Finite Group Databasebeta

← #25698

Subgroups — #25698

Order 256.

475 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 123 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#3507C211yesmaximal, characteristic
2#3166C211yesmaximal, characteristic, complemented
3#3142C211yesmaximal, characteristic
4#2565C211yesmaximal, characteristic, complemented
5#2038C212yesmaximal, complemented
6#2038C212yesmaximal, complemented
7#1986C211yesmaximal, characteristic, complemented
8#1980C211yesmaximal, characteristic
9#1860C212yesmaximal, complemented
10#1859C212yesmaximal, complemented
11C4×#101C211yesmaximal, characteristic, complemented
12#559—21no—
13#557—21no—
14#556—21no—
15#556—22no—
16#554—21no—
17#552—22no—
18#539—22no—
19#536C2212yes—
20#533C2211yescharacteristic
21#532C2211yescharacteristic
22#529—22no—
23C2×#125C2211yescharacteristic
24#520C2212yes—
25C2×C4×Q8C2211yescharacteristic
26#501—21no—
27#501—21no—
28#499—22no—
29#491—21no—
30#490—21no—
31#488—22no—
32#428C2212yes—
33#428C2212yes—
34C2×#104C2212yes—
35C2×#104C2212yes—
36#417C2212yes—
37#417C2212yes—
38C2×#101C2212yes—
39C4×M16C2211yescharacteristic
40C4×M16C2211yescharacteristic
41#398—22no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#387—22no—
45#384C2211yescharacteristic
46#382C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48#343C2212yes—
49#339C2212yes—
50#339C2212yes—
51C2×#41C2312yes—
52C22×Q8C2311yescharacteristic
53C2×M16C2312yes—
54C2×M16C2312yes—
55C4⋊Q8—22no—
56C4⋊Q8D411yescharacteristic
57C4⋊Q8D411yescharacteristic
58C4⋊Q8D411yescharacteristic
59#127D411yescharacteristic
60#126—24no—
61#126—24no—
62#125—22no—
63#125—24no—
64#125—21no—
65#125—21no—
66#125—21no—
67#125—21no—
68#125D412yes—
69#124—22no—
70#124—22no—
71#124D412yes—
72#123—24no—
73#123—24no—
74C22⋊Q8—24no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79#121—21no—
80#121—21no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—24no—
92C4×D4—21no—
93C4×D4—21no—
94#117—24no—
95#117—22no—
96#117C2312yes—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4D412yes—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C42D412yes—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107—22no—
112#106—22no—
113#106—22no—
114#104—24no—
115#104—24no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#101—24no—
121#97—21no—
122#97—21no—
123C4×C8—21no—
124C4×C8—21no—
125#95—21no—
126#95—22no—
127#95—22no—
128#95—22no—
129#41—22no—
130#41—22no—
131#41—22no—
132#41—22no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8C2×D412yes—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C2×D412yes—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C2×D412yes—
151C22×C4C2×D412yes—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155M16—22no—
156M16—22no—
157M16—22no—
158M16#4114yes—
159C2×C8—22no—
160C2×C8—22no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C42—24no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C23—22no—
203C23C22×D411yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209Q8—81no—
210D4—22no—
211D4—22no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—81no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—84no—
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C2×#4112yes—
246C2×C42−1+412yes—
247C8—24no—
248C8—42no—
249C22—21no—
250C22—21no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×#12011yescharacteristic
255C22#53211yescharacteristic
256C22#53611yescharacteristic
257C4—21no—
258C4—21no—
259C4—21no—
260C4—21no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4#54712yes—
271C4#54812yes—
272C2—21no—
273C2—21no—
274C2—82no—
275C2#243211yescharacteristic
276C2#301811yescharacteristic
277C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
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. #3166C2semidirect
4. #2565C2semidirect
5. #2038C2semidirect
6. #2038C2semidirect
7. #1986C2semidirect
9. #1860C2semidirect
10. #1859C2semidirect
11. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#31662
#31422
#25652
#20382
#19862
#19802
#18602
#18592
C4×#1012