Finite Group Databasebeta

← #20154

Subgroups — #20154

Order 256.

314 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3396C211yesmaximal, characteristic, complemented
3#3361C211yesmaximal, characteristic, complemented
4#3333C211yesmaximal, characteristic, complemented
5#1721C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1687C211yesmaximal, characteristic, complemented
8#1678C211yesmaximal, characteristic, complemented
9#1668C211yesmaximal, characteristic, complemented
10#1630C211yesmaximal, characteristic, complemented
11#1622C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1620C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1587C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#559C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#496—21no—
21#495—21no—
22#493C2211yescharacteristic
23#493C2211yescharacteristic
24#493C2211yescharacteristic
25#492C2211yescharacteristic
26#488—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#480—21no—
30#478—21no—
31#477C2211yescharacteristic
32#477C2211yescharacteristic
33#468—21no—
34#463—21no—
35#463—21no—
36#463—21no—
37#463—21no—
38#460C2211yescharacteristic
39#460C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#459—21no—
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#442—21no—
50C4×Q16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52C4×D8—21no—
53C4×D8—21no—
54#436C2211yescharacteristic
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#331C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16D411yescharacteristic
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—41no—
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—41no—
82C2×M16—21no—
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#127C2311yescharacteristic
88#125—21no—
89#124—21no—
90#124—21no—
91#124—21no—
92#124—21no—
93#123—41no—
94#123—41no—
95C22⋊Q8—21no—
96C22⋊Q8—41no—
97#121—21no—
98#121—41no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108C4×D4C2311yescharacteristic
109C2×C4⋊C4—21no—
110#107—21no—
111#107—21no—
112#107D411yescharacteristic
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#103D411yescharacteristic
124#103D411yescharacteristic
125#103D411yescharacteristic
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—
135#102—21no—
136#102D411yescharacteristic
137C22⋊C8—21no—
138#97—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8C2×D411yescharacteristic
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—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×D4—41no—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—41no—
168SD16—41no—
169SD16—82no—
170SD16—81no—
171SD16—81no—
172SD16—81no—
173SD16—81no—
174D8—42no—
175D8—42no—
176D8—42no—
177D8—81no—
178D8—81no—
179D8—82no—
180D8—81no—
181M16—81no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
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—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42C2411yescharacteristic
215C23—21no—
216C23—41no—
217C23—41no—
218C23—41no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—21no—
223Q8—21no—
224Q8—21no—
225Q8—81no—
226D4—21no—
227D4—21no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—81no—
241D4—81no—
242D4—81no—
243D4—82no—
244D4—161no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—161no—
257C2×C4—161no—
258C2×C4—161no—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C8—22no—
263C8—21no—
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22D4211yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4#57511yescharacteristic
294C4#57511yescharacteristic
295C4#57611yescharacteristic
296C4#57711yescharacteristic
297C2—81no—
298C2—161no—
299C2—161no—
300C2—161no—
301C2#326811yescharacteristic
302C2#327111yescharacteristic
303C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
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. #3399C2semidirect
2. #3396C2semidirect
3. #3361C2semidirect
4. #3333C2semidirect
5. #1721C2semidirect
6. #1698C2semidirect
7. #1687C2semidirect
8. #1678C2semidirect
9. #1668C2semidirect
10. #1630C2semidirect
11. #1622C2semidirect
12. #1621C2semidirect
13. #1620C2semidirect
14. #1588C2semidirect
15. #1587C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33962
#33612
#33332
#17212
#16982
#16872
#16782
#16682
#16302
#16222
#16212
#16202
#15882
#15872