Finite Group Databasebeta

← #20305

Subgroups — #20305

Order 256.

302 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#3384C211yesmaximal, characteristic, complemented
2#3354C211yesmaximal, characteristic
3#3331C211yesmaximal, characteristic, complemented
4#3329C211yesmaximal, characteristic, complemented
5#1707C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1703C211yesmaximal, characteristic, complemented
8#1692C211yesmaximal, characteristic, complemented
9#1691C211yesmaximal, characteristic, complemented
10#1649C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1634C211yesmaximal, characteristic, complemented
13#1632C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#554C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#483—21no—
29#482—21no—
30#481—21no—
31#478—21no—
32#477C2211yescharacteristic
33#476—21no—
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#474C2211yescharacteristic
38#474C2211yescharacteristic
39#470—21no—
40#469—21no—
41#468—21no—
42#462C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#448—21no—
46#446C2211yescharacteristic
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#428—21no—
54#419—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×Q16—41no—
71C2×SD16—21no—
72C2×SD16D411yescharacteristic
73C2×D8D411yescharacteristic
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#125—21no—
83#124—41no—
84#123—41no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#121—21no—
93C4×Q8C2311yescharacteristic
94C4×D4—21no—
95C4×D4—21no—
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—41no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8C2×D411yescharacteristic
154C2×D4—41no—
155C2×D4—41no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—82no—
166Q16—81no—
167SD16—41no—
168SD16—81no—
169D8—22no—
170M16—42no—
171M16—81no—
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—21no—
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—21no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—81no—
220Q8—81no—
221Q8—82no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—82no—
229D4—161no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
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—82no—
247C2×C4—161no—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4H(5,𝔽2)11yescharacteristic
251C8—22no—
252C8—21no—
253C8—21no—
254C8—21no—
255C8—41no—
256C8—41no—
257C8—41no—
258C8—41no—
259C8#13712yes—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22D4211yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4C2×#13711yescharacteristic
281C4#57511yescharacteristic
282C4#57611yescharacteristic
283C4#57811yescharacteristic
284C2—81no—
285C2—81no—
286C2—81no—
287C2#327311yescharacteristic
288C2#327611yescharacteristic
289C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3354C2
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
1. #3384C2semidirect
3. #3331C2semidirect
4. #3329C2semidirect
5. #1707C2semidirect
6. #1703C2semidirect
7. #1703C2semidirect
8. #1692C2semidirect
9. #1691C2semidirect
10. #1649C2semidirect
11. #1647C2semidirect
12. #1634C2semidirect
13. #1632C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect

Maximal subgroups

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

subgroupindex
#33842
#33542
#33312
#33292
#17072
#17032
#16922
#16912
#16492
#16472
#16342
#16322
#15902
#15892