Finite Group Databasebeta

← #9037

Subgroups — #9037

Order 256.

338 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 61 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#3002C211yesmaximal, characteristic
2#2976C211yesmaximal, characteristic, complemented
3#1789C211yesmaximal, characteristic, complemented
4#1780C211yesmaximal, characteristic, complemented
5#1501C211yesmaximal, characteristic, complemented
6#1298C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic
9C2×#135—21no—
10#525C2211yescharacteristic
11#452—21no—
12#451C411yescharacteristic, complemented
13#450C411yescharacteristic, complemented
14#449—21no—
15#448C411yescharacteristic, complemented
16#447C411yescharacteristic, complemented
17#436—21no—
18#435—21no—
19#435—21no—
20C8×D4—21no—
21#433—21no—
22C2×#104—21no—
23C2×#104—21no—
24#416—21no—
25#408—21no—
26#407C2211yescharacteristic
27C2×C22⋊C8—21no—
28C2×C22⋊C8—21no—
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C4×C22⋊C4C2211yescharacteristic
33#328—21no—
34#327—21no—
35C2×#41—21no—
36C2×#41—21no—
37C2×#41—21no—
38C2×#41—21no—
39C2×#41—42no—
40C2×#41—42no—
41C2×#41D411yescharacteristic
42C22×Q8C2×C411yescharacteristic
43C22×D4—21no—
44C22×D4C2×C411yescharacteristic
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47#135—42no—
48#135—42no—
49C2×Q16—21no—
50C2×SD16—21no—
51C2×SD16—21no—
52C2×D8—21no—
53C2×M16—21no—
54C2×M16D411yescharacteristic
55C2×M16D411yescharacteristic
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8D411yescharacteristic
60C22×C8D411yescharacteristic
61#123—21no—
62C22⋊Q8C2×C411yescharacteristic
63C22⋊Q8D411yescharacteristic
64#121C2×C411yescharacteristic
65#121D411yescharacteristic
66#120—21no—
67C4×D4—21no—
68C4×D4—21no—
69#117D411yescharacteristic
70C2×C22⋊C4—21no—
71C2×C42—21no—
72C4⋊C8—21no—
73C4⋊C8—21no—
74#104—42no—
75#104—42no—
76#103—21no—
77#103—21no—
78#102—21no—
79#102—21no—
80C22⋊C8—21no—
81C22⋊C8—21no—
82C22⋊C8—21no—
83C22⋊C8—21no—
84C22⋊C8—21no—
85C22⋊C8—21no—
86C22⋊C8—21no—
87C22⋊C8—21no—
88C22⋊C8—21no—
89C22⋊C8—42no—
90C22⋊C8—42no—
91C22⋊C8C2×C411yescharacteristic
92C22⋊C8C2×C411yescharacteristic
93#97—21no—
94C4×C8—21no—
95#95—21no—
96C24—21no—
97C24#4111yescharacteristic
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102#41—41no—
103#41—41no—
104#41—41no—
105#41—41no—
106#41—41no—
107#41—41no—
108#41—42no—
109#41—42no—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—42no—
115C2×Q8C22⋊C411yescharacteristic
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4C22⋊C411yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C22×C411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4#4111yescharacteristic
144Q16—42no—
145SD16—42no—
146SD16—42no—
147D8—42no—
148M16—41no—
149M16—41no—
150M16—81no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—81no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4C22⋊C411yescharacteristic
168C4⋊C4C22⋊C411yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C42—21no—
177C42—41no—
178C42—41no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23C4×D411yescharacteristic
192C23#12111yescharacteristic
193C23#12311yescharacteristic
194Q8—22no—
195Q8—22no—
196Q8—41no—
197Q8—41no—
198Q8—41no—
199D4—22no—
200D4—22no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—42no—
208D4—41no—
209D4—41no—
210D4—81no—
211D4—81no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4C2×C22⋊C411yescharacteristic
237C2×C4C4×D411yescharacteristic
238C2×C4#12011yescharacteristic
239C2×C4#12111yescharacteristic
240C8—41no—
241C8—41no—
242C8—41no—
243C8—41no—
244C8—41no—
245C8—41no—
246C8—81no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#38611yescharacteristic
268C22#44311yescharacteristic
269C22#44411yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—82no—
278C4B(4,𝔽2)11yescharacteristic
279C4B(4,𝔽2)11yescharacteristic
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—81no—
285C2—82no—
286C2#187611yescharacteristic
287C2#188111yescharacteristic
288C2#188911yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
socleC22#386
prime radical#3002C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2976C2semidirect
3. #1789C2semidirect
4. #1780C2semidirect
5. #1501C2semidirect
6. #1298C2semidirect
7. #1282C2semidirect
12. #451C4semidirect
13. #450C4semidirect
15. #448C4semidirect
16. #447C4semidirect

Maximal subgroups

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

subgroupindex
#30022
#29762
#17892
#17802
#15012
#12982
#12822