Finite Group Databasebeta

← #30439

Subgroups — #30439

Order 256.

441 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 105 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, complemented
2#3043C211yesmaximal, characteristic, complemented
3#3019C211yesmaximal, characteristic
4#3009C211yesmaximal, characteristic, complemented
5#2098C211yesmaximal, characteristic, complemented
6#2081C211yesmaximal, characteristic, complemented
7#2080C211yesmaximal, characteristic, complemented
8#2053C211yesmaximal, characteristic, complemented
9#2046C211yesmaximal, characteristic, complemented
10#2009C211yesmaximal, characteristic, complemented
11#1912C211yesmaximal, characteristic, complemented
12#1910C211yesmaximal, characteristic, complemented
13#1905C211yesmaximal, characteristic, complemented
14#1881C211yesmaximal, characteristic, complemented
15#1795C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17#575C2211yescharacteristic
18#575C2211yescharacteristic
19C2×#137C2211yescharacteristic
20C2×#137C2211yescharacteristic, complemented
21C22×SD16C2211yescharacteristic
22#568C2211yescharacteristic, complemented
23#536C2211yescharacteristic
24#535C2211yescharacteristic
25#519C2211yescharacteristic, complemented
26#473—22no—
27#471—22no—
28#464—22no—
29#460—22no—
30#455—22no—
31#453—22no—
32#452—22no—
33#450—22no—
34#430C2211yescharacteristic, complemented
35#424C2211yescharacteristic, complemented
36#424C2211yescharacteristic, complemented
37#421C2211yescharacteristic
38#421C2211yescharacteristic
39#421C2211yescharacteristic
40#421C2211yescharacteristic
41#419C2211yescharacteristic, complemented
42#418C2211yescharacteristic
43#418C2211yescharacteristic
44#418C2211yescharacteristic
45#416C2211yescharacteristic
46#416C2211yescharacteristic, complemented
47#413C2211yescharacteristic
48C2×#101C2211yescharacteristic, complemented
49C2×#100C2211yescharacteristic
50#410C2211yescharacteristic
51#410C2211yescharacteristic, complemented
52#408C2211yescharacteristic
53#344C2211yescharacteristic, complemented
54#344C2211yescharacteristic, complemented
55#341C2211yescharacteristic, complemented
56H(3,ℤ/4)C2211yescharacteristic, complemented
57H(3,ℤ/4)C2211yescharacteristic, complemented
58H(3,ℤ/4)C2211yescharacteristic, complemented
59H(5,𝔽2)—22no—
60H(5,𝔽2)—41no—
61H(5,𝔽2)—41no—
62H(5,𝔽2)—41no—
63C2×#41—21no—
64C2×#41—21no—
65C2×#41C2311yescharacteristic
66C2×#41C2311yescharacteristic
67C2×#41C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4C2311yescharacteristic
73#137—22no—
74#137—21no—
75#137—21no—
76#137—21no—
77#137—21no—
78#137—41no—
79#137—41no—
80#137—41no—
81AGL(1,ℤ/8)—21no—
82AGL(1,ℤ/8)—21no—
83AGL(1,ℤ/8)—21no—
84AGL(1,ℤ/8)—21no—
85#135—41no—
86#135—41no—
87C2×Q16—21no—
88C2×Q16D412yescomplemented
89C2×SD16—22no—
90C2×SD16—22no—
91C2×SD16—21no—
92C2×SD16—41no—
93C2×SD16D412yescomplemented
94#131—22no—
95#131—41no—
96C2×M16—41no—
97C2×M16C2311yescharacteristic
98C2×M16C2311yescharacteristic
99C2×M16C2311yescharacteristic
100C2×M16C2311yescharacteristic
101C2×M16C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C4⋊Q8—21no—
104#127D412yes—
105#124—21no—
106#124D412yes—
107#123—41no—
108C22⋊Q8—22no—
109#121—41no—
110#120—41no—
111C4×Q8—22no—
112C4×Q8—41no—
113#117—41no—
114#117C2311yescharacteristic
115#117C2311yescharacteristic
116#117C2311yescharacteristic
117#117C2311yescharacteristic
118#108—22no—
119C4⋊C8—22no—
120C4⋊C8—41no—
121#104—22no—
122#104—22no—
123#104—41no—
124#104—41no—
125#103—21no—
126#103—21no—
127#103—41no—
128#102—21no—
129#102—21no—
130#102—22no—
131#102—22no—
132#102—41no—
133#101—22no—
134#101—21no—
135#100—22no—
136#100—21no—
137#99—41no—
138#99—41no—
139C22⋊C8—22no—
140C24—21no—
141C24—41no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152#41—21no—
153#41—21no—
154#41—42no—
155#41—42no—
156#41—42no—
157#41—42no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8C2×D411yescharacteristic
164C2×Q8C2×D412yes—
165C2×Q8C2×D411yescharacteristic
166C2×Q8#4111yescharacteristic
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4C2×D412yes—
185C2×D4C2×D411yescharacteristic
186C2×D4C2×D411yescharacteristic
187C2×D4#4111yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4C2411yescharacteristic
194Q16—42no—
195Q16—42no—
196Q16—41no—
197Q16—41no—
198SD16—42no—
199SD16—44no—
200SD16—42no—
201SD16—41no—
202SD16—41no—
203SD16—41no—
204SD16—41no—
205D8—41no—
206D8—41no—
207M16—21no—
208M16—21no—
209M16—21no—
210M16—21no—
211M16—21no—
212M16—21no—
213M16—21no—
214M16—21no—
215M16—21no—
216M16—21no—
217M16—42no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—42no—
226C2×C8C2×D412yes—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—42no—
243C42C2×D412yes—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—44no—
249C23—41no—
250C23—41no—
251C23—161no—
252C23—162no—
253C23H(5,𝔽2)11yescharacteristic
254Q8—22no—
255Q8—21no—
256Q8—21no—
257Q8—22no—
258Q8—21no—
259Q8—21no—
260Q8—21no—
261Q8—21no—
262Q8—41no—
263Q8—41no—
264D4—21no—
265D4—21no—
266D4—21no—
267D4—21no—
268D4—22no—
269D4—21no—
270D4—21no—
271D4—22no—
272D4—44no—
273D4—44no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—162no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4C22×D411yescharacteristic
301C2×C4C22×D411yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C8—22no—
307C8—41no—
308C8—41no—
309C8—41no—
310C8—41no—
311C8—41no—
312C22—41no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22—82no—
320C22—81no—
321C22—82no—
322C22—161no—
323C22—164no—
324C22#53411yescharacteristic
325C22#54611yescharacteristic
326C22#56011yescharacteristic
327C4—42no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—42no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4#53411yescharacteristic
336C4#53811yescharacteristic
337C4D4211yescharacteristic
338C4#54611yescharacteristic
339C2—21no—
340C2—21no—
341C2—21no—
342C2—81no—
343C2—81no—
344C2—82no—
345C2—81no—
346C2#266811yescharacteristic
subgroupG/H
center, socleC2#2668
commutator, Frattini, cosocleC22×C4C24
prime radical#3019C2
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
1. #3069C2semidirect
2. #3043C2semidirect
4. #3009C2semidirect
5. #2098C2semidirect
6. #2081C2semidirect
7. #2080C2semidirect
8. #2053C2semidirect
9. #2046C2semidirect
10. #2009C2semidirect
11. #1912C2semidirect
12. #1910C2semidirect
13. #1905C2semidirect
14. #1881C2semidirect
15. #1795C2semidirect
20. C2×#137C22semidirect
22. #568C22semidirect
25. #519C22semidirect
34. #430C22semidirect
35. #424C22semidirect
36. #424C22semidirect
41. #419C22semidirect
46. #416C22semidirect
48. C2×#101C22semidirect
51. #410C22semidirect
53. #344C22semidirect
54. #344C22semidirect
55. #341C22semidirect
56. H(3,ℤ/4)C22semidirect
57. H(3,ℤ/4)C22semidirect
58. H(3,ℤ/4)C22semidirect
88. C2×Q16D4semidirect
93. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
#30692
#30432
#30192
#30092
#20982
#20812
#20802
#20532
#20462
#20092
#19122
#19102
#19052
#18812
#17952