Finite Group Databasebeta

← #31419

Subgroups — #31419

Order 256.

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

normal subgroups only

This group has 326 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#3069C211yesmaximal, characteristic
2#3044C211yesmaximal, characteristic, complemented
3#2103C211yesmaximal, characteristic, complemented
4#2082C211yesmaximal, characteristic, complemented
5#2080C211yesmaximal, characteristic, complemented
6#2047C211yesmaximal, characteristic, complemented
7#2020C211yesmaximal, characteristic, complemented
8#2009C211yesmaximal, characteristic
9#1915C211yesmaximal, characteristic, complemented
10#1905C211yesmaximal, characteristic, complemented
11#1902C211yesmaximal, characteristic, complemented
12#1902C211yesmaximal, characteristic, complemented
13#1881C211yesmaximal, characteristic
14#1868C211yesmaximal, characteristic, complemented
15#1830C211yesmaximal, characteristic, complemented
16#575C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic
19C2×#137C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21#568C2211yescharacteristic
22#568C2211yescharacteristic
23#536C2211yescharacteristic
24#519C2211yescharacteristic
25#473—22no—
26#471—22no—
27#464—22no—
28#462—22no—
29#430C2211yescharacteristic
30#424C2211yescharacteristic
31#424C2211yescharacteristic
32#421C2211yescharacteristic
33#421C2211yescharacteristic
34#419C2211yescharacteristic
35#419C2211yescharacteristic
36#419C2211yescharacteristic
37#416C2211yescharacteristic
38#413C2211yescharacteristic
39C2×#101C2211yescharacteristic
40C2×#101C2211yescharacteristic
41C2×#101C2211yescharacteristic
42C2×#101C2211yescharacteristic
43#410C2211yescharacteristic
44#408C2211yescharacteristic
45#344C2211yescharacteristic
46#344C2211yescharacteristic
47#344C2211yescharacteristic
48#344C2211yescharacteristic
49#344C2211yescharacteristic
50#344C2211yescharacteristic
51#341C2211yescharacteristic
52#341C2211yescharacteristic
53#338C2211yescharacteristic
54H(3,ℤ/4)C2211yescharacteristic
55C2×#41C2311yescharacteristic
56C2×#41C2311yescharacteristic
57C22×Q8C2311yescharacteristic
58C22×Q8C2311yescharacteristic
59#137—22no—
60#137—21no—
61#137—21no—
62#137—41no—
63#137—41no—
64#137—41no—
65#137—41no—
66#137—41no—
67AGL(1,ℤ/8)—21no—
68AGL(1,ℤ/8)—21no—
69#135—41no—
70C2×Q16—22no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16D412yes—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D412yes—
79#131—22no—
80#131—41no—
81#131—41no—
82#131—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16C2311yescharacteristic
86C2×M16C2311yescharacteristic
87C2×M16C2311yescharacteristic
88C2×M16C2311yescharacteristic
89C2×M16C2311yescharacteristic
90C2×M16C2311yescharacteristic
91C2×M16C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C4⋊Q8—21no—
94#124—21no—
95#123—22no—
96C22⋊Q8—41no—
97C4×Q8—22no—
98C4×Q8—41no—
99#117—22no—
100#117C2311yescharacteristic
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103#108—22no—
104C4⋊C8—22no—
105C4⋊C8—41no—
106#104—41no—
107#104—41no—
108#103—21no—
109#103—22no—
110#103—41no—
111#103—41no—
112#102—21no—
113#101—22no—
114#101—21no—
115#101—41no—
116#101—41no—
117#101—41no—
118#100—21no—
119#99—22no—
120C22⋊C8—41no—
121#41—21no—
122#41—21no—
123#41—21no—
124#41—21no—
125#41—21no—
126#41—21no—
127#41—21no—
128#41—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8C2×D411yescharacteristic
141C2×Q8C2×D412yes—
142C2×Q8#4111yescharacteristic
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×D411yescharacteristic
146C2×D4#4111yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4C2411yescharacteristic
152Q16—42no—
153Q16—42no—
154Q16—41no—
155Q16—44no—
156Q16—42no—
157SD16—42no—
158SD16—42no—
159SD16—41no—
160SD16—41no—
161SD16—42no—
162D8—41no—
163M16—21no—
164M16—21no—
165M16—21no—
166M16—21no—
167M16—21no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—42no—
178M16—42no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8C2×D412yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—42no—
199C22⋊C4#4112yes—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—42no—
204C23—21no—
205C23—21no—
206C23C2×#4111yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—21no—
210Q8—21no—
211Q8—22no—
212Q8—21no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4C22×D411yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4H(5,𝔽2)11yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C2×C4H(5,𝔽2)11yescharacteristic
248C8—22no—
249C8—41no—
250C8—41no—
251C8—41no—
252C8—41no—
253C8—41no—
254C8—41no—
255C8—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—81no—
260C22—81no—
261C22#54011yescharacteristic
262C22#54611yescharacteristic
263C22#55311yescharacteristic
264C4—42no—
265C4—41no—
266C4—41no—
267C4—42no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4#53411yescharacteristic
272C4#53811yescharacteristic
273C4#54611yescharacteristic
274C4#56011yescharacteristic
275C2—21no—
276C2—21no—
277C2—21no—
278C2—81no—
279C2—81no—
280C2#268811yescharacteristic
subgroupG/H
center, socleC2#2688
commutator, Frattini, cosocleC22×C4C24
prime radical#575C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3044C2semidirect
3. #2103C2semidirect
4. #2082C2semidirect
5. #2080C2semidirect
6. #2047C2semidirect
7. #2020C2semidirect
9. #1915C2semidirect
10. #1905C2semidirect
11. #1902C2semidirect
12. #1902C2semidirect
14. #1868C2semidirect
15. #1830C2semidirect

Maximal subgroups

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

subgroupindex
#30692
#30442
#21032
#20822
#20802
#20472
#20202
#20092
#19152
#19052
#19022
#18812
#18682
#18302