Finite Group Databasebeta

← #19744

Subgroups — #19744

Order 256.

335 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3231C211yesmaximal, characteristic, complemented
2#3230C211yesmaximal, characteristic, complemented
3#3228C211yesmaximal, characteristic
4#3226C211yesmaximal, characteristic
5#3210C211yesmaximal, characteristic
6#3209C211yesmaximal, characteristic, complemented
7#1723C211yesmaximal, characteristic, complemented
8#1723C211yesmaximal, characteristic, complemented
9#1722C211yesmaximal, characteristic, complemented
10#1720C211yesmaximal, characteristic, complemented
11#1719C211yesmaximal, characteristic, complemented
12#1718C211yesmaximal, characteristic, complemented
13#1713C211yesmaximal, characteristic, complemented
14#1711C211yesmaximal, characteristic, complemented
15#1454C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#500C2211yescharacteristic
24#500C2211yescharacteristic
25#500C2211yescharacteristic
26#498C2211yescharacteristic
27#498C2211yescharacteristic
28#494C2211yescharacteristic
29#494C2211yescharacteristic
30#494C2211yescharacteristic
31#492C2211yescharacteristic
32#487C2211yescharacteristic
33#487C2211yescharacteristic
34#487C2211yescharacteristic
35#487C2211yescharacteristic
36#487C2211yescharacteristic
37#487C2211yescharacteristic
38#487C2211yescharacteristic
39#486C2211yescharacteristic
40#486C2211yescharacteristic
41#486C2211yescharacteristic
42#486C2211yescharacteristic
43#486C2211yescharacteristic
44#485—22no—
45#484—22no—
46#482—22no—
47#480—22no—
48#479—22no—
49#478—22no—
50#477—22no—
51#474—22no—
52#464—22no—
53#462—22no—
54#452—22no—
55#451—22no—
56#431C2211yescharacteristic
57#431C2211yescharacteristic
58#431C2211yescharacteristic
59#322C2211yescharacteristic
60#322C2211yescharacteristic
61#322C2211yescharacteristic
62C82C2211yescharacteristic
63C2×#41—21no—
64C22×Q8—21no—
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×SD16—22no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2311yescharacteristic
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79#127C2311yescharacteristic
80#125—21no—
81#125—21no—
82#124—21no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87#121—21no—
88#121—21no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×D4—21no—
94C4×D4—21no—
95#117—21no—
96#117—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C42C2311yescharacteristic
100#107—22no—
101#107—22no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#106—22no—
107#106—22no—
108C4⋊C8D412yes—
109C4⋊C8D412yes—
110C4⋊C8D412yes—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123C22⋊C8D412yes—
124C22⋊C8D412yes—
125C22⋊C8D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132#41—82no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—41no—
141C2×Q8—82no—
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—82no—
152Q16—82no—
153Q16—82no—
154SD16—82no—
155C2×C8C2×D412yes—
156C2×C8C2×D412yes—
157C2×C8C2×D412yes—
158C2×C8C2×D412yes—
159C2×C8C2×D412yes—
160C2×C8C2×D412yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
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—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C42—41no—
186C42—41no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2411yescharacteristic
191C23—41no—
192C23#12711yescharacteristic
193Q8—42no—
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—82no—
201D4—42no—
202D4—42no—
203D4—42no—
204D4—82no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#12711yescharacteristic
218C2×C4#12711yescharacteristic
219C2×C4#12711yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C8—22no—
224C8—22no—
225C8—22no—
226C8—22no—
227C8—22no—
228C8—22no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22C2×#12711yescharacteristic
236C4—21no—
237C4—21no—
238C4—21no—
239C4—81no—
240C4—81no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4#57612yes—
247C4#57812yes—
248C4#57812yes—
249C2—41no—
250C2—161no—
251C2#315511yescharacteristic
252C2#315611yescharacteristic
253C2#315611yescharacteristic
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. #3231C2semidirect
2. #3230C2semidirect
6. #3209C2semidirect
7. #1723C2semidirect
8. #1723C2semidirect
9. #1722C2semidirect
10. #1720C2semidirect
11. #1719C2semidirect
12. #1718C2semidirect
13. #1713C2semidirect
14. #1711C2semidirect
15. #1454C2semidirect

Maximal subgroups

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

subgroupindex
#32312
#32302
#32282
#32262
#32102
#32092
#17232
#17222
#17202
#17192
#17182
#17132
#17112
#14542