Finite Group Databasebeta

← #18627

Subgroups — #18627

Order 256.

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

normal subgroups only

This group has 496 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#3511C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3C2×#494C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1634C214yesmaximal, complemented
6#1632C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C22×Q16—22no—
11#556—21no—
12#555C2212yes—
13#554C2212yes—
14#553—22no—
15#548—22no—
16#547—21no—
17#542—22no—
18#537—21no—
19#533—21no—
20#532—21no—
21#532C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#520—21no—
25C4×#41C2211yescharacteristic
26#498—21no—
27#494—22no—
28#494C2214yes—
29#486—21no—
30#481—22no—
31#477C2212yes—
32#476C2212yes—
33#476C2214yes—
34#475—21no—
35#474—21no—
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#417—21no—
40C2×C4×C8C2211yescharacteristic
41#333C2214yes—
42Q8⋊C8C2212yes—
43D4⋊C8C2214yes—
44D4⋊C8C2212yes—
45C2×#41—22no—
46C2×#41—21no—
47C2×#41D411yescharacteristic
48C22×Q8—21no—
49C22×Q8D412yes—
50C2×Q16—28no—
51C2×Q16—48no—
52C2×M16—21no—
53C22×C8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8D412yes—
60C4⋊Q8D411yescharacteristic
61C4⋊Q8C2312yes—
62C4⋊Q8C2312yes—
63#127D411yescharacteristic
64#126—24no—
65#125—22no—
66#124—22no—
67#124—24no—
68#124—21no—
69#123—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—24no—
74#121—22no—
75#121—24no—
76#121—22no—
77#121—22no—
78#121—21no—
79#121D412yes—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8D411yescharacteristic
83C4×Q8C2311yescharacteristic
84C4×D4—22no—
85C4×D4—24no—
86C4×D4—22no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4D411yescharacteristic
91C4×D4C2311yescharacteristic
92C4×D4C2312yes—
93#117—22no—
94#117—22no—
95#117—21no—
96#117D411yescharacteristic
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104#107—22no—
105#106—22no—
106#106—24no—
107C4⋊C8—21no—
108C4⋊C8C2312yes—
109C4⋊C8C2312yes—
110#103—22no—
111#102—22no—
112#102—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117#41—22no—
118#41—22no—
119#41—42no—
120#41—42no—
121#41—84no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—41no—
128C2×Q8C2×D411yescharacteristic
129C2×Q8C2×D414yes—
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—24no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4C2×D412yes—
136C2×D4C2×D411yescharacteristic
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—41no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147Q16—416no—
148M16—42no—
149C2×C8—24no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4C2×D411yescharacteristic
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D411yescharacteristic
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—22no—
175C22⋊C4—21no—
176C22⋊C4—22no—
177C42—24no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2411yescharacteristic
185C23—21no—
186C23—22no—
187C23#12011yescharacteristic
188Q8—21no—
189Q8—21no—
190Q8—44no—
191Q8—44no—
192Q8—42no—
193Q8—44no—
194Q8—82no—
195D4—21no—
196D4—21no—
197D4—24no—
198D4—42no—
199D4—44no—
200D4—42no—
201D4—42no—
202D4—84no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4#13512yes—
223C2×C4#13712yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—24no—
228C8—42no—
229C8—42no—
230C22—21no—
231C22—21no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—42no—
238C22#45212yes—
239C22C2×#12011yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—44no—
247C4—82no—
248C4#45212yes—
249C4#45312yes—
250C4C2×#13511yescharacteristic
251C4C2×#13711yescharacteristic
252C4#57812yes—
253C2—22no—
254C2—81no—
255C2—82no—
256C2C2×#45211yescharacteristic
257C2#301111yescharacteristic
258C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3511C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3227C2semidirect
3. C2×#494C2semidirect
4. #3079C2semidirect
5. #1634C2semidirect
6. #1632C2semidirect
7. #1548C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#32272
C2×#4942
#30792
#16342
#16322
#15482
#14922
#14762