Finite Group Databasebeta

← #26476

Subgroups — #26476

Order 256.

422 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic
3#2551C211yesmaximal, characteristic, complemented
4#2367C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic
10#1959C211yesmaximal, characteristic
11#1948C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic
15#1776C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#476—24no—
23#475—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#395—21no—
41#395—21no—
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#385—21no—
50#384C2211yescharacteristic
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×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C4⋊Q8—44no—
75#124—22no—
76#124Q814yescomplemented
77C4×Q8—22no—
78C4×Q8—41no—
79C4×D4—22no—
80C4×D4—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109#106—22no—
110#106D414yes—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—41no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—41no—
122#97—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8C2×Q812yes—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C2×Q812yes—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—81no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8C2×D412yes—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4C2×D412yes—
198C4⋊C4C2×D412yes—
199C4⋊C4#4112yes—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—81no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—42no—
211C42—81no—
212C42C2×Q812yes—
213C42#4112yes—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23C22×D411yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226Q8—84no—
227Q8—84no—
228D4—22no—
229D4—22no—
230D4—41no—
231D4—41no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
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—84no—
268C2×C4C22⋊Q812yes—
269C2×C4C22⋊Q812yes—
270C2×C4AGL(1,ℤ/8)12yes—
271C2×C4#13712yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×Q811yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C2×C42−1+411yescharacteristic
278C8—22no—
279C8—41no—
280C8—41no—
281C8—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22C2×C22⋊Q811yescharacteristic
290C22D4211yescharacteristic
291C22#54811yescharacteristic
292C22C2×AGL(1,ℤ/8)11yescharacteristic
293C22C2×#13711yescharacteristic
294C22#57711yescharacteristic
295C22#57711yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—42no—
304C4—41no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4#53211yescharacteristic
310C4#54211yescharacteristic
311C4D4×Q811yescharacteristic
312C4#55411yescharacteristic
313C2—82no—
314C2#245211yescharacteristic
315C2#307911yescharacteristic
316C2#308411yescharacteristic
317C2#326511yescharacteristic
318C2#326611yescharacteristic
319C2#336911yescharacteristic
320C2#337311yescharacteristic
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×#475C2semidirect
3. #2551C2semidirect
5. #2068C2semidirect
7. #2024C2semidirect
11. #1948C2semidirect
12. #1940C2semidirect
13. #1865C2semidirect
15. #1776C2semidirect
76. #124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4762
#25512
#23672
#20682
#20672
#20242
#20232
#19832
#19592
#19482
#19402
#18652
#18642
#17762