Finite Group Databasebeta

← #19412

Subgroups — #19412

Order 256.

325 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3207C211yesmaximal, characteristic, complemented
2#3115C211yesmaximal, characteristic, complemented
3#3106C211yesmaximal, characteristic, complemented
4#2988C211yesmaximal, characteristic, complemented
5#1684C211yesmaximal, characteristic, complemented
6#1683C211yesmaximal, characteristic, complemented
7#1678C211yesmaximal, characteristic, complemented
8#1677C211yesmaximal, characteristic, complemented
9#1668C211yesmaximal, characteristic, complemented
10#1667C211yesmaximal, characteristic, complemented
11#1666C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13#1572C211yesmaximal, characteristic, complemented
14#1551C211yesmaximal, characteristic, complemented
15#1540C211yesmaximal, characteristic, complemented
16#532C2211yescharacteristic
17#532C2211yescharacteristic
18C4×#41C2211yescharacteristic
19#493C2211yescharacteristic
20#492C2211yescharacteristic
21#487C2211yescharacteristic
22#486C2211yescharacteristic
23#482—21no—
24#482—21no—
25#477C2211yescharacteristic
26#476C2211yescharacteristic
27#475C2211yescharacteristic
28#474C2211yescharacteristic
29#463—21no—
30#463—21no—
31#460C2211yescharacteristic
32#460C2211yescharacteristic
33#459C2211yescharacteristic
34#459C2211yescharacteristic
35#449—21no—
36#449—21no—
37#449—21no—
38#449—21no—
39C8×Q8C2211yescharacteristic
40C8×Q8C2211yescharacteristic
41#442—21no—
42#440—21no—
43C4×SD16—21no—
44C4×D8—21no—
45C8×D4—21no—
46C8×D4—21no—
47C8×D4C2211yescharacteristic
48C8×D4C2211yescharacteristic
49#432C2211yescharacteristic
50#431—21no—
51#431C2211yescharacteristic
52#431C2211yescharacteristic
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#331C2211yescharacteristic
64#331C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41D411yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×D8—21no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#124—21no—
86#124—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91C4×Q8D411yescharacteristic
92C4×Q8C2311yescharacteristic
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4D411yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102#117D411yescharacteristic
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105#107—21no—
106#107—21no—
107#106—21no—
108#106—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8D411yescharacteristic
113C4⋊C8D411yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#102—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—
136C22⋊C8D411yescharacteristic
137C22⋊C8D411yescharacteristic
138#97—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—22no—
145#41—22no—
146#41—81no—
147#41—81no—
148#41—82no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
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×D4—41no—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4#4111yescharacteristic
170SD16—42no—
171SD16—42no—
172SD16—81no—
173SD16—81no—
174D8—42no—
175D8—42no—
176D8—81no—
177D8—81no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42C2×D411yescharacteristic
210C42C2×D411yescharacteristic
211C42#4111yescharacteristic
212C42C2411yescharacteristic
213C23—21no—
214C23—41no—
215C23—41no—
216C23#12111yescharacteristic
217Q8—21no—
218Q8—41no—
219Q8—41no—
220Q8—81no—
221Q8#13512yes—
222D4—24no—
223D4—21no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—81no—
233D4—81no—
234D4—82no—
235D4—82no—
236D4#13512yes—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#12111yescharacteristic
254C2×C4#12111yescharacteristic
255C2×C4#12111yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C8—21no—
260C8—21no—
261C8—21no—
262C8—21no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#12111yescharacteristic
282C4—21no—
283C4—21no—
284C4—22no—
285C4—21no—
286C4—21no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4C2×#13511yescharacteristic
292C4C2×#13511yescharacteristic
293C4#57611yescharacteristic
294C4#57711yescharacteristic
295C2—41no—
296C2—42no—
297C2—161no—
298C2—161no—
299C2#304511yescharacteristic
300C2#305011yescharacteristic
301C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
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
1. #3207C2semidirect
2. #3115C2semidirect
3. #3106C2semidirect
4. #2988C2semidirect
5. #1684C2semidirect
6. #1683C2semidirect
7. #1678C2semidirect
8. #1677C2semidirect
9. #1668C2semidirect
10. #1667C2semidirect
11. #1666C2semidirect
12. #1665C2semidirect
13. #1572C2semidirect
14. #1551C2semidirect
15. #1540C2semidirect

Maximal subgroups

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

subgroupindex
#32072
#31152
#31062
#29882
#16842
#16832
#16782
#16772
#16682
#16672
#16662
#16652
#15722
#15512
#15402