Finite Group Databasebeta

← #11716

Subgroups — #11716

Order 256.

296 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 63 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#2626C211yesmaximal, characteristic
2C2×#342C211yesmaximal, characteristic
3C2×#323C211yesmaximal, characteristic
4#1352C212yesmaximal, complemented
5#1351C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic, complemented
7C22×C22⋊C4—21no—
8C22×C22⋊C4C2211yescharacteristic
9C2×C22⋊C8—21no—
10#399C411yescharacteristic, complemented
11#399C2211yescharacteristic
12#396—21no—
13#396—21no—
14#393C411yescharacteristic, complemented
15#393C2211yescharacteristic
16#385—21no—
17#380C411yescharacteristic, complemented
18#379—21no—
19#342—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C25D411yescharacteristic
24C22×Q8C2×C411yescharacteristic, complemented
25C23×C4—41no—
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C22×C8—41no—
29C22⋊Q8—41no—
30C22⋊Q8—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic, complemented
36C2×C22⋊C4—24no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—41no—
45C2×C22⋊C4—41no—
46C2×C22⋊C4—41no—
47C2×C22⋊C4—42no—
48C2×C22⋊C4C2×C411yescharacteristic
49C2×C22⋊C4D411yescharacteristic
50C2×C22⋊C4D411yescharacteristic
51C2×C22⋊C4C2311yescharacteristic
52C22⋊C8—22no—
53C22⋊C8—41no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic, complemented
57#95C2×C411yescharacteristic, complemented
58C24—21no—
59C24—21no—
60C24—22no—
61C24—81no—
62C24—81no—
63C24C22⋊C411yescharacteristic
64C24C22⋊C411yescharacteristic
65C24SD1612yes—
66C24Q1612yes—
67C24C2×D411yescharacteristic
68C2×Q8—41no—
69C2×Q8—81no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—44no—
79C22×C4—42no—
80C22×C4—81no—
81C22×C4—81no—
82C22×C4—82no—
83C22×C4—82no—
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—42no—
89C2×C8—82no—
90C4⋊C4—41no—
91C4⋊C4—42no—
92C4⋊C4—81no—
93C4⋊C4—81no—
94C4⋊C4—81no—
95C22⋊C4—24no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—44no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—81no—
106C22⋊C4—81no—
107C22⋊C4—81no—
108C22⋊C4—81no—
109C23—22no—
110C23—22no—
111C23—21no—
112C23—21no—
113C23—22no—
114C23—22no—
115C23—21no—
116C23—21no—
117C23—41no—
118C23—41no—
119C23—42no—
120C23—42no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—81no—
126C23—81no—
127C23—82no—
128C23—82no—
129C23—84no—
130C23#9911yescharacteristic
131C23#9911yescharacteristic
132C23#10312yes—
133C23#10312yes—
134C23#10412yes—
135C23C2×C22⋊C411yescharacteristic
136C23C2×SD1611yescharacteristic
137C23C2×Q1611yescharacteristic
138Q8—82no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—88no—
165C2×C4—82no—
166C2×C4—82no—
167C8—82no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—21no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—44no—
183C22—44no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—82no—
191C22—82no—
192C22#32812yes—
193C22#32812yes—
194C22C2×#9911yescharacteristic
195C22C2×#10311yescharacteristic
196C22C2×#10411yescharacteristic
197C4—41no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—81no—
202C4—81no—
203C4—84no—
204C4—81no—
205C4—81no—
206C2—21no—
207C2—22no—
208C2—21no—
209C2—22no—
210C2—41no—
211C2—42no—
212C2—41no—
213C2C2×#32811yescharacteristic
214C2#212111yescharacteristic
215C2#212711yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1352C2semidirect
5. #1351C2semidirect
6. C2×C22⋊Q8C4semidirect
10. #399C4semidirect
14. #393C4semidirect
17. #380C4semidirect
24. C22×Q8C2×C4semidirect
35. C2×C4⋊C4C2×C4semidirect
56. #95C2×C4semidirect
57. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#26262
C2×#3422
C2×#3232
#13522
#13512