Finite Group Databasebeta

← #8502

Subgroups — #8502

Order 256.

336 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 69 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#3547C211yesmaximal, characteristic
2#1553C211yesmaximal, characteristic, complemented
3#1535C211yesmaximal, characteristic, complemented
4#1521C211yesmaximal, characteristic, complemented
5#1515C211yesmaximal, characteristic, complemented
6#1514C211yesmaximal, characteristic, complemented
7#1497C211yesmaximal, characteristic, complemented
8#562—22no—
9#555—21no—
10#555C411yescharacteristic, complemented
11#554C411yescharacteristic, complemented
12#553—21no—
13#552—21no—
14D4×Q8C411yescharacteristic, complemented
15#548—21no—
16#547C411yescharacteristic, complemented
17#544—21no—
18#542—22no—
19#540—21no—
20#537—21no—
21#536—21no—
22#533C2211yescharacteristic
23#532C2211yescharacteristic
24#529C2211yescharacteristic
25#432C2211yescharacteristic
26#432C2211yescharacteristic
27#431C2211yescharacteristic
28#383C2211yescharacteristic
29#332—21no—
30#331—21no—
31#330—21no—
32#329—21no—
33#328—22no—
34#328—21no—
35#327—21no—
36#327—21no—
37#327—21no—
38Q8⋊C8—22no—
39C2×#41—21no—
40C2×#41—21no—
41C22×Q8—21no—
42C4⋊Q8—22no—
43C4⋊Q8—21no—
44C4⋊Q8C2×C411yescharacteristic
45C4⋊Q8C2×C411yescharacteristic
46C4⋊Q8D411yescharacteristic
47#127D411yescharacteristic
48#126—22no—
49#126—21no—
50#125—22no—
51#125—21no—
52#125—21no—
53#125D411yescharacteristic
54#124—22no—
55#124—22no—
56#124—22no—
57#124—21no—
58#124D411yescharacteristic
59#123—22no—
60#123—22no—
61#123—22no—
62#123—21no—
63#123—22no—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8—21no—
67C22⋊Q8—22no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8—22no—
72C22⋊Q8D411yescharacteristic
73C22⋊Q8D412yes—
74#121—22no—
75#121—22no—
76#121—22no—
77#121—22no—
78#121—21no—
79#121—21no—
80#121—21no—
81#121—21no—
82#121D411yescharacteristic
83#121D411yescharacteristic
84#121D411yescharacteristic
85C4×Q8—21no—
86C4×Q8C2×C411yescharacteristic
87C4×Q8D412yes—
88C4×D4—22no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4C2×C411yescharacteristic
94C4×D4C2×C411yescharacteristic
95C4×D4C2×C411yescharacteristic
96#117—22no—
97#117—21no—
98#117—21no—
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C42C2311yescharacteristic
104C4⋊C8—21no—
105C4⋊C8—21no—
106C4⋊C8—21no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110#97—21no—
111#97—21no—
112C4×C8—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#41—82no—
117#41—82no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—82no—
122C2×Q8C22⋊C411yescharacteristic
123C2×Q8C22⋊C412yes—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4C22⋊C411yescharacteristic
132C2×D4C22⋊C411yescharacteristic
133C2×D4C22⋊C411yescharacteristic
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4C22⋊C411yescharacteristic
162C4⋊C4C22⋊C411yescharacteristic
163C4⋊C4C22⋊C411yescharacteristic
164C4⋊C4C22⋊C411yescharacteristic
165C4⋊C4C22⋊C412yes—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C42—22no—
182C42—21no—
183C42—22no—
184C42C22×C411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C23—21no—
189C23—21no—
190C23—21no—
191C23#12011yescharacteristic
192Q8—41no—
193Q8—42no—
194Q8—42no—
195Q8—81no—
196Q8—82no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—81no—
201D4—81no—
202D4—81no—
203D4—81no—
204D4—81no—
205D4—82no—
206D4—82no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4C2×C22⋊C411yescharacteristic
227C2×C4C2×C22⋊C411yescharacteristic
228C2×C4C2×C22⋊C411yescharacteristic
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C8—81no—
233C8—81no—
234C8—81no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22#37911yescharacteristic
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—42no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—161no—
262C2—41no—
263C2—81no—
264C2—81no—
265C2—81no—
266C2#179411yescharacteristic
267C2#179511yescharacteristic
268C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3547C2
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. #1553C2semidirect
3. #1535C2semidirect
4. #1521C2semidirect
5. #1515C2semidirect
6. #1514C2semidirect
7. #1497C2semidirect
10. #555C4semidirect
11. #554C4semidirect
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
#35472
#15532
#15352
#15212
#15152
#15142
#14972