Finite Group Databasebeta

← #18650

Subgroups — #18650

Order 256.

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

normal subgroups only

This group has 573 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#3226C211yesmaximal, characteristic, complemented
3C2×#500C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic, complemented
5#1649C214yesmaximal, complemented
6#1647C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×2−1+4—21no—
11C2×#137—21no—
12D4×Q8—21no—
13D4×Q8C2212yes—
14#547—22no—
15#547—21no—
16#547—22no—
17#547C2212yes—
18#536—22no—
19#536—21no—
20#532—21no—
21#532C2211yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C4×#41C2211yescharacteristic
26#500—22no—
27#500—21no—
28#500C2214yes—
29#486—21no—
30#480—22no—
31#474C2214yes—
32#462—21no—
33#462C2212yes—
34#461—21no—
35#461C2212yes—
36#431C2211yescharacteristic
37C2×#107—22no—
38#423C2211yescharacteristic
39#419—21no—
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—
56#137—44no—
57C2×Q16—22no—
58C2×SD16—22no—
59C2×M16—21no—
60C22×C8—21no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8D411yescharacteristic
70C4⋊Q8D412yes—
71C4⋊Q8C2312yes—
72C4⋊Q8C2312yes—
73#127D411yescharacteristic
74#124—22no—
75#124—21no—
76#123—24no—
77#123—24no—
78#123—24no—
79C22⋊Q8—24no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—22no—
83#121—22no—
84#121—22no—
85#121—21no—
86#121D412yes—
87C4×Q8D411yescharacteristic
88C4×Q8C2311yescharacteristic
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4D411yescharacteristic
95C4×D4C2311yescharacteristic
96C4×D4C2312yes—
97#117—21no—
98#117D411yescharacteristic
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D412yes—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107#107—28no—
108#107—24no—
109C4⋊C8—21no—
110C4⋊C8C2312yes—
111C4⋊C8C2312yes—
112#103—22no—
113#103—22no—
114#102—24no—
115C22⋊C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8C2312yes—
119#41—22no—
120#41—48no—
121#41—416no—
122#41—44no—
123#41—44no—
124#41—42no—
125#41—42no—
126#41D814yes—
127C2×Q8—24no—
128C2×Q8—22no—
129C2×Q8—21no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—48no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—41no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8C2×D411yescharacteristic
141C2×D4—21no—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C2×D412yes—
147C2×D4C2×D411yescharacteristic
148C22×C4—21no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159Q16—44no—
160SD16—44no—
161M16—42no—
162C2×C8—24no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4C2×D411yescharacteristic
179C4⋊C4C2×D411yescharacteristic
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D414yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42C2×D411yescharacteristic
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2411yescharacteristic
196C23—21no—
197C23—22no—
198C23#12011yescharacteristic
199Q8—21no—
200Q8—48no—
201Q8—42no—
202Q8—42no—
203Q8—42no—
204Q8—44no—
205Q8—42no—
206Q8—42no—
207Q8—82no—
208Q8C2×D812yes—
209D4—24no—
210D4—21no—
211D4—24no—
212D4—48no—
213D4—44no—
214D4—42no—
215D4—42no—
216D4C2×D812yes—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—48no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—82no—
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4#12011yescharacteristic
237C2×C4C2×D812yes—
238C2×C4AGL(1,ℤ/8)12yes—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C8—24no—
243C8—42no—
244C8—42no—
245C22—21no—
246C22—21no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22#44712yes—
254C22C2×#12011yescharacteristic
255C4—22no—
256C4—21no—
257C4—21no—
258C4—44no—
259C4—42no—
260C4—41no—
261C4—41no—
262C4—82no—
263C4#44712yes—
264C4#45612yes—
265C4C22×D811yescharacteristic
266C4C2×AGL(1,ℤ/8)11yescharacteristic
267C4#57812yes—
268C2—22no—
269C2—42no—
270C2—82no—
271C2C2×#44711yescharacteristic
272C2#301111yescharacteristic
273C2C2×#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. #3226C2semidirect
3. C2×#500C2semidirect
4. #3039C2semidirect
5. #1649C2semidirect
6. #1647C2semidirect
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
#32262
C2×#5002
#30392
#16492
#16472
#15462
#14922
#14762