Finite Group Databasebeta

← #35217383

Subgroups — #35217383

Order 20864.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104693C211yesmaximal, characteristic, complemented
2C22×(C2×D163)⋊C4C211yesmaximal, characteristic
3C2×C22⋊C4×D163C211yesmaximal, characteristic
4#23781557C211yesmaximal, characteristic, complemented
5#23781557C211yesmaximal, characteristic, complemented
6#23781555C211yesmaximal, characteristic, complemented
7#23781555C211yesmaximal, characteristic, complemented
8#23781540C211yesmaximal, characteristic, complemented
9#23781527C211yesmaximal, characteristic, complemented
10#23781522C211yesmaximal, characteristic, complemented
11#23781520C211yesmaximal, characteristic
12C22⋊C4×Dic163C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic163C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#385×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20#19104693—24no—
21C2×C4⋊Dic163C2211yescharacteristic, complemented
22C2×C4⋊Dic163C2211yescharacteristic, complemented
23C2×C22⋊Dic163C2211yescharacteristic
24C2×C22⋊Dic163C2211yescharacteristic
25C2×(C2×D163)⋊C4—24no—
26C2×(C2×D163)⋊C4C2211yescharacteristic
27C2×(C2×D163)⋊C4C2211yescharacteristic
28C2×(C2×D163)⋊C4C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C22⋊C4×D163C418yescomplemented
37C2×C4×Dic163C2211yescharacteristic, complemented
38C2×C4×Dic163C2211yescharacteristic, complemented
39C2×C4×Dic163C2211yescharacteristic, complemented
40C2×C4×Dic163C2211yescharacteristic, complemented
41C163⋊#95C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C23×C4×C163C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic, complemented
49C2×C4⋊C4×C163C2211yescharacteristic, complemented
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic, complemented
53#95×C163C2211yescharacteristic, complemented
54C23×D163—22no—
55C23×D163—22no—
56C23×D163—22no—
57C23×D163—22no—
58C23×D163—41no—
59C23×D163—41no—
60C23×D163C2×C414yes—
61C23×D163C2311yescharacteristic
62C23×D163C2311yescharacteristic
63C22×Dic163—22no—
64C22×Dic163—22no—
65C22×Dic163—41no—
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163—41no—
75C2×C4×D163—41no—
76C2×C4×D163C2×C414yescomplemented
77C2×C4×D163C2×C414yescomplemented
78C4⋊Dic163—22no—
79C4⋊Dic163—41no—
80C22⋊Dic163—22no—
81C22⋊Dic163C2×C414yes—
82(C2×D163)⋊C4—22no—
83(C2×D163)⋊C4—22no—
84(C2×D163)⋊C4—22no—
85(C2×D163)⋊C4—22no—
86(C2×D163)⋊C4—22no—
87(C2×D163)⋊C4—41no—
88(C2×D163)⋊C4—41no—
89(C2×D163)⋊C4C2×C414yescomplemented
90(C2×D163)⋊C4C2×C414yescomplemented
91(C2×D163)⋊C4D414yes—
92C4×Dic163—22no—
93C4×Dic163—22no—
94C4×Dic163—41no—
95C4×Dic163—41no—
96C24×C163C2311yescharacteristic
97C22×C4×C163—22no—
98C22×C4×C163—22no—
99C22×C4×C163—41no—
100C22×C4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102C22×C4×C163C2311yescharacteristic
103C22×C4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105C22×C4×C163C2311yescharacteristic
106C4⋊C4×C163—22no—
107C4⋊C4×C163—41no—
108C22⋊C4×C163—22no—
109C22⋊C4×C163C2×C414yes—
110C22×D163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—28no—
115C22×D163—21no—
116C22×D163—21no—
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—28no—
122C22×D163—44no—
123C22×D163—44no—
124C22×D163—44no—
125C22×D163—44no—
126C22×D163—44no—
127C22×D163C22×C412yes—
128C22×D163C22×C412yes—
129C22×D163C2×D412yes—
130C22×D163#4112yes—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—21no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—44no—
147C2×Dic163C22×C412yes—
148C2×Dic163C22×C412yes—
149C2×Dic163C2×D412yes—
150C4×D163—28no—
151C4×D163—44no—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163C22×C412yes—
157C23×C163#4112yes—
158C23×C163#4112yes—
159C23×C163C2411yescharacteristic
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163—21no—
173C2×C4×C163—21no—
174C2×C4×C163—21no—
175C2×C4×C163—44no—
176C2×C4×C163C22×C412yes—
177C2×C4×C163C22×C412yes—
178C2×C4×C163C2×D412yes—
179C22×C163—22no—
180C22×C163—22no—
181C22×C163—22no—
182C22×C163—22no—
183C22×C163—22no—
184C22×C163—22no—
185C22×C163#11714yescomplemented
186C22×C163C23×C411yescharacteristic
187C22×C163C22×D411yescharacteristic
188C22×C163C2×#4111yescharacteristic
189C22×C163C2×#4111yescharacteristic
190C22×C163C2×#4111yescharacteristic
191C22×C163H(5,𝔽2)11yescharacteristic
192C22×C163H(5,𝔽2)11yescharacteristic
193C2×D163—22no—
194C2×D163—22no—
195C2×D163—22no—
196C2×D163—22no—
197C2×D163—22no—
198C2×D163—22no—
199C2×D163—22no—
200C2×D163—22no—
201C2×D163—22no—
202C2×D163—416no—
203C2×D163—41no—
204C2×D163—41no—
205C2×D163—41no—
206C2×D163—41no—
207C2×D163C4×D414yes—
208C4×C163—22no—
209C4×C163—22no—
210C4×C163—22no—
211C4×C163—41no—
212C4×C163—41no—
213C4×C163—41no—
214Dic163—22no—
215Dic163—22no—
216Dic163—22no—
217Dic163—41no—
218Dic163—41no—
219Dic163—41no—
220C2×C163—24no—
221C2×C163C2×#11711yescharacteristic
222C2×C163C2×C4×D411yescharacteristic
223C2×C163#51811yescharacteristic
224C2×C163#54611yescharacteristic
225C2×C163#54611yescharacteristic
226C2×C163#55111yescharacteristic
227C2×C163#55111yescharacteristic
228D163—24no—
229D163—42no—
230C163#234211yescharacteristic, complemented, Sylow
231#2342—1631nomaximal, Sylow
232C2×#117—1631no—
233C22×C22⋊C4—1631no—
234C22×C22⋊C4—1631no—
235#388—1631no—
236#388—1631no—
237#388—1631no—
238#388—1631no—
239#385—1631no—
240#385D16311yescharacteristic, complemented
241#381—1631no—
242#380—1631no—
243#379—1631no—
244C4×C22⋊C4—1631no—
245C4×C22⋊C4—1631no—
246C4×C22⋊C4—1631no—
247C25—1631no—
248C23×C4—1631no—
249C23×C4—1631no—
250C23×C4—1631no—
251C23×C4C2×D16311yescharacteristic
252#117—3264no—
253C2×C4⋊C4—1631no—
254C2×C4⋊C4—1631no—
255C2×C4⋊C4C2×D16311yescharacteristic, complemented
256C2×C4⋊C4C2×D16311yescharacteristic, complemented
257C2×C22⋊C4—3264no—
258C2×C22⋊C4—1631no—
259C2×C22⋊C4—1631no—
260C2×C22⋊C4—1631no—
261C2×C22⋊C4—1631no—
262C2×C22⋊C4—1631no—
263C2×C22⋊C4—1631no—
264C2×C22⋊C4—1631no—
265C2×C22⋊C4—1631no—
266C2×C22⋊C4—1631no—
267C2×C22⋊C4—1631no—
268C2×C22⋊C4—1631no—
269C2×C22⋊C4—1631no—
270C2×C22⋊C4—1638no—
271C2×C22⋊C4C2×D16311yescharacteristic
272C2×C22⋊C4C2×D16311yescharacteristic
273C2×C42—1631no—
274C2×C42—1631no—
275C2×C42—1631no—
276C2×C42—1631no—
277#95—1631no—
278#95—1631no—
279#95—1631no—
280#95—1631no—
281#95—1631no—
282#95—1631no—
283#95C2×D16311yescharacteristic, complemented
284#95C2×D16311yescharacteristic, complemented
285C24—3262no—
286C24—3262no—
287C24—3262no—
288C24—3262no—
289C24—6521no—
290C24—6521no—
291C24—1631no—
292C24—1634no—
293C24—1631no—
294C24C22×D16311yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—3262no—
299C22×C4—3262no—
300C22×C4—3262no—
301C22×C4—3262no—
302C22×C4—6521no—
303C22×C4—6521no—
304C22×C4—6521no—
305C22×C4—1631no—
306C22×C4—1631no—
307C22×C4—1631no—
308C22×C4—1631no—
309C22×C4—1631no—
310C22×C4—1631no—
311C22×C4—1634no—
312C22×C4—1634no—
313C22×C4C22×D16311yescharacteristic
314C22×C4C22×D16311yescharacteristic
315C22×C4C22×D16311yescharacteristic
316C22×C4C22×D16311yescharacteristic
317C22×C4C22×D16311yescharacteristic
318C22×C4C22×D16311yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—3262no—
322C4⋊C4—6521no—
323C22⋊C4—22no—
324C22⋊C4—3262no—
325C22⋊C4—3262no—
326C22⋊C4—3262no—
327C22⋊C4—3262no—
328C22⋊C4—3262no—
329C22⋊C4—3262no—
330C22⋊C4—6521no—
331C22⋊C4—6521no—
332C22⋊C4—1634no—
333C22⋊C4—1634no—
334C22⋊C4—1634no—
335C22⋊C4—1634no—
336C22⋊C4C4×D16314yes—
337C42—3262no—
338C42—3262no—
339C42—6521no—
340C42—6521no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—3268no—
346C23—3261no—
347C23—3261no—
348C23—3261no—
349C23—3261no—
350C23—3261no—
351C23—3261no—
352C23—3261no—
353C23—3261no—
354C23—3261no—
355C23—3261no—
356C23—3268no—
357C23—6524no—
358C23—6524no—
359C23—6524no—
360C23—6524no—
361C23—6524no—
362C23—1632no—
363C23—1632no—
364C23—1632no—
365C23—1632no—
366C23C2×C4×D16312yes—
367C23#1457969812yes—
368C23#1457970012yes—
369C23C23×D16311yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—3268no—
387C2×C4—3261no—
388C2×C4—3261no—
389C2×C4—3261no—
390C2×C4—3261no—
391C2×C4—3261no—
392C2×C4—3261no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—3261no—
397C2×C4—3261no—
398C2×C4—3261no—
399C2×C4—3261no—
400C2×C4—3261no—
401C2×C4—3261no—
402C2×C4—6524no—
403C2×C4—6524no—
404C2×C4—1632no—
405C2×C4—1632no—
406C2×C4—1632no—
407C2×C4C2×C4×D16312yes—
408C2×C4C2×C4×D16312yes—
409C2×C4D4×D16312yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—3262no—
417C22—3262no—
418C22—3262no—
419C22—3262no—
420C22—3262no—
421C22—3262no—
422C22—3262no—
423C22—3262no—
424C22—3262no—
425C22—65216no—
426C22—6521no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22—1634no—
431C22#1910469314yescomplemented
432C22C22×C4×D16311yescharacteristic
433C22C2×D4×D16311yescharacteristic
434C22#41×D16311yescharacteristic
435C22C2×#1457969811yescharacteristic
436C22C2×#1457970011yescharacteristic
437C22#1910478011yescharacteristic
438C22#1910478111yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—3262no—
446C4—3262no—
447C4—3262no—
448C4—6521no—
449C4—6521no—
450C4—6521no—
451C2—24no—
452C2—3264no—
453C2—6522no—
454C2C2×#1910469311yescharacteristic
455C2C4×D4×D16311yescharacteristic
456C2#2378213711yescharacteristic
457C2#2378230211yescharacteristic
458C2#2378230611yescharacteristic
459C2#2378233811yescharacteristic
460C2#2378234211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#385×C163C2
socle, cosocleC23×C163C24
prime radicalC24×D163C22
hypercenter#385D163
hypocenterC163#2342
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#19104693C2semidirect
4. #23781557C2semidirect
5. #23781557C2semidirect
6. #23781555C2semidirect
7. #23781555C2semidirect
8. #23781540C2semidirect
9. #23781527C2semidirect
10. #23781522C2semidirect
12. C22⋊C4×Dic163C2semidirect
13. C4×C22⋊Dic163C2semidirect
14. #23781512C2semidirect
15. #385×C163C2semidirect
21. C2×C4⋊Dic163C22semidirect
22. C2×C4⋊Dic163C22semidirect
36. C22⋊C4×D163C4semidirect
37. C2×C4×Dic163C22semidirect
38. C2×C4×Dic163C22semidirect
39. C2×C4×Dic163C22semidirect
40. C2×C4×Dic163C22semidirect
41. C163⋊#95C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
48. C2×C4⋊C4×C163C22semidirect
49. C2×C4⋊C4×C163C22semidirect
52. #95×C163C22semidirect
53. #95×C163C22semidirect
76. C2×C4×D163C2×C4semidirect
77. C2×C4×D163C2×C4semidirect
89. (C2×D163)⋊C4C2×C4semidirect
90. (C2×D163)⋊C4C2×C4semidirect
185. C22×C163#117semidirect
230. C163#2342semidirect
240. #385D163semidirect
255. C2×C4⋊C4C2×D163semidirect
256. C2×C4⋊C4C2×D163semidirect
283. #95C2×D163semidirect
284. #95C2×D163semidirect
431. C22#19104693semidirect

Maximal subgroups

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

subgroupindex
C2×#191046932
C22×(C2×D163)⋊C42
C2×C22⋊C4×D1632
#237815572
#237815552
#237815402
#237815272
#237815222
#237815202
C22⋊C4×Dic1632
C4×C22⋊Dic1632
#237815122
#385×C1632
#2342163