Finite Group Databasebeta

← #20129

Subgroups — #20129

Order 256.

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

normal subgroups only

This group has 279 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#3398C211yesmaximal, characteristic
2#3374C211yesmaximal, characteristic, complemented
3#3364C211yesmaximal, characteristic
4#3342C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1694C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic
9#1670C211yesmaximal, characteristic, complemented
10#1653C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic
12#1650C211yesmaximal, characteristic
13#1648C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic
16#557C2211yescharacteristic
17#557C2211yescharacteristic
18#556C2211yescharacteristic
19#556C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#497—21no—
24#494C2211yescharacteristic
25#491—21no—
26#488—21no—
27#487C2211yescharacteristic
28#487C2211yescharacteristic
29#485—21no—
30#485—21no—
31#479—21no—
32#479—21no—
33#475C2211yescharacteristic
34#475C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#467—21no—
38#464—21no—
39#464—21no—
40#462—21no—
41#462—21no—
42#462C2211yescharacteristic
43#462C2211yescharacteristic
44#462C2211yescharacteristic
45#461C2211yescharacteristic
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#441—21no—
50#441—21no—
51C4×Q16—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16C2211yescharacteristic
54C8×D4C2211yescharacteristic
55#427—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×Q16—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16D411yescharacteristic
73C2×SD16—21no—
74C2×SD16D411yescharacteristic
75C22×C8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#126—41no—
83#126—41no—
84#125—21no—
85#125—21no—
86#125—41no—
87#125—41no—
88#124—21no—
89#124—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—41no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—21no—
112#106—41no—
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#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103D411yescharacteristic
133#103D411yescharacteristic
134#103D411yescharacteristic
135#102—21no—
136#102D411yescharacteristic
137C22⋊C8—21no—
138#97—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×D4C2×D411yescharacteristic
152C22×C4—21no—
153Q16—41no—
154Q16—42no—
155Q16—42no—
156Q16—42no—
157Q16—81no—
158Q16—81no—
159SD16—41no—
160SD16—81no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8C2×D411yescharacteristic
168C2×C8C2×D411yescharacteristic
169C2×C8C2×D411yescharacteristic
170C2×C8C2×D411yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4C2×D411yescharacteristic
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—41no—
206C42C2411yescharacteristic
207C23—21no—
208Q8—21no—
209Q8—21no—
210Q8—21no—
211Q8—21no—
212Q8—21no—
213Q8—21no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222D4—21no—
223D4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4H(5,𝔽2)11yescharacteristic
240C8—22no—
241C8—21no—
242C8—21no—
243C8—21no—
244C8—21no—
245C8—41no—
246C8—41no—
247C8—41no—
248C8—41no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22D4211yescharacteristic
253C4—21no—
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4#57511yescharacteristic
266C4#57511yescharacteristic
267C4#57711yescharacteristic
268C4#57811yescharacteristic
269C2—81no—
270C2#326911yescharacteristic
271C2#327111yescharacteristic
272C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalC2×D4C2×D4
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. #3374C2semidirect
5. #1723C2semidirect
6. #1700C2semidirect
7. #1694C2semidirect
9. #1670C2semidirect
10. #1653C2semidirect
13. #1648C2semidirect
14. #1587C2semidirect

Maximal subgroups

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

subgroupindex
#33982
#33742
#33642
#33422
#17232
#17002
#16942
#16792
#16702
#16532
#16522
#16502
#16482
#15872
#15852