Finite Group Databasebeta

← #19369

Subgroups — #19369

Order 256.

322 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3253C211yesmaximal, characteristic
2#3235C211yesmaximal, characteristic, complemented
3#3182C211yesmaximal, characteristic, complemented
4#2935C211yesmaximal, characteristic, complemented
5#1703C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1695C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1677C211yesmaximal, characteristic, complemented
11#1675C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13#1569C211yesmaximal, characteristic, complemented
14#1551C211yesmaximal, characteristic, complemented
15#1550C211yesmaximal, characteristic, complemented
16C2×#131—21no—
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19#532C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#492C2211yescharacteristic
23#490—21no—
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#482—21no—
27#482—21no—
28#476—21no—
29#476C2211yescharacteristic
30#476C2211yescharacteristic
31#476C2211yescharacteristic
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#469—21no—
36#469—21no—
37#468—21no—
38#468—21no—
39#459—21no—
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#440—21no—
43#440—21no—
44#436C2211yescharacteristic
45#436C2211yescharacteristic
46#435—21no—
47#435—21no—
48C8×D4C2211yescharacteristic
49C8×D4C2211yescharacteristic
50#432C2211yescharacteristic
51#423C2211yescharacteristic
52C2×C4⋊C8C2211yescharacteristic
53#408—21no—
54C4×M16C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#333C2211yescharacteristic
62#333C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×SD16—21no—
70C2×SD16—21no—
71C2×D8—21no—
72C2×D8—21no—
73#131—44no—
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—21no—
77C2×M16D411yescharacteristic
78C2×M16D411yescharacteristic
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8D411yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127D411yescharacteristic
87#127C2311yescharacteristic
88#125—21no—
89#124—21no—
90#124—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121D411yescharacteristic
97#121D411yescharacteristic
98C4×Q8—21no—
99C4×Q8—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106C2×C42C2311yescharacteristic
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141#97D411yescharacteristic
142#97D411yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—82no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4#4111yescharacteristic
168SD16—42no—
169SD16—42no—
170D8—81no—
171D8—81no—
172M16—22no—
173M16—41no—
174M16—42no—
175M16—42no—
176M16—81no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—41no—
209C42C2×D411yescharacteristic
210C42C2×D411yescharacteristic
211C42#4111yescharacteristic
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216C23#12111yescharacteristic
217Q8—42no—
218Q8—41no—
219Q8—41no—
220Q8—81no—
221Q8—81no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—42no—
227D4—42no—
228D4—42no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—81no—
234D4—81no—
235D4—82no—
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—41no—
250C2×C4—82no—
251C2×C4#12111yescharacteristic
252C2×C4#12111yescharacteristic
253C2×C4#12111yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C8—21no—
258C8—21no—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8AGL(1,ℤ/8)12yes—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22C2×#12111yescharacteristic
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4C2×AGL(1,ℤ/8)11yescharacteristic
290C4#57511yescharacteristic
291C4#57711yescharacteristic
292C4#57811yescharacteristic
293C2—41no—
294C2—81no—
295C2—81no—
296C2—161no—
297C2#303611yescharacteristic
298C2#306311yescharacteristic
299C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3253C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3235C2semidirect
3. #3182C2semidirect
4. #2935C2semidirect
5. #1703C2semidirect
6. #1697C2semidirect
7. #1695C2semidirect
8. #1685C2semidirect
9. #1683C2semidirect
10. #1677C2semidirect
11. #1675C2semidirect
12. #1665C2semidirect
13. #1569C2semidirect
14. #1551C2semidirect
15. #1550C2semidirect

Maximal subgroups

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

subgroupindex
#32532
#32352
#31822
#29352
#17032
#16972
#16952
#16852
#16832
#16772
#16752
#16652
#15692
#15512
#15502