Finite Group Databasebeta

← #8358

Subgroups — #8358

Order 256.

397 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 81 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#3506C211yesmaximal, characteristic
2#1954C211yesmaximal, characteristic, complemented
3#1800C211yesmaximal, characteristic, complemented
4#1550C211yesmaximal, characteristic, complemented
5#1547C211yesmaximal, characteristic, complemented
6#1493C211yesmaximal, characteristic, complemented
7#1492C211yesmaximal, characteristic, complemented
8#559—22no—
9#556—21no—
10#555—21no—
11#553—22no—
12#550—21no—
13#547—21no—
14#538C414yes—
15#535—21no—
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#525—22no—
19#520—21no—
20C2×C4×D4C2211yescharacteristic
21#423C2211yescharacteristic
22#423C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#100—21no—
25C2×C22⋊C8—22no—
26C2×C4×C8C2211yescharacteristic
27#343—21no—
28#333—21no—
29#332—21no—
30#332—21no—
31#331—21no—
32Q8⋊C8—22no—
33D4⋊C8—22no—
34C2×#41—22no—
35C2×#41—21no—
36C22×D4D411yescharacteristic
37C23×C4D412yes—
38C2×M16—21no—
39C2×M16—21no—
40C22×C8—21no—
41C22×C8—21no—
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44C4⋊Q8D411yescharacteristic
45#127—22no—
46#127D411yescharacteristic
47#126—24no—
48#125—21no—
49#124—22no—
50#124—24no—
51#124—21no—
52#123—24no—
53#123—24no—
54C22⋊Q8—22no—
55C22⋊Q8—21no—
56C22⋊Q8C2×C412yes—
57#121—22no—
58#121—22no—
59#121—22no—
60#121—24no—
61#121—21no—
62#121C2×C412yes—
63#120—24no—
64C4×Q8—22no—
65C4×Q8—21no—
66C4×Q8D412yes—
67C4×D4—22no—
68C4×D4—24no—
69C4×D4—24no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4D412yes—
76#117—22no—
77#117—22no—
78#117—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4D411yescharacteristic
81C2×C22⋊C4C2×C412yes—
82C2×C42C2311yescharacteristic
83C4⋊C8—21no—
84C4⋊C8—21no—
85C4⋊C8—21no—
86C4⋊C8—21no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89#100—42no—
90C22⋊C8—44no—
91C4×C8—21no—
92C4×C8—21no—
93C24C22⋊C412yes—
94#41—41no—
95#41—41no—
96#41—41no—
97#41—41no—
98#41—82no—
99#41—82no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8C22⋊C412yes—
103C2×D4—22no—
104C2×D4—24no—
105C2×D4—24no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4C22⋊C412yes—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4C22⋊C412yes—
122C22×C4C22×C411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125M16—41no—
126M16—41no—
127M16—41no—
128M16—41no—
129C2×C8—22no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4C22⋊C412yes—
146C4⋊C4C22⋊C412yes—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C42—24no—
156C42—22no—
157C42C2×D411yescharacteristic
158C42C2×D411yescharacteristic
159C42C2×D411yescharacteristic
160C42C2×D411yescharacteristic
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—42no—
166C23—42no—
167C23C2×C22⋊C411yescharacteristic
168Q8—42no—
169Q8—41no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—84no—
185D4—84no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—84no—
200C2×C4C2×C22⋊C411yescharacteristic
201C2×C4C2×C22⋊C411yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#13511yescharacteristic
207C2×C4#13511yescharacteristic
208C2×C4AGL(1,ℤ/8)11yescharacteristic
209C2×C4#13711yescharacteristic
210C8—42no—
211C8—81no—
212C8—81no—
213C8—81no—
214C22—21no—
215C22—21no—
216C22—44no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—82no—
224C22#37911yescharacteristic
225C22#41611yescharacteristic
226C22#41711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4#44911yescharacteristic
235C4#44911yescharacteristic
236C4#45211yescharacteristic
237C4#45211yescharacteristic
238C4#45311yescharacteristic
239C4#45511yescharacteristic
240C2—21no—
241C2—21no—
242C2—82no—
243C2—82no—
244C2#178811yescharacteristic
245C2#179111yescharacteristic
246C2#179211yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3506C2
LCSG ▷ C22×C4 ▷ 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. #1954C2semidirect
3. #1800C2semidirect
4. #1550C2semidirect
5. #1547C2semidirect
6. #1493C2semidirect
7. #1492C2semidirect

Maximal subgroups

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

subgroupindex
#35062
#19542
#18002
#15502
#15472
#14932
#14922