Finite Group Databasebeta

← #35227127

Subgroups — #35227127

Order 20864.

386 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#23782378C211yesmaximal, characteristic, complemented
2#23782316C211yesmaximal, characteristic, complemented
3#23782062C211yesmaximal, characteristic, complemented
4#23781992C211yesmaximal, characteristic, complemented
5#23781989C211yesmaximal, characteristic, complemented
6#23781985C211yesmaximal, characteristic, complemented
7#23781983C211yesmaximal, characteristic, complemented
8#23781954C211yesmaximal, characteristic, complemented
9#23781944C211yesmaximal, characteristic, complemented
10#23781919C211yesmaximal, characteristic, complemented
11#23781918C211yesmaximal, characteristic, complemented
12#23781747C211yesmaximal, characteristic, complemented
13#23781741C211yesmaximal, characteristic, complemented
14#23781706C211yesmaximal, characteristic, complemented
15#428×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×D1304C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104723—21no—
22#19104721—21no—
23#19104720—21no—
24#19104718—21no—
25#19104713—21no—
26#19104710C2211yescharacteristic
27#19104710C2211yescharacteristic
28#19104709C2211yescharacteristic
29#19104709C2211yescharacteristic
30#19104708—21no—
31#19104697C2211yescharacteristic
32#19104697C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34#19104693C2211yescharacteristic
35#19104692—21no—
36C4⋊C4×D163C2211yescharacteristic
37#19104668C2211yescharacteristic
38#19104667C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41#19104666C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104655C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#106×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×D652—21no—
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163D411yescharacteristic
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73D1304—22no—
74#14579682—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4D411yescharacteristic
83C22⋊Dic163D411yescharacteristic
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4D411yescharacteristic
92C4×Dic163—21no—
93C4×Dic163C2311yescharacteristic
94C22×C4×C163—21no—
95C22×C4×C163C2311yescharacteristic
96M16×C163—22no—
97C2×C8×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C4⋊C4×C163—21no—
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C42×C163—21no—
106C22×D163—21no—
107C22×D163—21no—
108C22×D163C2×D411yescharacteristic
109Dic326—42no—
110Dic326—81no—
111D652—21no—
112D652—21no—
113D652—21no—
114D652—21no—
115D652—42no—
116D652—81no—
117C22⋊D163—22no—
118C22⋊D163—42no—
119C22⋊D163—81no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C4×D163—21no—
126C4×D163—21no—
127C4×D163—42no—
128C4×D163—42no—
129C4×D163—42no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163C2×D411yescharacteristic
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—42no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2411yescharacteristic
142C8×C163—21no—
143C8×C163#4112yescomplemented
144C22×C163—21no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163C22×D411yescharacteristic
148C2×D163—21no—
149C2×D163—21no—
150C2×D163—21no—
151C2×D163—41no—
152C2×D163—41no—
153C2×D163—41no—
154C2×D163—41no—
155C2×D163—41no—
156C2×D163—41no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C2×#4111yescharacteristic
163C4×C1632−1+411yescharacteristic
164Dic163—21no—
165Dic163—21no—
166Dic163—41no—
167Dic163—41no—
168Dic163AGL(1,ℤ/8)12yes—
169C2×C163—41no—
170C2×C163#54711yescharacteristic
171C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C163#57711yescharacteristic
173D163—41no—
174D163—81no—
175D163—81no—
176C163#335411yescharacteristic, complemented, Sylow
177#3354—1631nomaximal, Sylow
178#555—1631no—
179#547—1631no—
180#498—1631no—
181#482—1631no—
182#482—1631no—
183#481—1631no—
184#481—1631no—
185#476—1631no—
186#474—1631no—
187#469—1631no—
188#469—1631no—
189C4×D8—1631no—
190#436—1631no—
191#428—1631no—
192#428D16311yescharacteristic, complemented
193C2×#41—3261no—
194C2×D8—1631no—
195C2×M16—1631no—
196C2×M16C2×D16311yescharacteristic
197C4⋊Q8—1631no—
198C4⋊Q8—1631no—
199#124—3261no—
200#124—3261no—
201#123—3261no—
202#123—3261no—
203C22⋊Q8—3261no—
204#121—3261no—
205#121—1631no—
206#121—1631no—
207#121—1631no—
208#121—1631no—
209C4×D4—1631no—
210C4×D4—1631no—
211C4×D4—1631no—
212#117—3261no—
213#117—1631no—
214#117C2×D16311yescharacteristic
215C2×C4⋊C4—1631no—
216C2×C4⋊C4C2×D16311yescharacteristic
217#107—1631no—
218#107C2×D16311yescharacteristic
219#107C2×D16311yescharacteristic
220#106—1631no—
221#106—1631no—
222#106—1631no—
223#106—1631no—
224#106C2×D16311yescharacteristic
225#106C2×D16311yescharacteristic
226C4⋊C8—1631no—
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233C22⋊C8—1631no—
234C22⋊C8—1631no—
235C4×C8—1631no—
236#41—6524no—
237C2×Q8—3261no—
238C2×Q8—3261no—
239C2×D4—3261no—
240C2×D4—3261no—
241C2×D4—3261no—
242C2×D4—3261no—
243C2×D4—1631no—
244C2×D4—1631no—
245C2×D4—1631no—
246C22×C4—21no—
247C22×C4—3261no—
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—1631no—
251C22×C4C22×D16311yescharacteristic
252D8—3262no—
253M16—22no—
254M16—3262no—
255C2×C8—1631no—
256C2×C8—1631no—
257C2×C8C22×D16311yescharacteristic
258C2×C8C22×D16311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4C22×D16311yescharacteristic
267C4⋊C4C22×D16311yescharacteristic
268C4⋊C4C22×D16311yescharacteristic
269C4⋊C4C22×D16311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—1631no—
279C22⋊C4—1631no—
280C42—21no—
281C42—3261no—
282C42—1631no—
283C23—3261no—
284C23—3261no—
285C23—1631no—
286C23D4×D16311yescharacteristic
287Q8—13041no—
288Q8—6522no—
289D4—13041no—
290D4—13041no—
291D4—3262no—
292D4—3261no—
293D4—3261no—
294D4—3261no—
295D4—3261no—
296D4—6522no—
297D4—6522no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—6522no—
310C2×C4—6522no—
311C2×C4—6522no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4—1631no—
315C2×C4D4×D16311yescharacteristic
316C2×C4C23×D16311yescharacteristic
317C8—21no—
318C8—3261no—
319C8—3261no—
320C8#1457969912yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—3261no—
325C22—3261no—
326C22—3261no—
327C22—6521no—
328C22—6521no—
329C22—6521no—
330C22—6521no—
331C22—6521no—
332C22—6521no—
333C22C2×D4×D16311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3261no—
340C4—3261no—
341C4—6521no—
342C4—6521no—
343C4—1632no—
344C4C2×#1457969911yescharacteristic
345C4#1910478311yescharacteristic
346C2—41no—
347C2—13041no—
348C2—13041no—
349C2—6521no—
350C2#2378231611yescharacteristic
351C2AGL(1,ℤ/8)×D16311yescharacteristic
352C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#428×C163C2
socleC22×C163C22×D4
hypercenter#428D163
hypocenterC163#3354
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #23782378C2semidirect
2. #23782316C2semidirect
3. #23782062C2semidirect
4. #23781992C2semidirect
5. #23781989C2semidirect
6. #23781985C2semidirect
7. #23781983C2semidirect
8. #23781954C2semidirect
9. #23781944C2semidirect
10. #23781919C2semidirect
11. #23781918C2semidirect
12. #23781747C2semidirect
13. #23781741C2semidirect
14. #23781706C2semidirect
15. #428×C163C2semidirect
143. C8×C163#41semidirect
176. C163#3354semidirect
192. #428D163semidirect
320. C8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823782
#237823162
#237820622
#237819922
#237819892
#237819852
#237819832
#237819542
#237819442
#237819192
#237819182
#237817472
#237817412
#237817062
#428×C1632
#3354163