Finite Group Databasebeta

← #24664

Subgroups — #24664

Order 256.

407 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#462C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic
3#2621C211yesmaximal, characteristic
4#2323C211yesmaximal, characteristic
5#2036C211yesmaximal, characteristic
6#2035C211yesmaximal, characteristic
7#2001C211yesmaximal, characteristic
8#2001C211yesmaximal, characteristic
9#1981C211yesmaximal, characteristic, complemented
10#1971C211yesmaximal, characteristic, complemented
11#1969C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1935C211yesmaximal, characteristic, complemented
14#1933C211yesmaximal, characteristic, complemented
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#462—24no—
25#433—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#398—21no—
43#398—21no—
44#397—21no—
45#397—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#389—21no—
50#389C2211yescharacteristic
51#388—21no—
52#388—21no—
53#387—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—41no—
70C2×Q16—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C4×Q8—22no—
84C4×Q8—22no—
85#117—22no—
86#117—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—41no—
112#106—41no—
113C4⋊C8—22no—
114C4⋊C8D414yes—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—41no—
120#103—41no—
121#103—41no—
122#103—41no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125#97—22no—
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
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
139C2×Q8—21no—
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×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8#4112yes—
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×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177Q16—44no—
178Q16—44no—
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—21no—
189C2×C8C2×D412yes—
190C2×C8C2×D412yes—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—21no—
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⋊C4—42no—
208C4⋊C4—81no—
209C4⋊C4#4112yes—
210C22⋊C4—24no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—81no—
216C42—21no—
217C42—21no—
218C42—42no—
219C42—81no—
220C42C2×D412yes—
221C42#4112yes—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23C22×D411yescharacteristic
230Q8—22no—
231Q8—22no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4#12112yes—
284C2×C4#12112yes—
285C2×C4C22×D411yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C8—22no—
292C8—22no—
293C8—41no—
294C8—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22C2×#12111yescharacteristic
303C22#52511yescharacteristic
304C22#53511yescharacteristic
305C22#57711yescharacteristic
306C22#57811yescharacteristic
307C22#57811yescharacteristic
308C22#57811yescharacteristic
309C4—22no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4#54611yescharacteristic
323C4#55011yescharacteristic
324C4#55311yescharacteristic
325C4#55511yescharacteristic
326C2—81no—
327C2#241411yescharacteristic
328C2#305911yescharacteristic
329C2#306311yescharacteristic
330C2#312011yescharacteristic
331C2#312711yescharacteristic
332C2#322311yescharacteristic
333C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#462C2semidirect
9. #1981C2semidirect
10. #1971C2semidirect
11. #1969C2semidirect
12. #1949C2semidirect
13. #1935C2semidirect
14. #1933C2semidirect
15. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×#4332
#26212
#23232
#20362
#20352
#20012
#19812
#19712
#19692
#19492
#19352
#19332
#18052