Finite Group Databasebeta

← #44549415

Subgroups — #44549415

Order 32128.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632275C211yesmaximal, characteristic
2#30632023C211yesmaximal, characteristic, complemented
3#30632006C211yesmaximal, characteristic, complemented
4#30631623C211yesmaximal, characteristic, complemented
5#30631425C211yesmaximal, characteristic, complemented
6#30631422C211yesmaximal, characteristic, complemented
7#323×C251C211yesmaximal, characteristic, complemented
8#22548708—21no—
9#22548701—21no—
10#22548697—21no—
11#22548694C2211yescharacteristic
12#22548689C2211yescharacteristic
13#22548688—21no—
14#22548684—21no—
15C4×C22⋊D251—21no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×Dic251⋊C4—21no—
18C2×C22⋊Dic251—21no—
19#22548649—21no—
20#22548647—21no—
21#22548640—21no—
22#22548637—21no—
23C251⋊#95C2211yescharacteristic
24C2×C22⋊C4×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26C22⋊C8×C251C2211yescharacteristic
27C2×Dic502D411yescharacteristic
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—21no—
30C2×C22⋊D251—21no—
31C2×C22⋊D251—21no—
32C22×Dic251—21no—
33C22×Dic251—21no—
34C2×C4×D251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251D411yescharacteristic
38C4⋊Dic251D411yescharacteristic
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41Dic251⋊C4—42no—
42C22⋊Dic251—21no—
43C22⋊Dic251—21no—
44C22⋊Dic251—21no—
45C22⋊Dic251—42no—
46(C2×D251)⋊C4—21no—
47(C2×D251)⋊C4—21no—
48C4×Dic251—21no—
49C24×C251D411yescharacteristic
50C22×C4×C251D411yescharacteristic
51C22×C4×C251C2311yescharacteristic
52C2×C8×C251—21no—
53C2×C8×C251—21no—
54C22⋊C4×C251—21no—
55C22⋊C4×C251—21no—
56C22×D251—21no—
57Dic502—41no—
58D1004—41no—
59C22⋊D251—41no—
60C22⋊D251—41no—
61C22⋊D251—81no—
62C22⋊D251—81no—
63C2×Dic251—22no—
64C2×Dic251—21no—
65C2×Dic251—21no—
66C2×Dic251—21no—
67C2×Dic251—41no—
68C2×Dic251—41no—
69C2×Dic251—41no—
70C2×Dic251—41no—
71C2×Dic251—41no—
72C4×D251—81no—
73C23×C251—21no—
74C23×C251—21no—
75C23×C251—21no—
76C23×C251—41no—
77C23×C251—41no—
78C23×C251C2×D411yescharacteristic
79C2×C4×C251—21no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C8×C251—41no—
85C8×C251—41no—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251—42no—
89C22×C251—41no—
90C22×C251—41no—
91C22×C251—41no—
92C22×C251—81no—
93C22×C251#12011yescharacteristic
94C22×C251#13511yescharacteristic
95C22×C251#13711yescharacteristic
96C2×D251—41no—
97C2×D251—41no—
98C2×D251—41no—
99C4×C251—21no—
100C4×C251—21no—
101C4×C251—41no—
102Dic251—42no—
103Dic251—41no—
104Dic251—41no—
105Dic251—41no—
106Dic251—81no—
107C2×C251—21no—
108C2×C251—21no—
109C2×C251—81no—
110C2×C251#45211yescharacteristic
111C2×C251#45311yescharacteristic
112C2×C251#45811yescharacteristic
113D251—81no—
114C251#161811yescharacteristic, complemented, Sylow
115#1618—2511nomaximal, Sylow
116#538—2511no—
117#484—2511no—
118#481—2511no—
119#399—2511no—
120#328—2511no—
121#327—2511no—
122#323D25111yescharacteristic, complemented
123#126—5021no—
124#124—5021no—
125#123—5021no—
126C22⋊Q8—2511no—
127#121—5021no—
128#121—2511no—
129#120—5021no—
130C4×D4—5021no—
131C2×C4⋊C4—5021no—
132C2×C4⋊C4—2511no—
133C2×C22⋊C4—5021no—
134C2×C22⋊C4C2×D25111yescharacteristic
135#107—5021no—
136#106—5021no—
137#103—5021no—
138#102—5021no—
139C22⋊C8C2×D25111yescharacteristic
140C22⋊C8C2×D25111yescharacteristic
141#95—2511no—
142C24D100411yescharacteristic
143C2×Q8—2511no—
144C2×D4—5021no—
145C2×D4—5021no—
146C2×D4—5021no—
147C2×D4—2511no—
148C22×C4—5021no—
149C22×C4—5021no—
150C22×C4—5021no—
151C22×C4D100411yescharacteristic
152C22×C4C22×D25111yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—10042no—
156C4⋊C4—5021no—
157C4⋊C4—5021no—
158C4⋊C4—5021no—
159C4⋊C4—5021no—
160C4⋊C4—2511no—
161C4⋊C4—2511no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—10042no—
165C22⋊C4—5021no—
166C22⋊C4—5021no—
167C22⋊C4—5021no—
168C22⋊C4—5021no—
169C22⋊C4—5021no—
170C42—5021no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—5021no—
177C23C2×D100411yescharacteristic
178Q8—10041no—
179D4—20081no—
180D4—20081no—
181D4—10041no—
182D4—10041no—
183D4—10041no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—20081no—
188C2×C4—10041no—
189C2×C4—10041no—
190C2×C4—10041no—
191C2×C4—10041no—
192C2×C4—10041no—
193C2×C4—5022no—
194C2×C4—5021no—
195C2×C4—5021no—
196C2×C4—5021no—
197C2×C4D4×D25111yescharacteristic
198C2×C4D4×D25111yescharacteristic
199C8—41no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—10041no—
209C22—10041no—
210C22—10041no—
211C22#2254868511yescharacteristic
212C22#2254873711yescharacteristic
213C22#2254874811yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—20081no—
218C4—10042no—
219C4—10041no—
220C4—10041no—
221C4—10041no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—20081no—
226C2#3063185111yescharacteristic
227C2#3063185711yescharacteristic
228C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251#120
prime radical#30632275C2
hypercenter#323D251
hypocenterC251#1618
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #30632023C2semidirect
3. #30632006C2semidirect
4. #30631623C2semidirect
5. #30631425C2semidirect
6. #30631422C2semidirect
7. #323×C251C2semidirect
114. C251#1618semidirect
122. #323D251semidirect

Maximal subgroups

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

subgroupindex
#306322752
#306320232
#306320062
#306316232
#306314252
#306314222
#323×C2512
#1618251