Finite Group Databasebeta

← #35226595

Subgroups — #35226595

Order 20864.

380 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
1#23782386C211yesmaximal, characteristic, complemented
2#23782326C211yesmaximal, characteristic
3#23782026C211yesmaximal, characteristic, complemented
4#23782004C211yesmaximal, characteristic, complemented
5#23781990C211yesmaximal, characteristic, complemented
6#23781958C211yesmaximal, characteristic, complemented
7#23781913C211yesmaximal, characteristic, complemented
8#23781903C211yesmaximal, characteristic, complemented
9#23781889C211yesmaximal, characteristic, complemented
10#23781826C211yesmaximal, characteristic, complemented
11#23781772C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic, complemented
13#23781674C211yesmaximal, characteristic, complemented
14#23781660C211yesmaximal, characteristic, complemented
15#451×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579683C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104730—21no—
22#19104728—21no—
23#19104724—21no—
24#19104724C2211yescharacteristic
25#19104721—21no—
26#19104711—21no—
27#19104711C2211yescharacteristic
28#19104711C2211yescharacteristic
29#19104709—21no—
30#19104709C2211yescharacteristic
31Q8×Dic163C2211yescharacteristic
32Q8×Dic163C2211yescharacteristic
33#19104700—21no—
34C4×C22⋊D163—21no—
35C4×C22⋊D163C2211yescharacteristic
36#19104694C2211yescharacteristic
37#19104693C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104667C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C2×Q16×C163C2211yescharacteristic
54C2×Q16×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579700—22no—
60#14579700—41no—
61C2×Dic326—21no—
62C2×D652—21no—
63C2×D652C2311yescharacteristic
64C2×C22⋊D163—21no—
65C2×C22⋊D163D411yescharacteristic
66C2×C4×D163—21no—
67C2×C4×D163C2311yescharacteristic
68#14579686—41no—
69#14579683—41no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C8×D163—41no—
73C4⋊Dic163—21no—
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4D411yescharacteristic
82C22⋊Dic163—21no—
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)⋊C4D411yescharacteristic
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163—21no—
97C2×Q8×C163—22no—
98C2×Q8×C163—41no—
99C2×Q8×C163C2311yescharacteristic
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163C2311yescharacteristic
104Q16×C163—22no—
105Q16×C163—41no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163C2×D411yescharacteristic
113Dic326—42no—
114D652—21no—
115D652—21no—
116D652—42no—
117C22⋊D163—41no—
118C22⋊D163—42no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C4×D163—21no—
126C4×D163—21no—
127C4×D163—42no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163C2×D411yescharacteristic
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—21no—
134Q8×C163—21no—
135Q8×C163—21no—
136Q8×C163—42no—
137Q8×C163#4112yes—
138C2×C4×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—42no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163C22×D411yescharacteristic
153C2×D163—21no—
154C2×D163—21no—
155C2×D163—21no—
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—22no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163—41no—
164C4×C163C2×#4111yescharacteristic
165C4×C163H(5,𝔽2)11yescharacteristic
166Dic163—21no—
167Dic163—21no—
168Dic163—21no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—41no—
173C2×C163#54611yescharacteristic
174C2×C163#57511yescharacteristic
175C2×C163#57811yescharacteristic
176D163—41no—
177D163—81no—
178C163#331311yescharacteristic, complemented, Sylow
179#3313—1631nomaximal, Sylow
180#556—1631no—
181#548—1631no—
182#489—1631no—
183#485—1631no—
184#482—1631no—
185#477—1631no—
186#468—1631no—
187#467—1631no—
188#464—1631no—
189#452—1631no—
190#451D16311yescharacteristic, complemented
191#441—1631no—
192#435—1631no—
193#419—1631no—
194#416—1631no—
195C2×#41—1631no—
196C22×Q8C2×D16311yescharacteristic
197C2×Q16C2×D16311yescharacteristic
198C2×Q16C2×D16311yescharacteristic
199C2×SD16—1631no—
200C2×M16—1631no—
201C22×C8—1631no—
202C4⋊Q8—1631no—
203#126—3261no—
204#126—3261no—
205#124—3261no—
206#124—3261no—
207#124—1631no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8C2×D16311yescharacteristic
212#121—3261no—
213#121—1631no—
214C4×Q8—3261no—
215C4×Q8—1631no—
216C4×Q8—1631no—
217C4×D4—3261no—
218C4×D4—1631no—
219#117—1631no—
220#117—1631no—
221#107—1631no—
222#107—1631no—
223#106—1631no—
224C4⋊C8—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic
231#103C2×D16311yescharacteristic
232#102—1631no—
233#102—1631no—
234#102—1631no—
235C22⋊C8—1631no—
236C22⋊C8C2×D16311yescharacteristic
237#97—1631no—
238#41—3262no—
239#41—6521no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—3261no—
244C2×Q8C22×D16311yescharacteristic
245C2×Q8C22×D16311yescharacteristic
246C2×Q8C22×D16311yescharacteristic
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4—1631no—
250C2×D4—1631no—
251C22×C4—21no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—6521no—
258M16—6521no—
259C2×C8—6521no—
260C2×C8—1631no—
261C2×C8—1631no—
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—1631no—
284C22⋊C4—1631no—
285C42—3261no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C23—3261no—
291C23—1631no—
292C23D4×D16311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—6522no—
300Q8#1457969812yes—
301D4—3261no—
302D4—3261no—
303D4—6522no—
304D4—6521no—
305D4—6522no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—6522no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4D4×D16311yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3261no—
327C8—3261no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—3261no—
332C22—3261no—
333C22—3261no—
334C22—6521no—
335C22—6521no—
336C22—6521no—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3261no—
344C4—3261no—
345C4—3261no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4C2×#1457969811yescharacteristic
350C4#1910478011yescharacteristic
351C2—41no—
352C2—13041no—
353C2—6521no—
354C2#2378230211yescharacteristic
355C2#2378246411yescharacteristic
356C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#451×C163C2
socleC22×C163C22×D4
prime radical#23782326C2
hypercenter#451D163
hypocenterC163#3313
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #23782386C2semidirect
3. #23782026C2semidirect
4. #23782004C2semidirect
5. #23781990C2semidirect
6. #23781958C2semidirect
7. #23781913C2semidirect
8. #23781903C2semidirect
9. #23781889C2semidirect
10. #23781826C2semidirect
11. #23781772C2semidirect
12. #23781739C2semidirect
13. #23781674C2semidirect
14. #23781660C2semidirect
15. #451×C163C2semidirect
178. C163#3313semidirect
190. #451D163semidirect

Maximal subgroups

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

subgroupindex
#237823862
#237823262
#237820262
#237820042
#237819902
#237819582
#237819132
#237819032
#237818892
#237818262
#237817722
#237817392
#237816742
#237816602
#451×C1632
#3313163