Finite Group Databasebeta

← #25829

Subgroups — #25829

Order 256.

491 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 147 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#3503C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic
4#2862C211yesmaximal, characteristic, complemented
5#2061C214yesmaximal, complemented
6#1997C211yesmaximal, characteristic, complemented
7#1994C211yesmaximal, characteristic, complemented
8#1860C214yesmaximal
9C4×#104C211yesmaximal, characteristic
10Q82—21no—
11#557—21no—
12#554—21no—
13#554—22no—
14D4×Q8—22no—
15D4×Q8—21no—
16#544—22no—
17#544C2212yes—
18#533—21no—
19#533C2211yescharacteristic
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23#520C2212yes—
24C4×#41C2211yescharacteristic
25#501—22no—
26#500—21no—
27#498—21no—
28#477—21no—
29#476—21no—
30#475—21no—
31#474—21no—
32#428C2212yes—
33#428C2214yes—
34#423C2211yescharacteristic
35C2×#104C2212yes—
36#417C2212yes—
37C4×M16C2211yescharacteristic
38#391—22no—
39#391—22no—
40#382C2214yes—
41C43C2211yescharacteristic
42#343C2214yes—
43#339C2212yes—
44#339C2214yes—
45C2×#41C2311yescharacteristic
46C22×Q8—22no—
47C22×Q8—21no—
48C22×Q8D412yes—
49C2×M16C2312yes—
50C2×M16C2312yes—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—24no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66#125—24no—
67#125—22no—
68#125—22no—
69#125D412yes—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76C22⋊Q8—24no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—21no—
81C4×Q8—21no—
82C4×Q8D411yescharacteristic
83C4×Q8D411yescharacteristic
84C4×D4—22no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—24no—
88C4×D4D411yescharacteristic
89C4×D4D411yescharacteristic
90#117—22no—
91#117—21no—
92#117C2311yescharacteristic
93#117C2312yes—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D412yes—
98C2×C4⋊C4C2312yes—
99C2×C4⋊C4C2312yes—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2312yes—
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—24no—
107#106—22no—
108#106—24no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111#104Q818yescomplemented
112#103—22no—
113#102—22no—
114#97—21no—
115C4×C8—21no—
116#95—28no—
117#41—22no—
118#41C2×Q814yes—
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—84no—
129C2×Q8C2×D411yescharacteristic
130C2×D4—21no—
131C2×D4C2×D411yescharacteristic
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D414yes—
141C22×C4C2411yescharacteristic
142M16—22no—
143M16—24no—
144M16C2×Q814yescomplemented
145C2×C8—22no—
146C2×C8—22no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D411yescharacteristic
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D411yescharacteristic
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C42—24no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—22no—
179C42—22no—
180C42—42no—
181C42—42no—
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2×Q814yescomplemented
187C23—21no—
188C23C22×D411yescharacteristic
189Q8—21no—
190Q8—42no—
191Q8—44no—
192Q8—42no—
193Q8—82no—
194Q8—82no—
195Q8—84no—
196Q8—82no—
197Q8C4⋊Q812yes—
198D4—24no—
199D4—21no—
200D4C4⋊Q812yes—
201C2×C4—21no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—84no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C22×Q812yes—
226C2×C42−1+412yes—
227C8—24no—
228C8—42no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22C2×#12011yescharacteristic
235C22D4×Q812yes—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—44no—
240C4—41no—
241C4—44no—
242C4—42no—
243C4—41no—
244C4—42no—
245C4—82no—
246C4#45412yes—
247C4#45512yes—
248C4C2×C4⋊Q811yescharacteristic
249C4#53611yescharacteristic
250C4D4×Q812yes—
251C2—22no—
252C2—42no—
253C2#243411yescharacteristic
254C2C2×#45511yescharacteristic
255C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
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. #3165C2semidirect
4. #2862C2semidirect
5. #2061C2semidirect
6. #1997C2semidirect
7. #1994C2semidirect
111. #104Q8semidirect
144. M16C2×Q8semidirect
186. C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35032
#31652
#30792
#28622
#20612
#19972
#19942
#18602
C4×#1042