Finite Group Databasebeta

← #9040

Subgroups — #9040

Order 256.

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

normal subgroups only

This group has 375 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
1C2×#451C211yesmaximal, characteristic
2#2329C211yesmaximal, characteristic
3#1787C211yesmaximal, characteristic
4C2×#328C211yesmaximal, characteristic
5#1446C211yesmaximal, characteristic
6#1294C211yesmaximal, characteristic
7#1280C211yesmaximal, characteristic
8C23×Q8C2211yescharacteristic
9C22×Q16—21no—
10C2×C22⋊Q8C2211yescharacteristic
11#451—22no—
12#451C414yescomplemented
13C2×#103—21no—
14C2×#103—21no—
15C2×#103—21no—
16C2×C22⋊C8C2211yescharacteristic
17C2×C22⋊C8C2211yescharacteristic
18#391—21no—
19#389—21no—
20#387—21no—
21#387—21no—
22#385—21no—
23#380C2211yescharacteristic
24C4×C22⋊C4—21no—
25C2×#95—21no—
26C2×#95—21no—
27C2×#95C2211yescharacteristic
28C2×#95C2211yescharacteristic
29#328—22no—
30C22×Q8—21no—
31C22×Q8—21no—
32C22×Q8—21no—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8—42no—
36C22×Q8—42no—
37C22×Q8C2×C412yes—
38C22×Q8D411yescharacteristic
39C23×C4—21no—
40C23×C4—21no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C2×Q16—24no—
45C2×Q16—42no—
46C2×Q16—42no—
47C22×C8—21no—
48C22×C8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8C2×C412yes—
52C22⋊Q8D412yes—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42—21no—
63#103—24no—
64#103—42no—
65#103—42no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8C2×C412yes—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77#95—21no—
78#95—21no—
79#95—42no—
80#95—42no—
81C24C2×D411yescharacteristic
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—44no—
90C2×Q8—41no—
91C2×Q8—41no—
92C2×Q8—41no—
93C2×Q8—41no—
94C2×Q8—44no—
95C2×Q8—41no—
96C2×Q8—41no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8C22⋊C412yes—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4Q1612yescomplemented
132C22×C4C22×C411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4#4111yescharacteristic
137C22×C4#4111yescharacteristic
138Q16—48no—
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C4⋊C4C22⋊C412yes—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—41no—
156C22⋊C4—41no—
157C42—81no—
158C42—81no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23C4×D411yescharacteristic
164C23#12011yescharacteristic
165C23#12111yescharacteristic
166C23C2×Q1611yescharacteristic
167C23#13511yescharacteristic
168C23#13711yescharacteristic
169C23#13711yescharacteristic
170Q8—24no—
171Q8—24no—
172Q8—42no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178Q8—42no—
179Q8—81no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4C2×C22⋊C411yescharacteristic
225C2×C4C4×D411yescharacteristic
226C2×C4#12111yescharacteristic
227C2×C4#12311yescharacteristic
228C8—42no—
229C8—81no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22#38611yescharacteristic
243C22C4×Q1611yescharacteristic
244C22#44111yescharacteristic
245C22#45111yescharacteristic
246C22#45211yescharacteristic
247C22#45511yescharacteristic
248C22#45611yescharacteristic
249C22B(4,𝔽2)11yescharacteristic
250C22B(4,𝔽2)11yescharacteristic
251C22#46211yescharacteristic
252C22#46411yescharacteristic
253C4—21no—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—42no—
261C4—82no—
262C4—81no—
263C4—81no—
264C4—81no—
265C2—21no—
266C2—21no—
267C2—21no—
268C2—21no—
269C2#160211yescharacteristic
270C2#160311yescharacteristic
271C2#160511yescharacteristic
272C2#160611yescharacteristic
273C2#187811yescharacteristic
274C2#188211yescharacteristic
275C2#188911yescharacteristic
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
12. #451C4semidirect
131. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
#23292
#17872
C2×#3282
#14462
#12942
#12802