Finite Group Databasebeta

← #44550886

Subgroups — #44550886

Order 32128.

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

normal subgroups only

This group has 312 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#30632453C211yesmaximal, characteristic
2#30632263C211yesmaximal, characteristic
3#30631697C211yesmaximal, characteristic, complemented
4#30631695C211yesmaximal, characteristic, complemented
5#30631660C211yesmaximal, characteristic
6#30631443C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×Dic502C2211yescharacteristic
10#22548723—22no—
11#22548721—22no—
12C2×#16860794C2211yescharacteristic
13M16×D251—22no—
14#22548715C412yescomplemented
15#22548703C412yescomplemented
16#22548699—22no—
17#22548695—22no—
18#22548674C2211yescharacteristic
19#22548640—22no—
20#22548638—22no—
21#22548629—22no—
22C2×M16×C251C2211yescharacteristic
23#117×C251C2211yescharacteristic
24#117×C251C2211yescharacteristic
25#16860810—21no—
26#16860810—21no—
27#16860810—21no—
28#16860810—21no—
29C2×Dic502—21no—
30C2×Dic502—21no—
31C2×Dic502—21no—
32C2×Dic502—21no—
33C2×Dic502—21no—
34C2×Dic502C2×C412yes—
35C2×Dic502C2×C411yescharacteristic
36C2×D1004C2×C411yescharacteristic
37C2×C22⋊D251D412yes—
38C22×Dic251—21no—
39C2×C4×D251D412yes—
40#16860794—21no—
41#16860794—21no—
42#16860793—24no—
43C2×C251⋊C8D412yes—
44C8×D251—24no—
45C4⋊Dic251—24no—
46C4⋊Dic251—21no—
47C22⋊Dic251—21no—
48(C2×D251)⋊C4—24no—
49C4×Dic251—21no—
50C22×C4×C251C2311yescharacteristic
51M16×C251—21no—
52M16×C251—21no—
53C2×C8×C251D412yes—
54C4⋊C4×C251—21no—
55C4⋊C4×C251—21no—
56C22⋊C4×C251—21no—
57C22⋊C4×C251—21no—
58C42×C251—21no—
59C42×C251C2×C412yes—
60C22×D251C22⋊C412yes—
61Dic502—21no—
62Dic502—21no—
63Dic502—22no—
64Dic502—22no—
65Dic502—41no—
66Dic502—41no—
67D1004—21no—
68D1004—21no—
69C22⋊D251—22no—
70C22⋊D251—22no—
71C2×Dic251—22no—
72C2×Dic251—41no—
73C2×Dic251—41no—
74C2×Dic251—41no—
75C2×Dic251C22⋊C412yes—
76C4×D251—22no—
77C4×D251—22no—
78C251⋊C8—22no—
79C23×C251C2×D411yescharacteristic
80C2×C4×C251—22no—
81C2×C4×C251—41no—
82C2×C4×C251C22×C411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251#4111yescharacteristic
87C2×C4×C251#4111yescharacteristic
88C8×C251—22no—
89C22×C251—41no—
90C22×C251C2×C22⋊C411yescharacteristic
91C22×C251#12111yescharacteristic
92C22×C251#12111yescharacteristic
93C2×D251—22no—
94C2×D251—42no—
95C4×C251—42no—
96C4×C251—81no—
97C4×C251C4×D411yescharacteristic
98C4×C251C4×D411yescharacteristic
99C4×C251#12011yescharacteristic
100C4×C251#12311yescharacteristic
101Dic251—22no—
102Dic251—42no—
103Dic251—81no—
104C2×C251—21no—
105C2×C251—21no—
106C2×C251—21no—
107C2×C251#38611yescharacteristic
108D251—42no—
109C251#190911yescharacteristic, complemented, Sylow
110#1909—2511nomaximal, Sylow
111#568—2511no—
112#536—2511no—
113#419—2511no—
114#419—2511no—
115#410—2511no—
116H(3,ℤ/4)—2511no—
117H(3,ℤ/4)D25111yescharacteristic, complemented
118C2×#41—2511no—
119C22×Q8—2511no—
120#131—5022no—
121#131—5022no—
122C2×M16—5022no—
123C2×M16—2511no—
124C2×M16C2×D25111yescharacteristic
125C4⋊Q8—2512no—
126#124—2512no—
127#123—5022no—
128C22⋊Q8—5022no—
129#117—2511no—
130#117C2×D25111yescharacteristic
131#117C2×D25111yescharacteristic
132#103—5022no—
133#103—5022no—
134#99—5022no—
135#41—5021no—
136#41—5021no—
137#41—5021no—
138#41—5021no—
139C2×Q8—5021no—
140C2×Q8—5021no—
141C2×Q8—5021no—
142C2×Q8—5021no—
143C2×Q8—5021no—
144C2×Q8—2512no—
145C2×Q8—2511no—
146C2×D4—2512no—
147C2×D4—2511no—
148C22×C4—5021no—
149C22×C4—2512no—
150C22×C4C22×D25111yescharacteristic
151M16—21no—
152M16—21no—
153M16—5024no—
154M16—5021no—
155M16—5021no—
156C2×C8—5024no—
157C2×C8—2512no—
158C2×C8D100412yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—5024no—
162C4⋊C4—5021no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—5024no—
166C22⋊C4—5021no—
167C42—21no—
168C42—5021no—
169C42C4×D25112yes—
170C23—2512no—
171C23D4×D25111yescharacteristic
172Q8—10041no—
173Q8—10041no—
174Q8—5022no—
175Q8—5022no—
176Q8—5021no—
177Q8—5021no—
178D4—5022no—
179D4—5022no—
180D4—5021no—
181D4—5021no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—10041no—
185C2×C4—10041no—
186C2×C4—10041no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—2512no—
191C2×C4C2×C4×D25111yescharacteristic
192C2×C4D4×D25111yescharacteristic
193C2×C4D4×D25111yescharacteristic
194C2×C4C2×D100411yescharacteristic
195C2×C4#1686081011yescharacteristic
196C2×C4#1686081111yescharacteristic
197C8—22no—
198C8—5022no—
199C22—41no—
200C22—10042no—
201C22—5022no—
202C22C22⋊C4×D25111yescharacteristic
203C22#2254868811yescharacteristic
204C22#2254869011yescharacteristic
205C4—42no—
206C4—81no—
207C4—20081no—
208C4—10042no—
209C4—5022no—
210C4#2254867711yescharacteristic
211C4C4×D100411yescharacteristic
212C4#2254868511yescharacteristic
213C4#2254870011yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—10042no—
218C2#3063156511yescharacteristic
subgroupG/H
centerC2#30631565
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingH(3,ℤ/4)×C251C2
socleC2×C251#386
prime radicalC2×#16860810C22
cosocleC22×C4×C251C23
hypercenterH(3,ℤ/4)D251
hypocenterC251#1909
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
3. #30631697C2semidirect
4. #30631695C2semidirect
6. #30631443C2semidirect
7. H(3,ℤ/4)×C251C2semidirect
14. #22548715C4semidirect
15. #22548703C4semidirect
109. C251#1909semidirect
117. H(3,ℤ/4)D251semidirect

Maximal subgroups

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

subgroupindex
#306324532
#306322632
#306316972
#306316952
#306316602
#306314432
H(3,ℤ/4)×C2512
#1909251