Finite Group Databasebeta

← #30778

Subgroups — #30778

Order 256.

397 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#500C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic
3#2491C211yesmaximal, characteristic
4#2439C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic
10#2023C211yesmaximal, characteristic, complemented
11#1949C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic, complemented
14#1921C211yesmaximal, characteristic, complemented
15#1854C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124—21no—
19C2×#121C2211yescharacteristic
20C2×#117C2211yescharacteristic
21#500—24no—
22#482—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#396—21no—
38#394—21no—
39#393—21no—
40#393—21no—
41#391—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#387—21no—
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74#124—44no—
75#121—22no—
76#121—41no—
77#117—22no—
78#117—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107—22no—
107#107D414yes—
108#106—22no—
109#106—41no—
110C4⋊C8—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114#102—41no—
115C22⋊C8—22no—
116C22⋊C8—22no—
117C4×C8—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24C2×D411yescharacteristic
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—81no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2411yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8C2×D412yes—
174C2×C8#4112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4C2×D412yes—
192C4⋊C4C2×D412yes—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—42no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23C22×D411yescharacteristic
224Q8—84no—
225Q8—84no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
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#13512yes—
270C2×C4AGL(1,ℤ/8)12yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4H(5,𝔽2)11yescharacteristic
274C2×C42−1+411yescharacteristic
275C2×C42−1+411yescharacteristic
276C2×C42−1+411yescharacteristic
277C8—22no—
278C8—22no—
279C8—41no—
280C8—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22#53611yescharacteristic
296C22D4211yescharacteristic
297C22#54711yescharacteristic
298C22C2×#13511yescharacteristic
299C22C2×AGL(1,ℤ/8)11yescharacteristic
300C22#57611yescharacteristic
301C22#57811yescharacteristic
302C4—22no—
303C4—22no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4#53611yescharacteristic
315C4#54211yescharacteristic
316C4#54811yescharacteristic
317C4#56311yescharacteristic
318C2—81no—
319C2—161no—
320C2#267411yescharacteristic
321C2#322811yescharacteristic
322C2#324311yescharacteristic
323C2#326411yescharacteristic
324C2#328511yescharacteristic
325C2#334111yescharacteristic
326C2#335311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#500C2semidirect
4. #2439C2semidirect
5. #2092C2semidirect
6. #2076C2semidirect
7. #2067C2semidirect
8. #2058C2semidirect
10. #2023C2semidirect
11. #1949C2semidirect
12. #1934C2semidirect
13. #1923C2semidirect
14. #1921C2semidirect
15. #1854C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5002
C2×#4822
#24912
#24392
#20922
#20762
#20672
#20582
#20332
#20232
#19492
#19342
#19232
#19212
#18542