Finite Group Databasebeta

← #24575

Subgroups — #24575

Order 256.

457 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2#3006C211yesmaximal, characteristic
3#2896C211yesmaximal, characteristic, complemented
4#2880C211yesmaximal, characteristic, complemented
5#2079C211yesmaximal, characteristic, complemented
6#2078C211yesmaximal, characteristic, complemented
7#2051C211yesmaximal, characteristic, complemented
8#2051C211yesmaximal, characteristic, complemented
9#1907C211yesmaximal, characteristic, complemented
10#1907C211yesmaximal, characteristic, complemented
11#1900C211yesmaximal, characteristic, complemented
12#1855C211yesmaximal, characteristic, complemented
13#1822C211yesmaximal, characteristic, complemented
14#1816C211yesmaximal, characteristic, complemented
15#1815C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×#131C2211yescharacteristic
20C2×#131C2211yescharacteristic
21C22×M16C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#482—21no—
25#482—21no—
26#482—21no—
27#482—21no—
28#449—21no—
29#449—21no—
30#448—21no—
31#447—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×#102C2211yescharacteristic, complemented
47C2×#102C2211yescharacteristic, complemented
48C2×#100C2211yescharacteristic, complemented
49C2×#100C2211yescharacteristic, 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×D4—21no—
79C22×D4—21no—
80C22×D4D411yescharacteristic
81C22×D4C2311yescharacteristic
82C23×C4—21no—
83C23×C4C2311yescharacteristic
84AGL(1,ℤ/8)—22no—
85AGL(1,ℤ/8)—22no—
86AGL(1,ℤ/8)—41no—
87AGL(1,ℤ/8)—41no—
88C2×SD16—21no—
89C2×SD16—21no—
90C2×D8—21no—
91C2×D8—21no—
92#131—22no—
93#131—22no—
94#131—41no—
95#131—41no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16C2311yescharacteristic
102C2×M16C2311yescharacteristic
103C2×M16C2311yescharacteristic
104C2×M16C2311yescharacteristic
105C2×M16C2311yescharacteristic
106C2×M16C2311yescharacteristic
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112#121—21no—
113#121—21no—
114#121—21no—
115#121—21no—
116#117—21no—
117#117—21no—
118#117—21no—
119#117—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#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#102—21no—
148#100—41no—
149#100—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—21no—
164C24#4111yescharacteristic
165#41—21no—
166#41—21no—
167#41—22no—
168#41—22no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—42no—
174#41—42no—
175#41—41no—
176#41—41no—
177#41—42no—
178#41—42no—
179#41#4112yes—
180#41#4112yes—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D411yescharacteristic
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—82no—
204C2×D4C2×D411yescharacteristic
205C2×D4C2×D411yescharacteristic
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—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4#4111yescharacteristic
225C22×C4C2411yescharacteristic
226SD16—42no—
227SD16—42no—
228D8—42no—
229D8—42no—
230M16—21no—
231M16—21no—
232M16—21no—
233M16—21no—
234M16—21no—
235M16—21no—
236M16—21no—
237M16—21no—
238M16—21no—
239M16—21no—
240M16—21no—
241M16—21no—
242M16—42no—
243M16—42no—
244M16—42no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8C2×D411yescharacteristic
259C2×C8C2×D411yescharacteristic
260C2×C8C2×D411yescharacteristic
261C2×C8C2×D411yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23#12111yescharacteristic
289C23#12111yescharacteristic
290C23C2×#4111yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—42no—
295Q8#12312yes—
296D4—22no—
297D4—22no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—82no—
311D4—82no—
312D4#12312yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4#12111yescharacteristic
339C2×C4#12111yescharacteristic
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C8—21no—
347C8—21no—
348C8—21no—
349C8—21no—
350C8—41no—
351C8—41no—
352C8—41no—
353C8—41no—
354C8—41no—
355C8—41no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—84no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—161no—
377C22—161no—
378C22C2×#12111yescharacteristic
379C22#55111yescharacteristic
380C22#55111yescharacteristic
381C4—21no—
382C4—21no—
383C4—22no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4C2×#12311yescharacteristic
389C4#53411yescharacteristic
390C4#54611yescharacteristic
391C4#54611yescharacteristic
392C2—21no—
393C2—21no—
394C2—41no—
395C2—41no—
396C2—42no—
397C2—81no—
398C2—161no—
399C2#241211yescharacteristic
400C2#306711yescharacteristic
401C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#3006C2
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×#482C2semidirect
3. #2896C2semidirect
4. #2880C2semidirect
5. #2079C2semidirect
6. #2078C2semidirect
7. #2051C2semidirect
8. #2051C2semidirect
9. #1907C2semidirect
10. #1907C2semidirect
11. #1900C2semidirect
12. #1855C2semidirect
13. #1822C2semidirect
14. #1816C2semidirect
15. #1815C2semidirect
32. C2×#108C22semidirect
33. C2×#108C22semidirect
34. C2×#107C22semidirect
35. C2×#106C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#100C22semidirect
49. C2×#100C22semidirect
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×#4822
#30062
#28962
#28802
#20792
#20782
#20512
#19072
#19002
#18552
#18222
#18162
#18152