Finite Group Databasebeta

← #35226388

Subgroups — #35226388

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782361C211yesmaximal, characteristic, complemented
2#23782311C211yesmaximal, characteristic, complemented
3#23782020C211yesmaximal, characteristic, complemented
4#23782001C211yesmaximal, characteristic, complemented
5#23781990C211yesmaximal, characteristic, complemented
6#23781966C211yesmaximal, characteristic, complemented
7#23781901C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781881C211yesmaximal, characteristic, complemented
10#23781812C211yesmaximal, characteristic, complemented
11#23781801C211yesmaximal, characteristic, complemented
12C4×D1304C211yesmaximal, characteristic, complemented
13#23781662C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15C8×D4×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579688C2211yescharacteristic, complemented
19C2×D1304C2211yescharacteristic
20C2×D1304C2211yescharacteristic
21#19104730—21no—
22#19104727C2211yescharacteristic, complemented
23#19104723C2211yescharacteristic
24#19104720—21no—
25#19104717—21no—
26#19104714C2211yescharacteristic
27#19104709C2211yescharacteristic
28#19104709C2211yescharacteristic
29#19104709C2211yescharacteristic, complemented
30#19104708—21no—
31#19104707—21no—
32#19104703—21no—
33C4×D652C2211yescharacteristic
34C4×D652C2211yescharacteristic, complemented
35(C22⋊D163)⋊C4—21no—
36#19104694C2211yescharacteristic, complemented
37C2×C4⋊Dic163C2211yescharacteristic
38C22⋊C4×D163—21no—
39#19104669C2211yescharacteristic, complemented
40#19104669C2211yescharacteristic, complemented
41#19104667C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic, complemented
52C22×C8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic, complemented
54C4×D4×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic, complemented
56C22⋊C8×C163C2211yescharacteristic
57C22⋊C8×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326C2311yescharacteristic
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—41no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C22×Dic163—21no—
74C2×C4×D163—21no—
75C2×C4×D163—21no—
76C2×C4×D163—21no—
77#14579688—41no—
78D1304—22no—
79D1304—41no—
80C4⋊Dic163—21no—
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—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—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—42no—
115C22×D163—42no—
116C22×D163—81no—
117Dic326—21no—
118Dic326—21no—
119D652—21no—
120D652—21no—
121D652—21no—
122D652—21no—
123D652—21no—
124D652—42no—
125D652#4112yes—
126C22⋊D163—42no—
127C22⋊D163—42no—
128C22⋊D163—81no—
129C22⋊D163—81no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—81no—
136C4×D163—42no—
137C4×D163—81no—
138C4×D163—81no—
139C23×C163C2×D411yescharacteristic
140C23×C163C2×D411yescharacteristic
141D4×C163—22no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2411yescharacteristic
151C8×C163—21no—
152C8×C163—21no—
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—21no—
159C22×C163—21no—
160C22×C163#13512yes—
161C22×C163C22×D411yescharacteristic
162C2×D163—22no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—41no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—84no—
172C4×C163—21no—
173C4×C163—21no—
174C4×C163—21no—
175C4×C163—21no—
176C4×C163C2×#4111yescharacteristic
177C4×C163H(5,𝔽2)11yescharacteristic
178Dic163—41no—
179Dic163—41no—
180Dic163—41no—
181Dic163—41no—
182Dic163—41no—
183C2×C163—22no—
184C2×C163—41no—
185C2×C163#54611yescharacteristic
186C2×C163C2×#13511yescharacteristic
187C2×C163#57611yescharacteristic
188D163—42no—
189D163—81no—
190D163—81no—
191C163#329711yescharacteristic, complemented, Sylow
192#3297—1631nomaximal, Sylow
193#553—1631no—
194#546—1631no—
195#488—1631no—
196#484—1631no—
197#482—1631no—
198#478—1631no—
199#466—1631no—
200#465—1631no—
201#463—1631no—
202#449—1631no—
203#447—1631no—
204C4×D8—1631no—
205C8×D4D16311yescharacteristic, complemented
206#416—1631no—
207C2×#102—1631no—
208C2×#41—1631no—
209C22×D4—1631no—
210C2×SD16—1631no—
211C2×D8—1631no—
212C2×D8—1631no—
213C22×C8C2×D16311yescharacteristic
214C22×C8C2×D16311yescharacteristic, complemented
215#126—3261no—
216#125—1631no—
217#124—1631no—
218#123—3261no—
219#123—3261no—
220C22⋊Q8—1631no—
221#121—3261no—
222#121—3261no—
223#121—1631no—
224#121—1631no—
225#121—1631no—
226#120—3261no—
227C4×D4—3261no—
228C4×D4—1631no—
229C4×D4—1631no—
230C4×D4C2×D16311yescharacteristic
231#117—1631no—
232C2×C4⋊C4—1631no—
233C2×C22⋊C4—3261no—
234#107—1631no—
235#107—1631no—
236#106—1631no—
237C4⋊C8C2×D16311yescharacteristic, complemented
238#103—1631no—
239#103—1631no—
240#103—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248C22⋊C8C2×D16311yescharacteristic
249C22⋊C8C2×D16311yescharacteristic, complemented
250C4×C8C2×D16311yescharacteristic, complemented
251C24—3261no—
252#41—3262no—
253#41—6521no—
254C2×Q8—1631no—
255C2×D4—3262no—
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—6521no—
262C2×D4—1631no—
263C2×D4—1631no—
264C2×D4—1631no—
265C2×D4D65211yescharacteristic
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4C22×D16311yescharacteristic
271C22×C4C22×D16311yescharacteristic
272SD16—6521no—
273D8—3262no—
274D8—6521no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D16311yescharacteristic
278C2×C8C22×D16311yescharacteristic
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4—1631no—
288C4⋊C4—1631no—
289C4⋊C4D65211yescharacteristic
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—3261no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4D65211yescharacteristic
300C22⋊C4D65211yescharacteristic
301C42—3261no—
302C42C22×D16311yescharacteristic
303C23—13041no—
304C23—3261no—
305C23—3261no—
306C23—3261no—
307C23—6522no—
308C23—6522no—
309C23C2×D65211yescharacteristic
310C23C2×D65211yescharacteristic
311Q8—3261no—
312Q8—3261no—
313D4—22no—
314D4—13041no—
315D4—13041no—
316D4—3261no—
317D4—3261no—
318D4—3261no—
319D4—3261no—
320D4—3261no—
321D4—6522no—
322D4—6522no—
323D4—6522no—
324D4—1632no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—13041no—
330C2×C4—13041no—
331C2×C4—13041no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—6522no—
338C2×C4C2×D65211yescharacteristic
339C2×C4C2×D65211yescharacteristic
340C2×C4C2×D65211yescharacteristic
341C2×C4C2×D65211yescharacteristic
342C2×C4C23×D16311yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—13044no—
353C22—3262no—
354C22—6521no—
355C22—6521no—
356C22—6521no—
357C22—6521no—
358C22—6521no—
359C22—6521no—
360C22—6521no—
361C22—6521no—
362C22#1910475712yes—
363C22C22×D65211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—6521no—
369C4—6521no—
370C4—6521no—
371C4—6521no—
372C4—6521no—
373C4#41×D16311yescharacteristic
374C4#1910478011yescharacteristic
375C2—22no—
376C2—41no—
377C2—13041no—
378C2—13041no—
379C2—6522no—
380C2#2378231011yescharacteristic
381C2C2×#1910475711yescharacteristic
382C2#2378247011yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC8×D4×C163C2
socleC22×C163C22×D4
hypercenterC8×D4D163
hypocenterC163#3297
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. #23782361C2semidirect
2. #23782311C2semidirect
3. #23782020C2semidirect
4. #23782001C2semidirect
5. #23781990C2semidirect
6. #23781966C2semidirect
7. #23781901C2semidirect
8. #23781894C2semidirect
9. #23781881C2semidirect
10. #23781812C2semidirect
11. #23781801C2semidirect
12. C4×D1304C2semidirect
13. #23781662C2semidirect
14. C2×#19104657C2semidirect
15. C8×D4×C163C2semidirect
18. C2×#14579688C22semidirect
22. #19104727C22semidirect
29. #19104709C22semidirect
34. C4×D652C22semidirect
36. #19104694C22semidirect
39. #19104669C22semidirect
40. #19104669C22semidirect
41. #19104667C22semidirect
42. #19104660C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
51. #19104657C22semidirect
53. C22×C8×C163C22semidirect
55. C4⋊C8×C163C22semidirect
57. C22⋊C8×C163C22semidirect
58. C4×C8×C163C22semidirect
191. C163#3297semidirect
205. C8×D4D163semidirect
214. C22×C8C2×D163semidirect
237. C4⋊C8C2×D163semidirect
249. C22⋊C8C2×D163semidirect
250. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823612
#237823112
#237820202
#237820012
#237819902
#237819662
#237819012
#237818942
#237818812
#237818122
#237818012
C4×D13042
#237816622
C2×#191046572
C8×D4×C1632
#3297163