Finite Group Databasebeta

← #18392

Subgroups — #18392

Order 256.

467 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 121 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#3507C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#1603C218yesmaximal, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1502C211yesmaximal, characteristic
8#1461C211yesmaximal, characteristic, complemented
9#559—21no—
10#557—21no—
11#556—21no—
12#556—22no—
13#554—21no—
14#552—22no—
15#539C2214yes—
16#536—21no—
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#529—22no—
21#520—21no—
22C2×C4×Q8C2211yescharacteristic
23#494—21no—
24#492—21no—
25#487—21no—
26#487—21no—
27#462—22no—
28#461—22no—
29#451C2214yes—
30#448C2214yes—
31#431C2211yescharacteristic
32#431C2211yescharacteristic
33#395C2214yes—
34#391—21no—
35#389—22no—
36#384—21no—
37C4×C4⋊C4C2211yescharacteristic
38#328C2214yes—
39#327C2214yes—
40#324C2214yes—
41C2×#41—21no—
42C22×Q8D411yescharacteristic
43C22×Q8C2312yes—
44C2×Q16—24no—
45C2×SD16—24no—
46C4⋊Q8—22no—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8D411yescharacteristic
52C4⋊Q8D411yescharacteristic
53C4⋊Q8D411yescharacteristic
54#127D411yescharacteristic
55#126—24no—
56#126—24no—
57#125—22no—
58#125—24no—
59#125—21no—
60#124—22no—
61#124—22no—
62#124—21no—
63#123—24no—
64#123—24no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—24no—
68C22⋊Q8—21no—
69C22⋊Q8C2312yes—
70#121—21no—
71#121C2312yes—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8D412yes—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—24no—
83C4×D4—21no—
84C4×D4D412yes—
85#117—22no—
86#117—24no—
87#117—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C42—21no—
96C2×C42D412yes—
97C2×C42C2311yescharacteristic
98C4⋊C8—21no—
99C4⋊C8—21no—
100#103—24no—
101#103—24no—
102C22⋊C8C2312yes—
103C22⋊C8C2312yes—
104C4×C8—21no—
105C4×C8—21no—
106#95—24no—
107#95C2312yes—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—82no—
119C2×Q8C2×D412yes—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4C2×D412yes—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133Q16—84no—
134SD16—84no—
135C2×C8—22no—
136C2×C8—22no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—84no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—24no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—42no—
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C23—22no—
174C23C22×D411yescharacteristic
175Q8—42no—
176Q8—44no—
177Q8—44no—
178Q8—42no—
179Q8—42no—
180Q8—42no—
181Q8—42no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—42no—
186D4—42no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—84no—
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#12011yescharacteristic
207C2×C4#12011yescharacteristic
208C2×C4#13512yes—
209C2×C4#13712yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C8—42no—
213C8—42no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×#12011yescharacteristic
220C22C2×#13511yescharacteristic
221C22C2×#13711yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—44no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4#45212yes—
233C4#45212yes—
234C4#45612yes—
235C2—21no—
236C2—21no—
237C2—82no—
238C2C2×#45211yescharacteristic
239C2C2×#45611yescharacteristic
240C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3209C2semidirect
4. #2567C2semidirect
5. #1603C2semidirect
6. #1503C2semidirect
8. #1461C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32092
#32062
#25672
#16032
#15032
#15022
#14612