Finite Group Databasebeta

← #24867

Subgroups — #24867

Order 256.

521 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C22×#103C211yesmaximal, characteristic, complemented
3#2409C211yesmaximal, characteristic
4D4×C4⋊C4C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1928C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic
13#1852C211yesmaximal, characteristic, complemented
14#1808C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#484—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#103—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#395—21no—
42#393—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#389—21no—
46#386—21no—
47#385—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—44no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4D411yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4D411yescharacteristic
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116#107—22no—
117#107—22no—
118#107—41no—
119#107—41no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
127#103D414yescomplemented
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24C2×D411yescharacteristic
140C24C2×D411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—28no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8C2×D412yes—
159C2×Q8#4112yes—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4Q1614yes—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4#4112yes—
195C22×C4#4112yes—
196C22×C4C2411yescharacteristic
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—44no—
209C2×C8C2×D412yes—
210C4⋊C4—22no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4C2×D412yes—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C42—21no—
237C42—21no—
238C42—42no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23#12312yes—
250C23C2×Q1612yes—
251C23C2×Q1612yes—
252C23AGL(1,ℤ/8)12yes—
253C23C22×D411yescharacteristic
254Q8—22no—
255Q8—22no—
256Q8—22no—
257Q8—22no—
258Q8—48no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263D4—28no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—48no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4#12312yes—
310C2×C4C2×Q1612yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C8—22no—
318C8—41no—
319C8—41no—
320C8—41no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22#48414yescomplemented
335C22C2×#12311yescharacteristic
336C22D4211yescharacteristic
337C22#54611yescharacteristic
338C22C22×Q1611yescharacteristic
339C22C2×AGL(1,ℤ/8)11yescharacteristic
340C22#57711yescharacteristic
341C22#57811yescharacteristic
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4#52511yescharacteristic
355C4#53811yescharacteristic
356C4#54611yescharacteristic
357C4#55111yescharacteristic
358C2—24no—
359C2—42no—
360C2#241811yescharacteristic
361C2C2×#48411yescharacteristic
362C2#309711yescharacteristic
363C2#326511yescharacteristic
364C2D4×Q1611yescharacteristic
365C2#329911yescharacteristic
366C2#330811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
2. C22×#103C2semidirect
5. #2095C2semidirect
6. #2072C2semidirect
7. #2036C2semidirect
8. #2036C2semidirect
9. #1928C2semidirect
10. #1928C2semidirect
11. #1872C2semidirect
13. #1852C2semidirect
14. #1808C2semidirect
15. #1787C2semidirect
127. #103D4semidirect
334. C22#484semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C22×#1032
#24092
D4×C4⋊C42
#20952
#20722
#20362
#19282
#18722
#18692
#18522
#18082
#17872