Finite Group Databasebeta

← #28568

Subgroups — #28568

Order 256.

446 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#436C211yesmaximal, characteristic
3#2417C211yesmaximal, characteristic
4#2415C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#1933C211yesmaximal, characteristic, complemented
10#1933C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1894C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#470—24no—
27#436—24no—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#398—21no—
40#397—21no—
41#396—21no—
42#394—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#389—21no—
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#386—21no—
50#380—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4D411yescharacteristic
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×Q16—22no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×M16—22no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16D414yes—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C4×D4—22no—
89C4×D4—22no—
90#117—22no—
91#117—41no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119#106—22no—
120C4⋊C8—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C4×C8—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24C2×D411yescharacteristic
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
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×D4—22no—
153C2×D4—22no—
154C2×D4—41no—
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—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D412yes—
180C22×C4C2×D411yescharacteristic
181C22×C4#4112yes—
182C22×C4C2411yescharacteristic
183Q16—44no—
184M16—44no—
185M16—44no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8C2×D412yes—
197C2×C8C2×D412yes—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—81no—
216C4⋊C4#4112yes—
217C4⋊C4#4112yes—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—81no—
227C42—21no—
228C42—21no—
229C42—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23#12112yes—
244C23C22×D411yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
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—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—84no—
297C2×C4#12112yes—
298C2×C4#13712yes—
299C2×C4C22×D411yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C8—22no—
306C8—22no—
307C8—41no—
308C8—41no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22C2×#12111yescharacteristic
324C22#54611yescharacteristic
325C22#54611yescharacteristic
326C22C2×#13711yescharacteristic
327C22#57511yescharacteristic
328C22#57711yescharacteristic
329C22#57811yescharacteristic
330C4—22no—
331C4—22no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—42no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#53211yescharacteristic
343C4#53411yescharacteristic
344C4#55311yescharacteristic
345C4#55311yescharacteristic
346C2—42no—
347C2—81no—
348C2#254311yescharacteristic
349C2#305611yescharacteristic
350C2#306311yescharacteristic
351C2#332311yescharacteristic
352C2#332711yescharacteristic
353C2#332811yescharacteristic
354C2#333111yescharacteristic
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×#470C2semidirect
5. #2088C2semidirect
6. #2077C2semidirect
7. #2056C2semidirect
8. #2001C2semidirect
9. #1933C2semidirect
10. #1933C2semidirect
11. #1894C2semidirect
12. #1894C2semidirect
13. #1873C2semidirect
14. #1872C2semidirect
15. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4362
#24172
#24152
#20882
#20772
#20562
#20012
#19332
#18942
#18732
#18722
#17362