Finite Group Databasebeta

← #44550896

Subgroups — #44550896

Order 32128.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#30632256C211yesmaximal, characteristic
2#30632161C211yesmaximal, characteristic, complemented
3#30631705C211yesmaximal, characteristic, complemented
4#30631703C211yesmaximal, characteristic, complemented
5C2×#22548631C211yesmaximal, characteristic, complemented
6#30631443C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×D1004C2211yescharacteristic
10C2×#16860794C2211yescharacteristic
11#22548712C412yescomplemented
12#22548703C412yescomplemented
13#22548690—22no—
14#22548685—22no—
15C4×D1004—22no—
16#22548677—22no—
17#22548674C2211yescharacteristic
18C22⋊C4×D251—22no—
19#22548643—22no—
20#22548641—22no—
21#22548631—22no—
22C2×M16×C251C2211yescharacteristic
23#117×C251C2211yescharacteristic
24#117×C251C2211yescharacteristic
25C23×D251D412yes—
26#16860810—21no—
27#16860810—21no—
28#16860810—21no—
29#16860810—21no—
30C2×Dic502C2×C411yescharacteristic, complemented
31C2×D1004—24no—
32C2×D1004—21no—
33C2×D1004—21no—
34C2×D1004—21no—
35C2×D1004—21no—
36C2×D1004C2×C412yes—
37C2×D1004C2×C411yescharacteristic, complemented
38C2×D1004D412yes—
39C2×C22⋊D251—21no—
40C2×C4×D251—24no—
41C2×C4×D251—21no—
42#16860794—21no—
43#16860794—21no—
44C2×C251⋊C8—21no—
45C4⋊Dic251—21no—
46C22⋊Dic251—21no—
47(C2×D251)⋊C4—24no—
48(C2×D251)⋊C4—24no—
49C4×Dic251D412yescomplemented
50C22×C4×C251C2311yescharacteristic
51M16×C251—21no—
52M16×C251—21no—
53C2×C8×C251—21no—
54C4⋊C4×C251—21no—
55C4⋊C4×C251—21no—
56C22⋊C4×C251—21no—
57C22⋊C4×C251—21no—
58C42×C251C2×C412yescomplemented
59C42×C251D412yescomplemented
60C22×D251—22no—
61C22×D251—22no—
62C22×D251—21no—
63C22×D251—82no—
64C22×D251C22⋊C414yes—
65Dic502—21no—
66Dic502—21no—
67D1004—22no—
68D1004—22no—
69D1004—21no—
70D1004—21no—
71D1004—22no—
72D1004—22no—
73D1004—84no—
74C22⋊D251—41no—
75C22⋊D251—41no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C4×D251—41no—
79C4×D251—41no—
80C4×D251—84no—
81C251⋊C8—41no—
82C23×C251C2×D411yescharacteristic
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251C22×C411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2×D411yescharacteristic
89C2×C4×C251#4111yescharacteristic
90C2×C4×C251#4111yescharacteristic
91C8×C251—41no—
92C22×C251—41no—
93C22×C251C2×C22⋊C411yescharacteristic
94C22×C251#12011yescharacteristic
95C22×C251#12311yescharacteristic
96C2×D251—24no—
97C2×D251—44no—
98C2×D251—44no—
99C2×D251—42no—
100C2×D251—41no—
101C2×D251—82no—
102C4×C251—42no—
103C4×C251—42no—
104C4×C251C4×D411yescharacteristic
105C4×C251C4×D411yescharacteristic
106C4×C251#12111yescharacteristic
107C4×C251#12111yescharacteristic
108Dic251—42no—
109Dic251—41no—
110C2×C251—21no—
111C2×C251—21no—
112C2×C251—21no—
113C2×C251#38611yescharacteristic
114D251—44no—
115D251—81no—
116C251#191111yescharacteristic, complemented, Sylow
117#1911—2511nomaximal, Sylow
118#535—2511no—
119#518—2511no—
120#421—2511no—
121#421—2511no—
122C2×#100—2511no—
123H(3,ℤ/4)—2511no—
124H(3,ℤ/4)D25111yescharacteristic, complemented
125C2×#41—2511no—
126C22×D4—2511no—
127C2×M16—2511no—
128C2×M16C2×D25111yescharacteristic
129#127—2512no—
130#124—2512no—
131#121—5022no—
132#120—5022no—
133C4×D4—5022no—
134C4×D4—5022no—
135#117—2511no—
136#117C2×D25111yescharacteristic
137#117C2×D25111yescharacteristic
138C2×C22⋊C4—5022no—
139#104—5022no—
140#104—5022no—
141#100—5022no—
142C24—2512no—
143#41—5021no—
144#41—5021no—
145#41—5021no—
146#41—5021no—
147C2×Q8—2511no—
148C2×D4—5024no—
149C2×D4—5021no—
150C2×D4—5021no—
151C2×D4—5021no—
152C2×D4—5021no—
153C2×D4—5021no—
154C2×D4—2511no—
155C2×D4—2512no—
156C2×D4—2512no—
157C22×C4—5024no—
158C22×C4—5021no—
159C22×C4C22×D25111yescharacteristic
160M16—21no—
161M16—21no—
162M16—5021no—
163M16—5021no—
164C2×C8—21no—
165C2×C8—5021no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—5021no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—5024no—
172C22⋊C4—5024no—
173C22⋊C4—5021no—
174C42—2512no—
175C42C4×D25112yescomplemented
176C42D100412yescomplemented
177C23—20082no—
178C23—5022no—
179C23—5021no—
180C23—5022no—
181C23—2514no—
182C23D4×D25111yescharacteristic
183Q8—5021no—
184Q8—5021no—
185D4—20084no—
186D4—10041no—
187D4—10041no—
188D4—5021no—
189D4—5022no—
190D4—5021no—
191D4—5022no—
192D4—5022no—
193D4—5022no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—20084no—
197C2×C4—10041no—
198C2×C4—10041no—
199C2×C4—5021no—
200C2×C4—5022no—
201C2×C4C2×C4×D25111yescharacteristic
202C2×C4D4×D25111yescharacteristic
203C2×C4D4×D25111yescharacteristic
204C2×C4C2×D100411yescharacteristic
205C2×C4#1686081011yescharacteristic
206C2×C4#1686081111yescharacteristic
207C8—41no—
208C8—10041no—
209C22—41no—
210C22—20082no—
211C22—10042no—
212C22—10044no—
213C22—10044no—
214C22—10041no—
215C22—5024no—
216C22C22⋊C4×D25111yescharacteristic
217C22#2254868511yescharacteristic
218C22#2254870011yescharacteristic
219C4—42no—
220C4—42no—
221C4—10042no—
222C4—10041no—
223C4#2254867711yescharacteristic
224C4C4×D100411yescharacteristic
225C4#2254868811yescharacteristic
226C4#2254869011yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—20081no—
231C2—10044no—
232C2#3063156511yescharacteristic
subgroupG/H
centerC2#30631565
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingH(3,ℤ/4)×C251C2
socleC2×C251#386
prime radical#30632256C2
cosocleC22×C4×C251C23
hypercenterH(3,ℤ/4)D251
hypocenterC251#1911
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #30632161C2semidirect
3. #30631705C2semidirect
4. #30631703C2semidirect
5. C2×#22548631C2semidirect
6. #30631443C2semidirect
7. H(3,ℤ/4)×C251C2semidirect
11. #22548712C4semidirect
12. #22548703C4semidirect
30. C2×Dic502C2×C4semidirect
37. C2×D1004C2×C4semidirect
49. C4×Dic251D4semidirect
58. C42×C251C2×C4semidirect
59. C42×C251D4semidirect
116. C251#1911semidirect
124. H(3,ℤ/4)D251semidirect
175. C42C4×D251semidirect
176. C42D1004semidirect

Maximal subgroups

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

subgroupindex
#306322562
#306321612
#306317052
#306317032
C2×#225486312
#306314432
H(3,ℤ/4)×C2512
#1911251