Finite Group Databasebeta

← #23134

Subgroups — #23134

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic
2#2361C211yesmaximal, characteristic
3#2360C211yesmaximal, characteristic, complemented
4#1936C211yesmaximal, characteristic, complemented
5#1935C211yesmaximal, characteristic, complemented
6#1920C211yesmaximal, characteristic, complemented
7#1919C211yesmaximal, characteristic, complemented
8#1880C211yesmaximal, characteristic, complemented
9#1877C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic
11#1872C211yesmaximal, characteristic, complemented
12#1763C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14#1752C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#465C418yescomplemented
26C2×#106C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#391C2211yescharacteristic
37#390C2211yescharacteristic
38#389—21no—
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#387—21no—
43#386—21no—
44#385—21no—
45#385—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C2×SD16—41no—
68C2×SD16—41no—
69C2×SD16C2×C414yescomplemented
70C22×C8—41no—
71C22×C8—41no—
72C22×C8C2×C414yescomplemented
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22⋊Q8—22no—
78C22⋊Q8C2×C414yescomplemented
79#121—22no—
80#121C2×C414yes—
81C4×Q8—22no—
82C4×Q8—41no—
83C4×D4—22no—
84C4×D4—41no—
85C4×D4—44no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#106C2×C414yescomplemented
115#103—22no—
116#103—41no—
117#103C2×C414yescomplemented
118#102—22no—
119#102—41no—
120#102C2×C414yescomplemented
121C22⋊C8—22no—
122#97—41no—
123#97—41no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C22×C412yes—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4C22×C412yes—
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—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4C22×C412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4#4112yes—
176C22×C4C2411yescharacteristic
177SD16—44no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—44no—
189C2×C8C22×C412yes—
190C2×C8C22×C412yes—
191C4⋊C4—24no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—81no—
205C4⋊C4C22×C412yes—
206C4⋊C4C22×C412yes—
207C4⋊C4#4112yes—
208C4⋊C4#4112yes—
209C22⋊C4—24no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—24no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—81no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—42no—
221C42—42no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23C4×D412yes—
235C23AGL(1,ℤ/8)12yes—
236C23#13712yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—41no—
241Q8—41no—
242D4—22no—
243D4—22no—
244D4—48no—
245D4—41no—
246D4—41no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4C4×D412yes—
290C2×C4C23×C411yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C8—22no—
297C8—22no—
298C8—41no—
299C8—41no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#44014yes—
314C22C2×C4×D411yescharacteristic
315C22#54611yescharacteristic
316C22#54611yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22C2×#13711yescharacteristic
319C22#57711yescharacteristic
320C22#57711yescharacteristic
321C4—22no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4C4×#4111yescharacteristic
334C4#51811yescharacteristic
335C4#55311yescharacteristic
336C4#55311yescharacteristic
337C2—24no—
338C2—82no—
339C2#236411yescharacteristic
340C2C2×#44011yescharacteristic
341C2#295211yescharacteristic
342C2#330611yescharacteristic
343C2#331011yescharacteristic
344C2#331611yescharacteristic
345C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×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
3. #2360C2semidirect
4. #1936C2semidirect
5. #1935C2semidirect
6. #1920C2semidirect
7. #1919C2semidirect
8. #1880C2semidirect
9. #1877C2semidirect
11. #1872C2semidirect
12. #1763C2semidirect
13. #1762C2semidirect
14. #1752C2semidirect
15. C2×#340C2semidirect
25. #465C4semidirect
69. C2×SD16C2×C4semidirect
72. C22×C8C2×C4semidirect
78. C22⋊Q8C2×C4semidirect
114. #106C2×C4semidirect
117. #103C2×C4semidirect
120. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
#23612
#23602
#19362
#19352
#19202
#19192
#18802
#18772
#18742
#18722
#17632
#17622
#17522
C2×#3402