Finite Group Databasebeta

← #25281

Subgroups — #25281

Order 256.

424 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3#2544C211yesmaximal, characteristic
4#2370C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic, complemented
10#1983C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic
12#1939C211yesmaximal, characteristic, complemented
13#1927C211yesmaximal, characteristic, complemented
14#1925C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C2×#127C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#488—24no—
22#478—24no—
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#394—21no—
42#394—21no—
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#387—21no—
49#387—21no—
50#386—21no—
51#386—21no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#127—22no—
72#127—41no—
73#125—22no—
74#125D414yescomplemented
75#124—22no—
76#124—22no—
77#124—22no—
78#124—41no—
79C4×D4—22no—
80C4×D4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—22no—
107C4⋊C8—22no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103—41no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—41no—
122C4×C8—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4#4112yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2411yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4C2×D412yes—
200C4⋊C4C2×D412yes—
201C4⋊C4#4112yes—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—81no—
215C42—81no—
216C42C2×D412yes—
217C42#4112yes—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C22×D411yescharacteristic
233Q8—41no—
234Q8—41no—
235Q8—41no—
236Q8—41no—
237D4—22no—
238D4—22no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—84no—
246D4—84no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
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—84no—
281C2×C4#12112yes—
282C2×C4#12112yes—
283C2×C4#13512yes—
284C2×C4C22×D411yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C42−1+411yescharacteristic
290C8—41no—
291C8—41no—
292C8—41no—
293C8—41no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22C2×#12111yescharacteristic
309C22#54611yescharacteristic
310C22#54811yescharacteristic
311C22C2×#13511yescharacteristic
312C22#57511yescharacteristic
313C22#57611yescharacteristic
314C22#57711yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—42no—
320C4—41no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4#52911yescharacteristic
328C4#54211yescharacteristic
329C4#54811yescharacteristic
330C4#55011yescharacteristic
331C2—82no—
332C2—161no—
333C2#242611yescharacteristic
334C2#303511yescharacteristic
335C2#304511yescharacteristic
336C2#329611yescharacteristic
337C2#332211yescharacteristic
338C2#336111yescharacteristic
339C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#488C2semidirect
2. C2×#478C2semidirect
4. #2370C2semidirect
5. #2068C2semidirect
6. #2065C2semidirect
7. #2042C2semidirect
9. #1983C2semidirect
10. #1983C2semidirect
12. #1939C2semidirect
13. #1927C2semidirect
14. #1925C2semidirect
15. C4×#102C2semidirect
74. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4782
#25442
#23702
#20682
#20652
#20422
#20412
#19832
#19822
#19392
#19272
#19252
C4×#1022