Finite Group Databasebeta

← #60754

Subgroups — #60754

Order 256.

1065 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3449C211yesmaximal, characteristic
2C2×#536C211yesmaximal, characteristic, complemented
3#3202C214yesmaximal, complemented
4#3201C218yesmaximal, complemented
5#3200C214yesmaximal, complemented
6#3095C212yesmaximal, complemented
7#3093C212yesmaximal, complemented
8#3056C214yesmaximal, complemented
9#3007C212yesmaximal, complemented
10#3004C212yesmaximal, complemented
11#2879C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C23×Q8C2211yescharacteristic, complemented
15C2×#137C2214yes—
16C22×M16C2212yescomplemented
17#537C2214yes—
18#536—22no—
19#536—22no—
20#536—21no—
21#536C2214yes—
22#536C2212yes—
23#536C2214yescomplemented
24#535—21no—
25#535C2212yes—
26#534—21no—
27#534—21no—
28#532—22no—
29C2×C4⋊Q8—21no—
30C2×#124—21no—
31#525—22no—
32#525C2212yes—
33#525C2212yes—
34C2×#123—21no—
35C2×#123—21no—
36C2×C22⋊Q8C2212yes—
37C2×C22⋊Q8C2212yescomplemented
38C2×#117C2211yescharacteristic
39C2×#117C2211yescharacteristic, complemented
40#485C2218yescomplemented
41#485C2218yescomplemented
42#484C2214yescomplemented
43#483C2214yescomplemented
44#482C2218yescomplemented
45#470C2218yescomplemented
46#468C2218yescomplemented
47#467C2218yescomplemented
48#465C2218yescomplemented
49#452C2218yescomplemented
50#451C2218yescomplemented
51#451C2214yescomplemented
52#450C2214yescomplemented
53#448C2218yescomplemented
54#428C2214yescomplemented
55#419C2214yescomplemented
56#417C2214yes—
57#408C2214yes—
58#408C2214yescomplemented
59#407C2212yes—
60#407C2212yescomplemented
61C2×#41—21no—
62C2×#41—21no—
63C2×#41—44no—
64C2×#41—44no—
65C2×#41D412yes—
66C2×#41D412yes—
67C2×#41C2312yes—
68C2×#41C2312yes—
69C2×#41C2312yes—
70C2×#41C2312yes—
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—44no—
74C22×Q8D411yescharacteristic
75C22×Q8C2312yes—
76C22×Q8C2311yescharacteristic
77C22×Q8C2314yes—
78C22×D4—21no—
79C22×D4D411yescharacteristic
80C22×D4D411yescharacteristic
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88#137—48no—
89C2×Q16C2318yes—
90C2×SD16C2318yes—
91C2×M16—44no—
92C2×M16C2314yes—
93C2×M16C2314yes—
94C22×C8C2314yes—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—22no—
98#127—22no—
99#124—22no—
100#124—24no—
101#124—24no—
102#124—22no—
103#123—24no—
104#123—24no—
105#123—24no—
106#123—24no—
107#123—22no—
108#123—22no—
109#123—22no—
110#123—22no—
111C22⋊Q8—24no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—24no—
115C22⋊Q8C2314yes—
116C22⋊Q8C2314yes—
117C22⋊Q8C2318yes—
118C22⋊Q8C2314yes—
119C22⋊Q8C2314yes—
120#121—24no—
121#121—24no—
122#121—24no—
123#121C2314yes—
124#121C2314yes—
125#120—22no—
126#120—22no—
127#120—22no—
128#120—22no—
129C4×D4—24no—
130C4×D4—24no—
131#117—21no—
132#117—21no—
133#117—22no—
134#117C2312yes—
135#117C2312yes—
136#117C2314yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C42—21no—
146C2×C42—21no—
147#107C2318yes—
148#106C2318yes—
149#103C2318yes—
150#103C2318yes—
151#103C2318yes—
152#102C2318yes—
153C22⋊C8C2314yes—
154C22⋊C8C2314yes—
155C22⋊C8C2318yes—
156C24—21no—
157C24C2×D411yescharacteristic
158C24C2×D411yescharacteristic
159C24C2×D411yescharacteristic
160#41—28no—
161#41—44no—
162#41—44no—
163#41—42no—
164#41—42no—
165#41—42no—
166#41—42no—
167#41—44no—
168#41—44no—
169#41—44no—
170C2×Q8—22no—
171C2×Q8—24no—
172C2×Q8—21no—
173C2×Q8—28no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—48no—
180C2×Q8C2×D412yes—
181C2×Q8C2×D412yes—
182C2×Q8C2414yes—
183C2×Q8C2412yes—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—21no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4C2×D412yes—
195C2×D4C2×D412yes—
196C2×D4C2×D412yes—
197C2×D4C2×D412yes—
198C2×D4C2×D412yes—
199C2×D4C2×D412yes—
200C2×D4C2412yes—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D412yes—
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D412yes—
220C22×C4C2411yescharacteristic
221C22×C4C2412yes—
222C22×C4C2411yescharacteristic
223C22×C4C2412yes—
224C22×C4C2411yescharacteristic
225Q16—48no—
226SD16—48no—
227M16—28no—
228M16—44no—
229C2×C8—44no—
230C2×C8C2418yes—
231C4⋊C4—24no—
232C4⋊C4—28no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4C2414yes—
239C4⋊C4C2414yes—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C42—22no—
251C42—22no—
252C42—24no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—82no—
262C23C22×D411yescharacteristic
263C23C22×D412yes—
264C23C22×D411yescharacteristic
265C23C22×D412yes—
266C23C22×D411yescharacteristic
267Q8—28no—
268Q8—24no—
269Q8—42no—
270Q8—42no—
271Q8—48no—
272Q8—42no—
273Q8—44no—
274D4—24no—
275D4—42no—
276D4—42no—
277D4—42no—
278D4—42no—
279D4—42no—
280D4—42no—
281D4—42no—
282D4—44no—
283D4—44no—
284D4—84no—
285D4—84no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—82no—
310C2×C4C22×D412yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C22×D412yes—
313C2×C4C22×D411yescharacteristic
314C2×C4C22×D411yescharacteristic
315C2×C4H(5,𝔽2)14yes—
316C2×C4C2511yescharacteristic
317C8—28no—
318C22—22no—
319C22—22no—
320C22—24no—
321C22—24no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—44no—
328C22—44no—
329C22—42no—
330C22—82no—
331C22—82no—
332C22—84no—
333C22#53412yes—
334C22C23×D411yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—44no—
341C4—42no—
342C4—44no—
343C4#53412yes—
344C4C2×H(5,𝔽2)12yes—
345C2—22no—
346C2—42no—
347C2—41no—
348C2—42no—
349C2—41no—
350C2—82no—
351C2C2×#53411yescharacteristic
352C2#358612yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#536C2semidirect
3. #3202C2semidirect
4. #3201C2semidirect
5. #3200C2semidirect
6. #3095C2semidirect
7. #3093C2semidirect
8. #3056C2semidirect
9. #3007C2semidirect
10. #3004C2semidirect
11. #2879C2semidirect
14. C23×Q8C22semidirect
16. C22×M16C22semidirect
23. #536C22semidirect
37. C2×C22⋊Q8C22semidirect
39. C2×#117C22semidirect
40. #485C22semidirect
41. #485C22semidirect
42. #484C22semidirect
43. #483C22semidirect
44. #482C22semidirect
45. #470C22semidirect
46. #468C22semidirect
47. #467C22semidirect
48. #465C22semidirect
49. #452C22semidirect
50. #451C22semidirect
51. #451C22semidirect
52. #450C22semidirect
53. #448C22semidirect
54. #428C22semidirect
55. #419C22semidirect
58. #408C22semidirect
60. #407C22semidirect

Maximal subgroups

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

subgroupindex
#34492
C2×#5362
#32022
#32012
#32002
#30952
#30932
#30562
#30072
#30042
#28792