Finite Group Databasebeta

← #26850

Subgroups — #26850

Order 256.

388 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3136C211yesmaximal, characteristic, complemented
2#3096C211yesmaximal, characteristic, complemented
3#3081C211yesmaximal, characteristic, complemented
4#2946C211yesmaximal, characteristic, complemented
5#2063C211yesmaximal, characteristic, complemented
6#2060C211yesmaximal, characteristic, complemented
7#2051C211yesmaximal, characteristic, complemented
8#2002C211yesmaximal, characteristic
9#1957C211yesmaximal, characteristic, complemented
10#1937C211yesmaximal, characteristic, complemented
11#1931C211yesmaximal, characteristic, complemented
12#1931C211yesmaximal, characteristic, complemented
13#1887C211yesmaximal, characteristic, complemented
14#1859C211yesmaximal, characteristic, complemented
15#1772C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)C2211yescharacteristic
17#544C2211yescharacteristic
18#535C2211yescharacteristic
19C2×#127C2211yescharacteristic
20#520C2211yescharacteristic
21#518C2211yescharacteristic
22#490—21no—
23#489—21no—
24#488—21no—
25#486—21no—
26#482—21no—
27#482—21no—
28#481—21no—
29#480—21no—
30#478—21no—
31#478—21no—
32#476—21no—
33#474—21no—
34#442—21no—
35#440—21no—
36C4×SD16—21no—
37C4×D8—21no—
38C2×#108C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#428C2211yescharacteristic
42#427C2211yescharacteristic
43#424C2211yescharacteristic
44C2×#104C2211yescharacteristic
45#418C2211yescharacteristic
46#418C2211yescharacteristic
47#418C2211yescharacteristic
48#417C2211yescharacteristic
49#416C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#102C2211yescharacteristic
54C2×#100C2211yescharacteristic
55C2×#100C2211yescharacteristic
56#408C2211yescharacteristic
57#405C2211yescharacteristic
58#405C2211yescharacteristic
59C2×C4×C8C2211yescharacteristic
60#384C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×D4—41no—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71AGL(1,ℤ/8)—22no—
72AGL(1,ℤ/8)—22no—
73C2×SD16—21no—
74C2×D8—21no—
75C2×M16C2311yescharacteristic
76C2×M16C2311yescharacteristic
77C2×M16C2311yescharacteristic
78C2×M16C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83#127—21no—
84#127—22no—
85#127—22no—
86#125—21no—
87#125—21no—
88#124—21no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#120—41no—
93C4×Q8—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—41no—
100#117—21no—
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103#117C2311yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—41no—
109C2×C42C2311yescharacteristic
110#108—22no—
111#107—21no—
112#107—21no—
113#107—21no—
114#107D411yescharacteristic, complemented
115#107D411yescharacteristic, complemented
116#106—21no—
117#106—21no—
118#106—21no—
119#106D411yescharacteristic, complemented
120#106D411yescharacteristic, complemented
121C4⋊C8—21no—
122C4⋊C8—21no—
123#104—22no—
124#103—21no—
125#103D411yescharacteristic, complemented
126#103D411yescharacteristic, complemented
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102—21no—
137#102—21no—
138#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102D411yescharacteristic, complemented
143#102D411yescharacteristic, complemented
144#100—22no—
145#100—22no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148#97—21no—
149#97—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—22no—
153C24—21no—
154C24—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159C2×Q8—41no—
160C2×Q8C2×D411yescharacteristic
161C2×D4—21no—
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—44no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4C2×D411yescharacteristic
179C2×D4#4112yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2411yescharacteristic
188SD16—24no—
189D8—24no—
190M16—21no—
191M16—21no—
192M16—21no—
193M16—21no—
194M16—21no—
195M16—21no—
196M16—21no—
197M16#4112yes—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C2×C8C2×D411yescharacteristic
209C2×C8C2×D411yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4C2×D411yescharacteristic
220C4⋊C4C2×D411yescharacteristic
221C4⋊C4C2×D411yescharacteristic
222C4⋊C4C2×D411yescharacteristic
223C4⋊C4C2×D411yescharacteristic
224C4⋊C4C2×D411yescharacteristic
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—41no—
238C42#4112yes—
239C23—21no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23H(5,𝔽2)11yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—161no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—88no—
267D4—161no—
268D4—161no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—161no—
286C2×C4—161no—
287C2×C4—161no—
288C2×C4C22×D411yescharacteristic
289C2×C4C22×D411yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C42−1+411yescharacteristic
294C8—22no—
295C8—21no—
296C8—21no—
297C8—21no—
298C8—21no—
299C8—41no—
300C8—41no—
301C8—41no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—82no—
309C22—82no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—161no—
316C22—161no—
317C22#54211yescharacteristic
318C22D4211yescharacteristic
319C22#55111yescharacteristic
320C4—24no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4#52511yescharacteristic
331C4#53211yescharacteristic
332C4#54711yescharacteristic
333C4#54811yescharacteristic
334C2—21no—
335C2—21no—
336C2—81no—
337C2—82no—
338C2—161no—
339C2#246111yescharacteristic
340C2#329111yescharacteristic
341C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#2002C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3136C2semidirect
2. #3096C2semidirect
3. #3081C2semidirect
4. #2946C2semidirect
5. #2063C2semidirect
6. #2060C2semidirect
7. #2051C2semidirect
9. #1957C2semidirect
10. #1937C2semidirect
11. #1931C2semidirect
12. #1931C2semidirect
13. #1887C2semidirect
14. #1859C2semidirect
15. #1772C2semidirect
114. #107D4semidirect
115. #107D4semidirect
119. #106D4semidirect
120. #106D4semidirect
125. #103D4semidirect
126. #103D4semidirect
142. #102D4semidirect
143. #102D4semidirect

Maximal subgroups

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

subgroupindex
#31362
#30962
#30812
#29462
#20632
#20602
#20512
#20022
#19572
#19372
#19312
#18872
#18592
#17722