Finite Group Databasebeta

← #18641

Subgroups — #18641

Order 256.

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

normal subgroups only

This group has 557 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#3227C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1637C214yesmaximal, complemented
6#1635C214yesmaximal, 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—21no—
26#487—22no—
27#487C2214yes—
28#486—21no—
29#481—22no—
30#477C2212yes—
31#476C2212yes—
32#476C2214yes—
33#475—21no—
34#474—21no—
35#431C2211yescharacteristic
36#428—21no—
37#423C2211yescharacteristic
38#417—21no—
39C2×#103—22no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yes—
42Q8⋊C8C2212yes—
43D4⋊C8C2212yes—
44D4⋊C8C2214yes—
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×M16—21no—
57C22×C8—21no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D411yescharacteristic
67C4⋊Q8D412yes—
68C4⋊Q8C2312yes—
69C4⋊Q8C2312yes—
70#127D411yescharacteristic
71#124—22no—
72#124—21no—
73#123—24no—
74#123—24no—
75#123—24no—
76C22⋊Q8—24no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C22⋊Q8—22no—
80#121—22no—
81#121—22no—
82#121—21no—
83#121D412yes—
84C4×Q8D411yescharacteristic
85C4×Q8C2311yescharacteristic
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4D411yescharacteristic
92C4×D4C2311yescharacteristic
93C4×D4C2312yes—
94#117—21no—
95#117D411yescharacteristic
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D412yes—
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#103—28no—
112#102—22no—
113#102—24no—
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#41SD1614yes—
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—44no—
135C2×Q8—41no—
136C2×Q8—42no—
137C2×Q8—42no—
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
157M16—42no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—42no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4C2×D414yes—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D411yescharacteristic
177C4⋊C4C2×D411yescharacteristic
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—22no—
183C22⋊C4—21no—
184C22⋊C4—22no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2411yescharacteristic
192C23—21no—
193C23—22no—
194C23#12011yescharacteristic
195Q8—21no—
196Q8—48no—
197Q8—44no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—44no—
202Q8—82no—
203Q8C2×SD1612yes—
204D4—24no—
205D4—21no—
206D4—24no—
207D4—48no—
208D4—44no—
209D4—42no—
210D4—42no—
211D4C2×SD1612yes—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—82no—
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4C2×SD1612yes—
233C2×C4AGL(1,ℤ/8)12yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C8—42no—
238C8—42no—
239C8—42no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22#45012yes—
249C22C2×#12011yescharacteristic
250C4—22no—
251C4—21no—
252C4—21no—
253C4—44no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—82no—
258C4#45012yes—
259C4#45412yes—
260C4C22×SD1611yescharacteristic
261C4C2×AGL(1,ℤ/8)11yescharacteristic
262C4#57812yes—
263C2—22no—
264C2—42no—
265C2—82no—
266C2C2×#45011yescharacteristic
267C2#301111yescharacteristic
268C2C2×#45411yescharacteristic
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. #3227C2semidirect
3. C2×#487C2semidirect
4. #3079C2semidirect
5. #1637C2semidirect
6. #1635C2semidirect
7. #1546C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect

Maximal subgroups

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

subgroupindex
#34992
#32272
C2×#4872
#30792
#16372
#16352
#15462
#14922
#14762