Finite Group Databasebeta

← #23034

Subgroups — #23034

Order 256.

513 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2362C211yesmaximal, characteristic, complemented
5#2361C211yesmaximal, characteristic
6#1973C211yesmaximal, characteristic, complemented
7#1942C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic
11#1872C211yesmaximal, characteristic, complemented
12#1865C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1776C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#468C418yescomplemented
26C4×SD16—24no—
27#428—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#388—21no—
48#388—21no—
49#387—21no—
50#386—21no—
51#385—21no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16C2×C414yescomplemented
71C2×M16—22no—
72C2×M16—22no—
73C2×M16C2×C414yescomplemented
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22⋊Q8—22no—
79C22⋊Q8C2×C414yescomplemented
80#121—22no—
81#121C2×C414yes—
82C4×Q8—22no—
83C4×Q8—41no—
84C4×D4—22no—
85C4×D4—41no—
86C4×D4—44no—
87#117—22no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
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⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#107—22no—
117#107—22no—
118#107C2×C414yescomplemented
119#106—22no—
120#106D414yescomplemented
121#103—22no—
122#103—41no—
123#103C2×C414yescomplemented
124#102—22no—
125#102—41no—
126#102C2×C414yescomplemented
127C22⋊C8—22no—
128#97—22no—
129C4×C8—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8C22×C412yes—
146C2×D4—24no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4C22×C412yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4C22×C412yes—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4#4112yes—
179C22×C4C2411yescharacteristic
180SD16—28no—
181M16—28no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8C22×C412yes—
191C2×C8C22×C412yes—
192C2×C8C2×D412yes—
193C2×C8#4112yes—
194C4⋊C4—24no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4C22×C412yes—
208C4⋊C4C22×C412yes—
209C4⋊C4C2×D412yes—
210C4⋊C4C2×D412yes—
211C22⋊C4—24no—
212C22⋊C4—24no—
213C22⋊C4—22no—
214C22⋊C4—41no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—42no—
224C42—42no—
225C42—42no—
226C42—42no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C4×D412yes—
241C23C22×D411yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—41no—
245Q8—41no—
246D4—22no—
247D4—22no—
248D4—48no—
249D4—41no—
250D4—41no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
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—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4C4×D412yes—
292C2×C4AGL(1,ℤ/8)12yes—
293C2×C4C23×C411yescharacteristic
294C2×C4C22×D411yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C2×C42−1+411yescharacteristic
299C8—22no—
300C8—41no—
301C8—41no—
302C8C4×D414yescomplemented
303C22—22no—
304C22—22no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22C2×C4×D411yescharacteristic
318C22D4211yescharacteristic
319C22#54711yescharacteristic
320C22C2×AGL(1,ℤ/8)11yescharacteristic
321C22#57511yescharacteristic
322C22#57711yescharacteristic
323C22#57811yescharacteristic
324C4—22no—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—41no—
333C4—42no—
334C4—81no—
335C4—81no—
336C4C2×C4×D411yescharacteristic
337C4#52011yescharacteristic
338C4#54811yescharacteristic
339C4#55311yescharacteristic
340C2—42no—
341C2—82no—
342C2#236111yescharacteristic
343C2C4×AGL(1,ℤ/8)11yescharacteristic
344C2#294711yescharacteristic
345C2#326511yescharacteristic
346C2#326911yescharacteristic
347C2#335111yescharacteristic
348C2#335311yescharacteristic
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
2. C2×C4×SD16C2semidirect
3. C2×#428C2semidirect
4. #2362C2semidirect
6. #1973C2semidirect
7. #1942C2semidirect
8. #1928C2semidirect
9. #1926C2semidirect
11. #1872C2semidirect
12. #1865C2semidirect
13. #1862C2semidirect
14. #1776C2semidirect
15. #1753C2semidirect
25. #468C4semidirect
70. C2×SD16C2×C4semidirect
73. C2×M16C2×C4semidirect
79. C22⋊Q8C2×C4semidirect
118. #107C2×C4semidirect
120. #106D4semidirect
123. #103C2×C4semidirect
126. #102C2×C4semidirect
302. C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×C4×SD162
C2×#4282
#23622
#23612
#19732
#19422
#19282
#19262
#18752
#18722
#18652
#18622
#17762
#17532