C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C251C211yesmaximal, characteristic, complemented
9C2×#16860810—28no—
10C2×#16860810—21no—
11C2×#16860810—21no—
12C2×#16860810C2212yes—
13C2×#16860810C2212yes—
14C22×Dic502C2211yescharacteristic
15C22×D1004C2211yescharacteristic
16C22×C22⋊D251—21no—
17C22×C4×D251—21no—
18#22548746—28no—
19C2×#16860800C2214yescomplemented
20C2×#16860797C2214yescomplemented
21C2×#16860794C2212yescomplemented
22#22548638C2214yescomplemented
23#22548635C2214yescomplemented
24C22⋊(C251⋊C8)C2212yes—
25C22⋊(C251⋊C8)C2212yescomplemented
26C22×Q8×C251C2211yescharacteristic, complemented
27C23×C4×C251C2211yescharacteristic
28C22⋊Q8×C251—21no—
29C22⋊Q8×C251—21no—
30C22⋊Q8×C251C2212yes—
31C22⋊Q8×C251C2212yescomplemented
32C2×C4⋊C4×C251—21no—
33C2×C4⋊C4×C251C2211yescharacteristic, complemented
34C2×C22⋊C4×C251—21no—
35C23×D251—21no—
36#16860810—28no—
37#16860810—28no—
38#16860810—24no—
39#16860810—24no—
40#16860810—24no—
41#16860810D418yes—
42C2×Dic502—24no—
43C2×Dic502—21no—
44C2×Dic502C2312yes—
45C2×D1004—24no—
46C2×D1004—21no—
47C2×D1004C2312yes—
48C2×C22⋊D251—28no—
49C2×C22⋊D251—22no—
50C2×C22⋊D251—22no—
51C22×Dic251—21no—
52C2×C4×D251—28no—
53C2×C4×D251—22no—
54C2×C4×D251—22no—
55#16860800—28no—
56#16860797—28no—
57#16860794—24no—
58C2×C251⋊C8C2314yes—
59C24×C251D411yescharacteristic
60C2×Q8×C251—21no—
61C2×Q8×C251—24no—
62C2×Q8×C251C2312yes—
63C22×C4×C251—24no—
64C22×C4×C251—21no—
65C22×C4×C251—21no—
66C22×C4×C251D411yescharacteristic
67C22×C4×C251D411yescharacteristic
68C22×C4×C251D411yescharacteristic
69C22×C4×C251C2311yescharacteristic
70C22×C4×C251C2311yescharacteristic
71C22×C4×C251C2311yescharacteristic
72C4⋊C4×C251—22no—
73C4⋊C4×C251—22no—
74C4⋊C4×C251—21no—
75C4⋊C4×C251C2312yes—
76C22⋊C4×C251—22no—
77C22⋊C4×C251—22no—
78C22×D251—24no—
79C22×D251—22no—
80C22×D251—42no—
81C22×D251—42no—
82Dic502—24no—
83Dic502—22no—
84Dic502C2×D414yes—
85D1004—24no—
86D1004—22no—
87D1004C2×D414yes—
88C22⋊D251—28no—
89C22⋊D251—28no—
90C2×Dic251—24no—
91C2×Dic251—22no—
92C4×D251—28no—
93C4×D251—28no—
94C251⋊C8—24no—
95C23×C251—22no—
96C23×C251—22no—
97C23×C251—41no—
98C23×C251C2×D411yescharacteristic
99C23×C251C2×D411yescharacteristic
100C23×C251C2×D411yescharacteristic
101Q8×C251—24no—
102Q8×C251—42no—
103C2×C4×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—42no—
111C2×C4×C251—42no—
112C2×C4×C251C2×D414yes—
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2411yescharacteristic
117C22×C251—21no—
118C22×C251—21no—
119C22×C251—21no—
120C22×C251—21no—
121C22×C251—21no—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—44no—
125C22×C251#12012yes—
126C22×C251#13712yescomplemented
127C22×C251C22×D411yescharacteristic
128C2×D251—24no—
129C2×D251—48no—
130C2×D251—42no—
131C2×D251—42no—
132C4×C251—21no—
133C4×C251—21no—
134C4×C251—42no—
135C4×C251—42no—
136C4×C251#12012yes—
137C4×C251C22×D412yes—
138Dic251—24no—
139C2×C251—22no—
140C2×C251—22no—
141C2×C251—41no—
142C2×C251C2×#12011yescharacteristic
143C2×C251C2×#13711yescharacteristic
144C2×C251#57511yescharacteristic
145D251—44no—
146C251#300711yescharacteristic, complemented, Sylow
147#3007—2511nomaximal, Sylow
148C22×#41—2511no—
149C2×#137—2512no—
150C2×C22⋊Q8D25111yescharacteristic, complemented
151#452—2514no—
152#451—2512no—
153#450—2512no—
154#417—2512no—
155#407—2511no—
156C2×#41—5028no—
157C2×#41—5021no—
158C2×#41—5021no—
159C2×#41—2512no—
160C2×#41—2512no—
161C22×Q8—2511no—
162C22×Q8C2×D25111yescharacteristic, complemented
163C22×D4—5021no—
164C22×D4—2511no—
165C23×C4—5021no—
166C23×C4C2×D25111yescharacteristic
167#137—5028no—
168C2×Q16—2514no—
169C2×SD16—2514no—
170C2×M16—2512no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D25112yes—
174C22⋊Q8C2×D25112yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D25111yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—2514no—
179#102—2514no—
180C22⋊C8—2512no—
181C22⋊C8—2512no—
182C24—5021no—
183C24C22⋊D25111yescharacteristic
184#41—5028no—
185#41—5028no—
186#41—5024no—
187#41—5024no—
188#41—5024no—
189#41—2518no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—5024no—
193C2×Q8—5021no—
194C2×Q8—2512no—
195C2×Q8C22×D25112yes—
196C2×D4—5024no—
197C2×D4—5028no—
198C2×D4—5022no—
199C2×D4—5022no—
200C2×D4—5021no—
201C2×D4—2512no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—5028no—
206C22×C4—5021no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4C22⋊D25111yescharacteristic
210C22×C4C22⋊D25111yescharacteristic
211C22×C4C22⋊D25111yescharacteristic
212C22×C4C22×D25111yescharacteristic
213C22×C4C22×D25111yescharacteristic
214C22×C4C22×D25111yescharacteristic
215Q16—5028no—
216SD16—5028no—
217M16—5024no—
218C2×C8—2514no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D25112yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—10042no—
229C23—10042no—
230C23—5024no—
231C23—5022no—
232C23C2×C22⋊D25111yescharacteristic
233C23C2×C22⋊D25111yescharacteristic
234C23C2×C22⋊D25111yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—5024no—
238Q8—5022no—
239Q8—2514no—
240D4—5024no—
241D4—5028no—
242D4—5028no—
243D4—5022no—
244D4—2514no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—5028no—
255C2×C4—5024no—
256C2×C4—5022no—
257C2×C4—5028no—
258C2×C4D4×D25114yes—
259C2×C4C2×C22⋊D25111yescharacteristic
260C2×C4C2×C22⋊D25111yescharacteristic
261C2×C4C2×C22⋊D25111yescharacteristic
262C2×C4C23×D25111yescharacteristic
263C8—5024no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—10048no—
273C22—10042no—
274C22—10042no—
275C22—5024no—
276C22#2254868312yes—
277C22#2254874612yescomplemented
278C22C22×C22⋊D25111yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—5024no—
284C4#2254868312yes—
285C4C2×D4×D25112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—10044no—
290C2C2×#2254868311yescharacteristic
291C2C2×#2254874611yescharacteristic
292C2#3063248511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C22⋊Q8×C251C2
socleC22×C251C22×D4
prime radicalC22×#16860810C2
hypercenterC2×C22⋊Q8D251
hypocenterC251#3007
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. #30631847C2semidirect
4. #30631842C2semidirect
5. #30631837C2semidirect
6. #30631686C2semidirect
7. #30631651C2semidirect
8. C2×C22⋊Q8×C251C2semidirect
19. C2×#16860800C22semidirect
20. C2×#16860797C22semidirect
21. C2×#16860794C22semidirect
22. #22548638C22semidirect
23. #22548635C22semidirect
25. C22⋊(C251⋊C8)C22semidirect
26. C22×Q8×C251C22semidirect
31. C22⋊Q8×C251C22semidirect
33. C2×C4⋊C4×C251C22semidirect
126. C22×C251#137semidirect
146. C251#3007semidirect
150. C2×C22⋊Q8D251semidirect
162. C22×Q8C2×D251semidirect
174. C22⋊Q8C2×D251semidirect
176. C2×C4⋊C4C2×D251semidirect
277. C22#22548746semidirect

Maximal subgroups

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

subgroupindex
C22×#168608102
C2×#225487462
#306318472
#306318422
#306318372
#306316862
#306316512
C2×C22⋊Q8×C2512
#3007251