Finite Group Databasebeta

← #35226425

Subgroups — #35226425

Order 20864.

386 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#23782355C211yesmaximal, characteristic, complemented
2#23782324C211yesmaximal, characteristic
3#23782020C211yesmaximal, characteristic, complemented
4#23782007C211yesmaximal, characteristic, complemented
5#23781979C211yesmaximal, characteristic, complemented
6#23781973C211yesmaximal, characteristic, complemented
7#23781906C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781888C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12C4×Dic652C211yesmaximal, characteristic, complemented
13#23781662C211yesmaximal, characteristic, complemented
14C2×#19104660C211yesmaximal, characteristic, complemented
15C8×D4×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C2×Dic652C2211yescharacteristic
19C2×Dic652C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21#19104728—21no—
22#19104727C2211yescharacteristic
23#19104725—21no—
24#19104723C2211yescharacteristic
25#19104721—21no—
26#19104716—21no—
27#19104714C2211yescharacteristic
28#19104714C2211yescharacteristic
29#19104714C2211yescharacteristic
30#19104713—21no—
31#19104709C2211yescharacteristic
32#19104707—21no—
33C4×Dic326C2211yescharacteristic
34C4×Dic326C2211yescharacteristic
35(C22⋊D163)⋊C4—21no—
36#19104694C2211yescharacteristic
37#19104691—21no—
38C2×C4⋊Dic163C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104669C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104660C2211yescharacteristic
47#19104660C2211yescharacteristic
48#19104660C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic
52C22×C8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic
56C22⋊C8×C163C2211yescharacteristic
57C22⋊C8×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—41no—
61C2×Dic326—22no—
62C2×Dic326—21no—
63C2×Dic326—41no—
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C22×Dic163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73Dic652—22no—
74Dic652—41no—
75#14579688—41no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C2×D4×C163D411yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C2×C8×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C22⋊C4×C163D411yescharacteristic
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—21no—
117Dic326—42no—
118Dic326#4112yes—
119D652—21no—
120D652—21no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—42no—
131C2×Dic163—42no—
132C4×D163—42no—
133C23×C163C2×D411yescharacteristic
134C23×C163C2×D411yescharacteristic
135D4×C163—22no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2411yescharacteristic
145C8×C163—21no—
146C8×C163—21no—
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163#13512yes—
155C22×C163C22×D411yescharacteristic
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—21no—
163C4×C163C2×#4111yescharacteristic
164C4×C163H(5,𝔽2)11yescharacteristic
165Dic163—22no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—22no—
173C2×C163—41no—
174C2×C163#54611yescharacteristic
175C2×C163C2×#13511yescharacteristic
176C2×C163#57811yescharacteristic
177D163—81no—
178C163#330011yescharacteristic, complemented, Sylow
179#3300—1631nomaximal, Sylow
180#552—1631no—
181#548—1631no—
182#488—1631no—
183#485—1631no—
184#480—1631no—
185#479—1631no—
186#467—1631no—
187#465—1631no—
188#464—1631no—
189#452—1631no—
190#451—1631no—
191C4×Q16—1631no—
192C8×D4D16311yescharacteristic, complemented
193#416—1631no—
194C2×#103—1631no—
195C2×#41—1631no—
196C22×Q8—1631no—
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16—1631no—
200C22×C8C2×D16311yescharacteristic
201C22×C8C2×D16311yescharacteristic
202#126—3261no—
203#125—3261no—
204#125—1631no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8—1631no—
212#121—3261no—
213#121—1631no—
214C4×Q8—1631no—
215C4×Q8—1631no—
216C4×D4—3261no—
217C4×D4C2×D16311yescharacteristic
218#117—3261no—
219#117—1631no—
220C2×C4⋊C4—1631no—
221#107—1631no—
222#107—1631no—
223#106—1631no—
224C4⋊C8C2×D16311yescharacteristic
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#103—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235C22⋊C8C2×D16311yescharacteristic
236C22⋊C8C2×D16311yescharacteristic
237C4×C8C2×D16311yescharacteristic
238#41—3262no—
239#41—6521no—
240C2×Q8—3262no—
241C2×Q8—3261no—
242C2×Q8—6521no—
243C2×Q8—1631no—
244C2×Q8—1631no—
245C2×Q8—1631no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4D65211yescharacteristic
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4C22×D16311yescharacteristic
254C22×C4C22×D16311yescharacteristic
255Q16—3262no—
256Q16—6521no—
257SD16—6521no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D16311yescharacteristic
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4D65211yescharacteristic
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4D65211yescharacteristic
285C22⋊C4D65211yescharacteristic
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42C22×D16311yescharacteristic
290C23—3261no—
291C23C2×D65211yescharacteristic
292C23C2×D65211yescharacteristic
293Q8—3261no—
294Q8—3261no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—6522no—
299Q8—1632no—
300D4—22no—
301D4—3261no—
302D4—3261no—
303D4—6522no—
304D4—6522no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3261no—
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—6522no—
319C2×C4C2×D65211yescharacteristic
320C2×C4C2×D65211yescharacteristic
321C2×C4C2×D65211yescharacteristic
322C2×C4C2×D65211yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—6521no—
334C22—6521no—
335C22—6521no—
336C22#1910475712yes—
337C22C22×D65211yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—3262no—
343C4—6521no—
344C4—6521no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4#41×D16311yescharacteristic
350C4#1910478011yescharacteristic
351C2—22no—
352C2—41no—
353C2—13041no—
354C2#2378231011yescharacteristic
355C2C2×#1910475711yescharacteristic
356C2#2378248211yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC8×D4×C163C2
socleC22×C163C22×D4
prime radical#23782324C2
hypercenterC8×D4D163
hypocenterC163#3300
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23782355C2semidirect
3. #23782020C2semidirect
4. #23782007C2semidirect
5. #23781979C2semidirect
6. #23781973C2semidirect
7. #23781906C2semidirect
8. #23781894C2semidirect
9. #23781888C2semidirect
10. #23781829C2semidirect
11. #23781823C2semidirect
12. C4×Dic652C2semidirect
13. #23781662C2semidirect
14. C2×#19104660C2semidirect
15. C8×D4×C163C2semidirect
178. C163#3300semidirect
192. C8×D4D163semidirect

Maximal subgroups

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

subgroupindex
#237823552
#237823242
#237820202
#237820072
#237819792
#237819732
#237819062
#237818942
#237818882
#237818292
#237818232
C4×Dic6522
#237816622
C2×#191046602
C8×D4×C1632
#3300163