Finite Group Databasebeta

← #25924

Subgroups — #25924

Order 256.

569 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 131 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, complemented
2#3138C211yesmaximal, characteristic, complemented
3#3038C211yesmaximal, characteristic
4#2858C211yesmaximal, characteristic, complemented
5#2027C214yesmaximal, complemented
6#1997C211yesmaximal, characteristic, complemented
7#1994C211yesmaximal, characteristic, complemented
8#1883C214yesmaximal, complemented
9C4×#104C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)C2212yes—
11Q82—21no—
12#557—21no—
13#554—21no—
14#554—22no—
15D4×Q8—22no—
16D4×Q8—21no—
17#544—22no—
18#544C2212yes—
19#533—21no—
20#533C2211yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#127C2211yescharacteristic
24C2×#124—22no—
25#520C2212yes—
26C4×#41C2211yescharacteristic
27#490—22no—
28#486—21no—
29#486—21no—
30#460—21no—
31#460—21no—
32#459—21no—
33#459—21no—
34#423C2211yescharacteristic
35C2×#104C2212yes—
36#418C2214yes—
37#418C2212yes—
38C4×M16C2211yescharacteristic
39#390—22no—
40#390C2214yes—
41C43C2211yescharacteristic
42#343C2214yes—
43#339C2214yes—
44#339C2212yes—
45C2×#41C2311yescharacteristic
46C22×Q8—22no—
47C22×Q8D412yes—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4C2312yes—
51AGL(1,ℤ/8)—28no—
52C2×SD16—22no—
53C2×D8—22no—
54C2×M16C2312yes—
55C2×M16C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—24no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8D411yescharacteristic
66C4⋊Q8D411yescharacteristic
67#127—21no—
68#127—21no—
69#127—21no—
70#127—21no—
71#125—24no—
72#125—22no—
73#125—22no—
74#125D412yes—
75#124—48no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C22⋊Q8—24no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—24no—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8D411yescharacteristic
89C4×Q8D411yescharacteristic
90C4×D4—22no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—24no—
94C4×D4D411yescharacteristic
95C4×D4D411yescharacteristic
96#117—22no—
97#117—21no—
98#117C2311yescharacteristic
99#117C2312yes—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4D412yes—
103C2×C4⋊C4C2312yes—
104C2×C22⋊C4—28no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2312yes—
109C2×C42C2311yescharacteristic
110C4⋊C8—21no—
111C4⋊C8—21no—
112#104—24no—
113#102—22no—
114#102—22no—
115#102—24no—
116#102—24no—
117#97—21no—
118C4×C8—21no—
119C24—22no—
120#41—22no—
121#41#4114yes—
122C2×Q8—24no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—84no—
129C2×Q8C2×D411yescharacteristic
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—44no—
136C2×D4—44no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4C2×D411yescharacteristic
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D414yes—
150C22×C4C2411yescharacteristic
151SD16—44no—
152D8—44no—
153M16—22no—
154M16—24no—
155M16—22no—
156C2×C8—22no—
157C2×C8—22no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D411yescharacteristic
175C4⋊C4C2×D411yescharacteristic
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—416no—
182C42—24no—
183C42—24no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—42no—
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C23—21no—
196C23—44no—
197C23—44no—
198C23—42no—
199C23—44no—
200C23C22×D411yescharacteristic
201Q8—21no—
202Q8—44no—
203Q8—42no—
204Q8—42no—
205Q8—82no—
206Q8—84no—
207Q8#12412yescomplemented
208D4—24no—
209D4—21no—
210D4—42no—
211D4—42no—
212D4—44no—
213D4—82no—
214D4—82no—
215D4—82no—
216D4—82no—
217D4#12412yescomplemented
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—84no—
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4#12011yescharacteristic
237C2×C4#12011yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4C22×D411yescharacteristic
240C2×C4C2×#4112yes—
241C2×C42−1+412yes—
242C8—42no—
243C8—42no—
244C22—21no—
245C22—21no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—88no—
250C22—82no—
251C22—82no—
252C22—82no—
253C22C2×#12011yescharacteristic
254C22#54812yes—
255C4—22no—
256C4—21no—
257C4—21no—
258C4—44no—
259C4—41no—
260C4—44no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4#45312yes—
265C4#45412yes—
266C4C2×#12411yescharacteristic
267C4#53611yescharacteristic
268C4#54812yes—
269C2—22no—
270C2—42no—
271C2—162no—
272C2#243611yescharacteristic
273C2C2×#45311yescharacteristic
274C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3038C2
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
1. #3503C2semidirect
2. #3138C2semidirect
4. #2858C2semidirect
5. #2027C2semidirect
6. #1997C2semidirect
7. #1994C2semidirect
8. #1883C2semidirect
9. C4×#104C2semidirect
207. Q8#124semidirect
217. D4#124semidirect

Maximal subgroups

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

subgroupindex
#35032
#31382
#30382
#28582
#20272
#19972
#19942
#18832
C4×#1042