Finite Group Databasebeta

← #44560494

Subgroups — #44560494

Order 32128.

380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#30632242C211yesmaximal, characteristic
2C4×Q8×D251C211yesmaximal, characteristic
3#30632016C212yesmaximal
4#30631973C211yesmaximal, characteristic, complemented
5#30631972C211yesmaximal, characteristic, complemented
6#30631894C211yesmaximal, characteristic, complemented
7#30631841C212yesmaximal
8#30631794C212yesmaximal, complemented
9#30631791C212yesmaximal, complemented
10#30631743C211yesmaximal, characteristic
11#462×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C2×#16860800C2212yes—
14#22548715C2211yescharacteristic
15#22548706—21no—
16#22548695—21no—
17#22548691C2212yes—
18Q8×Dic251C2212yes—
19Q8×Dic251C2211yescharacteristic
20C4×Dic502—21no—
21C4×Dic502C2211yescharacteristic
22#22548680—21no—
23#22548680—21no—
24#22548680C2211yescharacteristic
25#22548673—21no—
26C4⋊C4×D251—21no—
27C4⋊C4×D251C2211yescharacteristic
28C42×D251—21no—
29C42×D251C2211yescharacteristic
30#22548647C2212yes—
31#22548646C2212yes—
32C4⋊(C251⋊C8)C2211yescharacteristic
33#22548640C2212yes—
34#22548639C2212yes—
35#22548639C2212yes—
36(C2×D251)⋊C8C2212yes—
37#22548625C2212yes—
38C2×Q16×C251C2212yes—
39C4⋊Q8×C251C2211yescharacteristic
40C4×Q8×C251C2211yescharacteristic
41C4⋊C8×C251C2211yescharacteristic
42#103×C251C2212yes—
43C2×Dic502—21no—
44C2×Dic502—21no—
45C2×Dic502C2311yescharacteristic
46Q8×D251—22no—
47Q8×D251—22no—
48C2×C4×D251—21no—
49C2×C4×D251—21no—
50C2×C4×D251D411yescharacteristic
51C2×C4×D251D411yescharacteristic
52C2×C4×D251C2311yescharacteristic
53#16860800—24no—
54C2×C251⋊C8C2312yes—
55C4⋊Dic251—21no—
56C4⋊Dic251—22no—
57C4⋊Dic251—21no—
58C4⋊Dic251C2311yescharacteristic
59C4⋊Dic251C2312yes—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—22no—
63Dic251⋊C4—21no—
64(C2×D251)⋊C4—22no—
65C4×Dic251—21no—
66C4×Dic251—21no—
67C4×Dic251—22no—
68C4×Dic251—21no—
69C4×Dic251D411yescharacteristic
70C4×Dic251D411yescharacteristic
71C4×Dic251C2311yescharacteristic
72C2×Q8×C251C2312yes—
73C2×Q8×C251C2311yescharacteristic
74Q16×C251—24no—
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251—21no—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2311yescharacteristic
80C42×C251—21no—
81C42×C251C2311yescharacteristic
82C22×D251C2×D411yescharacteristic
83Dic502—24no—
84Dic502—21no—
85Dic502—42no—
86Dic502#4112yes—
87C2×Dic251—21no—
88C2×Dic251—21no—
89C2×Dic251—22no—
90C2×Dic251C2×D411yescharacteristic
91C2×Dic251C2×D411yescharacteristic
92C2×Dic251C2×D411yescharacteristic
93C4×D251—24no—
94C4×D251—22no—
95C4×D251—22no—
96C4×D251—21no—
97C4×D251—21no—
98C4×D251—42no—
99C251⋊C8—22no—
100Q8×C251—21no—
101Q8×C251—22no—
102Q8×C251—22no—
103Q8×C251#4112yes—
104C2×C4×C251—21no—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C8×C251—22no—
111C22×C251C22×D411yescharacteristic
112C2×D251—21no—
113C2×D251—21no—
114C2×D251#13712yes—
115C4×C251—22no—
116C4×C251—21no—
117C4×C251—42no—
118C4×C251—41no—
119C4×C251#13712yes—
120C4×C251C2×#4111yescharacteristic
121C4×C251C2×#4111yescharacteristic
122Dic251—22no—
123Dic251—21no—
124Dic251—21no—
125Dic251—21no—
126Dic251—41no—
127Dic251—42no—
128C2×C251#52511yescharacteristic
129C2×C251C2×#13711yescharacteristic
130C2×C251C2×#13711yescharacteristic
131D251—22no—
132C251#311311yescharacteristic, complemented, Sylow
133#3113—2511nomaximal, Sylow
134#533—2511no—
135C2×C4×Q8—2511no—
136#483—2512no—
137#475—2511no—
138#475—2511no—
139#462—2511no—
140#462D25111yescharacteristic, complemented
141#451—2512no—
142#441—2512no—
143#441—2512no—
144#432—2511no—
145C22×Q8—2511no—
146C2×Q16—2512no—
147C2×Q16C2×D25112yes—
148C4⋊Q8—2511no—
149C4⋊Q8C2×D25111yescharacteristic
150#125—5021no—
151C22⋊Q8—5021no—
152C22⋊Q8—2512no—
153C4×Q8—5021no—
154C4×Q8—5021no—
155C4×Q8—5021no—
156C4×Q8—2511no—
157C4×Q8—2512no—
158C4×Q8—2511no—
159C4×Q8—2511no—
160C4×Q8C2×D25111yescharacteristic
161#117—5021no—
162C2×C4⋊C4—5021no—
163C2×C4⋊C4—2511no—
164C2×C42—5021no—
165C2×C42—2511no—
166#106—2512no—
167#106—2512no—
168C4⋊C8—2511no—
169C4⋊C8C2×D25111yescharacteristic
170#103—2512no—
171#103—2512no—
172#103—2512no—
173#103C2×D25112yes—
174C22⋊C8—2512no—
175#97—2512no—
176C2×Q8—5021no—
177C2×Q8—5022no—
178C2×Q8—5022no—
179C2×Q8—5021no—
180C2×Q8—2511no—
181C2×Q8C22×D25112yes—
182C2×Q8C22×D25111yescharacteristic
183C22×C4—5021no—
184C22×C4—5021no—
185C22×C4—2511no—
186C22×C4—2511no—
187C22×C4—2511no—
188Q16—24no—
189Q16—5024no—
190C2×C8—2512no—
191C2×C8C22×D25112yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—5021no—
196C4⋊C4—5021no—
197C4⋊C4—5021no—
198C4⋊C4—5022no—
199C4⋊C4—5022no—
200C4⋊C4—5021no—
201C4⋊C4—5021no—
202C4⋊C4—2511no—
203C4⋊C4—2512no—
204C4⋊C4C22×D25111yescharacteristic
205C22⋊C4—5022no—
206C42—21no—
207C42—5021no—
208C42—5021no—
209C42—5022no—
210C42—5021no—
211C42—2511no—
212C42—2511no—
213C42—2511no—
214C42C22×D25111yescharacteristic
215C23—2511no—
216Q8—21no—
217Q8—22no—
218Q8—22no—
219Q8—10042no—
220Q8—5024no—
221Q8—5021no—
222Q8—2512no—
223Q8#1686080912yes—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—10042no—
228C2×C4—5024no—
229C2×C4—5022no—
230C2×C4—5021no—
231C2×C4—5022no—
232C2×C4—5021no—
233C2×C4—5022no—
234C2×C4—5021no—
235C2×C4—5021no—
236C2×C4—2511no—
237C2×C4—2511no—
238C2×C4—2511no—
239C2×C4D4×D25111yescharacteristic
240C2×C4D4×D25111yescharacteristic
241C2×C4C23×D25111yescharacteristic
242C8—22no—
243C8—5022no—
244C22—5021no—
245C22—5021no—
246C22—2512no—
247C22C2×D4×D25111yescharacteristic
248C4—22no—
249C4—21no—
250C4—42no—
251C4—41no—
252C4—10041no—
253C4—10042no—
254C4—5022no—
255C4—5021no—
256C4—5021no—
257C4—5021no—
258C4#2254874712yes—
259C4#41×D25111yescharacteristic
260C4C2×#1686080911yescharacteristic
261C2—5022no—
262C2#3063219711yescharacteristic
263C2#137×D25111yescharacteristic
264C2C2×#2254874711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#462×C251C2
socleC22×C251C22×D4
prime radicalC22×D251C2×D4
hypercenter#462D251
hypocenterC251#3113
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
4. #30631973C2semidirect
5. #30631972C2semidirect
6. #30631894C2semidirect
8. #30631794C2semidirect
9. #30631791C2semidirect
11. #462×C251C2semidirect
132. C251#3113semidirect
140. #462D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
C4×Q8×D2512
#306320162
#306319732
#306319722
#306318942
#306318412
#306317942
#306317912
#306317432
#462×C2512
#3113251