Finite Group Databasebeta

← #28355

Subgroups — #28355

Order 256.

511 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic
4C2×#419C211yesmaximal, characteristic
5#2499C211yesmaximal, characteristic
6#2434C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2096C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2016C211yesmaximal, characteristic, complemented
11#1971C211yesmaximal, characteristic, complemented
12#1969C211yesmaximal, characteristic, complemented
13#1960C211yesmaximal, characteristic, complemented
14#1956C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#485—24no—
27#470—24no—
28#451—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32#419—24no—
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#397—22no—
47#395—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#387—22no—
53#385—22no—
54#384C2211yescharacteristic
55#382C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16D414yes—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96#117—22no—
97#117—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D412yes—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—22no—
115#106—22no—
116#106—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127C22⋊C8D414yes—
128#97—41no—
129C4×C8—41no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95D412yes—
136C24C2×D411yescharacteristic
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—28no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—42no—
166C2×Q8#4112yes—
167C2×Q8#4112yes—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4C2×D412yes—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C22×C4C2411yescharacteristic
189Q16—44no—
190Q16—44no—
191Q16—44no—
192M16—44no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8C2×D412yes—
202C2×C8C2×D412yes—
203C2×C8C2×D412yes—
204C2×C8C2×D412yes—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C22⋊C4—24no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23#12712yes—
232C23C22×D411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—48no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—48no—
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—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4#12712yes—
283C2×C4#13714yes—
284C2×C4C22×D411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C8—22no—
291C8—22no—
292C8—22no—
293C8—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22C2×#12711yescharacteristic
300C22#54611yescharacteristic
301C22#54611yescharacteristic
302C22C2×#13712yes—
303C22#57812yes—
304C4—22no—
305C4—24no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4#52511yescharacteristic
316C4#53411yescharacteristic
317C4#54611yescharacteristic
318C4#54611yescharacteristic
319C2—42no—
320C2#253911yescharacteristic
321C2#315211yescharacteristic
322C2#315611yescharacteristic
323C2#331212yes—
324C2#332812yes—
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
7. #2097C2semidirect
8. #2096C2semidirect
9. #2016C2semidirect
10. #2016C2semidirect
11. #1971C2semidirect
12. #1969C2semidirect
13. #1960C2semidirect
14. #1956C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4702
C2×#4512
C2×#4192
#24992
#24342
#20972
#20962
#20162
#19712
#19692
#19602
#19562
#17532