Finite Group Databasebeta

← #8501

Subgroups — #8501

Order 256.

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

normal subgroups only

This group has 308 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#3555C211yesmaximal, characteristic
2#1554C211yesmaximal, characteristic, complemented
3#1535C211yesmaximal, characteristic
4#1521C211yesmaximal, characteristic
5#1515C211yesmaximal, characteristic, complemented
6#1515C211yesmaximal, characteristic, complemented
7#1497C211yesmaximal, characteristic, complemented
8#563—22no—
9Q82C411yescharacteristic, complemented
10#557—21no—
11#557C411yescharacteristic, complemented
12#556—21no—
13#556—21no—
14#556C411yescharacteristic, complemented
15#554C411yescharacteristic, complemented
16#552—21no—
17#544—21no—
18#542—21no—
19#541—21no—
20#541—21no—
21#541—22no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#529C2211yescharacteristic
25#432C2211yescharacteristic
26#432C2211yescharacteristic
27#431C2211yescharacteristic
28#383C2211yescharacteristic
29#333—21no—
30#332—21no—
31#330—21no—
32#329—21no—
33#328—21no—
34#328—21no—
35#328—22no—
36#328—21no—
37#327—21no—
38Q8⋊C8—22no—
39C4⋊Q8—22no—
40C4⋊Q8—21no—
41C4⋊Q8—22no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8C2×C411yescharacteristic
45C4⋊Q8C2×C411yescharacteristic
46C4⋊Q8D411yescharacteristic
47C4⋊Q8D411yescharacteristic
48#126—22no—
49#126—22no—
50#126—22no—
51#126—22no—
52#126—22no—
53#126—22no—
54#126—21no—
55#125—22no—
56#125—22no—
57#125—22no—
58#125—22no—
59#125—21no—
60#125—22no—
61#125—21no—
62#125—21no—
63#125—21no—
64#125D411yescharacteristic
65#124—22no—
66#124—21no—
67#124D411yescharacteristic
68#123—22no—
69#123—21no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—21no—
73C22⋊Q8—21no—
74C22⋊Q8—22no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8D411yescharacteristic
78C22⋊Q8D411yescharacteristic
79C22⋊Q8D411yescharacteristic
80C22⋊Q8D412yes—
81#121D411yescharacteristic
82C4×Q8—22no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8C2×C411yescharacteristic
91C4×Q8C2×C411yescharacteristic
92C4×Q8C2×C411yescharacteristic
93C4×Q8D412yes—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4C2×C411yescharacteristic
97#117—22no—
98#117—21no—
99#117—21no—
100#117—21no—
101#117—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—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8C22⋊C411yescharacteristic
120C2×Q8C22⋊C411yescharacteristic
121C2×Q8C22⋊C411yescharacteristic
122C2×Q8C22⋊C412yes—
123C2×D4—21no—
124C2×D4C22⋊C411yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4C22⋊C411yescharacteristic
159C4⋊C4C22⋊C411yescharacteristic
160C4⋊C4C22⋊C411yescharacteristic
161C4⋊C4C22⋊C411yescharacteristic
162C4⋊C4C22⋊C412yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—21no—
167C22⋊C4—22no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C42—22no—
175C42—22no—
176C42—22no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—22no—
181C42C22×C411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C23—21no—
186C23#12011yescharacteristic
187Q8—42no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—81no—
193Q8—82no—
194D4—41no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4C2×C22⋊C411yescharacteristic
214C2×C4C2×C22⋊C411yescharacteristic
215C2×C4C2×C22⋊C411yescharacteristic
216C2×C4#12011yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C8—81no—
220C8—81no—
221C8—81no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22#37911yescharacteristic
229C4—21no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—42no—
244C4—161no—
245C2—41no—
246C2—81no—
247C2#179411yescharacteristic
248C2#179611yescharacteristic
249C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#121D4
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. #1554C2semidirect
5. #1515C2semidirect
6. #1515C2semidirect
7. #1497C2semidirect
9. Q82C4semidirect
11. #557C4semidirect
14. #556C4semidirect
15. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35552
#15542
#15352
#15212
#15152
#14972