Finite Group Databasebeta

← #20307

Subgroups — #20307

Order 256.

304 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic
2#3384C211yesmaximal, characteristic, complemented
3#3333C211yesmaximal, characteristic, complemented
4#3328C211yesmaximal, characteristic, complemented
5#1708C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1692C211yesmaximal, characteristic, complemented
9#1686C211yesmaximal, characteristic, complemented
10#1643C211yesmaximal, characteristic, complemented
11#1641C211yesmaximal, characteristic, complemented
12#1638C211yesmaximal, characteristic, complemented
13#1627C211yesmaximal, characteristic, complemented
14#1591C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#501—21no—
21#498C2211yescharacteristic
22#496—21no—
23#492C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#480—21no—
30#478—21no—
31#477C2211yescharacteristic
32#476—21no—
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#474C2211yescharacteristic
36#470—21no—
37#468—21no—
38#463—21no—
39#462C2211yescharacteristic
40#461C2211yescharacteristic
41#460—21no—
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#451—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#442—21no—
49#440—21no—
50C4×Q16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#419—21no—
55#417—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×Q16D411yescharacteristic
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×D8—21no—
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C4⋊Q8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127—21no—
84#127C2311yescharacteristic
85#126—41no—
86#125—21no—
87#124—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93#121—21no—
94#121—41no—
95C4×Q8—21no—
96C4×Q8C2311yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103#117—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107D411yescharacteristic
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—21no—
112#106—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#103—21no—
124#103—21no—
125#103—41no—
126#103D411yescharacteristic
127#103D411yescharacteristic
128#102—21no—
129#102—21no—
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—
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—41no—
150C2×Q8—42no—
151C2×Q8—81no—
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166Q16—41no—
167Q16—81no—
168SD16—42no—
169SD16—81no—
170SD16—81no—
171SD16—81no—
172D8—22no—
173D8—81no—
174M16—81no—
175M16—81no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
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—21no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42C2411yescharacteristic
212C23—21no—
213C23—21no—
214C23—41no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—81no—
224Q8—82no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—81no—
234D4—82no—
235D4—161no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—161no—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8#13712yes—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22D4211yescharacteristic
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4C2×#13711yescharacteristic
285C4#57511yescharacteristic
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—81no—
289C2—81no—
290C2—161no—
291C2#327311yescharacteristic
292C2#327811yescharacteristic
293C2#329011yescharacteristic
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. #3384C2semidirect
3. #3333C2semidirect
4. #3328C2semidirect
5. #1708C2semidirect
6. #1699C2semidirect
7. #1697C2semidirect
8. #1692C2semidirect
9. #1686C2semidirect
10. #1643C2semidirect
11. #1641C2semidirect
12. #1638C2semidirect
13. #1627C2semidirect
14. #1591C2semidirect
15. #1589C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33842
#33332
#33282
#17082
#16992
#16972
#16922
#16862
#16432
#16412
#16382
#16272
#15912
#15892