Finite Group Databasebeta

← #30442

Subgroups — #30442

Order 256.

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

normal subgroups only

This group has 439 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#3043C211yesmaximal, characteristic, complemented
2#3042C211yesmaximal, characteristic, complemented
3#3022C211yesmaximal, characteristic, complemented
4#3021C211yesmaximal, characteristic
5#2098C211yesmaximal, characteristic, complemented
6#2080C211yesmaximal, characteristic, complemented
7#2080C211yesmaximal, characteristic, complemented
8#2049C211yesmaximal, characteristic, complemented
9#2048C211yesmaximal, characteristic, complemented
10#2019C211yesmaximal, characteristic, complemented
11#1912C211yesmaximal, characteristic, complemented
12#1911C211yesmaximal, characteristic, complemented
13#1905C211yesmaximal, characteristic, complemented
14#1904C211yesmaximal, characteristic, complemented
15#1794C211yesmaximal, characteristic, complemented
16#585C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C2×AGL(1,ℤ/8)C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22#536C2211yescharacteristic
23#535C2211yescharacteristic
24#519C2211yescharacteristic, complemented
25#518C2211yescharacteristic, complemented
26#464—22no—
27#463—22no—
28#461—22no—
29#460—22no—
30#456—22no—
31#455—22no—
32#454—22no—
33#453—22no—
34#430C2211yescharacteristic, complemented
35#424C2211yescharacteristic, complemented
36#424C2211yescharacteristic, complemented
37#421C2211yescharacteristic
38#421C2211yescharacteristic
39#421C2211yescharacteristic
40#421C2211yescharacteristic
41#421C2211yescharacteristic
42#421C2211yescharacteristic
43#419C2211yescharacteristic, complemented
44#418C2211yescharacteristic, complemented
45#416C2211yescharacteristic
46#416C2211yescharacteristic
47#413C2211yescharacteristic
48#413C2211yescharacteristic
49C2×#101C2211yescharacteristic, complemented
50C2×#100C2211yescharacteristic, complemented
51#410C2211yescharacteristic
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
592−1+4—21no—
602−1+4—21no—
612−1+4—21no—
62H(5,𝔽2)—21no—
63H(5,𝔽2)—21no—
64H(5,𝔽2)—21no—
65H(5,𝔽2)—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41C2311yescharacteristic
70C2×#41C2311yescharacteristic
71C2×#41C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×D4C2311yescharacteristic
74#137—22no—
75#137—22no—
76#137—41no—
77#137—41no—
78AGL(1,ℤ/8)—22no—
79AGL(1,ℤ/8)—22no—
80AGL(1,ℤ/8)—41no—
81AGL(1,ℤ/8)—41no—
82C2×Q16—21no—
83C2×Q16—21no—
84C2×SD16—22no—
85C2×SD16—21no—
86C2×SD16—21no—
87C2×SD16—22no—
88C2×SD16—21no—
89C2×SD16—21no—
90C2×SD16—41no—
91C2×D8—21no—
92C2×D8—21no—
93C2×M16C2311yescharacteristic
94C2×M16C2311yescharacteristic
95C2×M16C2311yescharacteristic
96C2×M16C2311yescharacteristic
97C2×M16C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C4⋊Q8D412yescomplemented
100#127D412yes—
101#124D412yes—
102#124D412yescomplemented
103#123—41no—
104C22⋊Q8—41no—
105#121—41no—
106#120—41no—
107C4×Q8—22no—
108C4×Q8—41no—
109C4×D4—22no—
110C4×D4—41no—
111#117—41no—
112#117C2311yescharacteristic
113#117C2311yescharacteristic
114#117C2311yescharacteristic
115#117C2311yescharacteristic
116C2×C22⋊C4—41no—
117#108—41no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120#104—22no—
121#104—22no—
122#104—22no—
123#104—22no—
124#104—41no—
125#104—41no—
126#103—21no—
127#103—21no—
128#103—22no—
129#102—21no—
130#102—21no—
131#102—22no—
132#101—21no—
133#101—21no—
134#101—41no—
135#100—21no—
136#100—21no—
137#100—41no—
138#99—41no—
139C22⋊C8—41no—
140C24—21no—
141#41—21no—
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—42no—
154#41—42no—
155#41—42no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162#41—42no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8C2×D411yescharacteristic
172C2×Q8C2×D411yescharacteristic
173C2×Q8C2×D412yes—
174C2×Q8#4111yescharacteristic
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4C2×D411yescharacteristic
191C2×D4C2×D411yescharacteristic
192C2×D4C2×D412yes—
193C2×D4#4111yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C2411yescharacteristic
201Q16—42no—
202Q16—42no—
203SD16—44no—
204SD16—42no—
205SD16—42no—
206SD16—42no—
207SD16—42no—
208D8—42no—
209D8—42no—
210M16—21no—
211M16—21no—
212M16—21no—
213M16—21no—
214M16—21no—
215M16—21no—
216M16—21no—
217M16—21no—
218M16—21no—
219M16—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—21no—
242C42—21no—
243C42—42no—
244C42C2×D412yes—
245C42C2×D412yes—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23—161no—
254C23H(5,𝔽2)11yescharacteristic
255Q8—22no—
256Q8—21no—
257Q8—21no—
258Q8—21no—
259Q8—22no—
260Q8—21no—
261Q8—21no—
262Q8—21no—
263Q8—42no—
264Q8—41no—
265Q8—41no—
266D4—21no—
267D4—22no—
268D4—21no—
269D4—21no—
270D4—22no—
271D4—21no—
272D4—21no—
273D4—21no—
274D4—42no—
275D4—42no—
276D4—42no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—162no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—162no—
306C2×C4C22×D411yescharacteristic
307C2×C4C22×D411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C8—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—81no—
324C22—82no—
325C22—81no—
326C22—81no—
327C22—82no—
328C22—161no—
329C22—162no—
330C22#53411yescharacteristic
331C22#53411yescharacteristic
332C22#53811yescharacteristic
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—81no—
340C4—81no—
341C4D4211yescharacteristic
342C4#54611yescharacteristic
343C4#54611yescharacteristic
344C4#56011yescharacteristic
345C2—21no—
346C2—21no—
347C2—21no—
348C2—82no—
349C2—81no—
350C2—81no—
351C2—81no—
352C2#266811yescharacteristic
subgroupG/H
center, socleC2#2668
commutator, Frattini, cosocleC22×C4C24
prime radical#3021C2
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. #3043C2semidirect
2. #3042C2semidirect
3. #3022C2semidirect
5. #2098C2semidirect
6. #2080C2semidirect
7. #2080C2semidirect
8. #2049C2semidirect
9. #2048C2semidirect
10. #2019C2semidirect
11. #1912C2semidirect
12. #1911C2semidirect
13. #1905C2semidirect
14. #1904C2semidirect
15. #1794C2semidirect
21. C22×SD16C22semidirect
24. #519C22semidirect
25. #518C22semidirect
34. #430C22semidirect
35. #424C22semidirect
36. #424C22semidirect
43. #419C22semidirect
44. #418C22semidirect
49. C2×#101C22semidirect
50. C2×#100C22semidirect
53. #344C22semidirect
54. #344C22semidirect
55. #341C22semidirect
56. H(3,ℤ/4)C22semidirect
57. H(3,ℤ/4)C22semidirect
58. H(3,ℤ/4)C22semidirect
99. C4⋊Q8D4semidirect
102. #124D4semidirect

Maximal subgroups

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

subgroupindex
#30432
#30422
#30222
#30212
#20982
#20802
#20492
#20482
#20192
#19122
#19112
#19052
#19042
#17942