Finite Group Databasebeta

← #18651

Subgroups — #18651

Order 256.

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

normal subgroups only

This group has 563 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#3499C211yesmaximal, characteristic
2#3206C211yesmaximal, characteristic, complemented
3C2×#498C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1654C214yesmaximal, complemented
6#1651C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×2−1+4—21no—
11D4×Q8—21no—
12D4×Q8C2212yes—
13#547—22no—
14#547—21no—
15#547—22no—
16#547C2212yes—
17#536—22no—
18#536—21no—
19#532—21no—
20#532C2211yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C4×#41C2211yescharacteristic
25#498—22no—
26#498C2214yes—
27#492—21no—
28#487—21no—
29#477—21no—
30#476—21no—
31#475C2212yes—
32#474C2212yes—
33#461C2214yes—
34#448—22no—
35#431C2211yescharacteristic
36#428—21no—
37C2×#106—22no—
38#423C2211yescharacteristic
39#417—21no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yes—
42Q8⋊C8C2212yes—
43D4⋊C8C2214yes—
44D4⋊C8C2212yes—
452−1+4—44no—
462−1+4—44no—
47C2×#41—22no—
48C2×#41—24no—
49C2×#41—22no—
50C2×#41—21no—
51C2×#41D411yescharacteristic
52C22×Q8—22no—
53C22×Q8—22no—
54C22×Q8—21no—
55C22×Q8—21no—
56C2×SD16—24no—
57C2×M16—21no—
58C22×C8—21no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8D412yes—
69C4⋊Q8C2312yes—
70C4⋊Q8C2312yes—
71#127D411yescharacteristic
72#124—22no—
73#124—21no—
74#123—24no—
75#123—24no—
76#123—24no—
77C22⋊Q8—24no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80C22⋊Q8—22no—
81#121—22no—
82#121—22no—
83#121—21no—
84#121D412yes—
85C4×Q8D411yescharacteristic
86C4×Q8C2311yescharacteristic
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4D411yescharacteristic
93C4×D4C2311yescharacteristic
94C4×D4C2312yes—
95#117—21no—
96#117D411yescharacteristic
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D412yes—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—28no—
107#106—22no—
108C4⋊C8—21no—
109C4⋊C8C2312yes—
110C4⋊C8C2312yes—
111#103—22no—
112#103—24no—
113#102—22no—
114C22⋊C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C2312yes—
118#41—22no—
119#41—48no—
120#41—416no—
121#41—44no—
122#41—44no—
123#41—42no—
124#41—42no—
125#41SD1614yescomplemented
126C2×Q8—24no—
127C2×Q8—22no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—48no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—41no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—82no—
138C2×Q8C2×D411yescharacteristic
139C2×D4—21no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C2×D412yes—
145C2×D4C2×D411yescharacteristic
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—41no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157SD16—84no—
158M16—42no—
159C2×C8—24no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D414yes—
177C4⋊C4C2×D411yescharacteristic
178C4⋊C4C2×D411yescharacteristic
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C22⋊C4—21no—
185C22⋊C4—22no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2411yescharacteristic
193C23—21no—
194C23—22no—
195C23#12011yescharacteristic
196Q8—21no—
197Q8—48no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—44no—
202Q8—44no—
203Q8—82no—
204Q8C2×SD1612yes—
205D4—24no—
206D4—21no—
207D4—24no—
208D4—48no—
209D4—44no—
210D4—42no—
211D4—42no—
212D4C2×SD1612yes—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C2×SD1612yes—
234C2×C4AGL(1,ℤ/8)12yes—
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C8—24no—
239C8—42no—
240C8—42no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22#45012yes—
250C22C2×#12011yescharacteristic
251C4—22no—
252C4—21no—
253C4—21no—
254C4—44no—
255C4—42no—
256C4—41no—
257C4—41no—
258C4—82no—
259C4#45012yes—
260C4#45612yes—
261C4C22×SD1611yescharacteristic
262C4C2×AGL(1,ℤ/8)11yescharacteristic
263C4#57712yes—
264C2—22no—
265C2—42no—
266C2—82no—
267C2C2×#45011yescharacteristic
268C2#301011yescharacteristic
269C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3499C2
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. #3206C2semidirect
3. C2×#498C2semidirect
4. #3079C2semidirect
5. #1654C2semidirect
6. #1651C2semidirect
7. #1546C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
125. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#34992
#32062
C2×#4982
#30792
#16542
#16512
#15462
#14922
#14762