Finite Group Databasebeta

← #19752

Subgroups — #19752

Order 256.

353 conjugacy classes of subgroups.

The subgroup lattice has 257 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3185C211yesmaximal, characteristic, complemented
2#3182C211yesmaximal, characteristic
3#3182C211yesmaximal, characteristic
4#3165C211yesmaximal, characteristic, complemented
5#3141C211yesmaximal, characteristic
6#3139C211yesmaximal, characteristic, complemented
7#1723C211yesmaximal, characteristic, complemented
8#1720C211yesmaximal, characteristic, complemented
9#1719C211yesmaximal, characteristic, complemented
10#1718C211yesmaximal, characteristic, complemented
11#1718C211yesmaximal, characteristic, complemented
12#1714C211yesmaximal, characteristic, complemented
13#1710C211yesmaximal, characteristic, complemented
14#1708C211yesmaximal, characteristic, complemented
15#1455C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#501—22no—
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#500C2211yescharacteristic
24#500C2211yescharacteristic
25#498C2211yescharacteristic
26#498C2211yescharacteristic
27#498C2211yescharacteristic
28#494C2211yescharacteristic
29#492C2211yescharacteristic
30#492C2211yescharacteristic
31#491—22no—
32#489—22no—
33#487C2211yescharacteristic
34#487C2211yescharacteristic
35#487C2211yescharacteristic
36#487C2211yescharacteristic
37#487C2211yescharacteristic
38#487C2211yescharacteristic
39#487C2211yescharacteristic
40#487C2211yescharacteristic
41#487C2211yescharacteristic
42#487C2211yescharacteristic
43#486C2211yescharacteristic
44#486C2211yescharacteristic
45#486C2211yescharacteristic
46#486C2211yescharacteristic
47#470—22no—
48#468—22no—
49#468—22no—
50#441—22no—
51#440—22no—
52#440—22no—
53#428—22no—
54#419—22no—
55#417—22no—
56C4×M16C2211yescharacteristic
57C4×M16C2211yescharacteristic
58C4×M16C2211yescharacteristic
59#322C2211yescharacteristic
60#322C2211yescharacteristic
61#322C2211yescharacteristic
62C82C2211yescharacteristic
63C2×#41—21no—
64C22×Q8—21no—
65C2×Q16—22no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×M16D412yes—
69C2×M16D412yes—
70C2×M16D412yes—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8—21no—
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#127C2311yescharacteristic
82#125—21no—
83#125—21no—
84#124—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89#121—21no—
90#121—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×D4—21no—
96C4×D4—21no—
97#117—21no—
98#117—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C42C2311yescharacteristic
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#106—22no—
107#106—22no—
108#106—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#97D412yes—
124#97D412yes—
125#97D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132#41—44no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—44no—
141C2×Q8—41no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151Q16—44no—
152SD16—44no—
153SD16—44no—
154M16—24no—
155M16—42no—
156M16—42no—
157C2×C8C2×D412yes—
158C2×C8C2×D412yes—
159C2×C8C2×D412yes—
160C2×C8C2×D412yes—
161C2×C8C2×D412yes—
162C2×C8C2×D412yes—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2411yescharacteristic
193C23—41no—
194C23#12711yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—82no—
203D4—42no—
204D4—42no—
205D4—42no—
206D4—82no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4#12711yescharacteristic
220C2×C4#12711yescharacteristic
221C2×C4#12711yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C8—22no—
226C8—22no—
227C8—22no—
228C8—22no—
229C8—22no—
230C8#13714yes—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22C2×#12711yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4—81no—
247C4—81no—
248C4C2×#13712yes—
249C4#57512yes—
250C4#57512yes—
251C2—41no—
252C2—161no—
253C2#315211yescharacteristic
254C2#315311yescharacteristic
255C2#315311yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
prime radical#532C22
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. #3185C2semidirect
4. #3165C2semidirect
6. #3139C2semidirect
7. #1723C2semidirect
8. #1720C2semidirect
9. #1719C2semidirect
10. #1718C2semidirect
11. #1718C2semidirect
12. #1714C2semidirect
13. #1710C2semidirect
14. #1708C2semidirect
15. #1455C2semidirect

Maximal subgroups

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

subgroupindex
#31852
#31822
#31652
#31412
#31392
#17232
#17202
#17192
#17182
#17142
#17102
#17082
#14552