Finite Group Databasebeta

← #32230

Subgroups — #32230

Order 256.

350 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2551C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic
6#2065C211yesmaximal, characteristic
7#2023C211yesmaximal, characteristic
8#2023C211yesmaximal, characteristic
9#1990C211yesmaximal, characteristic, complemented
10#1987C211yesmaximal, characteristic, complemented
11#1934C211yesmaximal, characteristic
12#1926C211yesmaximal, characteristic
13#1920C211yesmaximal, characteristic, complemented
14#1919C211yesmaximal, characteristic, complemented
15#1916C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#126—21no—
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#397—21no—
29#396—21no—
30#395—21no—
31#395—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389—21no—
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#384C2211yescharacteristic
41#384C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—21no—
61C22×D4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68C4⋊Q8—41no—
69C4⋊Q8—41no—
70C4⋊Q8—44no—
71#126—44no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#106—22no—
102#106—22no—
103C4⋊C8—41no—
104#102—22no—
105#102—22no—
106#102—41no—
107#102—41no—
108C4×C8—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95D411yescharacteristic
115#95D411yescharacteristic
116C24—21no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—81no—
122C2×Q8—81no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4SD1614yes—
178C4⋊C4#4112yes—
179C4⋊C4#4112yes—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—81no—
185C22⋊C4—81no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—42no—
193C42—42no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23C22×D411yescharacteristic
202Q8—84no—
203Q8—84no—
204D4—41no—
205D4—41no—
206D4—41no—
207D4—41no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
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×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4C2×SD1612yes—
244C2×C4C2×SD1612yes—
245C2×C4C2×SD1612yes—
246C2×C4AGL(1,ℤ/8)12yes—
247C2×C4C2×#4111yescharacteristic
248C2×C4C2×#4111yescharacteristic
249C2×C4H(5,𝔽2)11yescharacteristic
250C2×C42−1+411yescharacteristic
251C2×C42−1+411yescharacteristic
252C2×C42−1+411yescharacteristic
253C8—41no—
254C8—41no—
255C8—41no—
256C8—41no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22#53611yescharacteristic
265C22#54611yescharacteristic
266C22#54811yescharacteristic
267C22C22×SD1611yescharacteristic
268C22C2×AGL(1,ℤ/8)11yescharacteristic
269C22#57611yescharacteristic
270C22#57811yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—41no—
277C4—41no—
278C4—42no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#54111yescharacteristic
285C4#54211yescharacteristic
286C4#55611yescharacteristic
287C4#56311yescharacteristic
288C2—161no—
289C2#271611yescharacteristic
290C2#322711yescharacteristic
291C2#324311yescharacteristic
292C2#329811yescharacteristic
293C2#331811yescharacteristic
294C2#336211yescharacteristic
295C2#336711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2551C2semidirect
3. #2100C2semidirect
4. #2084C2semidirect
9. #1990C2semidirect
10. #1987C2semidirect
13. #1920C2semidirect
14. #1919C2semidirect
15. #1916C2semidirect

Maximal subgroups

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

subgroupindex
#25512
#25122
#21002
#20842
#20712
#20652
#20232
#19902
#19872
#19342
#19262
#19202
#19192
#19162