Finite Group Databasebeta

← #35224355

Subgroups — #35224355

Order 20864.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×D163C211yesmaximal, characteristic, complemented
2C4×#14579700C211yesmaximal, characteristic
3#23781974C211yesmaximal, characteristic, complemented
4#23781972C211yesmaximal, characteristic, complemented
5#23781970C211yesmaximal, characteristic, complemented
6#23781968C211yesmaximal, characteristic, complemented
7#23781966C211yesmaximal, characteristic, complemented
8#23781964C211yesmaximal, characteristic, complemented
9#23781962C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781687C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic
14#23781665C211yesmaximal, characteristic
15#479×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579682C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19#19104727C2211yescharacteristic
20#19104726C2211yescharacteristic
21#19104726C2211yescharacteristic
22#19104725—21no—
23#19104725—21no—
24Q8×Dic163C2211yescharacteristic
25C4×D652—21no—
26C4×D652C2211yescharacteristic
27#19104697—21no—
28#19104697C2211yescharacteristic
29#19104693—21no—
30#19104693C2211yescharacteristic
31C4⋊C4×D163—21no—
32C4⋊C4×D163—21no—
33C4⋊C4×D163C2211yescharacteristic
34C4⋊C4×D163C2211yescharacteristic
35C42×D163—21no—
36C42×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104667C2211yescharacteristic
41#19104666C2211yescharacteristic
42#19104666C2211yescharacteristic
43C4⋊(C163⋊C8)C2211yescharacteristic
44Dic163⋊C8C2211yescharacteristic
45Dic163⋊C8C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104659C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104655C2211yescharacteristic
51#19104655C2211yescharacteristic
52#125×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59#14579700—22no—
60#14579700Q814yes—
61C2×D652—21no—
62C2×D652C2311yescharacteristic
63C2×C4×D163—21no—
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163D411yescharacteristic
69C2×C4×D163D411yescharacteristic
70C2×C4×D163C2311yescharacteristic
71#14579682—22no—
72#14579682—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163D411yescharacteristic
93C4×Dic163D411yescharacteristic
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C2×C8×C163C2311yescharacteristic
97C2×C8×C163C2311yescharacteristic
98C4⋊C4×C163—21no—
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C42×C163—21no—
105C42×C163C2311yescharacteristic
106C22×D163—21no—
107C22×D163C2×D411yescharacteristic
108D652—24no—
109D652—21no—
110D652C2×Q812yes—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163C2×D411yescharacteristic
115C2×Dic163C2×D411yescharacteristic
116C2×Dic163C2×D411yescharacteristic
117C4×D163—24no—
118C4×D163—41no—
119C4×D163—41no—
120C4×D163—42no—
121C4×D163—42no—
122C4×D163—42no—
123C4×D163—42no—
124C4×D163C2×Q812yes—
125C4×D163#4112yes—
126C163⋊C8—21no—
127C163⋊C8—21no—
128Q8×C163—21no—
129Q8×C163C2×Q812yes—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2411yescharacteristic
137C8×C163—21no—
138C8×C163—21no—
139C22×C163C22×D411yescharacteristic
140C2×D163—21no—
141C2×D163—21no—
142C2×D163—22no—
143C2×D163—41no—
144C2×D163—41no—
145C2×D163C22⋊Q812yes—
146C4×C163—22no—
147C4×C163—21no—
148C4×C163—21no—
149C4×C163—41no—
150C4×C163—41no—
151C4×C163—41no—
152C4×C163C22×Q811yescharacteristic
153C4×C163C2×#4111yescharacteristic
154Dic163—21no—
155Dic163—21no—
156Dic163—41no—
157Dic163—41no—
158Dic163—41no—
159Dic163C22⋊Q812yes—
160C2×C163C2×C22⋊Q811yescharacteristic
161C2×C163#57511yescharacteristic
162C2×C163#57511yescharacteristic
163D163—22no—
164D163—42no—
165C163#308011yescharacteristic, complemented, Sylow
166#3080—1631nomaximal, Sylow
167C2×#125—1631no—
168C4×#41—1631no—
169#479—1631no—
170#479—1631no—
171#479—1631no—
172#479D16311yescharacteristic, complemented
173#478—1631no—
174#478—1631no—
175#478—1631no—
176#478—1631no—
177#428—1631no—
178#428—1631no—
179#423—1631no—
180#417—1631no—
181#417—1631no—
182C2×#41—1631no—
183C2×M16—1631no—
184C2×M16—1631no—
185#125—3261no—
186#125—3261no—
187#125—1631no—
188#125—1631no—
189#125—1631no—
190#125C2×D16311yescharacteristic
191C4×Q8—1631no—
192C4×Q8C2×D16311yescharacteristic
193C4×D4—3261no—
194C4×D4—3261no—
195C4×D4—1631no—
196C4×D4—1631no—
197#117—3261no—
198#117—1631no—
199C2×C4⋊C4—3261no—
200C2×C4⋊C4—3261no—
201C2×C4⋊C4—1631no—
202C2×C4⋊C4—1631no—
203C2×C42—3261no—
204C2×C42—1631no—
205#107—1631no—
206#107—1631no—
207#107—1631no—
208#107C2×D16311yescharacteristic
209#106—1631no—
210#106—1631no—
211#106—1631no—
212#106C2×D16311yescharacteristic
213C4⋊C8—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216C4⋊C8C2×D16311yescharacteristic
217#103—1631no—
218#103—1631no—
219#103C2×D16311yescharacteristic
220#103C2×D16311yescharacteristic
221#102—1631no—
222#102—1631no—
223#102—1631no—
224#102—1631no—
225#41—3262no—
226#41—1634no—
227C2×Q8C22×D16311yescharacteristic
228C2×D4—3261no—
229C2×D4—1631no—
230C22×C4—3261no—
231C22×C4—3261no—
232C22×C4—3261no—
233C22×C4—3261no—
234C22×C4—3261no—
235C22×C4—1631no—
236C22×C4—1631no—
237C22×C4—1631no—
238M16—3262no—
239M16—3262no—
240C2×C8—1631no—
241C2×C8—1631no—
242C2×C8C22×D16311yescharacteristic
243C2×C8C22×D16311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—1631no—
257C4⋊C4—1631no—
258C4⋊C4—1631no—
259C4⋊C4C22×D16311yescharacteristic
260C4⋊C4C22×D16311yescharacteristic
261C4⋊C4C22×D16311yescharacteristic
262C22⋊C4—3261no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C42—21no—
266C42—3261no—
267C42—3261no—
268C42—1631no—
269C42—1631no—
270C42—1631no—
271C42C22×D16311yescharacteristic
272C23—3261no—
273C23—1631no—
274Q8—21no—
275Q8Q8×D16312yes—
276D4—3264no—
277D4—3261no—
278D4—1632no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—3264no—
284C2×C4—3261no—
285C2×C4—3261no—
286C2×C4—3261no—
287C2×C4—6521no—
288C2×C4—6521no—
289C2×C4—6522no—
290C2×C4—6522no—
291C2×C4—6522no—
292C2×C4—6522no—
293C2×C4—1631no—
294C2×C4—1631no—
295C2×C4—1631no—
296C2×C4—1632no—
297C2×C4—1632no—
298C2×C4D4×D16311yescharacteristic
299C2×C4D4×D16311yescharacteristic
300C2×C4C23×D16311yescharacteristic
301C8—21no—
302C8—21no—
303C8—3261no—
304C8—3261no—
305C22—3261no—
306C22—3261no—
307C22—3262no—
308C22—6521no—
309C22—6521no—
310C22—1632no—
311C22C2×D4×D16311yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3261no—
319C4—3261no—
320C4—6521no—
321C4—6521no—
322C4—6521no—
323C4—1632no—
324C4C2×Q8×D16311yescharacteristic
325C4#41×D16311yescharacteristic
326C2—3262no—
327C2—6522no—
328C2C22⋊Q8×D16311yescharacteristic
329C2#2378246211yescharacteristic
330C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#479×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579700C22
hypercenter#479D163
hypocenterC163#3080
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #125×D163C2semidirect
3. #23781974C2semidirect
4. #23781972C2semidirect
5. #23781970C2semidirect
6. #23781968C2semidirect
7. #23781966C2semidirect
8. #23781964C2semidirect
9. #23781962C2semidirect
10. #23781706C2semidirect
11. #23781704C2semidirect
12. #23781687C2semidirect
15. #479×C163C2semidirect
165. C163#3080semidirect
172. #479D163semidirect

Maximal subgroups

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

subgroupindex
#125×D1632
C4×#145797002
#237819742
#237819722
#237819702
#237819682
#237819662
#237819642
#237819622
#237817062
#237817042
#237816872
#237816652
#479×C1632
#3080163