Finite Group Databasebeta

← #33764

Subgroups — #33764

Order 256.

298 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 97 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#3061C211yesmaximal, characteristic, complemented
2#3058C211yesmaximal, characteristic, complemented
3#2237C211yesmaximal, characteristic, complemented
4#2234C211yesmaximal, characteristic, complemented
5#2233C211yesmaximal, characteristic, complemented
6#2232C211yesmaximal, characteristic, complemented
7#2220C211yesmaximal, characteristic, complemented
8#2218C211yesmaximal, characteristic, complemented
9#2217C211yesmaximal, characteristic, complemented
10#2215C211yesmaximal, characteristic, complemented
11#2187C211yesmaximal, characteristic, complemented
12#2186C211yesmaximal, characteristic, complemented
13#2186C211yesmaximal, characteristic, complemented
14#2184C211yesmaximal, characteristic, complemented
15#2113C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic, complemented
17C22×D8C2211yescharacteristic
18C2×#131C2211yescharacteristic
19#537—21no—
20#535—21no—
21C2×SD32C2211yescharacteristic
22C2×SD32C2211yescharacteristic
23C2×D16C2211yescharacteristic
24C2×D16C2211yescharacteristic
25C2×M32C2211yescharacteristic, complemented
26C22×C16C2211yescharacteristic, complemented
27#469—21no—
28#469—21no—
29#468—21no—
30#468—21no—
31#467C2211yescharacteristic
32#467C2211yescharacteristic, complemented
33#466C2211yescharacteristic
34#466C2211yescharacteristic
35#466C2211yescharacteristic
36#466C2211yescharacteristic
37#466C2211yescharacteristic
38#466C2211yescharacteristic, complemented
39#427C2211yescharacteristic
40C2×#107C2211yescharacteristic, complemented
41#418—21no—
42#417—21no—
43#367C2211yescharacteristic, complemented
44#367C2211yescharacteristic, complemented
45#366C2211yescharacteristic, complemented
46#366C2211yescharacteristic, complemented
47#358C2211yescharacteristic, complemented
48#358C2211yescharacteristic, complemented
49#357C2211yescharacteristic
50#357C2211yescharacteristic
51#357C2211yescharacteristic
52#357C2211yescharacteristic
53#357C2211yescharacteristic, complemented
54#357C2211yescharacteristic, complemented
55C22⋊C16C2211yescharacteristic
56C22⋊C16C2211yescharacteristic
57C22⋊C16C2211yescharacteristic, complemented
58C22⋊C16C2211yescharacteristic, complemented
59C2×#41—21no—
60C2×#41D411yescharacteristic
61C22×D4—21no—
62#135—41no—
63#135—41no—
64C2×Q16C2311yescharacteristic
65C2×SD16—21no—
66C2×D8—21no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8C2311yescharacteristic
70C2×D8C2311yescharacteristic
71C2×D8C2311yescharacteristic
72#131—41no—
73#131—41no—
74C2×M16D411yescharacteristic
75C2×M16D411yescharacteristic
76C2×M16D411yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#127—41no—
81#124—41no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—41no—
85#121—21no—
86#121—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—41no—
92#121—41no—
93#121—41no—
94#120—41no—
95#120—41no—
96#117—21no—
97C2×C4⋊C4—21no—
98SD32—41no—
99SD32—41no—
100D16—41no—
101D16—41no—
102M32—41no—
103C2×C16—41no—
104C2×C16C2311yescharacteristic
105C2×C16C2311yescharacteristic
106C2×C16C2311yescharacteristic
107C2×C16C2311yescharacteristic
108#107—21no—
109#107C2311yescharacteristic
110#107C2311yescharacteristic
111#107C2311yescharacteristic
112#107C2311yescharacteristic
113#106—21no—
114#103—21no—
115#103—21no—
116#102—21no—
117#102—21no—
118#102—21no—
119#102—21no—
120#102—21no—
121#102—21no—
122C24—41no—
123#41—41no—
124#41—41no—
125#41—42no—
126#41—42no—
127C2×Q8—21no—
128C2×Q8D811yescharacteristic
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4D811yescharacteristic
143C2×D4D811yescharacteristic
144C2×D4D811yescharacteristic
145C22×C4—41no—
146C22×C4—41no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150Q16—21no—
151Q16—21no—
152SD16—42no—
153D8—21no—
154D8—21no—
155D8—21no—
156D8—21no—
157D8—21no—
158D8—21no—
159D8—42no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8C2×D411yescharacteristic
170C2×C8C2×D411yescharacteristic
171C2×C8C2×D411yescharacteristic
172C2×C8C2411yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C42—41no—
190C16—21no—
191C16—21no—
192C16—21no—
193C16—21no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—82no—
198C23—161no—
199C23—161no—
200C23C2×D811yescharacteristic
201C23C2×D811yescharacteristic
202C23C2×D811yescharacteristic
203Q8—41no—
204Q8—41no—
205Q8—41no—
206D4—41no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—82no—
216D4—82no—
217D4—161no—
218D4—161no—
219D4—161no—
220D4—161no—
221D4—161no—
222D4—161no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—82no—
235C2×C4—161no—
236C2×C4C2×D811yescharacteristic
237C2×C4C2×D811yescharacteristic
238C2×C4C2×D811yescharacteristic
239C2×C4C22×D411yescharacteristic
240C8—21no—
241C8—21no—
242C8—21no—
243C8H(5,𝔽2)11yescharacteristic
244C8H(5,𝔽2)11yescharacteristic
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—164no—
264C22C22×D811yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4#53411yescharacteristic
274C4#57511yescharacteristic
275C2—41no—
276C2—41no—
277C2—41no—
278C2—161no—
279C2—161no—
280C2—161no—
281C2#318611yescharacteristic
282C2#341511yescharacteristic
283C2#341611yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3061C2semidirect
2. #3058C2semidirect
3. #2237C2semidirect
4. #2234C2semidirect
5. #2233C2semidirect
6. #2232C2semidirect
7. #2220C2semidirect
8. #2218C2semidirect
9. #2217C2semidirect
10. #2215C2semidirect
11. #2187C2semidirect
12. #2186C2semidirect
13. #2186C2semidirect
14. #2184C2semidirect
15. #2113C2semidirect
16. C2×#135C22semidirect
25. C2×M32C22semidirect
26. C22×C16C22semidirect
32. #467C22semidirect
38. #466C22semidirect
40. C2×#107C22semidirect
43. #367C22semidirect
44. #367C22semidirect
45. #366C22semidirect
46. #366C22semidirect
47. #358C22semidirect
48. #358C22semidirect
53. #357C22semidirect
54. #357C22semidirect
57. C22⋊C16C22semidirect
58. C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#30612
#30582
#22372
#22342
#22332
#22322
#22202
#22182
#22172
#22152
#21872
#21862
#21842
#21132