Finite Group Databasebeta

← #44563060

Subgroups — #44563060

Order 32128.

348 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 115 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#30632422C211yesmaximal, characteristic
2#30632412C211yesmaximal, characteristic, complemented
3#30632043C212yesmaximal, complemented
4#30632035C211yesmaximal, characteristic
5#30631986C212yesmaximal, complemented
6#30631912C212yesmaximal, complemented
7#30631888C211yesmaximal, characteristic
8#30631811C211yesmaximal, characteristic, complemented
9C4×#16860800C211yesmaximal, characteristic, complemented
10#30631774C212yesmaximal, complemented
11#475×C251C211yesmaximal, characteristic, complemented
12C2×#16860800C2211yescharacteristic
13C2×#16860799C2212yes—
14#22548713—21no—
15#22548712C2211yescharacteristic
16#22548706C2212yes—
17#22548705—22no—
18#22548704C2212yes—
19#22548703—21no—
20#22548700—22no—
21Q8×Dic251—21no—
22C4×Dic502C2211yescharacteristic
23C4×Dic502C2211yescharacteristic
24#22548680C2212yes—
25C4×D1004—21no—
26#22548677C2212yes—
27#22548648C2211yescharacteristic
28Dic251⋊C8C2212yes—
29#22548639C2211yescharacteristic
30#22548638C2211yescharacteristic
31#22548637C2212yes—
32#22548635C2212yes—
33#22548635C2212yes—
34C4×C251⋊C8C2211yescharacteristic
35C8×Dic251C2212yes—
36C4⋊Q8×C251C2211yescharacteristic
37C4×Q8×C251C2211yescharacteristic
38#106×C251C2212yes—
39C4⋊C8×C251C2211yescharacteristic
40#103×C251C2212yes—
41C2×Dic502D411yescharacteristic
42C2×Dic502C2311yescharacteristic
43C2×D1004—21no—
44C2×D1004—21no—
45C2×D1004C2312yes—
46C2×C4×D251—22no—
47#16860800—22no—
48#16860799—24no—
49C2×C251⋊C8C2311yescharacteristic
50C2×C251⋊C8C2311yescharacteristic
51C4⋊Dic251—22no—
52C4⋊Dic251D411yescharacteristic
53C4⋊Dic251C2311yescharacteristic
54Dic251⋊C4—22no—
55Dic251⋊C4—22no—
56Dic251⋊C4—21no—
57Dic251⋊C4D412yes—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—22no—
61C4×Dic251—22no—
62C4×Dic251—21no—
63C4×Dic251C2312yes—
64C2×Q8×C251—21no—
65C2×Q8×C251C2311yescharacteristic
66C2×C8×C251C2312yes—
67C4⋊C4×C251—21no—
68C4⋊C4×C251—21no—
69C4⋊C4×C251C2312yes—
70C4⋊C4×C251C2311yescharacteristic
71C42×C251—21no—
72C42×C251C2311yescharacteristic
73C22×D251—22no—
74Dic502—22no—
75Dic502—21no—
76Dic502#4112yes—
77D1004—22no—
78D1004—22no—
79D1004—42no—
80D1004—81no—
81C2×Dic251—21no—
82C2×Dic251—21no—
83C2×Dic251C2×D412yes—
84C2×Dic251C2×D412yes—
85C4×D251—82no—
86C251⋊C8—21no—
87C251⋊C8—21no—
88Q8×C251—21no—
89Q8×C251—21no—
90Q8×C251—42no—
91C2×C4×C251—21no—
92C2×C4×C251—21no—
93C2×C4×C251—22no—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2411yescharacteristic
97C8×C251—22no—
98C22×C251C22×D411yescharacteristic
99C2×D251—42no—
100C2×D251—42no—
101C2×D251—42no—
102C4×C251—21no—
103C4×C251—41no—
104C4×C251—42no—
105C4×C251—41no—
106C4×C251#13512yes—
107C4×C251C2×#4111yescharacteristic
108C4×C2512−1+411yescharacteristic
109Dic251—22no—
110Dic251—22no—
111Dic251—22no—
112Dic251—41no—
113C2×C251#54811yescharacteristic
114C2×C251C2×#13511yescharacteristic
115C2×C251#57711yescharacteristic
116D251—82no—
117C251#336511yescharacteristic, complemented, Sylow
118#3365—2511nomaximal, Sylow
119#559—2511no—
120#557—2511no—
121#488—2512no—
122#486—2511no—
123#478—2512no—
124#475D25111yescharacteristic, complemented
125#464—2512no—
126#460—2511no—
127C8×Q8—2511no—
128C4×Q16—2511no—
129C4×SD16—2512no—
130C2×Q16—2511no—
131C2×SD16—2512no—
132C4⋊Q8—5021no—
133C4⋊Q8C2×D25111yescharacteristic
134#127—2511no—
135#125—5022no—
136#125—2512no—
137#124—5021no—
138#124—2512no—
139#123—5022no—
140C4×Q8—5021no—
141C4×Q8—2512no—
142C4×Q8—2511no—
143C4×Q8—2511no—
144C4×Q8C2×D25111yescharacteristic
145C4×D4—5021no—
146C4×D4—2512no—
147#107—2511no—
148#106C2×D25112yes—
149C4⋊C8—2512no—
150C4⋊C8C2×D25111yescharacteristic
151#103—2511no—
152#103—2511no—
153#103C2×D25112yes—
154#102—2512no—
155#102—2512no—
156#102—2512no—
157C4×C8—2512no—
158C4×C8—2511no—
159C2×Q8—21no—
160C2×Q8—2511no—
161C2×Q8—2511no—
162C2×Q8C22×D25111yescharacteristic
163C2×D4—5021no—
164C2×D4—5021no—
165C2×D4—2512no—
166C22×C4—5022no—
167Q16—5022no—
168SD16—5024no—
169C2×C8—2511no—
170C2×C8—2511no—
171C2×C8C22×D25112yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—5022no—
175C4⋊C4—5022no—
176C4⋊C4—5022no—
177C4⋊C4—5021no—
178C4⋊C4—2511no—
179C4⋊C4—2512no—
180C4⋊C4—2511no—
181C4⋊C4C22×D25112yes—
182C4⋊C4C22×D25111yescharacteristic
183C22⋊C4—5022no—
184C22⋊C4—5022no—
185C22⋊C4—5022no—
186C42—21no—
187C42—5022no—
188C42—5021no—
189C42—2512no—
190C42C22×D25111yescharacteristic
191C23—5022no—
192Q8—21no—
193Q8—21no—
194Q8—42no—
195Q8—5022no—
196Q8—5021no—
197Q8—2512no—
198D4—20081no—
199D4—10042no—
200D4—5022no—
201D4—5022no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—20082no—
206C2×C4—5021no—
207C2×C4—5021no—
208C2×C4—2512no—
209C2×C4—2512no—
210C2×C4D4×D25111yescharacteristic
211C2×C4D4×D25111yescharacteristic
212C2×C4C23×D25111yescharacteristic
213C8—22no—
214C8—5021no—
215C8—5021no—
216C22—10042no—
217C22—10042no—
218C22—10042no—
219C22C2×D4×D25111yescharacteristic
220C4—21no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—10041no—
225C4—5022no—
226C4—5022no—
227C4—5022no—
228C4#2254873612yes—
229C4#41×D25111yescharacteristic
230C4#2254876311yescharacteristic
231C2—20082no—
232C2#3063234711yescharacteristic
233C2C2×#2254873611yescharacteristic
234C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#475×C251C2
socleC22×C251C22×D4
prime radical#22548712C22
hypercenter#475D251
hypocenterC251#3365
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #30632412C2semidirect
3. #30632043C2semidirect
5. #30631986C2semidirect
6. #30631912C2semidirect
8. #30631811C2semidirect
9. C4×#16860800C2semidirect
10. #30631774C2semidirect
11. #475×C251C2semidirect
117. C251#3365semidirect
124. #475D251semidirect

Maximal subgroups

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

subgroupindex
#306324222
#306324122
#306320432
#306320352
#306319862
#306319122
#306318882
#306318112
C4×#168608002
#306317742
#475×C2512
#3365251