Finite Group Databasebeta

← #35226410

Subgroups — #35226410

Order 20864.

388 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#23782370C211yesmaximal, characteristic
2#23782328C211yesmaximal, characteristic
3#23782014C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23781959C211yesmaximal, characteristic, complemented
6#23781907C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781874C211yesmaximal, characteristic, complemented
9#23781823C211yesmaximal, characteristic, complemented
10#23781822C211yesmaximal, characteristic, complemented
11#23781759C211yesmaximal, characteristic, complemented
12(C22⋊D163)⋊C8C211yesmaximal, characteristic
13C2×#19104658C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15#484×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×Dic652C2211yescharacteristic
19C2×Dic652C2211yescharacteristic
20C2×#14579689C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22C2×C8×D163C2211yescharacteristic
23#19104734—21no—
24#19104734C2211yescharacteristic
25#19104733C2211yescharacteristic
26#19104725—21no—
27#19104715C2211yescharacteristic
28#19104715C2211yescharacteristic
29#19104714—21no—
30#19104714C2211yescharacteristic
31#19104713—21no—
32#19104713—21no—
33#19104712—21no—
34#19104700C2211yescharacteristic
35#19104700C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38(C22⋊D163)⋊C4—21no—
39C4⋊C4×D163C2211yescharacteristic
40#19104668C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49#19104658C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22⋊Q8×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326—22no—
62C2×Dic326—21no—
63C2×Dic326—41no—
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67Q8×D163—22no—
68Q8×D163—41no—
69C2×C22⋊D163D411yescharacteristic
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74Dic652—22no—
75Dic652—41no—
76#14579689—41no—
77C2×C163⋊C8—41no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163—41no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91Dic163⋊C4D411yescharacteristic
92C22⋊Dic163—21no—
93C22⋊Dic163D411yescharacteristic
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163—21no—
98C4×Dic163—21no—
99C4×Dic163C2311yescharacteristic
100C2×Q8×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C22×C4×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—21no—
117Dic326—42no—
118Dic326—42no—
119Dic326—81no—
120Dic326#4112yes—
121C22⋊D163Q1614yescomplemented
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—42no—
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—42no—
133C4×D163—81no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163C2×D411yescharacteristic
137Q8×C163—21no—
138Q8×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—81no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C8×C163—21no—
149C8×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163C2×Q1612yes—
153C22×C163C22×D411yescharacteristic
154C2×D163—21no—
155C2×D163—21no—
156C2×D163C2×Q1612yes—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C2×#4111yescharacteristic
163C4×C163H(5,𝔽2)11yescharacteristic
164Dic163—21no—
165Dic163—22no—
166Dic163—21no—
167Dic163—41no—
168Dic163—41no—
169Dic163—41no—
170Dic163C2×Q1612yes—
171C2×C163—22no—
172C2×C163#54611yescharacteristic
173C2×C163C22×Q1611yescharacteristic
174C2×C163#57711yescharacteristic
175D163—22no—
176C163#329911yescharacteristic, complemented, Sylow
177#3299—1631nomaximal, Sylow
178#554—1631no—
179D4×Q8—1631no—
180#487—1631no—
181#484—1631no—
182#484D16311yescharacteristic, complemented
183#477—1631no—
184#467—1631no—
185#467—1631no—
186#462—1631no—
187#451—1631no—
188#451—1631no—
189C4×Q16—1631no—
190C8×D4—1631no—
191C2×#103—1631no—
192C2×#103—1631no—
193C22×Q8—1631no—
194C22×Q8—1631no—
195C2×Q16—1631no—
196C2×Q16—1631no—
197C2×Q16—1631no—
198C22×C8—1631no—
199C22×C8—1631no—
200C4⋊Q8—3261no—
201C4⋊Q8—1631no—
202C4⋊Q8—1631no—
203#125—3261no—
204C22⋊Q8—3261no—
205C22⋊Q8—3261no—
206C22⋊Q8—3261no—
207C22⋊Q8—3261no—
208C22⋊Q8—1631no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8C2×D16311yescharacteristic
212C4×Q8—1631no—
213C4×Q8—1631no—
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—1631no—
217C2×C4⋊C4—1631no—
218C2×C4⋊C4C2×D16311yescharacteristic
219#107—1631no—
220#107C2×D16311yescharacteristic
221#107C2×D16311yescharacteristic
222C4⋊C8—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#103C2×D16311yescharacteristic
232#103C2×D16311yescharacteristic
233C22⋊C8—1631no—
234C22⋊C8C2×D16311yescharacteristic
235C4×C8—1631no—
236C2×Q8—3261no—
237C2×Q8—3261no—
238C2×Q8—3262no—
239C2×Q8—3261no—
240C2×Q8—3262no—
241C2×Q8—6521no—
242C2×Q8—6521no—
243C2×Q8—1631no—
244C2×Q8—1631no—
245C2×Q8—1631no—
246C2×Q8C22×D16311yescharacteristic
247C2×D4—1631no—
248C22×C4—21no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—1631no—
253C22×C4C22×D16311yescharacteristic
254Q16—3262no—
255Q16—6521no—
256Q16—6521no—
257C2×C8—6521no—
258C2×C8—6521no—
259C2×C8—1631no—
260C2×C8—1631no—
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4C22×D16311yescharacteristic
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—1631no—
284C22⋊C4—1631no—
285C42—3261no—
286C42—3261no—
287C42—1631no—
288C23—1631no—
289C23D4×D16311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—13041no—
293Q8—3261no—
294Q8—3261no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—6522no—
299Q8—6522no—
300Q8—1632no—
301D4—1634no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—13041no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—6522no—
317C2×C4—6522no—
318C2×C4—1631no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4D4×D16311yescharacteristic
322C2×C4C23×D16311yescharacteristic
323C8—21no—
324C8—21no—
325C8—3261no—
326C8—3261no—
327C22—21no—
328C22—21no—
329C22—3261no—
330C22—3261no—
331C22—1632no—
332C22Q16×D16312yes—
333C22C2×D4×D16311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3261no—
340C4—3261no—
341C4—3262no—
342C4—6521no—
343C4—6521no—
344C4—6521no—
345C4—1632no—
346C4#41×D16311yescharacteristic
347C4#1910478111yescharacteristic
348C2—22no—
349C2—3262no—
350C2#2378230711yescharacteristic
351C2C2×Q16×D16311yescharacteristic
352C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#484×C163C2
socleC22×C163C22×D4
prime radicalC2×C22⋊D163D4
hypercenter#484D163
hypocenterC163#3299
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #23782014C2semidirect
4. #23782002C2semidirect
5. #23781959C2semidirect
6. #23781907C2semidirect
7. #23781904C2semidirect
8. #23781874C2semidirect
9. #23781823C2semidirect
10. #23781822C2semidirect
11. #23781759C2semidirect
13. C2×#19104658C2semidirect
14. #103×D163C2semidirect
15. #484×C163C2semidirect
121. C22⋊D163Q16semidirect
176. C163#3299semidirect
182. #484D163semidirect

Maximal subgroups

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

subgroupindex
#237823702
#237823282
#237820142
#237820022
#237819592
#237819072
#237819042
#237818742
#237818232
#237818222
#237817592
(C22⋊D163)⋊C82
C2×#191046582
#103×D1632
#484×C1632
#3299163