Finite Group Databasebeta

← #28588

Subgroups — #28588

Order 256.

418 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic
2C2×#467C211yesmaximal, characteristic
3#2621C211yesmaximal, characteristic
4#2323C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#2043C211yesmaximal, characteristic, complemented
8#2016C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1927C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic
13#1878C211yesmaximal, characteristic, complemented
14#1878C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#470—24no—
27#467—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38#398—21no—
39#398—21no—
40#397—21no—
41#397—21no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#389—21no—
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#387—21no—
50#382C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
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—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—41no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8D414yes—
89C4×Q8—41no—
90C4×Q8—41no—
91#117—41no—
92#117—41no—
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⋊C4D411yescharacteristic
101C2×C4⋊C4D411yescharacteristic
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⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—22no—
118#106—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
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×Q8C2×D412yes—
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×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D412yes—
175C22×C4#4112yes—
176C22×C4C2411yescharacteristic
177Q16—44no—
178Q16—44no—
179M16—44no—
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×C8—21no—
190C2×C8—44no—
191C2×C8#4112yes—
192C2×C8#4112yes—
193C4⋊C4—24no—
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—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—81no—
211C4⋊C4C2×D412yes—
212C22⋊C4—24no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—81no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23#12112yes—
231C23C22×D411yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
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—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4#12112yes—
287C2×C4C22×D411yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C8—22no—
294C8—22no—
295C8—41no—
296C8—41no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22C2×#12111yescharacteristic
305C22#53211yescharacteristic
306C22#53411yescharacteristic
307C22#57711yescharacteristic
308C22#57811yescharacteristic
309C22#57811yescharacteristic
310C22#57811yescharacteristic
311C4—22no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4#54611yescharacteristic
325C4#54611yescharacteristic
326C4#55311yescharacteristic
327C4#55311yescharacteristic
328C2—42no—
329C2#254311yescharacteristic
330C2#304411yescharacteristic
331C2#304611yescharacteristic
332C2#317711yescharacteristic
333C2#317811yescharacteristic
334C2#320111yescharacteristic
335C2#320211yescharacteristic
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
5. #2085C2semidirect
6. #2074C2semidirect
7. #2043C2semidirect
8. #2016C2semidirect
9. #1928C2semidirect
10. #1927C2semidirect
13. #1878C2semidirect
14. #1878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4672
#26212
#23232
#20852
#20742
#20432
#20162
#19282
#19272
#18942
#18922
#18782
#17292