Finite Group Databasebeta

← #8499

Subgroups — #8499

Order 256.

338 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 71 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#3552C211yesmaximal, characteristic
2#1993C211yesmaximal, characteristic, complemented
3#1556C211yesmaximal, characteristic, complemented
4#1532C211yesmaximal, characteristic, complemented
5#1532C211yesmaximal, characteristic, complemented
6#1512C211yesmaximal, characteristic, complemented
7#1512C211yesmaximal, characteristic, complemented
8#563—22no—
9#557C411yescharacteristic, complemented
10#556C411yescharacteristic, complemented
11#554—21no—
12#553—21no—
13D4×Q8—21no—
14D4×Q8C411yescharacteristic, complemented
15#548—21no—
16#547C411yescharacteristic, complemented
17#544—21no—
18#542—21no—
19#541—21no—
20#539—22no—
21#536—21no—
22C2×C4⋊Q8C2211yescharacteristic
23#529C2211yescharacteristic
24#529C2211yescharacteristic
25#432C2211yescharacteristic
26#432C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#382C2211yescharacteristic
29#340—22no—
30#332—22no—
31#330—21no—
32#330—21no—
33#329—21no—
34#329—21no—
35#328—21no—
36#328—21no—
37#327—21no—
38#327—21no—
39C2×#41—21no—
40C22×Q8—21no—
41C22×Q8—21no—
42C22×C8—21no—
43C4⋊Q8—22no—
44C4⋊Q8—21no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8—21no—
48C4⋊Q8C2×C411yescharacteristic
49C4⋊Q8C2×C411yescharacteristic
50C4⋊Q8D412yes—
51#126—22no—
52#126—22no—
53#126—22no—
54#126—21no—
55#126—21no—
56#125—22no—
57#125—22no—
58#125—22no—
59#125—21no—
60#125D411yescharacteristic
61#125D411yescharacteristic
62#124—22no—
63#124—21no—
64#124D411yescharacteristic
65#124D411yescharacteristic
66#123—22no—
67#123—22no—
68#123—21no—
69#123—21no—
70#123—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8D411yescharacteristic
81C22⋊Q8D411yescharacteristic
82#121—22no—
83#121—21no—
84#121D411yescharacteristic
85#121D411yescharacteristic
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8C2×C411yescharacteristic
90C4×Q8C2×C411yescharacteristic
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4C2×C411yescharacteristic
97C4×D4C2×C411yescharacteristic
98#117—21no—
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D412yes—
105C2×C42C2311yescharacteristic
106C4⋊C8—21no—
107C4⋊C8—21no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110#97—21no—
111#97—21no—
112#97—21no—
113#97—21no—
114#95—21no—
115#95—21no—
116#41—82no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—82no—
122C2×Q8—82no—
123C2×Q8C22⋊C411yescharacteristic
124C2×Q8C22⋊C411yescharacteristic
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4C22⋊C411yescharacteristic
129C2×D4C22⋊C411yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—42no—
163C4⋊C4C22⋊C411yescharacteristic
164C4⋊C4C22⋊C412yes—
165C4⋊C4C22⋊C412yes—
166C4⋊C4C22⋊C411yescharacteristic
167C4⋊C4C22⋊C411yescharacteristic
168C4⋊C4C22⋊C411yescharacteristic
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C42—22no—
184C42—22no—
185C42—21no—
186C42—21no—
187C42C22×C411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C23—21no—
192C23—21no—
193C23#12011yescharacteristic
194Q8—41no—
195Q8—41no—
196Q8—42no—
197Q8—42no—
198Q8—82no—
199Q8—82no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—41no—
206D4—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4C2×C22⋊C411yescharacteristic
228C2×C4C2×C22⋊C411yescharacteristic
229C2×C4C2×C22⋊C411yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C8—42no—
234C8—81no—
235C8—81no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22#37911yescharacteristic
245C22#45411yescharacteristic
246C22#45611yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—161no—
262C2—21no—
263C2—21no—
264C2—81no—
265C2—81no—
266C2#179311yescharacteristic
267C2#179511yescharacteristic
268C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3552C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1993C2semidirect
3. #1556C2semidirect
4. #1532C2semidirect
5. #1532C2semidirect
6. #1512C2semidirect
7. #1512C2semidirect
9. #557C4semidirect
10. #556C4semidirect
14. D4×Q8C4semidirect
16. #547C4semidirect

Maximal subgroups

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

subgroupindex
#35522
#19932
#15562
#15322
#15122