Finite Group Databasebeta

← #18640

Subgroups — #18640

Order 256.

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

normal subgroups only

This group has 605 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, complemented
2#3205C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic, complemented
5#1627C214yesmaximal, complemented
6#1625C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×2−1+4—21no—
11C2×AGL(1,ℤ/8)—21no—
12D4×Q8—21no—
13D4×Q8C2212yescomplemented
14#547—22no—
15#547—21no—
16#547—22no—
17#547C2212yescomplemented
18#536—22no—
19#536—21no—
20#532—21no—
21#532C2211yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×#127C2211yescharacteristic
25C4×#41C2211yescharacteristic
26#493—21no—
27#486—22no—
28#486—21no—
29#486C2214yes—
30#460—21no—
31#460C2212yes—
32#459—21no—
33#459C2212yes—
34#459C2214yes—
35#447—22no—
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38#418—21no—
39C2×#102—22no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43D4⋊C8C2212yescomplemented
44D4⋊C8C2214yescomplemented
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×D4—21no—
56C22×D4—21no—
57C22×D4—21no—
58AGL(1,ℤ/8)—44no—
59C2×SD16—22no—
60C2×D8—22no—
61C2×D8—24no—
62C2×M16—21no—
63C22×C8—21no—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8D411yescharacteristic
70C4⋊Q8D412yes—
71C4⋊Q8C2312yes—
72#127—21no—
73#127—21no—
74#127—21no—
75#127D411yescharacteristic
76#127C2312yes—
77#124—22no—
78#124—21no—
79#123—24no—
80#123—24no—
81#123—24no—
82C22⋊Q8—24no—
83C22⋊Q8—22no—
84C22⋊Q8—24no—
85C22⋊Q8—22no—
86#121—22no—
87#121—22no—
88#121—21no—
89#121D412yes—
90C4×Q8D411yescharacteristic
91C4×Q8C2311yescharacteristic
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4D411yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2312yes—
100#117—21no—
101#117D411yescharacteristic
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4D412yes—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8C2312yes—
110C4⋊C8C2312yes—
111#102—28no—
112#102—22no—
113#102—22no—
114#102—24no—
115C22⋊C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8C2312yes—
119C24—41no—
120#41—22no—
121#41—48no—
122#41—416no—
123#41—44no—
124#41—44no—
125#41—42no—
126#41—42no—
127#41D814yescomplemented
128C2×Q8—24no—
129C2×Q8—22no—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—48no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8C2×D411yescharacteristic
138C2×D4—21no—
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—44no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—82no—
153C2×D4C2×D412yes—
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165SD16—44no—
166D8—44no—
167D8—84no—
168M16—42no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—42no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D414yes—
183C4⋊C4C2×D411yescharacteristic
184C4⋊C4C2×D411yescharacteristic
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42C2×D411yescharacteristic
196C42C2×D411yescharacteristic
197C42C2×D411yescharacteristic
198C42C2411yescharacteristic
199C23—21no—
200C23—22no—
201C23—42no—
202C23—82no—
203C23—82no—
204C23—82no—
205C23#12011yescharacteristic
206Q8—21no—
207Q8—48no—
208Q8—42no—
209Q8—42no—
210Q8—42no—
211Q8—44no—
212Q8C2×D812yes—
213D4—24no—
214D4—21no—
215D4—24no—
216D4—44no—
217D4—48no—
218D4—44no—
219D4—42no—
220D4—42no—
221D4—44no—
222D4—42no—
223D4—42no—
224D4—82no—
225D4—82no—
226D4—82no—
227D4C2×D812yes—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4#12011yescharacteristic
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4C2×D812yes—
247C2×C4AGL(1,ℤ/8)12yes—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C8—42no—
252C8—42no—
253C8—42no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—164no—
266C22#44712yes—
267C22C2×#12011yescharacteristic
268C4—22no—
269C4—21no—
270C4—21no—
271C4—44no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4#44712yes—
276C4#45412yes—
277C4C22×D811yescharacteristic
278C4C2×AGL(1,ℤ/8)11yescharacteristic
279C4#57712yes—
280C2—22no—
281C2—42no—
282C2—82no—
283C2—162no—
284C2C2×#44711yescharacteristic
285C2#301011yescharacteristic
286C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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
1. #3499C2semidirect
2. #3205C2semidirect
3. C2×#486C2semidirect
4. #3038C2semidirect
5. #1627C2semidirect
6. #1625C2semidirect
7. #1546C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect
13. D4×Q8C22semidirect
17. #547C22semidirect
41. #332C22semidirect
42. Q8⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. D4⋊C8C22semidirect
127. #41D8semidirect

Maximal subgroups

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

subgroupindex
#34992
#32052
C2×#4862
#30382
#16272
#16252
#15462
#14922
#14762