Finite Group Databasebeta

← #24574

Subgroups — #24574

Order 256.

444 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic, complemented
2#3007C211yesmaximal, characteristic
3#2896C211yesmaximal, characteristic
4#2880C211yesmaximal, characteristic
5#2079C211yesmaximal, characteristic, complemented
6#2078C211yesmaximal, characteristic, complemented
7#2052C211yesmaximal, characteristic, complemented
8#2052C211yesmaximal, characteristic, complemented
9#1908C211yesmaximal, characteristic, complemented
10#1908C211yesmaximal, characteristic, complemented
11#1901C211yesmaximal, characteristic, complemented
12#1855C211yesmaximal, characteristic, complemented
13#1822C211yesmaximal, characteristic, complemented
14#1816C211yesmaximal, characteristic, complemented
15#1815C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#137C2211yescharacteristic
18C2×#137C2211yescharacteristic
19C2×#131C2211yescharacteristic
20C2×#131C2211yescharacteristic
21C22×M16C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#485—21no—
25#485—21no—
26#485—21no—
27#485—21no—
28#452—21no—
29#452—21no—
30#451—21no—
31#450—21no—
32C2×#108C2211yescharacteristic, complemented
33C2×#108C2211yescharacteristic, complemented
34C2×#107C2211yescharacteristic, complemented
35C2×#106C2211yescharacteristic, complemented
36#419—21no—
37#418—21no—
38#417C2211yescharacteristic
39#417C2211yescharacteristic
40#417C2211yescharacteristic
41#417C2211yescharacteristic
42#417C2211yescharacteristic
43#417C2211yescharacteristic
44#416—21no—
45#416—21no—
46C2×#103C2211yescharacteristic, complemented
47C2×#103C2211yescharacteristic, complemented
48C2×#101C2211yescharacteristic, complemented
49C2×#101C2211yescharacteristic, complemented
50#408—21no—
51#408—21no—
52#408C2211yescharacteristic
53#408C2211yescharacteristic
54#407—21no—
55#407C2211yescharacteristic
56#407C2211yescharacteristic
57C2×C22⋊C8—21no—
58C2×C22⋊C8C2211yescharacteristic
59#343C2211yescharacteristic, complemented
60#343C2211yescharacteristic, complemented
61#343C2211yescharacteristic, complemented
62#343C2211yescharacteristic, complemented
63#343C2211yescharacteristic, complemented
64#343C2211yescharacteristic, complemented
65#341C2211yescharacteristic, complemented
66#341C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41D411yescharacteristic
74C2×#41D411yescharacteristic
75C2×#41C2311yescharacteristic
76C2×#41C2311yescharacteristic
77C22×Q8D411yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4D411yescharacteristic
81C23×C4—21no—
82C23×C4C2311yescharacteristic
83#137—22no—
84#137—22no—
85#137—41no—
86#137—41no—
87C2×Q16—21no—
88C2×Q16—21no—
89C2×SD16—21no—
90C2×SD16—21no—
91#131—22no—
92#131—22no—
93#131—41no—
94#131—41no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16C2311yescharacteristic
101C2×M16C2311yescharacteristic
102C2×M16C2311yescharacteristic
103C2×M16C2311yescharacteristic
104C2×M16C2311yescharacteristic
105C2×M16C2311yescharacteristic
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115#117—21no—
116#117—21no—
117#117—21no—
118#117—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C42—21no—
126#108—41no—
127#108—41no—
128#107—21no—
129#107—21no—
130#106—21no—
131#106—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103—21no—
137#103—21no—
138#103—21no—
139#103—21no—
140#103—21no—
141#103—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#102—21no—
148#101—41no—
149#101—41no—
150C22⋊C8—21no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8D411yescharacteristic
159C22⋊C8D411yescharacteristic
160C22⋊C8D411yescharacteristic
161C22⋊C8D411yescharacteristic
162C24—21no—
163C24#4111yescharacteristic
164#41—21no—
165#41—21no—
166#41—22no—
167#41—22no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—42no—
173#41—42no—
174#41—41no—
175#41—41no—
176#41—42no—
177#41—42no—
178#41#4112yes—
179#41#4112yes—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8C2×D411yescharacteristic
190C2×Q8C2×D411yescharacteristic
191C2×D4—21no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4C2×D411yescharacteristic
201C2×D4C2×D411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4#4111yescharacteristic
222C22×C4C2411yescharacteristic
223Q16—42no—
224Q16—42no—
225SD16—42no—
226SD16—42no—
227M16—21no—
228M16—21no—
229M16—21no—
230M16—21no—
231M16—21no—
232M16—21no—
233M16—21no—
234M16—21no—
235M16—21no—
236M16—21no—
237M16—21no—
238M16—21no—
239M16—42no—
240M16—42no—
241M16—42no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8C2×D411yescharacteristic
256C2×C8C2×D411yescharacteristic
257C2×C8C2×D411yescharacteristic
258C2×C8C2×D411yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C42—41no—
272C42—41no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23—81no—
282C23#12111yescharacteristic
283C23#12111yescharacteristic
284C23C2×#4111yescharacteristic
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—42no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8#12312yes—
294D4—22no—
295D4—22no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—42no—
300D4—42no—
301D4—42no—
302D4—41no—
303D4—41no—
304D4#12312yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4#12111yescharacteristic
334C2×C4#12111yescharacteristic
335C2×C4C22×D411yescharacteristic
336C2×C4C22×D411yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4C2×#4111yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C2×C4H(5,𝔽2)11yescharacteristic
341C8—21no—
342C8—21no—
343C8—21no—
344C8—21no—
345C8—41no—
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—41no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—84no—
368C22C2×#12111yescharacteristic
369C22#55111yescharacteristic
370C22#55111yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4C2×#12311yescharacteristic
380C4#53411yescharacteristic
381C4#54611yescharacteristic
382C4#54611yescharacteristic
383C2—21no—
384C2—21no—
385C2—41no—
386C2—41no—
387C2—42no—
388C2—81no—
389C2#241211yescharacteristic
390C2#306711yescharacteristic
391C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#485C2semidirect
5. #2079C2semidirect
6. #2078C2semidirect
7. #2052C2semidirect
8. #2052C2semidirect
9. #1908C2semidirect
10. #1908C2semidirect
11. #1901C2semidirect
12. #1855C2semidirect
13. #1822C2semidirect
14. #1816C2semidirect
15. #1815C2semidirect
32. C2×#108C22semidirect
33. C2×#108C22semidirect
34. C2×#107C22semidirect
35. C2×#106C22semidirect
46. C2×#103C22semidirect
47. C2×#103C22semidirect
48. C2×#101C22semidirect
49. C2×#101C22semidirect
59. #343C22semidirect
60. #343C22semidirect
61. #343C22semidirect
62. #343C22semidirect
63. #343C22semidirect
64. #343C22semidirect
65. #341C22semidirect
66. #341C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
#30072
#28962
#28802
#20792
#20782
#20522
#19082
#19012
#18552
#18222
#18162
#18152