Finite Group Databasebeta

← #25688

Subgroups — #25688

Order 256.

482 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 127 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#3509C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic
4#2862C211yesmaximal, characteristic, complemented
5#2038C214yesmaximal
6#1994C211yesmaximal, characteristic, complemented
7#1966C211yesmaximal, characteristic, complemented
8#1857C214yesmaximal, complemented
9C4×#104C211yesmaximal, characteristic
10#557—21no—
11#557—21no—
12#556—21no—
13#556—21no—
14#552—22no—
15#552—22no—
16#541—22no—
17#536C2212yes—
18#533—21no—
19#533—21no—
20#533C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23#519C2212yes—
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—
37C2×#101C2214yes—
38C4×M16C2211yescharacteristic
39#391—22no—
40#391C2214yes—
41#382—22no—
42C43C2211yescharacteristic
43#339C2212yes—
44#339C2214yes—
45C2×#41C2311yescharacteristic
46C22×Q8—21no—
47C22×Q8C2312yes—
48C2×M16C2312yes—
49C2×M16C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8D411yescharacteristic
60C4⋊Q8D411yescharacteristic
61#126—24no—
62#126—24no—
63#125—22no—
64#125—22no—
65#125—24no—
66#125—24no—
67#125D412yes—
68#124—22no—
69#123—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—21no—
73C22⋊Q8—21no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8D411yescharacteristic
85C4×Q8D411yescharacteristic
86C4×D4—21no—
87C4×D4—21no—
88C4×D4D411yescharacteristic
89C4×D4D411yescharacteristic
90#117—22no—
91#117—22no—
92#117—24no—
93#117—21no—
94#117C2311yescharacteristic
95#117C2312yes—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D412yes—
98C2×C4⋊C4D412yes—
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#104—24no—
112#103—22no—
113#102—22no—
114#101—28no—
115#97—21no—
116C4×C8—21no—
117#95—28no—
118#41—22no—
119#41—22no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—24no—
125C2×Q8—24no—
126C2×Q8—44no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8C2×D411yescharacteristic
130C2×D4—21no—
131C2×D4C2×D411yescharacteristic
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4C2×D414yes—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2411yescharacteristic
141M16—22no—
142M16—24no—
143M16#4114yes—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4C2×D411yescharacteristic
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D411yescharacteristic
167C4⋊C4C2×D412yes—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—21no—
171C22⋊C4—22no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C42—24no—
175C42—24no—
176C42—24no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—42no—
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C23—21no—
189C23C22×D411yescharacteristic
190Q8—21no—
191Q8—21no—
192Q8—42no—
193Q8—42no—
194Q8—44no—
195Q8—42no—
196Q8—42no—
197Q8—82no—
198Q8—82no—
199D4—21no—
200D4—21no—
201D4—42no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—21no—
211C2×C4—28no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4112yes—
226C2×C42−1+412yes—
227C8—24no—
228C8—42no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22C2×#12011yescharacteristic
235C22#54712yes—
236C4—21no—
237C4—21no—
238C4—44no—
239C4—44no—
240C4—41no—
241C4—41no—
242C4—42no—
243C4—41no—
244C4—42no—
245C4—82no—
246C4#45412yes—
247C4#45612yes—
248C4#53211yescharacteristic
249C4#53611yescharacteristic
250C4#54812yes—
251C2—22no—
252C2—81no—
253C2#243211yescharacteristic
254C2C2×#45411yescharacteristic
255C2C2×#45611yescharacteristic
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
6. #1994C2semidirect
7. #1966C2semidirect
8. #1857C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#31652
#30792
#28622
#20382
#19942
#19662
#18572
C4×#1042