Finite Group Databasebeta

← #19870

Subgroups — #19870

Order 256.

302 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic
2#3393C211yesmaximal, characteristic, complemented
3#3342C211yesmaximal, characteristic, complemented
4#3304C211yesmaximal, characteristic, complemented
5#1722C211yesmaximal, characteristic, complemented
6#1690C211yesmaximal, characteristic, complemented
7#1681C211yesmaximal, characteristic, complemented
8#1673C211yesmaximal, characteristic, complemented
9#1666C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1650C211yesmaximal, characteristic, complemented
12#1642C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1584C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#556C2211yescharacteristic
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#496—21no—
23#495—21no—
24#494C2211yescharacteristic
25#492C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#485—21no—
30#482—21no—
31#475C2211yescharacteristic
32#475C2211yescharacteristic
33#475C2211yescharacteristic
34#467—21no—
35#465—21no—
36#464—21no—
37#464—21no—
38#463—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#440—21no—
49C4×Q16—21no—
50C4×Q16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#427—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16D411yescharacteristic
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×D8—21no—
78C2×D8D411yescharacteristic
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#126—41no—
87#126—41no—
88#126—41no—
89#124—21no—
90#124—21no—
91#124—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95#121—21no—
96#121—41no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106#117—21no—
107#117—21no—
108#107—21no—
109#107—21no—
110#107—41no—
111#107D411yescharacteristic
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115#106—21no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
129#103D411yescharacteristic
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—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×Q8—21no—
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161Q16—41no—
162Q16—42no—
163Q16—81no—
164SD16—42no—
165SD16—81no—
166SD16—81no—
167SD16—81no—
168SD16—81no—
169SD16—81no—
170D8—22no—
171D8—81no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—42no—
178C2×C8—81no—
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4—161no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8#13512yes—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22D4211yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4C2×#13511yescharacteristic
284C4#57511yescharacteristic
285C4#57611yescharacteristic
286C4#57711yescharacteristic
287C2—81no—
288C2—81no—
289C2—161no—
290C2#326711yescharacteristic
291C2#328711yescharacteristic
292C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3397C2
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. #3393C2semidirect
3. #3342C2semidirect
4. #3304C2semidirect
5. #1722C2semidirect
6. #1690C2semidirect
7. #1681C2semidirect
8. #1673C2semidirect
9. #1666C2semidirect
10. #1650C2semidirect
11. #1650C2semidirect
12. #1642C2semidirect
13. #1628C2semidirect
14. #1584C2semidirect
15. #1580C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33932
#33422
#33042
#17222
#16902
#16812
#16732
#16662
#16502
#16422
#16282
#15842
#15802