Finite Group Databasebeta

← #33864

Subgroups — #33864

Order 256.

290 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 101 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#3339C211yesmaximal, characteristic, complemented
2#3280C211yesmaximal, characteristic, complemented
3#2242C211yesmaximal, characteristic, complemented
4#2236C211yesmaximal, characteristic, complemented
5#2234C211yesmaximal, characteristic, complemented
6#2228C211yesmaximal, characteristic, complemented
7#2217C211yesmaximal, characteristic, complemented
8#2215C211yesmaximal, characteristic, complemented
9#2210C211yesmaximal, characteristic, complemented
10#2186C211yesmaximal, characteristic, complemented
11#2184C211yesmaximal, characteristic, complemented
12C4×D16C211yesmaximal, characteristic, complemented
13D4×C16C211yesmaximal, characteristic, complemented
14#2138C211yesmaximal, characteristic, complemented
15C2×#357C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#553—21no—
19#546—21no—
20C2×SD32C2211yescharacteristic, complemented
21C2×D16C2211yescharacteristic
22C2×D16C2211yescharacteristic
23C22×C16C2211yescharacteristic
24C22×C16C2211yescharacteristic, complemented
25#499C2211yescharacteristic, complemented
26#495C2211yescharacteristic
27#482—21no—
28#481—21no—
29#478—21no—
30#467C2211yescharacteristic
31#466C2211yescharacteristic
32#466C2211yescharacteristic
33#466C2211yescharacteristic, complemented
34#463—21no—
35#450—21no—
36#449—21no—
37C4×D8C2211yescharacteristic
38C4×D8C2211yescharacteristic, complemented
39C8×D4C2211yescharacteristic
40#427C2211yescharacteristic, complemented
41C2×#107C2211yescharacteristic
42#367C2211yescharacteristic, complemented
43#366C2211yescharacteristic, complemented
44#366C2211yescharacteristic, complemented
45C4⋊C16C2211yescharacteristic, complemented
46#358C2211yescharacteristic, complemented
47#358C2211yescharacteristic, complemented
48#358C2211yescharacteristic, complemented
49#357C2211yescharacteristic
50#357C2211yescharacteristic
51#357C2211yescharacteristic
52#357C2211yescharacteristic
53#357C2211yescharacteristic
54#357C2211yescharacteristic
55#357C2211yescharacteristic, complemented
56C22⋊C16C2211yescharacteristic
57C22⋊C16C2211yescharacteristic, complemented
58C4×C16C2211yescharacteristic, complemented
59C2×#41—21no—
60C22×D4—21no—
61#135—22no—
62#135—41no—
63C2×Q16C2311yescharacteristic
64C2×SD16—21no—
65C2×SD16—21no—
66C2×D8—22no—
67C2×D8—21no—
68C2×D8—41no—
69C2×D8C2311yescharacteristic
70C2×D8C2311yescharacteristic
71C2×D8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#126—41no—
75#125—21no—
76#124—21no—
77#123—41no—
78#123—41no—
79C22⋊Q8—21no—
80#121—21no—
81#121—21no—
82#121—21no—
83#121—41no—
84#121—41no—
85#120—41no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4—41no—
89C4×D4D411yescharacteristic
90#117—21no—
91C2×C4⋊C4—21no—
92C2×C22⋊C4—41no—
93SD32—41no—
94D16—22no—
95D16—41no—
96C2×C16—41no—
97C2×C16—41no—
98C2×C16C2311yescharacteristic
99C2×C16C2311yescharacteristic
100C2×C16C2311yescharacteristic
101C2×C16C2311yescharacteristic
102#107—21no—
103#107C2311yescharacteristic
104#107C2311yescharacteristic
105#107C2311yescharacteristic
106#107C2311yescharacteristic
107#106—21no—
108#106—21no—
109C4⋊C8D411yescharacteristic
110#103—21no—
111#102—21no—
112#102—21no—
113#102—21no—
114#102—21no—
115#102—21no—
116C22⋊C8D411yescharacteristic
117C22⋊C8D411yescharacteristic
118C4×C8C2311yescharacteristic
119C24—41no—
120#41—42no—
121#41—81no—
122C2×Q8—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—42no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—81no—
133C2×D4D811yescharacteristic
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140Q16—21no—
141Q16—21no—
142SD16—42no—
143SD16—81no—
144D8—21no—
145D8—21no—
146D8—21no—
147D8—21no—
148D8—21no—
149D8—42no—
150D8#4112yes—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8C2×D411yescharacteristic
156C2×C8C2×D411yescharacteristic
157C2×C8C2×D411yescharacteristic
158C2×C8C2411yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4D811yescharacteristic
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4D811yescharacteristic
178C22⋊C4D811yescharacteristic
179C42—41no—
180C42C2×D411yescharacteristic
181C16—21no—
182C16—21no—
183C16—21no—
184C16—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—82no—
189C23—82no—
190C23—161no—
191C23C2×D811yescharacteristic
192C23C2×D811yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—22no—
196D4—22no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—41no—
202D4—82no—
203D4—82no—
204D4—82no—
205D4—161no—
206D4—161no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—82no—
217C2×C4—161no—
218C2×C4—161no—
219C2×C4—161no—
220C2×C4C2×D811yescharacteristic
221C2×C4C2×D811yescharacteristic
222C2×C4C2×D811yescharacteristic
223C2×C4C2×D811yescharacteristic
224C2×C4C22×D411yescharacteristic
225C8—21no—
226C8—21no—
227C8—21no—
228C8C2×#4111yescharacteristic
229C8H(5,𝔽2)11yescharacteristic
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—42no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—164no—
245C22#50812yes—
246C22C22×D811yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4#54611yescharacteristic
257C4#57711yescharacteristic
258C2—22no—
259C2—41no—
260C2—82no—
261C2—161no—
262C2—161no—
263C2#329411yescharacteristic
264C2C2×#50811yescharacteristic
265C2#341511yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3339C2semidirect
2. #3280C2semidirect
3. #2242C2semidirect
4. #2236C2semidirect
5. #2234C2semidirect
6. #2228C2semidirect
7. #2217C2semidirect
8. #2215C2semidirect
9. #2210C2semidirect
10. #2186C2semidirect
11. #2184C2semidirect
12. C4×D16C2semidirect
13. D4×C16C2semidirect
14. #2138C2semidirect
15. C2×#357C2semidirect
20. C2×SD32C22semidirect
24. C22×C16C22semidirect
25. #499C22semidirect
33. #466C22semidirect
38. C4×D8C22semidirect
40. #427C22semidirect
42. #367C22semidirect
43. #366C22semidirect
44. #366C22semidirect
45. C4⋊C16C22semidirect
46. #358C22semidirect
47. #358C22semidirect
48. #358C22semidirect
55. #357C22semidirect
57. C22⋊C16C22semidirect
58. C4×C16C22semidirect

Maximal subgroups

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

subgroupindex
#33392
#32802
#22422
#22362
#22342
#22282
#22172
#22152
#22102
#21862
#21842
C4×D162
D4×C162
#21382
C2×#3572