Finite Group Databasebeta

← #20253

Subgroups — #20253

Order 256.

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

normal subgroups only

This group has 296 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#3366C211yesmaximal, characteristic
2#3360C211yesmaximal, characteristic, complemented
3#3356C211yesmaximal, characteristic, complemented
4#3333C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1704C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1678C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1632C211yesmaximal, characteristic, complemented
12#1632C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1584C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#498C2211yescharacteristic
21#494C2211yescharacteristic
22#490—21no—
23#488—21no—
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#483—21no—
30#480—21no—
31#479—21no—
32#478—21no—
33#477C2211yescharacteristic
34#477C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#475C2211yescharacteristic
38#470—21no—
39#468—21no—
40#463—21no—
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#459—21no—
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#442—21no—
49C4×Q16C2211yescharacteristic
50C4×SD16—21no—
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#428—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—41no—
69C2×Q16D411yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×D8—21no—
73C2×D8D411yescharacteristic
74C2×M16—21no—
75C2×M16—21no—
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79#127—21no—
80#127C2311yescharacteristic
81#126—41no—
82#125—21no—
83#125—21no—
84#124—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90#121—21no—
91#121—41no—
92C4×Q8—21no—
93C4×Q8C2311yescharacteristic
94C4×Q8C2311yescharacteristic
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107D411yescharacteristic
106#107D411yescharacteristic
107#106—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
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—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163Q16—41no—
164Q16—82no—
165SD16—42no—
166SD16—81no—
167D8—41no—
168D8—42no—
169M16—42no—
170M16—81no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8C2×D411yescharacteristic
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—21no—
205C42—21no—
206C42—41no—
207C42C2411yescharacteristic
208C23—21no—
209C23—21no—
210C23—41no—
211Q8—21no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—81no—
218Q8—81no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—81no—
228D4—82no—
229D4—161no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—82no—
244C2×C4—161no—
245C2×C4—161no—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—41no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22D4211yescharacteristic
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4#57511yescharacteristic
279C4#57511yescharacteristic
280C4#57711yescharacteristic
281C4#57811yescharacteristic
282C2—81no—
283C2—81no—
284C2—161no—
285C2#326711yescharacteristic
286C2#327711yescharacteristic
287C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3366C2
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
2. #3360C2semidirect
3. #3356C2semidirect
4. #3333C2semidirect
5. #1720C2semidirect
6. #1704C2semidirect
7. #1697C2semidirect
8. #1691C2semidirect
9. #1678C2semidirect
10. #1642C2semidirect
11. #1632C2semidirect
12. #1632C2semidirect
13. #1628C2semidirect
14. #1594C2semidirect
15. #1584C2semidirect

Maximal subgroups

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

subgroupindex
#33662
#33602
#33562
#33332
#17202
#17042
#16972
#16912
#16782
#16422
#16322
#16282
#15942
#15842