Finite Group Databasebeta

← #20249

Subgroups — #20249

Order 256.

299 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic
2#3380C211yesmaximal, characteristic, complemented
3#3356C211yesmaximal, characteristic, complemented
4#3333C211yesmaximal, characteristic, complemented
5#1714C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1696C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1666C211yesmaximal, characteristic, complemented
10#1643C211yesmaximal, characteristic, complemented
11#1638C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1584C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#500C2211yescharacteristic
21#499—21no—
22#498C2211yescharacteristic
23#496—21no—
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#487C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#483—21no—
30#480—21no—
31#478—21no—
32#478—21no—
33#477C2211yescharacteristic
34#477C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#475C2211yescharacteristic
38#470—21no—
39#468—21no—
40#463—21no—
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#440—21no—
49C4×Q16C2211yescharacteristic
50C4×D8—21no—
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#428—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16D411yescharacteristic
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—41no—
74C2×D8—21no—
75C2×D8D411yescharacteristic
76C2×M16—21no—
77C2×M16—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#127—21no—
82#127C2311yescharacteristic
83#126—41no—
84#125—21no—
85#125—21no—
86#124—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92#121—21no—
93#121—41no—
94C4×Q8—21no—
95C4×Q8C2311yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107D411yescharacteristic
110#107D411yescharacteristic
111#106—21no—
112#106—21no—
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—41no—
127#103D411yescharacteristic
128#103D411yescharacteristic
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
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—
161C22×C4—41no—
162C22×C4—41no—
163Q16—41no—
164SD16—42no—
165SD16—81no—
166SD16—81no—
167SD16—81no—
168SD16—82no—
169D8—22no—
170D8—81no—
171M16—42no—
172M16—81no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—41no—
213Q8—21no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—41no—
218Q8—41no—
219Q8—81no—
220Q8—81no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—81no—
230D4—82no—
231D4—161no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—82no—
246C2×C4—161no—
247C2×C4—161no—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4H(5,𝔽2)11yescharacteristic
251C8—21no—
252C8—21no—
253C8—21no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—41no—
258C8—41no—
259C8#13712yes—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22D4211yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4C2×#13711yescharacteristic
281C4#57511yescharacteristic
282C4#57711yescharacteristic
283C4#57711yescharacteristic
284C2—81no—
285C2—81no—
286C2—161no—
287C2#326711yescharacteristic
288C2#327811yescharacteristic
289C2#328811yescharacteristic
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. #3380C2semidirect
3. #3356C2semidirect
4. #3333C2semidirect
5. #1714C2semidirect
6. #1697C2semidirect
7. #1696C2semidirect
8. #1691C2semidirect
9. #1666C2semidirect
10. #1643C2semidirect
11. #1638C2semidirect
12. #1631C2semidirect
13. #1622C2semidirect
14. #1594C2semidirect
15. #1584C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33802
#33562
#33332
#17142
#16972
#16962
#16912
#16662
#16432
#16382
#16312
#16222
#15942
#15842