Finite Group Databasebeta

← #35225213

Subgroups — #35225213

Order 20864.

374 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#23782270C211yesmaximal, characteristic, complemented
2#23782071C211yesmaximal, characteristic, complemented
3#23782055C211yesmaximal, characteristic, complemented
4#23782019C211yesmaximal, characteristic, complemented
5#23782013C211yesmaximal, characteristic, complemented
6#23781916C211yesmaximal, characteristic, complemented
7#23781913C211yesmaximal, characteristic, complemented
8#23781899C211yesmaximal, characteristic, complemented
9#23781896C211yesmaximal, characteristic, complemented
10#23781775C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12C4×#14579686C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14#23781621C211yesmaximal, characteristic, complemented
15#542×C163C211yesmaximal, characteristic, complemented
16C2×#14579686C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579683C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22#19104732C2211yescharacteristic
23#19104727C2211yescharacteristic
24#19104720—21no—
25#19104720—21no—
26#19104709C2211yescharacteristic
27#19104709C2211yescharacteristic
28#19104708—21no—
29#19104708—21no—
30C4×D652C2211yescharacteristic
31C4×D652C2211yescharacteristic
32#19104668C2211yescharacteristic
33#19104668C2211yescharacteristic
34#19104666C2211yescharacteristic
35#19104666C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104656C2211yescharacteristic
39#19104656C2211yescharacteristic
40#19104655C2211yescharacteristic
41#19104655C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104646C2211yescharacteristic
45#19104646C2211yescharacteristic
46C4×C163⋊C8C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48C4⋊Q8×C163C2211yescharacteristic
49#126×C163—21no—
50#124×C163—21no—
51#124×C163C2211yescharacteristic
52#123×C163—21no—
53C22⋊Q8×C163—21no—
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic
58#117×C163C2211yescharacteristic
59C2×D652—21no—
60C2×D652—21no—
61C2×D652C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67#14579686—22no—
68#14579686—22no—
69#14579684—22no—
70#14579684—22no—
71#14579683—22no—
72C2×C163⋊C8—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85C2×Q8×C163—21no—
86C2×Q8×C163C2311yescharacteristic
87C2×D4×C163—21no—
88C2×D4×C163C2311yescharacteristic
89C22×C4×C163—21no—
90C22×C4×C163C2311yescharacteristic
91C4⋊C4×C163—21no—
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163D411yescharacteristic
95C4⋊C4×C163D411yescharacteristic
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163D411yescharacteristic
104C22⋊C4×C163D411yescharacteristic
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—21no—
110D652—21no—
111D652—21no—
112D652—21no—
113D652—21no—
114D652—81no—
115D652—81no—
116C22⋊D163—81no—
117C22⋊D163—81no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C4×D163—81no—
121C4×D163—81no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C163⋊C8#4112yes—
125C163⋊C8#4112yes—
126C23×C163—21no—
127C23×C163C2×D411yescharacteristic
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163—81no—
131D4×C163—21no—
132D4×C163—21no—
133D4×C163—81no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—81no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C22×C163—21no—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163C22×D411yescharacteristic
153C2×D163—41no—
154C2×D163—41no—
155C2×D163—41no—
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—21no—
163C4×C163—21no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163C2×#4111yescharacteristic
168C4×C163C2×#4111yescharacteristic
169Dic163—41no—
170Dic163—41no—
171C2×C163—41no—
172C2×C163—81no—
173C2×C163#53211yescharacteristic
174C2×C163#57611yescharacteristic
175C2×C163#57711yescharacteristic
176D163—81no—
177D163—81no—
178C163#317911yescharacteristic, complemented, Sylow
179#3179—1631nomaximal, Sylow
180#542D16311yescharacteristic, complemented
181#540—1631no—
182#501—1631no—
183#496—1631no—
184#488—1631no—
185#486—1631no—
186#469—1631no—
187#468—1631no—
188#466—1631no—
189#465—1631no—
190#442—1631no—
191#440—1631no—
192C4×SD16—1631no—
193C4×D8—1631no—
194#405—1631no—
195C2×SD16—1631no—
196C2×SD16—1631no—
197C2×D8—1631no—
198C2×D8—1631no—
199C2×M16—1631no—
200C22×C8—1631no—
201C4⋊Q8C2×D16311yescharacteristic
202#127—1631no—
203#126—21no—
204#125—1631no—
205#124—21no—
206#124C2×D16311yescharacteristic
207#123—21no—
208#123—3261no—
209#123—3261no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D16311yescharacteristic
212#121—3261no—
213#121—3261no—
214#121—1631no—
215#121—1631no—
216#121C2×D16311yescharacteristic
217C4×Q8C2×D16311yescharacteristic
218C4×D4—1631no—
219C4×D4—1631no—
220C4×D4C2×D16311yescharacteristic
221#117C2×D16311yescharacteristic
222#107—1631no—
223#107—1631no—
224#106—1631no—
225#106—1631no—
226#103—1631no—
227#103—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#97—1631no—
235#97—1631no—
236C4×C8—1631no—
237C4×C8—1631no—
238C2×Q8—21no—
239C2×Q8C22×D16311yescharacteristic
240C2×D4—21no—
241C2×D4—3261no—
242C2×D4—3261no—
243C2×D4—3261no—
244C2×D4—3261no—
245C2×D4—1631no—
246C2×D4—1631no—
247C2×D4C22×D16311yescharacteristic
248C22×C4—21no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4C22×D16311yescharacteristic
252SD16—3262no—
253SD16—3262no—
254D8—3262no—
255D8—3262no—
256M16—3262no—
257C2×C8—3262no—
258C2×C8—1631no—
259C2×C8—1631no—
260C2×C8—1631no—
261C2×C8—1631no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4C22⋊D16311yescharacteristic
270C4⋊C4C22⋊D16311yescharacteristic
271C4⋊C4C22×D16311yescharacteristic
272C4⋊C4C22×D16311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4C22⋊D16311yescharacteristic
283C22⋊C4C22⋊D16311yescharacteristic
284C42—21no—
285C42C22×D16311yescharacteristic
286C42C22×D16311yescharacteristic
287C23—21no—
288C23—3261no—
289C23—3261no—
290C23C2×C22⋊D16311yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—81no—
297D4—13041no—
298D4—13041no—
299D4—13041no—
300D4—13041no—
301D4—3261no—
302D4—3261no—
303D4—3261no—
304D4—3261no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—13041no—
312C2×C4—13041no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4C2×C22⋊D16311yescharacteristic
316C2×C4C2×C22⋊D16311yescharacteristic
317C2×C4C2×C22⋊D16311yescharacteristic
318C2×C4C2×C22⋊D16311yescharacteristic
319C2×C4C2×C22⋊D16311yescharacteristic
320C2×C4C23×D16311yescharacteristic
321C8—3261no—
322C8—3261no—
323C8—1632no—
324C8—1632no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—6521no—
332C22—6521no—
333C22—6521no—
334C22—6521no—
335C22—6521no—
336C22—6521no—
337C22C22×C22⋊D16311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—6521no—
347C4—6521no—
348C4#41×D16311yescharacteristic
349C4#41×D16311yescharacteristic
350C2—41no—
351C2—81no—
352C2—13041no—
353C2—13041no—
354C2#2378221111yescharacteristic
355C2#2378246811yescharacteristic
356C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#542×C163C2
socleC22×C163C22×D4
hypercenter#542D163
hypocenterC163#3179
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #23782270C2semidirect
2. #23782071C2semidirect
3. #23782055C2semidirect
4. #23782019C2semidirect
5. #23782013C2semidirect
6. #23781916C2semidirect
7. #23781913C2semidirect
8. #23781899C2semidirect
9. #23781896C2semidirect
10. #23781775C2semidirect
11. #23781763C2semidirect
12. C4×#14579686C2semidirect
13. C4×#14579684C2semidirect
14. #23781621C2semidirect
15. #542×C163C2semidirect
178. C163#3179semidirect
180. #542D163semidirect

Maximal subgroups

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

subgroupindex
#237822702
#237820712
#237820552
#237820192
#237820132
#237819162
#237819132
#237818992
#237818962
#237817752
#237817632
C4×#145796862
C4×#145796842
#237816212
#542×C1632
#3179163