Finite Group Databasebeta

← #29708

Subgroups — #29708

Order 256.

465 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 119 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#3461C211yesmaximal, characteristic
2#3080C212yesmaximal, complemented
3#2565C211yesmaximal, characteristic, complemented
4#2061C211yesmaximal, characteristic, complemented
5#2060C211yesmaximal, characteristic, complemented
6#2040C211yesmaximal, characteristic
7#2039C211yesmaximal, characteristic
8#2038C212yesmaximal, complemented
9#2038C212yesmaximal, complemented
10#1806C212yesmaximal, complemented
11#1801C211yesmaximal, characteristic, complemented
12#544C2211yescharacteristic
13#542—22no—
14#541—22no—
15#539—22no—
16#537C2211yescharacteristic
17#536C2212yes—
18#533—22no—
19#529—22no—
20#529—22no—
21C2×#125C2211yescharacteristic
22C4×#41C2212yes—
23C2×C4×Q8C2211yescharacteristic
24#479—22no—
25#479—22no—
26#478—22no—
27#478—22no—
28#428C2212yes—
29#428C2212yes—
30#423C2212yes—
31C2×#104C2212yes—
32C2×#104C2212yes—
33#417C2212yes—
34#417C2212yes—
35C2×#101C2212yes—
36#398—22no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#387—22no—
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43#343C2211yescharacteristic
44#343C2211yescharacteristic
45#339C2212yes—
46#339C2212yes—
47C2×#41C2312yes—
48C22×Q8C2311yescharacteristic
49C2×M16C2312yes—
50C2×M16C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53#126—24no—
54#126—24no—
55#125—22no—
56#125—24no—
57#125—22no—
58#125—22no—
59#125—22no—
60#124—22no—
61#124—22no—
62#123—24no—
63#123—24no—
64C22⋊Q8—24no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69#121—22no—
70#121—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×Q8D412yes—
75C4×Q8D412yes—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4D412yes—
80C4×D4D412yes—
81#117—24no—
82#117—22no—
83#117C2312yes—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4D412yes—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C42—22no—
93C2×C42D412yes—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—22no—
98#107—22no—
99#106—22no—
100#106—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103#104—42no—
104#104—42no—
105#103—22no—
106#103—22no—
107#102—22no—
108#102—22no—
109#101—42no—
110#95—21no—
111#95—22no—
112#95—22no—
113#95—22no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8C2×D412yes—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4C2×D412yes—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4C2×D412yes—
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139M16—22no—
140M16—22no—
141M16—22no—
142M16—22no—
143C2×C8—22no—
144C2×C8—22no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—24no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42C2×D412yes—
179C42C2×D412yes—
180C23—22no—
181C23C22×D411yescharacteristic
182Q8—22no—
183Q8—22no—
184Q8—42no—
185Q8—42no—
186Q8—82no—
187D4—22no—
188D4—22no—
189D4—42no—
190D4—42no—
191D4—82no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—82no—
216C2×C4#12011yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4#12012yes—
219C2×C4C22×D412yes—
220C2×C4H(5,𝔽2)11yescharacteristic
221C2×C42−1+411yescharacteristic
222C2×C42−1+412yes—
223C8—42no—
224C8—42no—
225C22—21no—
226C22—21no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22C2×#12011yescharacteristic
231C22#53611yescharacteristic
232C22#53711yescharacteristic
233C4—22no—
234C4—21no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4#53612yes—
246C4#53712yes—
247C2—21no—
248C2—21no—
249C2—82no—
250C2#261611yescharacteristic
251C2#301811yescharacteristic
252C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3080C2semidirect
3. #2565C2semidirect
4. #2061C2semidirect
5. #2060C2semidirect
8. #2038C2semidirect
9. #2038C2semidirect
10. #1806C2semidirect
11. #1801C2semidirect

Maximal subgroups

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

subgroupindex
#34612
#30802
#25652
#20612
#20602
#20402
#20392
#20382
#18062
#18012