Finite Group Databasebeta

← #30437

Subgroups — #30437

Order 256.

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

normal subgroups only

This group has 453 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#3068C211yesmaximal, characteristic, complemented
2#3041C211yesmaximal, characteristic, complemented
3#3019C211yesmaximal, characteristic, complemented
4#3008C211yesmaximal, characteristic, complemented
5#2098C211yesmaximal, characteristic, complemented
6#2081C211yesmaximal, characteristic, complemented
7#2080C211yesmaximal, characteristic, complemented
8#2053C211yesmaximal, characteristic, complemented
9#2045C211yesmaximal, characteristic, complemented
10#2008C211yesmaximal, characteristic, complemented
11#1911C211yesmaximal, characteristic, complemented
12#1910C211yesmaximal, characteristic, complemented
13#1904C211yesmaximal, characteristic, complemented
14#1881C211yesmaximal, characteristic, complemented
15#1795C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17#575C2211yescharacteristic
18#575C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C2×AGL(1,ℤ/8)C2211yescharacteristic, complemented
21C22×D8C2211yescharacteristic
22#568C2211yescharacteristic, complemented
23#535C2211yescharacteristic
24#535C2211yescharacteristic
25#518C2211yescharacteristic, complemented
26#472—22no—
27#471—22no—
28#463—22no—
29#459—22no—
30#455—22no—
31#453—22no—
32#449—22no—
33#447—22no—
34#430C2211yescharacteristic, complemented
35#424C2211yescharacteristic, complemented
36#424C2211yescharacteristic, complemented
37#421C2211yescharacteristic
38#421C2211yescharacteristic
39#421C2211yescharacteristic
40#421C2211yescharacteristic
41#418C2211yescharacteristic
42#418C2211yescharacteristic
43#418C2211yescharacteristic
44#418C2211yescharacteristic, complemented
45#416C2211yescharacteristic
46#416C2211yescharacteristic, complemented
47#413C2211yescharacteristic
48C2×#100C2211yescharacteristic
49C2×#100C2211yescharacteristic, complemented
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×D4—21no—
69C22×D4—21no—
70C22×D4—41no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73#137—21no—
74#137—21no—
75#137—21no—
76#137—21no—
77AGL(1,ℤ/8)—22no—
78AGL(1,ℤ/8)—21no—
79AGL(1,ℤ/8)—21no—
80AGL(1,ℤ/8)—21no—
81AGL(1,ℤ/8)—21no—
82AGL(1,ℤ/8)—41no—
83AGL(1,ℤ/8)—41no—
84AGL(1,ℤ/8)—41no—
85#135—41no—
86#135—41no—
87C2×SD16—21no—
88C2×SD16D412yescomplemented
89C2×D8—22no—
90C2×D8—22no—
91C2×D8—21no—
92C2×D8—41no—
93C2×D8D412yescomplemented
94#131—22no—
95#131—41no—
96C2×M16—41no—
97C2×M16C2311yescharacteristic
98C2×M16C2311yescharacteristic
99C2×M16C2311yescharacteristic
100C2×M16C2311yescharacteristic
101C2×M16C2311yescharacteristic
102C22×C8C2311yescharacteristic
103#127—21no—
104#127D412yescomplemented
105#124—21no—
106#124D412yescomplemented
107#121—22no—
108#121—41no—
109#120—41no—
110#120—41no—
111C4×D4—22no—
112C4×D4—41no—
113#117C2311yescharacteristic
114#117C2311yescharacteristic
115#117C2311yescharacteristic
116#117C2311yescharacteristic
117C2×C22⋊C4—41no—
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#102—21no—
128#102—21no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133#101—21no—
134#100—22no—
135#100—22no—
136#100—21no—
137#99—41no—
138#99—41no—
139C22⋊C8—22no—
140C24—21no—
141C24—21no—
142C24—41no—
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—21no—
155#41—42no—
156#41—42no—
157#41—42no—
158#41—42no—
159C2×Q8C2×D411yescharacteristic
160C2×Q8C2×D411yescharacteristic
161C2×Q8#4111yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
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×D4—42no—
185C2×D4C2×D412yes—
186C2×D4C2×D411yescharacteristic
187C2×D4C2×D412yes—
188C2×D4C2×D411yescharacteristic
189C2×D4#4111yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4C2411yescharacteristic
196Q16—41no—
197Q16—41no—
198SD16—42no—
199SD16—42no—
200SD16—41no—
201SD16—41no—
202SD16—41no—
203SD16—41no—
204D8—42no—
205D8—44no—
206D8—42no—
207D8—41no—
208D8—41no—
209M16—21no—
210M16—21no—
211M16—21no—
212M16—21no—
213M16—21no—
214M16—21no—
215M16—21no—
216M16—21no—
217M16—21no—
218M16—21no—
219M16—42no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—42no—
228C2×C8C2×D412yes—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42C2×D412yes—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—44no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—161no—
255C23—161no—
256C23—162no—
257C23H(5,𝔽2)11yescharacteristic
258Q8—21no—
259Q8—21no—
260Q8—21no—
261Q8—21no—
262Q8—21no—
263Q8—21no—
264D4—21no—
265D4—22no—
266D4—21no—
267D4—22no—
268D4—21no—
269D4—22no—
270D4—21no—
271D4—21no—
272D4—21no—
273D4—22no—
274D4—44no—
275D4—44no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—162no—
287D4—162no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—162no—
303C2×C4C22×D411yescharacteristic
304C2×C4C22×D411yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C8—22no—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C8—41no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—81no—
323C22—82no—
324C22—82no—
325C22—81no—
326C22—82no—
327C22—82no—
328C22—161no—
329C22—161no—
330C22—164no—
331C22#53411yescharacteristic
332C22#54611yescharacteristic
333C22#56011yescharacteristic
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4#53411yescharacteristic
342C4#53811yescharacteristic
343C4D4211yescharacteristic
344C4#54611yescharacteristic
345C2—21no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—81no—
350C2—82no—
351C2—82no—
352C2—81no—
353C2#266811yescharacteristic
subgroupG/H
center, socleC2#2668
commutator, Frattini, cosocleC22×C4C24
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. #3068C2semidirect
2. #3041C2semidirect
3. #3019C2semidirect
4. #3008C2semidirect
5. #2098C2semidirect
6. #2081C2semidirect
7. #2080C2semidirect
8. #2053C2semidirect
9. #2045C2semidirect
10. #2008C2semidirect
11. #1911C2semidirect
12. #1910C2semidirect
13. #1904C2semidirect
14. #1881C2semidirect
15. #1795C2semidirect
20. C2×AGL(1,ℤ/8)C22semidirect
22. #568C22semidirect
25. #518C22semidirect
34. #430C22semidirect
35. #424C22semidirect
36. #424C22semidirect
44. #418C22semidirect
46. #416C22semidirect
49. C2×#100C22semidirect
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×SD16D4semidirect
93. C2×D8D4semidirect
104. #127D4semidirect
106. #124D4semidirect

Maximal subgroups

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

subgroupindex
#30682
#30412
#30192
#30082
#20982
#20812
#20802
#20532
#20452
#20082
#19112
#19102
#19042
#18812
#17952