Finite Group Databasebeta

← #27535

Subgroups — #27535

Order 256.

415 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic
2Q8×C22⋊C4C211yesmaximal, characteristic
3#2075C211yesmaximal, characteristic
4#2075C211yesmaximal, characteristic
5#2072C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#1935C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1924C211yesmaximal, characteristic
10#1863C211yesmaximal, characteristic, complemented
11#1863C211yesmaximal, characteristic, complemented
12#1803C211yesmaximal, characteristic, complemented
13#1787C211yesmaximal, characteristic
14#1787C211yesmaximal, characteristic
15#1763C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C2×#126—21no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#400—21no—
35#397—21no—
36#391—21no—
37#391—21no—
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#387—21no—
43#384C2211yescharacteristic
44#380—21no—
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—21no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#126—44no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—41no—
78C4×Q8—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⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—41no—
102#107—41no—
103C4⋊C8—41no—
104#103—22no—
105#103—22no—
106#103—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#103—41no—
111#103—41no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114#97—41no—
115#95—21no—
116#95—21no—
117#95—21no—
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
127C24C2×D411yescharacteristic
128C2×Q8—24no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×Q8—44no—
143C2×Q8#4112yes—
144C2×Q8#4112yes—
145C2×Q8#4112yes—
146C2×Q8#4112yes—
147C22×C4—21no—
148C22×C4—24no—
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×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2411yescharacteristic
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×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—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—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—42no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—42no—
211C42—42no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23AGL(1,ℤ/8)12yes—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—48no—
226Q8—41no—
227Q8—41no—
228Q8#12614yescomplemented
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—48no—
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—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4#13712yes—
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22#52511yescharacteristic
291C22#54611yescharacteristic
292C22#54611yescharacteristic
293C22C2×AGL(1,ℤ/8)11yescharacteristic
294C22C2×#13711yescharacteristic
295C22#57811yescharacteristic
296C22#57811yescharacteristic
297C4—24no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4C2×#12611yescharacteristic
311C4#53811yescharacteristic
312C4#55111yescharacteristic
313C4#55111yescharacteristic
314C2—42no—
315C2#248011yescharacteristic
316C2#311211yescharacteristic
317C2#312711yescharacteristic
318C2#330811yescharacteristic
319C2#330811yescharacteristic
320C2#331211yescharacteristic
321C2#332811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
5. #2072C2semidirect
6. #2068C2semidirect
7. #1935C2semidirect
8. #1928C2semidirect
10. #1863C2semidirect
11. #1863C2semidirect
12. #1803C2semidirect
15. #1763C2semidirect
228. Q8#126semidirect

Maximal subgroups

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

subgroupindex
#24962
Q8×C22⋊C42
#20752
#20722
#20682
#19352
#19282
#19242
#18632
#18032
#17872
#17632