Finite Group Databasebeta

← #19963

Subgroups — #19963

Order 256.

304 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic
2#3398C211yesmaximal, characteristic, complemented
3#3365C211yesmaximal, characteristic, complemented
4#3304C211yesmaximal, characteristic, complemented
5#1709C211yesmaximal, characteristic, complemented
6#1690C211yesmaximal, characteristic, complemented
7#1688C211yesmaximal, characteristic, complemented
8#1682C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1640C211yesmaximal, characteristic, complemented
12#1628C211yesmaximal, characteristic, complemented
13#1626C211yesmaximal, characteristic, complemented
14#1591C211yesmaximal, characteristic, complemented
15C8×SD16C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#557C2211yescharacteristic
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20#497—21no—
21#496—21no—
22#494C2211yescharacteristic
23#493C2211yescharacteristic
24#488—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#482—21no—
30#478—21no—
31#475C2211yescharacteristic
32#475C2211yescharacteristic
33#465—21no—
34#464—21no—
35#464—21no—
36#463—21no—
37#462—21no—
38#462C2211yescharacteristic
39#461C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#459—21no—
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#442—21no—
50#441—21no—
51C4×Q16C2211yescharacteristic
52C4×SD16—21no—
53C4×SD16C2211yescharacteristic
54C8×D4C2211yescharacteristic
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16D411yescharacteristic
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C22×C8—21no—
81C4⋊Q8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#126—41no—
88#125—21no—
89#125—41no—
90#124—21no—
91#124—21no—
92#124—21no—
93#123—41no—
94C22⋊Q8—21no—
95#121—21no—
96#121—41no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107#117—21no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106D411yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103D411yescharacteristic
124#103D411yescharacteristic
125#103D411yescharacteristic
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102—41no—
136#102D411yescharacteristic
137C22⋊C8—21no—
138#97—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163Q16—41no—
164Q16—42no—
165Q16—81no—
166Q16—81no—
167SD16—22no—
168SD16—42no—
169SD16—81no—
170SD16—81no—
171SD16—81no—
172D8—42no—
173D8—81no—
174D8—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—81no—
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—41no—
215C23—41no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226D4—21no—
227D4—21no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—81no—
237D4—81no—
238D4—82no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—161no—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8#13512yes—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22D4211yescharacteristic
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4C2×#13511yescharacteristic
286C4#57511yescharacteristic
287C4#57611yescharacteristic
288C4#57711yescharacteristic
289C2—81no—
290C2—161no—
291C2—161no—
292C2#327511yescharacteristic
293C2#328711yescharacteristic
294C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3399C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3398C2semidirect
3. #3365C2semidirect
4. #3304C2semidirect
5. #1709C2semidirect
6. #1690C2semidirect
7. #1688C2semidirect
8. #1682C2semidirect
9. #1681C2semidirect
10. #1642C2semidirect
11. #1640C2semidirect
12. #1628C2semidirect
13. #1626C2semidirect
14. #1591C2semidirect
15. C8×SD16C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33982
#33652
#33042
#17092
#16902
#16882
#16822
#16812
#16422
#16402
#16282
#16262
#15912
C8×SD162