Finite Group Databasebeta

← #8871

Subgroups — #8871

Order 256.

309 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 77 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#2329C211yesmaximal, characteristic
2#1872C211yesmaximal, characteristic
3#1853C211yesmaximal, characteristic
4C2×#328C211yesmaximal, characteristic
5#1444C211yesmaximal, characteristic
6#1294C211yesmaximal, characteristic
7#1280C211yesmaximal, characteristic
8C2×C22⋊Q8C2211yescharacteristic
9C22×C4⋊C4—21no—
10C22×C4⋊C4C2211yescharacteristic
11C2×#106—21no—
12C2×#103—21no—
13C2×C22⋊C8C2211yescharacteristic
14C2×C22⋊C8C2211yescharacteristic
15#391—21no—
16#389—21no—
17#387—21no—
18#387—21no—
19#385—21no—
20#380C2211yescharacteristic
21C4×C22⋊C4—21no—
22C2×#95—21no—
23C2×#95C2211yescharacteristic
24C2×#95C2211yescharacteristic
25#340—21no—
26#340—21no—
27#328C414yescomplemented
28C22×Q8D411yescharacteristic
29C23×C4—21no—
30C23×C4—21no—
31C23×C4D411yescharacteristic
32C23×C4D411yescharacteristic
33C23×C4C2311yescharacteristic
34C22×C8—21no—
35C22×C8—21no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C22⋊Q8C2×C412yes—
39C22⋊Q8Q812yes—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C42—21no—
55C2×C42—21no—
56#106—42no—
57#103—42no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C22⋊C8C2×C412yes—
62#95—21no—
63#95—21no—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—21no—
68#95—21no—
69#95—21no—
70#95—42no—
71#95C2×C412yes—
72C24C2×D411yescharacteristic
73C2×Q8—21no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8C4⋊C412yescomplemented
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4SD1612yes—
107C22×C4C22×C411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×Q811yescharacteristic
111C22×C4#4111yescharacteristic
112C22×C4#4111yescharacteristic
113C2×C8—22no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—48no—
129C4⋊C4—81no—
130C4⋊C4—81no—
131C4⋊C4C4⋊C412yescomplemented
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C42—81no—
137C42—81no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23C4×D411yescharacteristic
143C23#12011yescharacteristic
144C23C22⋊Q811yescharacteristic
145C23C2×SD1611yescharacteristic
146C23#13511yescharacteristic
147C23#13711yescharacteristic
148C23#13711yescharacteristic
149Q8—42no—
150Q8—81no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4C2×C4⋊C411yescharacteristic
196C2×C4C4×D411yescharacteristic
197C2×C4C22⋊Q811yescharacteristic
198C2×C4#12311yescharacteristic
199C8—42no—
200C8—81no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22#38511yescharacteristic
214C22C4×SD1611yescharacteristic
215C22#44111yescharacteristic
216C22#44811yescharacteristic
217C22#45211yescharacteristic
218C22#45311yescharacteristic
219C22#45611yescharacteristic
220C22B(4,𝔽2)11yescharacteristic
221C22#45811yescharacteristic
222C22#47511yescharacteristic
223C22#47911yescharacteristic
224C4—21no—
225C4—21no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—82no—
233C4—81no—
234C4—81no—
235C4—81no—
236C2—21no—
237C2—21no—
238C2—21no—
239C2—21no—
240C2#159611yescharacteristic
241C2#160311yescharacteristic
242C2#161111yescharacteristic
243C2#161811yescharacteristic
244C2#185911yescharacteristic
245C2#186511yescharacteristic
246C2#186711yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #328C4semidirect
76. C2×Q8C4⋊C4semidirect
131. C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23292
#18722
#18532
C2×#3282
#14442
#12942
#12802