Finite Group Databasebeta

← #26348

Subgroups — #26348

Order 256.

461 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C22×#106C211yesmaximal, characteristic, complemented
3#2442C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic
5#2092C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2059C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1919C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic
12#1874C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1850C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#478—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#400—21no—
39#398—21no—
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#386—21no—
46#385—21no—
47#385—21no—
48#382C2211yescharacteristic
49#380—21no—
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×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#125—22no—
75#125—41no—
76#121—22no—
77#121Q814yes—
78C4×D4—22no—
79C4×D4—41no—
80C4×D4—44no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—41no—
111#106—22no—
112#106—22no—
113#106—22no—
114#106D414yescomplemented
115C4⋊C8—22no—
116C4⋊C8—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—41no—
121C22⋊C8—22no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C24C2×D411yescharacteristic
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4C2×Q812yes—
142C22×C4—21no—
143C22×C4—21no—
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—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×Q812yes—
168C22×C4#4112yes—
169C22×C4C2411yescharacteristic
170C2×C8—28no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8C2×D412yes—
182C2×C8#4112yes—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4C2×D412yes—
202C4⋊C4C2×D412yes—
203C4⋊C4C2×Q812yes—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23C22⋊Q812yes—
228C23#13512yes—
229C23AGL(1,ℤ/8)12yes—
230C23C22×D411yescharacteristic
231D4—22no—
232D4—22no—
233D4—48no—
234D4—41no—
235D4—41no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
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—84no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4C22⋊Q812yes—
279C2×C4C22×D411yescharacteristic
280C2×C4C22×Q811yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C42−1+411yescharacteristic
285C8—22no—
286C8—22no—
287C8—41no—
288C8—41no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22#47814yescomplemented
303C22C2×C22⋊Q811yescharacteristic
304C22D4211yescharacteristic
305C22#54711yescharacteristic
306C22C2×#13511yescharacteristic
307C22C2×AGL(1,ℤ/8)11yescharacteristic
308C22#57711yescharacteristic
309C22#57711yescharacteristic
310C4—22no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4#52511yescharacteristic
323C4#53911yescharacteristic
324C4D4×Q811yescharacteristic
325C4#55411yescharacteristic
326C2—24no—
327C2—82no—
328C2#244911yescharacteristic
329C2C2×#47811yescharacteristic
330C2#308411yescharacteristic
331C2#326511yescharacteristic
332C2#328211yescharacteristic
333C2#333711yescharacteristic
334C2#334511yescharacteristic
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×#478C2semidirect
2. C22×#106C2semidirect
3. #2442C2semidirect
5. #2092C2semidirect
6. #2088C2semidirect
7. #2065C2semidirect
8. #2059C2semidirect
9. #1949C2semidirect
10. #1919C2semidirect
13. #1864C2semidirect
14. #1852C2semidirect
15. #1850C2semidirect
114. #106D4semidirect
302. C22#478semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C22×#1062
#24422
#23612
#20922
#20882
#20652
#20592
#19492
#19192
#18932
#18742
#18642
#18522
#18502