Finite Group Databasebeta

← #25693

Subgroups — #25693

Order 256.

519 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 125 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#3165C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic, complemented
4#2563C211yesmaximal, characteristic, complemented
5#2037C214yesmaximal, complemented
6#1986C211yesmaximal, characteristic, complemented
7#1980C211yesmaximal, characteristic, complemented
8#1860C214yesmaximal, complemented
9C4×#100C211yesmaximal, characteristic, complemented
10#559—22no—
11#556—21no—
12#555—21no—
13#553—22no—
14#550—21no—
15#547—21no—
16#538—22no—
17#535C2212yes—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21#525—22no—
22#520C2212yes—
23C2×C4×D4C2211yescharacteristic
24#501—22no—
25#500—21no—
26#498—21no—
27#489—22no—
28#487—21no—
29#486—21no—
30#428C2214yes—
31C2×#104C2214yes—
32#417C2214yes—
33C2×#100C2212yes—
34C4×M16C2211yescharacteristic
35C4×M16C2211yescharacteristic
36#393—22no—
37#390C2212yes—
38#382C2212yes—
39#380—22no—
40C4×C4⋊C4C2211yescharacteristic
41#343C2212yes—
42#339C2214yes—
43C2×#41—22no—
44C2×#41C2312yes—
45C22×Q8—21no—
46C22×D4C2311yescharacteristic
47C23×C4D412yes—
48C2×M16C2312yes—
49C2×M16C2312yes—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8D411yescharacteristic
55C4⋊Q8D411yescharacteristic
56C4⋊Q8D411yescharacteristic
57#127—22no—
58#127D411yescharacteristic
59#126—24no—
60#125D412yes—
61#124—24no—
62#124—22no—
63#124D412yes—
64#123—24no—
65#123—24no—
66C22⋊Q8—22no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69#121—22no—
70#121—24no—
71#121—21no—
72#121—21no—
73#121—22no—
74#121—22no—
75#120—24no—
76C4×Q8—22no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×D4—24no—
80C4×D4—22no—
81C4×D4—24no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88#117—22no—
89#117—22no—
90#117C2312yes—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—24no—
97C2×C22⋊C4D412yes—
98C2×C42C2312yes—
99C2×C42C2311yescharacteristic
100#107—24no—
101#106—24no—
102#104—28no—
103#103—24no—
104#102—24no—
105#100—24no—
106#97—21no—
107#97—21no—
108C4×C8—21no—
109C4×C8—21no—
110#95—21no—
111#95—24no—
112C24C2×D412yes—
113#41—24no—
114#41—24no—
115#41—84no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—82no—
121C2×Q8C2×D412yes—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4C2×D412yes—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—44no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143M16—22no—
144M16—22no—
145M16—22no—
146M16#4114yescomplemented
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—24no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—84no—
174C42—24no—
175C42—22no—
176C42—24no—
177C42C2×D411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—42no—
186C23—42no—
187C23C22×D411yescharacteristic
188Q8—24no—
189Q8—42no—
190Q8—81no—
191Q8—82no—
192Q8—82no—
193D4—24no—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—42no—
199D4—81no—
200D4—84no—
201D4—84no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C2×#4112yes—
225C2×C42−1+412yes—
226C8—24no—
227C8—42no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22C2×#12011yescharacteristic
239C22#53211yescharacteristic
240C22#53611yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—82no—
249C4—82no—
250C4#45512yes—
251C4#54712yes—
252C4#54812yes—
253C2—21no—
254C2—21no—
255C2—82no—
256C2—82no—
257C2#243211yescharacteristic
258C2C2×#45511yescharacteristic
259C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3506C2
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
3. #3141C2semidirect
4. #2563C2semidirect
5. #2037C2semidirect
6. #1986C2semidirect
7. #1980C2semidirect
8. #1860C2semidirect
9. C4×#100C2semidirect
146. M16#41semidirect

Maximal subgroups

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

subgroupindex
#35062
#31652
#31412
#25632
#20372
#19862
#19802
#18602
C4×#1002