Finite Group Databasebeta

← #19383

Subgroups — #19383

Order 256.

303 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#3251C211yesmaximal, characteristic
2#3218C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic, complemented
4#2969C211yesmaximal, characteristic, complemented
5#1702C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1700C211yesmaximal, characteristic, complemented
8#1699C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1681C211yesmaximal, characteristic, complemented
11#1680C211yesmaximal, characteristic, complemented
12#1679C211yesmaximal, characteristic, complemented
13#1569C211yesmaximal, characteristic, complemented
14#1553C211yesmaximal, characteristic, complemented
15#1546C211yesmaximal, characteristic, complemented
16#532C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18#529C2211yescharacteristic
19#497—21no—
20#491—21no—
21#489—21no—
22#489—21no—
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#478—21no—
28#475C2211yescharacteristic
29#474C2211yescharacteristic
30#469—21no—
31#468—21no—
32#463—21no—
33#462C2211yescharacteristic
34#462C2211yescharacteristic
35#461C2211yescharacteristic
36#461C2211yescharacteristic
37#460C2211yescharacteristic
38#459C2211yescharacteristic
39#451—21no—
40#448—21no—
41#446C2211yescharacteristic
42C8×Q8C2211yescharacteristic
43#436—21no—
44#436C2211yescharacteristic
45#435—21no—
46C8×D4C2211yescharacteristic
47#433—21no—
48#432C2211yescharacteristic
49#431C2211yescharacteristic
50#423C2211yescharacteristic
51#417—21no—
52#417—21no—
53#405—21no—
54C4×M16C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×D8—21no—
75C2×M16—21no—
76C2×M16—21no—
77C2×M16D411yescharacteristic
78C2×M16D411yescharacteristic
79C22×C8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127C2311yescharacteristic
87#126—21no—
88#125D411yescharacteristic
89#124—21no—
90#124D411yescharacteristic
91#123—21no—
92C22⋊Q8D411yescharacteristic
93#121—21no—
94#121—21no—
95#121D411yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4C2311yescharacteristic
101#117—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C42C2311yescharacteristic
105#107—21no—
106#106—21no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8C2311yescharacteristic
111C4⋊C8C2311yescharacteristic
112C4⋊C8C2311yescharacteristic
113C4⋊C8C2311yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116#103—21no—
117#103—21no—
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
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—
132C22⋊C8—21no—
133C22⋊C8—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136#97—21no—
137#97—21no—
138#97D411yescharacteristic
139#97D411yescharacteristic
140C4×C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4#4111yescharacteristic
165Q16—81no—
166Q16—81no—
167SD16—81no—
168SD16—81no—
169SD16—81no—
170D8—81no—
171M16—41no—
172M16—41no—
173M16—42no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42C2×D411yescharacteristic
209C42C2×D411yescharacteristic
210C42#4111yescharacteristic
211C42C2411yescharacteristic
212C23—21no—
213C23—41no—
214C23#12111yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—82no—
224D4—21no—
225D4—21no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—81no—
231D4—81no—
232D4—82no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4#12111yescharacteristic
249C2×C4#12111yescharacteristic
250C2×C4#12111yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22C2×#12111yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#57511yescharacteristic
285C4#57511yescharacteristic
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—41no—
289C2—81no—
290C2—161no—
291C2#304011yescharacteristic
292C2#306211yescharacteristic
293C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3251C2
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. #3218C2semidirect
3. #3141C2semidirect
4. #2969C2semidirect
5. #1702C2semidirect
6. #1701C2semidirect
7. #1700C2semidirect
8. #1699C2semidirect
9. #1682C2semidirect
10. #1681C2semidirect
11. #1680C2semidirect
12. #1679C2semidirect
13. #1569C2semidirect
14. #1553C2semidirect
15. #1546C2semidirect

Maximal subgroups

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

subgroupindex
#32512
#32182
#31412
#29692
#17022
#17012
#17002
#16992
#16822
#16812
#16802
#16792
#15692
#15532
#15462