Finite Group Databasebeta

← #17427

Subgroups — #17427

Order 256.

511 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic
2#2974C212yesmaximal, complemented
3#1473C212yesmaximal, complemented
4#1471C212yesmaximal, complemented
5#1468C212yesmaximal, complemented
6#1465C211yesmaximal, characteristic, complemented
7#1464C211yesmaximal, characteristic, complemented
8#1463C211yesmaximal, characteristic, complemented
9#1459C211yesmaximal, characteristic, complemented
10#1458C211yesmaximal, characteristic, complemented
11C2×#324C211yesmaximal, characteristic, complemented
12#553—22no—
13#553—22no—
14#553—22no—
15#553—22no—
16C2×#126C411yescharacteristic
17C2×#126C2211yescharacteristic
18C2×#125C411yescharacteristic
19C2×#123C411yescharacteristic
20C2×#123C2211yescharacteristic
21C2×#121C411yescharacteristic
22C2×#121C411yescharacteristic
23C2×#121C411yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C411yescharacteristic
26C2×C4×D4C411yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#436—22no—
32#435—22no—
33#435—22no—
34C8×D4—22no—
35#432C2212yes—
36#431C2212yes—
37#407C2212yes—
38#407C2212yes—
39C2×C22⋊C8C2212yes—
40C2×C22⋊C8C2212yes—
41#324C2211yescharacteristic
42#324C2211yescharacteristic
43#324C2212yes—
44#324C2211yescharacteristic
45#324C2211yescharacteristic
46#324C2212yes—
47#323C2212yes—
48#323C2212yes—
49#323C2212yes—
50#323C2212yes—
51C22×D4C2×C411yescharacteristic
52C22×D4C2×C411yescharacteristic
53C22×D4C2×C411yescharacteristic
54C22×D4C2×C411yescharacteristic
55C22×D4C2×C411yescharacteristic
56C22×D4C2×C411yescharacteristic
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2×C411yescharacteristic
60C23×C4C2×C411yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C2×M16—22no—
64C2×M16—22no—
65C22×C8—22no—
66C22×C8—22no—
67#126—22no—
68#126—22no—
69#126—22no—
70#126—22no—
71#125—22no—
72#125—22no—
73#123—22no—
74#123—22no—
75#123—22no—
76#123—22no—
77#121—22no—
78#121—22no—
79#121—22no—
80#121—22no—
81#121—22no—
82#121—22no—
83#121—22no—
84#121—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4D412yes—
96C4×D4D412yes—
97C4×D4D412yes—
98C4×D4D412yes—
99#117—22no—
100#117—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C2×C411yescharacteristic
104C2×C4⋊C4C2×C411yescharacteristic
105C2×C4⋊C4C2×C411yescharacteristic
106C2×C4⋊C4C2×C411yescharacteristic
107C2×C4⋊C4C2×C411yescharacteristic
108C2×C4⋊C4C2×C411yescharacteristic
109C2×C4⋊C4C2×C411yescharacteristic
110C2×C4⋊C4C2×C411yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
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—
138C24C22×C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C411yescharacteristic
141C24C22×C411yescharacteristic
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—81no—
155C2×D4—81no—
156C2×D4C22⋊C412yes—
157C2×D4C22⋊C412yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C22×C411yescharacteristic
175C22×C4C22×C411yescharacteristic
176C22×C4C22×C411yescharacteristic
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C22×C411yescharacteristic
181C22×C4C22×C411yescharacteristic
182C22×C4C22×C411yescharacteristic
183C22×C4C22×C411yescharacteristic
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D412yes—
188C22×C4C2411yescharacteristic
189M16—82no—
190M16—82no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4C22⋊C412yes—
209C4⋊C4C22⋊C412yes—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
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—22no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23C2×C22⋊C412yes—
251C23C2×C22⋊C411yescharacteristic
252C23C2×C22⋊C411yescharacteristic
253C23C23×C411yescharacteristic
254D4—24no—
255D4—42no—
256D4—44no—
257D4—42no—
258D4—42no—
259D4—42no—
260D4—42no—
261D4—82no—
262D4—82no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4C2×C22⋊C411yescharacteristic
295C2×C4C2×C22⋊C411yescharacteristic
296C2×C4C2×C22⋊C411yescharacteristic
297C2×C4C2×C22⋊C411yescharacteristic
298C2×C4C2×C22⋊C412yes—
299C2×C4C2×C22⋊C412yes—
300C2×C4#13112yes—
301C2×C4C22×D412yes—
302C8—42no—
303C8—42no—
304C8—42no—
305C8—42no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#41312yes—
331C22C22×C22⋊C411yescharacteristic
332C22C2×#13111yescharacteristic
333C22#56811yescharacteristic
334C4—22no—
335C4—21no—
336C4—21no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C2—21no—
348C2—21no—
349C2—22no—
350C2—41no—
351C2—42no—
352C2—81no—
353C2—81no—
354C2#288011yescharacteristic
355C2#288311yescharacteristic
356C2C2×#41311yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#553C2
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. #1471C2semidirect
5. #1468C2semidirect
6. #1465C2semidirect
7. #1464C2semidirect
8. #1463C2semidirect
9. #1459C2semidirect
10. #1458C2semidirect
11. C2×#324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
#29742
#14732
#14712
#14682
#14652
#14642
#14632
#14592
#14582
C2×#3242