Finite Group Databasebeta

← #22225

Subgroups — #22225

Order 256.

379 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#491C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3C4×#125C211yesmaximal, characteristic
4#1992C211yesmaximal, characteristic
5#1960C211yesmaximal, characteristic
6#1928C212yesmaximal
7#1927C212yesmaximal
8#1865C212yesmaximal
9#1863C212yesmaximal
10#1847C211yesmaximal, characteristic
11#1745C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×Q8C2212yes—
15#491C418yescomplemented
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#97C2211yescharacteristic
23C2×#97C2211yescharacteristic
24#391C2212yes—
25#389—21no—
26#389—21no—
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#387—21no—
30#387—21no—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C4⋊C4C2211yescharacteristic
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8C2×C414yes—
49#125—22no—
50#125C2×C414yes—
51C4×Q8—42no—
52C4×Q8—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—42no—
78#106—42no—
79#103—42no—
80#103—42no—
81#103C2×C418yes—
82#103C2×C418yes—
83#97—22no—
84#97C2×C414yes—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—21no—
92#95—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8C22×C414yes—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8C22×C414yes—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—44no—
132C4⋊C4—42no—
133C4⋊C4C22×C412yes—
134C4⋊C4C22×C412yes—
135C42—22no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—41no—
140C42—41no—
141C42—42no—
142C42—44no—
143C42—42no—
144C42C22×C412yes—
145C42#4112yes—
146C23C22×D411yescharacteristic
147Q8—24no—
148Q8—24no—
149Q8—42no—
150Q8—42no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4C4×D412yes—
184C2×C4C4×D412yes—
185C2×C4#13712yes—
186C2×C4#13712yes—
187C2×C4C23×C411yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4H(5,𝔽2)12yes—
190C2×C42−1+412yes—
191C8—24no—
192C8—42no—
193C22C2×C4×D411yescharacteristic
194C22#53511yescharacteristic
195C22#53611yescharacteristic
196C22C2×#13711yescharacteristic
197C22C2×#13711yescharacteristic
198C22#57511yescharacteristic
199C22#57511yescharacteristic
200C4—22no—
201C4—22no—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—42no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—81no—
210C4—82no—
211C4—81no—
212C4#52012yes—
213C4#54212yes—
214C2#232611yescharacteristic
215C2C4×#13711yescharacteristic
216C2C4×#13711yescharacteristic
217C2#321811yescharacteristic
218C2#321911yescharacteristic
219C2#323611yescharacteristic
220C2#323711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #491C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
C4×C4⋊Q82
C4×#1252
#19922
#19602
#19282
#19272
#18652
#18632
#18472
#17452