Finite Group Databasebeta

← #20299

Subgroups — #20299

Order 256.

304 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#3399C211yesmaximal, characteristic
2#3385C211yesmaximal, characteristic, complemented
3#3375C211yesmaximal, characteristic, complemented
4#3333C211yesmaximal, characteristic, complemented
5#1706C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1696C211yesmaximal, characteristic, complemented
9#1688C211yesmaximal, characteristic, complemented
10#1631C211yesmaximal, characteristic, complemented
11#1630C211yesmaximal, characteristic, complemented
12#1623C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1591C211yesmaximal, characteristic, complemented
15#1590C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17Q82C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#496—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#489—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#480—21no—
30#479—21no—
31#477C2211yescharacteristic
32#477C2211yescharacteristic
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#474—21no—
36#474—21no—
37#468—21no—
38#464—21no—
39#463—21no—
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#459—21no—
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49#442—21no—
50#440—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16C2211yescharacteristic
54#436C2211yescharacteristic
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×Q16D411yescharacteristic
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—41no—
73C2×SD16—41no—
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×M16—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—41no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#125—21no—
88#124—21no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93#121—21no—
94#121—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2311yescharacteristic
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107D411yescharacteristic
112#106—21no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103D411yescharacteristic
125#103D411yescharacteristic
126#103D411yescharacteristic
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102D411yescharacteristic
137C22⋊C8—21no—
138#97—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—41no—
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—41no—
166SD16—22no—
167SD16—42no—
168SD16—82no—
169SD16—82no—
170SD16—81no—
171SD16—81no—
172D8—42no—
173D8—81no—
174D8—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—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—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42C2411yescharacteristic
211C23—21no—
212C23—41no—
213C23—41no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—81no—
221Q8—81no—
222Q8—161no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—81no—
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—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—161no—
250C2×C4—161no—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C8—22no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8#13712yes—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22D4211yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—81no—
282C4—81no—
283C4C2×#13711yescharacteristic
284C4#57511yescharacteristic
285C4#57711yescharacteristic
286C4#57711yescharacteristic
287C2—81no—
288C2—161no—
289C2—161no—
290C2#327511yescharacteristic
291C2#327811yescharacteristic
292C2#329211yescharacteristic
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. #3385C2semidirect
3. #3375C2semidirect
4. #3333C2semidirect
5. #1706C2semidirect
6. #1698C2semidirect
7. #1697C2semidirect
8. #1696C2semidirect
9. #1688C2semidirect
10. #1631C2semidirect
11. #1630C2semidirect
12. #1623C2semidirect
13. #1622C2semidirect
14. #1591C2semidirect
15. #1590C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33852
#33752
#33332
#17062
#16982
#16972
#16962
#16882
#16312
#16302
#16232
#16222
#15912
#15902