Finite Group Databasebeta

← #31416

Subgroups — #31416

Order 256.

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

normal subgroups only

This group has 333 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#3039C211yesmaximal, characteristic, complemented
3#2102C211yesmaximal, characteristic, complemented
4#2082C211yesmaximal, characteristic, complemented
5#2082C211yesmaximal, characteristic, complemented
6#2049C211yesmaximal, characteristic, complemented
7#2009C211yesmaximal, characteristic
8#2005C211yesmaximal, characteristic, complemented
9#1909C211yesmaximal, characteristic, complemented
10#1906C211yesmaximal, characteristic
11#1905C211yesmaximal, characteristic, complemented
12#1905C211yesmaximal, characteristic, complemented
13#1886C211yesmaximal, characteristic, complemented
14#1861C211yesmaximal, characteristic, complemented
15#1827C211yesmaximal, characteristic, complemented
16#575C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic
19C2×#137C2211yescharacteristic
20#568C2211yescharacteristic
21#536C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23#519C2211yescharacteristic
24C4×#41C2211yescharacteristic
25#473—22no—
26#471—22no—
27#462—22no—
28#461—22no—
29#430C2211yescharacteristic
30#430C2211yescharacteristic
31#423C2211yescharacteristic
32#421C2211yescharacteristic
33#421C2211yescharacteristic
34C2×#104C2211yescharacteristic
35#419C2211yescharacteristic
36#419C2211yescharacteristic
37#419C2211yescharacteristic
38#419C2211yescharacteristic
39#419C2211yescharacteristic
40#413C2211yescharacteristic
41C2×#101C2211yescharacteristic
42C2×#101C2211yescharacteristic
43#410C2211yescharacteristic
44#410C2211yescharacteristic
45#344C2211yescharacteristic
46#344C2211yescharacteristic
47#344C2211yescharacteristic
48#344C2211yescharacteristic
49#339C2211yescharacteristic
50#339C2211yescharacteristic
51#339C2211yescharacteristic
52H(3,ℤ/4)C2211yescharacteristic
53H(3,ℤ/4)C2211yescharacteristic
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—21no—
71C2×Q16—21no—
72C2×Q16D412yes—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D412yes—
76#131—22no—
77#131—41no—
78C2×M16—41no—
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C2×M16C2311yescharacteristic
83C2×M16C2311yescharacteristic
84C2×M16C2311yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89#124—21no—
90#123—41no—
91C22⋊Q8—41no—
92C4×Q8—21no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—21no—
96C4×D4—41no—
97#117—21no—
98#117—41no—
99#117C2311yescharacteristic
100#117C2311yescharacteristic
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103C2×C4⋊C4—41no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106#108—22no—
107#108—41no—
108C4⋊C8—22no—
109#104—21no—
110#104—21no—
111#104—41no—
112#104—41no—
113#103—22no—
114#103—22no—
115#103—41no—
116#103—41no—
117#103—41no—
118#101—22no—
119#101—21no—
120#101—41no—
121#100—21no—
122#99—41no—
123#99—41no—
124#41—21no—
125#41—21no—
126#41—21no—
127#41—21no—
128#41—21no—
129#41—21no—
130#41—21no—
131#41—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×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C2×D412yes—
144C2×Q8C2×D411yescharacteristic
145C2×Q8#4111yescharacteristic
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C2×D411yescharacteristic
149C2×D4#4111yescharacteristic
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4C2411yescharacteristic
157Q16—42no—
158Q16—42no—
159Q16—41no—
160Q16—42no—
161SD16—42no—
162SD16—42no—
163SD16—41no—
164SD16—41no—
165SD16—42no—
166D8—41no—
167M16—21no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—21no—
178M16—21no—
179M16—42no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8C2×D412yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42#4111yescharacteristic
210C42#4111yescharacteristic
211C23—21no—
212C23—21no—
213C23H(5,𝔽2)11yescharacteristic
214Q8—22no—
215Q8—21no—
216Q8—21no—
217Q8—22no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C8—22no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—81no—
270C22—81no—
271C22#53811yescharacteristic
272C22#54611yescharacteristic
273C22#56011yescharacteristic
274C4—22no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—42no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#53411yescharacteristic
285C4#54011yescharacteristic
286C4#54611yescharacteristic
287C4#55311yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—81no—
292C2—81no—
293C2#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. #3039C2semidirect
3. #2102C2semidirect
4. #2082C2semidirect
5. #2082C2semidirect
6. #2049C2semidirect
8. #2005C2semidirect
9. #1909C2semidirect
11. #1905C2semidirect
12. #1905C2semidirect
13. #1886C2semidirect
14. #1861C2semidirect
15. #1827C2semidirect

Maximal subgroups

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

subgroupindex
#30692
#30392
#21022
#20822
#20492
#20092
#20052
#19092
#19062
#19052
#18862
#18612
#18272