Finite Group Databasebeta

← #30390

Subgroups — #30390

Order 256.

293 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2647C211yesmaximal, characteristic, complemented
2#2644C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2087C211yesmaximal, characteristic, complemented
7#2083C211yesmaximal, characteristic, complemented
8#2071C211yesmaximal, characteristic
9#2067C211yesmaximal, characteristic
10#2067C211yesmaximal, characteristic
11#1944C211yesmaximal, characteristic, complemented
12#1942C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic
14#1934C211yesmaximal, characteristic
15#1768C211yesmaximal, characteristic, complemented
16C2×#107C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—21no—
29#400—21no—
30#398—21no—
31#398—21no—
32#397—21no—
33#397—21no—
34#397—21no—
35#396—21no—
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#384C2211yescharacteristic
47#382C2211yescharacteristic
48#376C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89#107—22no—
90#107—41no—
91#107—41no—
92#106—22no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95#102—22no—
96#102—22no—
97#102—41no—
98#102—41no—
99#97—41no—
100C4×C8—41no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95D411yescharacteristic
111#95D411yescharacteristic
112#95D411yescharacteristic
113#95D411yescharacteristic
114C24—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2411yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8#4112yes—
146C2×C8#4112yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C22⋊C4—81no—
171C22⋊C4—81no—
172C22⋊C4—81no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23C22×D411yescharacteristic
190D4—41no—
191D4—41no—
192D4—41no—
193D4—41no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4C2×#4111yescharacteristic
234C2×C4C2×#4111yescharacteristic
235C2×C4H(5,𝔽2)11yescharacteristic
236C2×C42−1+411yescharacteristic
237C2×C42−1+411yescharacteristic
238C2×C42−1+411yescharacteristic
239C8—22no—
240C8—22no—
241C8—41no—
242C8—41no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22#53211yescharacteristic
251C22#53611yescharacteristic
252C22#53711yescharacteristic
253C22#57611yescharacteristic
254C22#57611yescharacteristic
255C22#57711yescharacteristic
256C22#57811yescharacteristic
257C4—41no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—81no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4#53911yescharacteristic
271C4#53911yescharacteristic
272C4#54111yescharacteristic
273C4#54111yescharacteristic
274C2—161no—
275C2#266411yescharacteristic
276C2#317511yescharacteristic
277C2#317911yescharacteristic
278C2#324111yescharacteristic
279C2#324311yescharacteristic
280C2#325811yescharacteristic
281C2#325911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #2647C2semidirect
3. #2101C2semidirect
5. #2088C2semidirect
6. #2087C2semidirect
7. #2083C2semidirect
11. #1944C2semidirect
12. #1942C2semidirect
15. #1768C2semidirect

Maximal subgroups

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

subgroupindex
#26472
#26442
#21012
#20922
#20882
#20872
#20832
#20712
#20672
#19442
#19422
#19362
#19342
#17682