Finite Group Databasebeta

← #26468

Subgroups — #26468

Order 256.

403 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic
3#2550C211yesmaximal, characteristic, complemented
4#2367C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2026C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic
10#1958C211yesmaximal, characteristic
11#1949C211yesmaximal, characteristic, complemented
12#1939C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic
15C4×#107C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#478—24no—
22#477—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#398—21no—
40#395—21no—
41#391—21no—
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#385—21no—
50#382C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71#125—22no—
72#125—41no—
73#124—22no—
74#124Q814yescomplemented
75C4×Q8—22no—
76C4×Q8—41no—
77C4×D4—22no—
78C4×D4—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107D414yes—
107#106—22no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—41no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—41no—
119C4×C8—22no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95D411yescharacteristic
129#95D411yescharacteristic
130C24—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8C2×Q812yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×Q812yes—
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—81no—
156C22×C4C2×D411yescharacteristic
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×C8—21no—
174C2×C8C2×D412yes—
175C4⋊C4—22no—
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—41no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4C2×D412yes—
193C4⋊C4C2×D412yes—
194C4⋊C4#4112yes—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—81no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C42—81no—
206C42—81no—
207C42C2×Q812yes—
208C42#4112yes—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—84no—
222D4—22no—
223D4—22no—
224D4—41no—
225D4—41no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
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—41no—
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—84no—
263C2×C4C22⋊Q812yes—
264C2×C4C22⋊Q812yes—
265C2×C4#13512yes—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×Q811yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C42−1+411yescharacteristic
272C8—22no—
273C8—41no—
274C8—41no—
275C8—41no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22C2×C22⋊Q811yescharacteristic
284C22D4211yescharacteristic
285C22#54811yescharacteristic
286C22C2×#13511yescharacteristic
287C22#57511yescharacteristic
288C22#57611yescharacteristic
289C22#57811yescharacteristic
290C4—22no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#53211yescharacteristic
304C4#54211yescharacteristic
305C4D4×Q811yescharacteristic
306C4#55411yescharacteristic
307C2—82no—
308C2#245211yescharacteristic
309C2#307611yescharacteristic
310C2#308311yescharacteristic
311C2#327311yescharacteristic
312C2#328511yescharacteristic
313C2#336111yescharacteristic
314C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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×#477C2semidirect
3. #2550C2semidirect
5. #2068C2semidirect
7. #2026C2semidirect
11. #1949C2semidirect
12. #1939C2semidirect
13. #1865C2semidirect
15. C4×#107C2semidirect
74. #124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4782
#25502
#23672
#20682
#20672
#20262
#20232
#19832
#19582
#19492
#19392
#18652
#18622
C4×#1072