Finite Group Databasebeta

← #19860

Subgroups — #19860

Order 256.

316 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#3360C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3335C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1691C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1673C211yesmaximal, characteristic, complemented
9#1670C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1643C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#488—21no—
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#484—21no—
29#483—21no—
30#482—21no—
31#481—21no—
32#479—21no—
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#475C2211yescharacteristic
37#474C2211yescharacteristic
38#468—21no—
39#467—21no—
40#465—21no—
41#463—21no—
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#461C2211yescharacteristic
45#452—21no—
46#452—21no—
47C8×Q8C2211yescharacteristic
48C4×SD16—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55C2×#106—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C22×C8—21no—
79C22×C8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#126—41no—
85#125—21no—
86#124—21no—
87#123—41no—
88#123—41no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—41no—
110#107D411yescharacteristic
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106D411yescharacteristic
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—41no—
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
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—42no—
166Q16—81no—
167SD16—22no—
168SD16—42no—
169SD16—42no—
170SD16—42no—
171D8—22no—
172M16—42no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C4⋊C4—21no—
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⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216Q8—21no—
217Q8—21no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—82no—
231D4—82no—
232D4AGL(1,ℤ/8)12yes—
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—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—161no—
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#13512yescomplemented
263C22—22no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22D4211yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4C2×#13511yescharacteristic
284C4C2×AGL(1,ℤ/8)11yescharacteristic
285C4#57711yescharacteristic
286C4#57811yescharacteristic
287C2—42no—
288C2—81no—
289C2—81no—
290C2#326511yescharacteristic
291C2#328311yescharacteristic
292C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3283C2
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. #3360C2semidirect
2. #3353C2semidirect
3. #3335C2semidirect
5. #1723C2semidirect
6. #1691C2semidirect
7. #1683C2semidirect
8. #1673C2semidirect
9. #1670C2semidirect
10. #1650C2semidirect
11. #1647C2semidirect
12. #1643C2semidirect
13. #1635C2semidirect
14. #1581C2semidirect
15. SD16⋊C8C2semidirect
262. C8#135semidirect

Maximal subgroups

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

subgroupindex
#33602
#33532
#33352
#32832
#17232
#16912
#16832
#16732
#16702
#16502
#16472
#16432
#16352
#15812
SD16⋊C82