Finite Group Databasebeta

← #27412

Subgroups — #27412

Order 256.

388 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2426C211yesmaximal, characteristic
2#2345C211yesmaximal, characteristic, complemented
3#2086C211yesmaximal, characteristic, complemented
4#2042C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic
7#1983C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1924C211yesmaximal, characteristic, complemented
11#1920C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic
13#1872C211yesmaximal, characteristic, complemented
14#1865C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#396—21no—
36#394—21no—
37#390—21no—
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#387—21no—
44#387—21no—
45#380—21no—
46#380C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#124—22no—
69#124—41no—
70C22⋊Q8—22no—
71C22⋊Q8—41no—
72#121—22no—
73#121—41no—
74C4×Q8—22no—
75C4×Q8—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
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×C22⋊C4—21no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—41no—
105#106—22no—
106C4⋊C8—41no—
107#103—22no—
108#103—22no—
109#103—22no—
110#103—41no—
111#102—22no—
112#102—22no—
113#102—41no—
114#102—41no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117#97—41no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8#4112yes—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—81no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4#4112yes—
193C4⋊C4#4112yes—
194C4⋊C4#4112yes—
195C22⋊C4—22no—
196C22⋊C4—41no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42—81no—
210C42—81no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23#13712yes—
225C23C22×D411yescharacteristic
226Q8—22no—
227Q8—22no—
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—84no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—84no—
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4H(5,𝔽2)11yescharacteristic
281C2×C4H(5,𝔽2)11yescharacteristic
282C8—41no—
283C8—41no—
284C8—41no—
285C8—41no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#52511yescharacteristic
301C22#54611yescharacteristic
302C22#54611yescharacteristic
303C22C2×#13711yescharacteristic
304C22#57511yescharacteristic
305C22#57611yescharacteristic
306C22#57711yescharacteristic
307C4—42no—
308C4—42no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4#52911yescharacteristic
320C4#54011yescharacteristic
321C4#55311yescharacteristic
322C4#55511yescharacteristic
323C2—42no—
324C2—161no—
325C2#247611yescharacteristic
326C2#311711yescharacteristic
327C2#311811yescharacteristic
328C2#330911yescharacteristic
329C2#331411yescharacteristic
330C2#331611yescharacteristic
331C2#332611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2345C2semidirect
3. #2086C2semidirect
4. #2042C2semidirect
5. #2036C2semidirect
7. #1983C2semidirect
8. #1936C2semidirect
9. #1925C2semidirect
10. #1924C2semidirect
11. #1920C2semidirect
13. #1872C2semidirect
14. #1865C2semidirect
15. #1763C2semidirect

Maximal subgroups

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

subgroupindex
#24262
#23452
#20862
#20422
#20362
#20342
#19832
#19362
#19252
#19242
#19202
#18742
#18722
#18652
#17632