Finite Group Databasebeta

← #17422

Subgroups — #17422

Order 256.

530 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#551C211yesmaximal, characteristic
2#2974C212yesmaximal, complemented
3#1473C212yesmaximal, complemented
4#1472C212yesmaximal, complemented
5#1468C212yesmaximal, complemented
6#1465C211yesmaximal, characteristic, complemented
7#1464C211yesmaximal, characteristic, complemented
8#1462C211yesmaximal, characteristic, complemented
9#1459C211yesmaximal, characteristic, complemented
10#1458C211yesmaximal, characteristic, complemented
11C2×#323C211yesmaximal, characteristic, complemented
12#551—22no—
13#551—22no—
14#551—22no—
15#551—22no—
16C2×#126C411yescharacteristic
17C2×#126C2211yescharacteristic
18C2×#124C411yescharacteristic
19C2×#123C411yescharacteristic
20C2×#123C411yescharacteristic
21C2×#123C2211yescharacteristic
22C2×C22⋊Q8C411yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#120C411yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×#117C411yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C22⋊C4C411yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31#436—22no—
32#435—22no—
33#435—22no—
34C8×D4—22no—
35#432C2212yescomplemented
36#431C2212yescomplemented
37#407C2212yes—
38#407C2212yescomplemented
39C2×C22⋊C8C2212yes—
40C2×C22⋊C8C2212yescomplemented
41#324C2212yescomplemented
42#324C2212yescomplemented
43#324C2212yescomplemented
44#324C2212yescomplemented
45#323C2211yescharacteristic
46#323C2211yescharacteristic
47#323C2212yes—
48#323C2211yescharacteristic
49#323C2211yescharacteristic
50#323C2212yes—
51C25C2×C411yescharacteristic
52C22×Q8C2×C411yescharacteristic
53C22×D4C2×C411yescharacteristic
54C22×D4C2×C411yescharacteristic
55C22×D4C2×C411yescharacteristic
56C23×C4C2×C411yescharacteristic
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C2×M16—22no—
62C2×M16—22no—
63C22×C8—22no—
64C22×C8—22no—
65#126—22no—
66#126—22no—
67#126—22no—
68#126—22no—
69#124—22no—
70#124—22no—
71#123—22no—
72#123—22no—
73#123—22no—
74#123—22no—
75#123—22no—
76#123—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81#120—22no—
82#120—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4D412yes—
86C4×D4D412yes—
87C4×D4D412yes—
88C4×D4D412yes—
89#117—21no—
90#117—21no—
91#117—21no—
92#117—21no—
93#117—22no—
94#117—22no—
95C2×C4⋊C4C2×C411yescharacteristic
96C2×C4⋊C4C2×C411yescharacteristic
97C2×C4⋊C4C2×C411yescharacteristic
98C2×C4⋊C4C2×C411yescharacteristic
99C2×C4⋊C4C2×C411yescharacteristic
100C2×C4⋊C4C2×C411yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4C2×C411yescharacteristic
110C2×C22⋊C4C2×C411yescharacteristic
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C22⋊C4C2×C411yescharacteristic
117C2×C22⋊C4C2×C411yescharacteristic
118C2×C22⋊C4C2×C411yescharacteristic
119C2×C22⋊C4C2×C411yescharacteristic
120C2×C22⋊C4C2×C411yescharacteristic
121C2×C22⋊C4C2311yescharacteristic
122C2×C22⋊C4C2311yescharacteristic
123C2×C42C2×C411yescharacteristic
124C2×C42C2×C411yescharacteristic
125C2×C42C2311yescharacteristic
126C4⋊C8—22no—
127C4⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8C2312yes—
133C22⋊C8C2312yes—
134C22⋊C8C2312yes—
135C22⋊C8C2312yes—
136#97—22no—
137C4×C8—22no—
138C24—21no—
139C24—21no—
140C24—22no—
141C24—41no—
142C24—42no—
143C24—81no—
144C24C22×C411yescharacteristic
145C24C22×C411yescharacteristic
146C24C22×C411yescharacteristic
147C24C22×C411yescharacteristic
148C2×Q8—22no—
149C2×Q8—81no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—42no—
155C2×D4—81no—
156C2×D4C22⋊C412yes—
157C2×D4C22⋊C412yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4C22×C411yescharacteristic
171C22×C4C22×C411yescharacteristic
172C22×C4C22×C411yescharacteristic
173C22×C4C22×C411yescharacteristic
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2411yescharacteristic
185M16—82no—
186M16—82no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—82no—
192C2×C8—82no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4C22⋊C412yes—
203C4⋊C4C22⋊C412yes—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—22no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4C22⋊C411yescharacteristic
221C22⋊C4C22⋊C411yescharacteristic
222C22⋊C4C22⋊C411yescharacteristic
223C22⋊C4C22⋊C411yescharacteristic
224C22⋊C4C22⋊C412yes—
225C22⋊C4C22⋊C412yes—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42C2×D412yes—
231C42C2×D412yes—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—81no—
254C23—81no—
255C23—84no—
256C23—84no—
257C23C2×C22⋊C411yescharacteristic
258C23C2×C22⋊C412yes—
259C23C2×C22⋊C411yescharacteristic
260C23#13112yes—
261C23C23×C411yescharacteristic
262Q8—82no—
263D4—24no—
264D4—42no—
265D4—44no—
266D4—82no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4C2×C22⋊C411yescharacteristic
298C2×C4C2×C22⋊C411yescharacteristic
299C2×C4C2×C22⋊C411yescharacteristic
300C2×C4C2×C22⋊C411yescharacteristic
301C2×C4C2×C22⋊C412yes—
302C2×C4C2×C22⋊C412yes—
303C2×C4C22×D412yes—
304C8—42no—
305C8—42no—
306C8—42no—
307C8—42no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22—84no—
331C22—81no—
332C22—81no—
333C22—84no—
334C22#41312yes—
335C22C22×C22⋊C411yescharacteristic
336C22C2×#13111yescharacteristic
337C22#56811yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C2—21no—
352C2—21no—
353C2—22no—
354C2—41no—
355C2—42no—
356C2—42no—
357C2—81no—
358C2#288011yescharacteristic
359C2#288311yescharacteristic
360C2C2×#41311yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#120C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2974C2semidirect
3. #1473C2semidirect
4. #1472C2semidirect
5. #1468C2semidirect
6. #1465C2semidirect
7. #1464C2semidirect
8. #1462C2semidirect
9. #1459C2semidirect
10. #1458C2semidirect
11. C2×#323C2semidirect
35. #432C22semidirect
36. #431C22semidirect
38. #407C22semidirect
40. C2×C22⋊C8C22semidirect
41. #324C22semidirect
42. #324C22semidirect
43. #324C22semidirect
44. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5512
#29742
#14732
#14722
#14682
#14652
#14642
#14622
#14592
#14582
C2×#3232