Finite Group Databasebeta

← #35217189

Subgroups — #35217189

Order 20864.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104705C211yesmaximal, characteristic
2C2×#19104697C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
4#23781567C211yesmaximal, characteristic, complemented
5#23781565C211yesmaximal, characteristic, complemented
6#23781558C211yesmaximal, characteristic, complemented
7#23781555C211yesmaximal, characteristic, complemented
8#23781554C211yesmaximal, characteristic, complemented
9#23781543C211yesmaximal, characteristic, complemented
10#23781543C211yesmaximal, characteristic, complemented
11#23781541C211yesmaximal, characteristic, complemented
12#23781539C211yesmaximal, characteristic, complemented
13#23781520C211yesmaximal, characteristic, complemented
14C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
15#388×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic
22C22×C4×D163C2211yescharacteristic, complemented
23#19104705C418yescomplemented
24#19104697—24no—
25C2×C4⋊Dic163C2211yescharacteristic, complemented
26C2×Dic163⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic163C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
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
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
40C22⋊C4×D163—24no—
41C2×C4×Dic163C2211yescharacteristic, complemented
42C2×C4×Dic163C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C2×C4⋊C4×C163C2211yescharacteristic, complemented
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C42×C163C2211yescharacteristic, complemented
52#95×C163C2211yescharacteristic, complemented
53#95×C163C2211yescharacteristic, complemented
54C23×D163—22no—
55C23×D163—22no—
56C23×D163—41no—
57C23×D163—41no—
58C23×D163—41no—
59C23×D163—41no—
60C23×D163C2×C414yescomplemented
61C23×D163C2311yescharacteristic
62C23×D163C2311yescharacteristic
63C23×D163C2311yescharacteristic
64C2×D652—22no—
65C2×D652—22no—
66C2×D652—41no—
67C2×D652—41no—
68C2×D652C2×C414yes—
69C2×D652C2×C414yescomplemented
70C2×C22⋊D163—41no—
71C2×C22⋊D163—41no—
72C2×C22⋊D163C2×C414yescomplemented
73C22×Dic163C2311yescharacteristic
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C22×Dic163C2311yescharacteristic
77C22×Dic163C2311yescharacteristic
78C2×C4×D163—22no—
79C2×C4×D163—22no—
80C2×C4×D163—41no—
81C2×C4×D163—41no—
82C2×C4×D163—41no—
83C2×C4×D163—41no—
84C2×C4×D163—41no—
85C2×C4×D163—41no—
86C2×C4×D163D414yescomplemented
87C4⋊Dic163—41no—
88Dic163⋊C4—41no—
89C22⋊Dic163—22no—
90C22⋊Dic163—41no—
91(C2×D163)⋊C4—22no—
92(C2×D163)⋊C4—22no—
93(C2×D163)⋊C4—22no—
94(C2×D163)⋊C4—22no—
95(C2×D163)⋊C4—22no—
96(C2×D163)⋊C4—41no—
97(C2×D163)⋊C4—41no—
98(C2×D163)⋊C4—41no—
99(C2×D163)⋊C4—41no—
100(C2×D163)⋊C4C2×C414yescomplemented
101(C2×D163)⋊C4C2×C414yescomplemented
102C4×Dic163—22no—
103C4×Dic163—41no—
104C24×C163C2311yescharacteristic
105C22×C4×C163C2311yescharacteristic
106C22×C4×C163C2311yescharacteristic
107C22×C4×C163C2311yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C4⋊C4×C163—22no—
112C22⋊C4×C163—22no—
113C22⋊C4×C163—41no—
114C22⋊C4×C163C2×C414yescomplemented
115C42×C163—41no—
116C22×D163—21no—
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—21no—
122C22×D163—21no—
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—21no—
128C22×D163—21no—
129C22×D163—21no—
130C22×D163—21no—
131C22×D163—21no—
132C22×D163—28no—
133C22×D163—44no—
134C22×D163—44no—
135C22×D163—44no—
136C22×D163—44no—
137C22×D163—44no—
138C22×D163—44no—
139C22×D163C22×C412yes—
140C22×D163C22×C412yes—
141C22×D163C22×C412yes—
142C22×D163C2×D412yes—
143C22×D163#4112yes—
144D652—28no—
145D652—44no—
146C22⋊D163—44no—
147C2×Dic163—21no—
148C2×Dic163—21no—
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163—21no—
152C2×Dic163—21no—
153C2×Dic163—21no—
154C2×Dic163—21no—
155C2×Dic163—21no—
156C2×Dic163—21no—
157C2×Dic163—21no—
158C2×Dic163—21no—
159C2×Dic163—21no—
160C2×Dic163C22×C412yes—
161C2×Dic163C2×D412yes—
162C4×D163—44no—
163C4×D163—44no—
164C4×D163—44no—
165C23×C163—21no—
166C23×C163—21no—
167C23×C163—21no—
168C23×C163—21no—
169C23×C163—21no—
170C23×C163—21no—
171C23×C163C22×C412yes—
172C23×C163C2411yescharacteristic
173C2×C4×C163—21no—
174C2×C4×C163—21no—
175C2×C4×C163—21no—
176C2×C4×C163—21no—
177C2×C4×C163—21no—
178C2×C4×C163—21no—
179C2×C4×C163—21no—
180C2×C4×C163—21no—
181C2×C4×C163—21no—
182C2×C4×C163—21no—
183C2×C4×C163—21no—
184C2×C4×C163—21no—
185C2×C4×C163—21no—
186C2×C4×C163—21no—
187C2×C4×C163—21no—
188C2×C4×C163C22×C412yes—
189C2×C4×C163C22×C412yes—
190C2×C4×C163C2×D412yes—
191C22×C163—22no—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—41no—
195C22×C163—41no—
196C22×C163—41no—
197C22×C163—41no—
198C22×C163C23×C411yescharacteristic
199C22×C163C22×D411yescharacteristic
200C22×C163C2×#4111yescharacteristic
201C22×C163H(5,𝔽2)11yescharacteristic
202C22×C163H(5,𝔽2)11yescharacteristic
203C22×C163H(5,𝔽2)11yescharacteristic
204C22×C163H(5,𝔽2)11yescharacteristic
205C2×D163—22no—
206C2×D163—22no—
207C2×D163—22no—
208C2×D163—22no—
209C2×D163—22no—
210C2×D163—22no—
211C2×D163—22no—
212C2×D163—22no—
213C2×D163—22no—
214C2×D163—22no—
215C2×D163—22no—
216C2×D163—22no—
217C2×D163—416no—
218C2×D163—41no—
219C2×D163—41no—
220C2×D163—41no—
221C2×D163—41no—
222C2×D163—41no—
223C2×D163—41no—
224C2×D163—41no—
225C2×D163—41no—
226C2×D163C4×D414yes—
227C4×C163—22no—
228C4×C163—22no—
229C4×C163—22no—
230C4×C163—41no—
231C4×C163—41no—
232C4×C163—41no—
233Dic163—22no—
234Dic163—22no—
235Dic163—41no—
236Dic163—41no—
237Dic163—41no—
238C2×C163—42no—
239C2×C163C2×C4×D411yescharacteristic
240C2×C163#51811yescharacteristic
241C2×C163#51811yescharacteristic
242C2×C163#53411yescharacteristic
243C2×C163#53511yescharacteristic
244C2×C163#53811yescharacteristic
245C2×C163#53811yescharacteristic
246D163—24no—
247D163—42no—
248D163—42no—
249C163#232111yescharacteristic, complemented, Sylow
250#2321—1631nomaximal, Sylow
251C2×#120—1631no—
252C2×C4×D4—1631no—
253C22×C22⋊C4—1631no—
254#390—1631no—
255#390—1631no—
256#388—1631no—
257#388—1631no—
258#388—1631no—
259#388D16311yescharacteristic, complemented
260#386—1631no—
261#386—1631no—
262#386—1631no—
263#385—1631no—
264#379—1631no—
265C4×C22⋊C4—1631no—
266C25—1631no—
267C22×D4—1631no—
268C22×D4—1631no—
269C22×D4—1631no—
270C23×C4—1631no—
271C23×C4—1631no—
272C23×C4—1631no—
273#120—1638no—
274C4×D4—3264no—
275C2×C4⋊C4—1631no—
276C2×C4⋊C4—1631no—
277C2×C4⋊C4C2×D16311yescharacteristic, complemented
278C2×C22⋊C4—3264no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4—1631no—
282C2×C22⋊C4—1631no—
283C2×C22⋊C4—1631no—
284C2×C22⋊C4—1631no—
285C2×C22⋊C4—1631no—
286C2×C22⋊C4—1631no—
287C2×C22⋊C4—1631no—
288C2×C22⋊C4—1631no—
289C2×C22⋊C4—1631no—
290C2×C22⋊C4—1631no—
291C2×C22⋊C4—1631no—
292C2×C22⋊C4C2×D16311yescharacteristic
293C2×C22⋊C4C2×D16311yescharacteristic
294C2×C22⋊C4C2×D16311yescharacteristic
295C2×C42—1631no—
296C2×C42—1631no—
297C2×C42C2×D16311yescharacteristic, complemented
298#95—1631no—
299#95—1631no—
300#95—1631no—
301#95—1631no—
302#95C2×D16311yescharacteristic, complemented
303#95C2×D16311yescharacteristic, complemented
304C24—3262no—
305C24—3262no—
306C24—6521no—
307C24—6521no—
308C24—6521no—
309C24—6521no—
310C24—1631no—
311C24—1631no—
312C24—1631no—
313C24—1634no—
314C24C22×D16311yescharacteristic
315C2×D4—3262no—
316C2×D4—3262no—
317C2×D4—6521no—
318C2×D4—6521no—
319C2×D4—6521no—
320C2×D4—6521no—
321C2×D4—1634no—
322C2×D4—1634no—
323C2×D4—1634no—
324C22×C4—3262no—
325C22×C4—3262no—
326C22×C4—6521no—
327C22×C4—6521no—
328C22×C4—6521no—
329C22×C4—6521no—
330C22×C4—6521no—
331C22×C4—6521no—
332C22×C4—1631no—
333C22×C4—1631no—
334C22×C4—1631no—
335C22×C4—1631no—
336C22×C4—1631no—
337C22×C4—1634no—
338C22×C4C22×D16311yescharacteristic
339C22×C4C22×D16311yescharacteristic
340C22×C4C22×D16311yescharacteristic
341C22×C4C22×D16311yescharacteristic
342C22×C4C22×D16311yescharacteristic
343C22×C4C22×D16311yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—6521no—
346C4⋊C4—6521no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—3262no—
350C22⋊C4—3262no—
351C22⋊C4—3262no—
352C22⋊C4—3262no—
353C22⋊C4—3262no—
354C22⋊C4—3262no—
355C22⋊C4—6521no—
356C22⋊C4—6521no—
357C22⋊C4—6521no—
358C22⋊C4—6521no—
359C22⋊C4—6521no—
360C22⋊C4—1634no—
361C22⋊C4—1634no—
362C22⋊C4C4×D16314yescomplemented
363C42—41no—
364C42—3262no—
365C42—6521no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—3268no—
373C23—3261no—
374C23—3261no—
375C23—3261no—
376C23—3261no—
377C23—3261no—
378C23—3261no—
379C23—3261no—
380C23—3261no—
381C23—3261no—
382C23—3261no—
383C23—3261no—
384C23—3261no—
385C23—3261no—
386C23—3261no—
387C23—3261no—
388C23—3261no—
389C23—6524no—
390C23—6524no—
391C23—6524no—
392C23—6524no—
393C23—6524no—
394C23—6524no—
395C23—1632no—
396C23—1632no—
397C23—1632no—
398C23—1632no—
399C23—1632no—
400C23C2×C4×D16312yes—
401C23C23×D16311yescharacteristic
402D4—3268no—
403D4—6524no—
404D4—6524no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—3261no—
421C2×C4—3261no—
422C2×C4—3261no—
423C2×C4—3261no—
424C2×C4—3261no—
425C2×C4—3261no—
426C2×C4—3261no—
427C2×C4—3261no—
428C2×C4—3261no—
429C2×C4—3261no—
430C2×C4—3261no—
431C2×C4—3261no—
432C2×C4—3261no—
433C2×C4—6524no—
434C2×C4—6524no—
435C2×C4—6524no—
436C2×C4—1632no—
437C2×C4—1632no—
438C2×C4C2×C4×D16312yes—
439C2×C4C2×C4×D16312yes—
440C2×C4D4×D16312yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—3262no—
449C22—3262no—
450C22—3262no—
451C22—3262no—
452C22—3262no—
453C22—3262no—
454C22—3262no—
455C22—3262no—
456C22—3262no—
457C22—3262no—
458C22—3262no—
459C22—3262no—
460C22—65216no—
461C22—6521no—
462C22—6521no—
463C22—6521no—
464C22—6521no—
465C22—6521no—
466C22—6521no—
467C22—6521no—
468C22—6521no—
469C22—1634no—
470C22C22×C4×D16311yescharacteristic
471C22C2×D4×D16311yescharacteristic
472C22#41×D16311yescharacteristic
473C22#1910478011yescharacteristic
474C22#1910478111yescharacteristic
475C22#1910478111yescharacteristic
476C22#1910478111yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—3262no—
484C4—3262no—
485C4—6521no—
486C4—6521no—
487C4—6521no—
488C2—42no—
489C2—3264no—
490C2—6522no—
491C2—6522no—
492C2C4×D4×D16311yescharacteristic
493C2#2378213711yescharacteristic
494C2#2378213911yescharacteristic
495C2#2378222311yescharacteristic
496C2#2378223311yescharacteristic
497C2#2378225111yescharacteristic
498C2#2378225411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#388×C163C2
socle, cosocleC23×C163C24
prime radicalC2×#19104705C2
hypercenter#388D163
hypocenterC163#2321
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#19104697C2semidirect
3. C2×C22⋊C4×D163C2semidirect
4. #23781567C2semidirect
5. #23781565C2semidirect
6. #23781558C2semidirect
7. #23781555C2semidirect
8. #23781554C2semidirect
9. #23781543C2semidirect
10. #23781543C2semidirect
11. #23781541C2semidirect
12. #23781539C2semidirect
13. #23781520C2semidirect
14. C4×(C2×D163)⋊C4C2semidirect
15. #388×C163C2semidirect
22. C22×C4×D163C22semidirect
23. #19104705C4semidirect
25. C2×C4⋊Dic163C22semidirect
26. C2×Dic163⋊C4C22semidirect
37. C2×(C2×D163)⋊C4C22semidirect
38. C2×(C2×D163)⋊C4C22semidirect
39. C2×(C2×D163)⋊C4C22semidirect
41. C2×C4×Dic163C22semidirect
42. C2×C4×Dic163C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C2×C4⋊C4×C163C22semidirect
51. C2×C42×C163C22semidirect
52. #95×C163C22semidirect
53. #95×C163C22semidirect
60. C23×D163C2×C4semidirect
69. C2×D652C2×C4semidirect
72. C2×C22⋊D163C2×C4semidirect
86. C2×C4×D163D4semidirect
100. (C2×D163)⋊C4C2×C4semidirect
101. (C2×D163)⋊C4C2×C4semidirect
114. C22⋊C4×C163C2×C4semidirect
249. C163#2321semidirect
259. #388D163semidirect
277. C2×C4⋊C4C2×D163semidirect
297. C2×C42C2×D163semidirect
302. #95C2×D163semidirect
303. #95C2×D163semidirect
362. C22⋊C4C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047052
C2×#191046972
C2×C22⋊C4×D1632
#237815672
#237815652
#237815582
#237815552
#237815542
#237815432
#237815412
#237815392
#237815202
C4×(C2×D163)⋊C42
#388×C1632
#2321163