Finite Group Databasebeta

← #44560820

Subgroups — #44560820

Order 32128.

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

normal subgroups only

This group has 552 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
1C2×#22548748C211yesmaximal, characteristic, complemented
2C2×#22548746C211yesmaximal, characteristic, complemented
3C2×#22548745C211yesmaximal, characteristic, complemented
4#30632233C211yesmaximal, characteristic
5C2×#22548713C211yesmaximal, characteristic, complemented
6#30632080C212yesmaximal, complemented
7#30632078C212yesmaximal, complemented
8#30632065C212yesmaximal, complemented
9#30632058C212yesmaximal, complemented
10M16×Dic251C211yesmaximal, characteristic, complemented
11C2×#137×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×Dic502C2211yescharacteristic
14#22548748—24no—
15#22548746—24no—
16#22548745—24no—
17C2×Dic1004C2212yes—
18C2×#16860800C2212yes—
19C2×#16860800C2212yes—
20C2×#16860799C2212yes—
21C2×#16860798C2212yes—
22C2×#16860797C2212yes—
23C2×#16860794C2211yescharacteristic
24#22548713—22no—
25#22548713C2212yes—
26#22548713C2212yes—
27#22548713C2211yescharacteristic
28#22548711C2211yescharacteristic
29#22548704C2211yescharacteristic
30#22548702C2211yescharacteristic
31#22548687—22no—
32C4×C22⋊D251—22no—
33C2×Dic251⋊C4—22no—
34C2×C4×Dic251C2211yescharacteristic
35#22548625C2212yes—
36C8×Dic251C2212yes—
37C2×#41×C251C2211yescharacteristic
38C22×Q8×C251C2211yescharacteristic
39#137×C251—24no—
40C2×Q16×C251C2212yes—
41C2×SD16×C251C2212yes—
42C2×M16×C251C2211yescharacteristic
43#16860810—22no—
44#16860810—22no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502—22no—
48C2×Dic502C2312yes—
49C2×Dic502C2311yescharacteristic
50C2×D1004C2311yescharacteristic
51C2×C22⋊D251—22no—
52C2×C22⋊D251—21no—
53C22×Dic251—21no—
54C22×Dic251D412yes—
55C2×C4×D251—21no—
56Dic1004—24no—
57#16860800—24no—
58#16860800—24no—
59#16860799—24no—
60#16860798—24no—
61#16860797—24no—
62#16860794D414yescomplemented
63C2×C251⋊C8C2312yes—
64Dic251⋊C4—24no—
65Dic251⋊C4—24no—
66Dic251⋊C4—22no—
67C22⋊Dic251—22no—
68(C2×D251)⋊C4—22no—
69C4×Dic251D412yes—
70C4×Dic251C2311yescharacteristic
71C4×Dic251C2311yescharacteristic
72#41×C251—22no—
73#41×C251—22no—
74C2×Q8×C251—21no—
75C2×Q8×C251—22no—
76C2×Q8×C251—22no—
77C2×Q8×C251C2312yes—
78C2×Q8×C251C2311yescharacteristic
79C2×D4×C251—21no—
80C2×D4×C251C2311yescharacteristic
81C22×C4×C251—21no—
82C22×C4×C251—21no—
83C22×C4×C251C2311yescharacteristic
84Q16×C251—24no—
85SD16×C251—24no—
86M16×C251D414yescomplemented
87C2×C8×C251C2312yes—
88C22×D251—21no—
89Dic502—22no—
90Dic502—22no—
91Dic502—21no—
92Dic502—21no—
93Dic502—42no—
94D1004—21no—
95D1004—21no—
96C22⋊D251—42no—
97C22⋊D251—44no—
98C2×Dic251—24no—
99C2×Dic251—21no—
100C2×Dic251—22no—
101C2×Dic251—42no—
102C2×Dic251C2×D412yes—
103C2×Dic251C2×D412yes—
104C4×D251—42no—
105C251⋊C8C2×D414yes—
106C23×C251—21no—
107C23×C251C2×D411yescharacteristic
108Q8×C251—22no—
109Q8×C251—21no—
110Q8×C251—22no—
111Q8×C251—21no—
112Q8×C251—42no—
113D4×C251—21no—
114D4×C251—21no—
115D4×C251—42no—
116C2×C4×C251—21no—
117C2×C4×C251—22no—
118C2×C4×C251—42no—
119C2×C4×C251—42no—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2411yescharacteristic
124C8×C251C2×D414yes—
125C22×C251—22no—
126C22×C251—41no—
127C22×C251—42no—
128C22×C251#12712yes—
129C22×C251C22×D411yescharacteristic
130C2×D251—41no—
131C2×D251—42no—
132C4×C251—41no—
133C4×C251—42no—
134C4×C251#12712yes—
135C4×C251C22×D411yescharacteristic
136C4×C251C22×D411yescharacteristic
137Dic251—22no—
138Dic251—41no—
139Dic251—42no—
140Dic251#13714yescomplemented
141C2×C251—22no—
142C2×C251—81no—
143C2×C251C2×#12711yescharacteristic
144C2×C251C2×#13712yes—
145D251—81no—
146C251#315211yescharacteristic, complemented, Sylow
147#3152—2511nomaximal, Sylow
148C2×#137—2511no—
149C2×#137—2511no—
150C2×#137—2511no—
151C2×#137D25111yescharacteristic, complemented
152#532—2511no—
153C2×C4⋊Q8—2511no—
154#497—2512no—
155#497—2512no—
156#494—2512no—
157#492—2512no—
158C4×M16—2511no—
159C2×#41—2511no—
160C2×#41C2×D25111yescharacteristic
161C22×Q8—2511no—
162C22×Q8C2×D25111yescharacteristic
163#137—24no—
164#137—5024no—
165#137—5024no—
166#137—5024no—
167C2×Q16—2512no—
168C2×Q16—2512no—
169C2×Q16—2512no—
170C2×Q16C2×D25112yes—
171C2×SD16—2512no—
172C2×SD16—2512no—
173C2×SD16—2512no—
174C2×SD16C2×D25112yes—
175C2×M16—2511no—
176C2×M16C2×D25111yescharacteristic
177C4⋊Q8—5022no—
178C4⋊Q8—2512no—
179C4⋊Q8—2512no—
180C4⋊Q8—2511no—
181#127—2511no—
182#124—2511no—
183#124—2511no—
184#121—5022no—
185C4×D4—5022no—
186C2×C4⋊C4—5022no—
187C2×C42—2511no—
188#97—2512no—
189C4×C8—2512no—
190#41—22no—
191#41—22no—
192#41—5022no—
193#41—5022no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—5022no—
198C2×Q8—5022no—
199C2×Q8—5021no—
200C2×Q8—2511no—
201C2×Q8—2512no—
202C2×Q8C22×D25111yescharacteristic
203C2×Q8C22×D25112yes—
204C2×D4—21no—
205C2×D4—5021no—
206C2×D4—5022no—
207C2×D4—2511no—
208C2×D4C22×D25111yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—5021no—
212C22×C4—5021no—
213C22×C4—2512no—
214C22×C4C22×D25111yescharacteristic
215Q16—24no—
216Q16—5024no—
217Q16—5024no—
218Q16—5024no—
219SD16—24no—
220SD16—5024no—
221SD16—5024no—
222SD16—5024no—
223M16—2514no—
224M16C22⋊D25114yescomplemented
225C2×C8—2512no—
226C2×C8C22×D25112yes—
227C4⋊C4—5024no—
228C4⋊C4—5024no—
229C4⋊C4—5022no—
230C22⋊C4—5022no—
231C22⋊C4—5022no—
232C42—2512no—
233C42—2511no—
234C42—2511no—
235C23—21no—
236C23—5021no—
237C23D4×D25111yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—10042no—
244Q8—5022no—
245Q8—5021no—
246Q8—5022no—
247Q8—5021no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—10042no—
252D4—10044no—
253D4—5021no—
254D4—5021no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—10042no—
260C2×C4—10042no—
261C2×C4—5021no—
262C2×C4—5022no—
263C2×C4—5024no—
264C2×C4—2512no—
265C2×C4—2512no—
266C2×C4D4×D25112yes—
267C2×C4D4×D25111yescharacteristic
268C2×C4C2×C22⋊D25112yes—
269C2×C4C23×D25111yescharacteristic
270C8—2514no—
271C8C2×C22⋊D25114yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—10042no—
276C22—10041no—
277C22#2254871112yes—
278C22C2×D4×D25111yescharacteristic
279C4—41no—
280C4—42no—
281C4—10041no—
282C4—10042no—
283C4—5022no—
284C4—2514no—
285C4#2254871112yes—
286C4C2×D4×D25111yescharacteristic
287C4C22×C22⋊D25111yescharacteristic
288C2—22no—
289C2—81no—
290C2—20081no—
291C2C2×#2254871111yescharacteristic
292C2#137×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
prime radical#30632233C2
hypercenterC2×#137D251
hypocenterC251#3152
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22548748C2semidirect
2. C2×#22548746C2semidirect
3. C2×#22548745C2semidirect
5. C2×#22548713C2semidirect
6. #30632080C2semidirect
7. #30632078C2semidirect
8. #30632065C2semidirect
9. #30632058C2semidirect
10. M16×Dic251C2semidirect
11. C2×#137×C251C2semidirect
62. #16860794D4semidirect
86. M16×C251D4semidirect
140. Dic251#137semidirect
146. C251#3152semidirect
151. C2×#137D251semidirect
224. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C2×#225487462
C2×#225487452
#306322332
C2×#225487132
#306320802
#306320782
#306320652
#306320582
M16×Dic2512
C2×#137×C2512
#3152251