Finite Group Databasebeta

← #35225222

Subgroups — #35225222

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#23782282C211yesmaximal, characteristic, complemented
2#23782268C211yesmaximal, characteristic, complemented
3#23782073C211yesmaximal, characteristic, complemented
4#23782053C211yesmaximal, characteristic, complemented
5#23782018C211yesmaximal, characteristic, complemented
6#23782010C211yesmaximal, characteristic, complemented
7#23781916C211yesmaximal, characteristic, complemented
8#23781909C211yesmaximal, characteristic, complemented
9#23781894C211yesmaximal, characteristic, complemented
10#23781777C211yesmaximal, characteristic, complemented
11#23781764C211yesmaximal, characteristic, complemented
12#23781756C211yesmaximal, characteristic, complemented
13#23781745C211yesmaximal, characteristic, complemented
14#23781620C211yesmaximal, characteristic, complemented
15#466×C163C211yesmaximal, characteristic, complemented
16C2×#14579688C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579683C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104731C2211yescharacteristic
22#19104728—21no—
23#19104726C2211yescharacteristic
24#19104722—21no—
25#19104722C2211yescharacteristic
26#19104719—21no—
27#19104717—21no—
28#19104716—21no—
29#19104714C2211yescharacteristic
30#19104712—21no—
31#19104709C2211yescharacteristic
32#19104707—21no—
33#19104706—21no—
34#19104700C2211yescharacteristic
35D4×Dic163C2211yescharacteristic
36D4×Dic163C2211yescharacteristic
37#19104697C2211yescharacteristic
38#19104694C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104667C2211yescharacteristic
41#19104666C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104645C2211yescharacteristic
49#19104645C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×D8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54#121×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C22×Dic163—21no—
66C22×Dic163—21no—
67C2×C4×D163—21no—
68#14579688—22no—
69#14579687—22no—
70#14579684—22no—
71#14579683—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163D411yescharacteristic
76C4⋊Dic163D411yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163D411yescharacteristic
89C22⋊Dic163D411yescharacteristic
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92C4×Dic163—21no—
93C4×Dic163C2311yescharacteristic
94C4×Dic163C2311yescharacteristic
95C2×D4×C163—21no—
96C2×D4×C163—21no—
97C2×D4×C163C2311yescharacteristic
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100D8×C163—22no—
101C2×C8×C163—22no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C22×D163—21no—
109Dic326—21no—
110Dic326—21no—
111Dic326—81no—
112D652—21no—
113D652—21no—
114C22⋊D163—81no—
115C22⋊D163—81no—
116C22⋊D163—81no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—81no—
121C2×Dic163—81no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—81no—
127C163⋊C8—21no—
128C163⋊C8#4112yes—
129C23×C163—21no—
130C23×C163—21no—
131C23×C163C2×D411yescharacteristic
132D4×C163—21no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—81no—
137D4×C163—81no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2411yescharacteristic
144C8×C163—21no—
145C8×C163#4112yes—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C4×C163—21no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C2×#4111yescharacteristic
163C4×C163C2×#4111yescharacteristic
164Dic163—21no—
165Dic163—21no—
166Dic163—21no—
167Dic163—21no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171C2×C163—41no—
172C2×C163—81no—
173C2×C163—81no—
174C2×C163#53211yescharacteristic
175C2×C163#57611yescharacteristic
176C2×C163#57711yescharacteristic
177D163—81no—
178C163#317911yescharacteristic, complemented, Sylow
179#3179—1631nomaximal, Sylow
180#542—1631no—
181#540—1631no—
182#501—1631no—
183#496—1631no—
184#488—1631no—
185#486—1631no—
186#469—1631no—
187#468—1631no—
188#466D16311yescharacteristic, complemented
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×D8C2×D16311yescharacteristic
199C2×M16—1631no—
200C22×C8C2×D16311yescharacteristic
201C4⋊Q8—1631no—
202#127—1631no—
203#126—3261no—
204#125—1631no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208#123—3261no—
209#123—3261no—
210C22⋊Q8—3261no—
211C22⋊Q8—1631no—
212#121—3261no—
213#121—3261no—
214#121—1631no—
215#121C2×D16311yescharacteristic
216#121C2×D16311yescharacteristic
217C4×Q8—1631no—
218C4×D4—1631no—
219C4×D4—1631no—
220C4×D4—1631no—
221#117—1631no—
222#107—1631no—
223#107C2×D16311yescharacteristic
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#102C2×D16311yescharacteristic
233#102C2×D16311yescharacteristic
234#97—1631no—
235#97—1631no—
236C4×C8—1631no—
237C4×C8—1631no—
238C2×Q8—3261no—
239C2×Q8—1631no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—3261no—
243C2×D4—3261no—
244C2×D4—3261no—
245C2×D4—1631no—
246C2×D4C22×D16311yescharacteristic
247C2×D4C22×D16311yescharacteristic
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4C22×D16311yescharacteristic
252SD16—3262no—
253SD16—3262no—
254D8—22no—
255D8—3262no—
256M16—3262no—
257C2×C8—22no—
258C2×C8—1631no—
259C2×C8—1631no—
260C2×C8C22×D16311yescharacteristic
261C2×C8C22×D16311yescharacteristic
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4C22×D16311yescharacteristic
272C4⋊C4C22×D16311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—1631no—
283C22⋊C4—1631no—
284C42—3261no—
285C42—1631no—
286C42—1631no—
287C23—21no—
288C23—21no—
289C23—3261no—
290C23D4×D16311yescharacteristic
291Q8—13041no—
292Q8—3261no—
293Q8—3261no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—81no—
299D4—81no—
300D4—13041no—
301D4—13041no—
302D4—13041no—
303D4—3261no—
304D4—3261no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—13041no—
310C2×C4—13041no—
311C2×C4—13041no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—1631no—
316C2×C4—1631no—
317C2×C4—1631no—
318C2×C4—1631no—
319C2×C4D4×D16311yescharacteristic
320C2×C4C23×D16311yescharacteristic
321C8—21no—
322C8—3261no—
323C8—1632no—
324C8#1457969812yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—6521no—
335C22—6521no—
336C22—6521no—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—3261no—
342C4—3261no—
343C4—3261no—
344C4—3261no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4#41×D16311yescharacteristic
349C4C2×#1457969811yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—13041no—
354C2#2378221011yescharacteristic
355C2#2378246611yescharacteristic
356C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#466×C163C2
socleC22×C163C22×D4
hypercenter#466D163
hypocenterC163#3179
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #23782282C2semidirect
2. #23782268C2semidirect
3. #23782073C2semidirect
4. #23782053C2semidirect
5. #23782018C2semidirect
6. #23782010C2semidirect
7. #23781916C2semidirect
8. #23781909C2semidirect
9. #23781894C2semidirect
10. #23781777C2semidirect
11. #23781764C2semidirect
12. #23781756C2semidirect
13. #23781745C2semidirect
14. #23781620C2semidirect
15. #466×C163C2semidirect
178. C163#3179semidirect
188. #466D163semidirect

Maximal subgroups

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

subgroupindex
#237822822
#237822682
#237820732
#237820532
#237820182
#237820102
#237819162
#237819092
#237818942
#237817772
#237817642
#237817562
#237817452
#237816202
#466×C1632
#3179163