Finite Group Databasebeta

← #35217384

Subgroups — #35217384

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
2C2×C22⋊C4×D163C211yesmaximal, characteristic
3C2×C22⋊C4×D163C211yesmaximal, characteristic
4#23781559C211yesmaximal, characteristic, complemented
5#23781558C211yesmaximal, characteristic, complemented
6#23781553C211yesmaximal, characteristic, complemented
7#23781539C211yesmaximal, characteristic, complemented
8#23781536C211yesmaximal, characteristic, complemented
9#23781527C211yesmaximal, characteristic, complemented
10#23781524C211yesmaximal, characteristic, complemented
11#23781520C211yesmaximal, characteristic
12C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
13C4×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#388×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104693—24no—
22C2×C4⋊Dic163C2211yescharacteristic, complemented
23C2×Dic163⋊C4C2211yescharacteristic, complemented
24C2×Dic163⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic163C2211yescharacteristic
26C2×C22⋊Dic163C2211yescharacteristic
27C2×C22⋊Dic163C2211yescharacteristic
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×D163—24no—
37C22⋊C4×D163C418yescomplemented
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
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—22no—
57C23×D163—22no—
58C23×D163—41no—
59C23×D163—41no—
60C23×D163C2×C414yes—
61C23×D163C2311yescharacteristic
62C23×D163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C2×C4×D163—22no—
70C2×C4×D163—22no—
71C2×C4×D163—22no—
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C2×C4×D163—41no—
76C2×C4×D163—41no—
77C2×C4×D163—41no—
78C2×C4×D163—41no—
79C2×C4×D163C2×C414yescomplemented
80C2×C4×D163C2×C414yescomplemented
81C4⋊Dic163—22no—
82Dic163⋊C4—41no—
83Dic163⋊C4—41no—
84C22⋊Dic163—22no—
85C22⋊Dic163—41no—
86C22⋊Dic163C2×C414yescomplemented
87(C2×D163)⋊C4—22no—
88(C2×D163)⋊C4—22no—
89(C2×D163)⋊C4—22no—
90(C2×D163)⋊C4—22no—
91(C2×D163)⋊C4—22no—
92(C2×D163)⋊C4C2×C414yes—
93(C2×D163)⋊C4C2×C414yes—
94(C2×D163)⋊C4D414yes—
95C4×Dic163—22no—
96C4×Dic163—22no—
97C4×Dic163—41no—
98C24×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102C22×C4×C163C2311yescharacteristic
103C22×C4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105C4⋊C4×C163—22no—
106C22⋊C4×C163—22no—
107C22⋊C4×C163—41no—
108C22⋊C4×C163C2×C414yescomplemented
109C42×C163—41no—
110C22×D163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—21no—
115C22×D163—21no—
116C22×D163—21no—
117C22×D163—21no—
118C22×D163—28no—
119C22×D163—28no—
120C22×D163—44no—
121C22×D163—44no—
122C22×D163—44no—
123C22×D163—44no—
124C22×D163—44no—
125C22×D163C22×C412yes—
126C22×D163C22×C412yes—
127C22×D163C2×D412yes—
128C22×D163#4112yes—
129C22×D163#4112yes—
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×Dic163C22×C412yes—
147C2×Dic163C22×C412yes—
148C2×Dic163C2×D412yes—
149C4×D163—28no—
150C4×D163—44no—
151C4×D163—44no—
152C4×D163—44no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—21no—
159C23×C163C22×C412yes—
160C23×C163C2411yescharacteristic
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—21no—
176C2×C4×C163C22×C412yes—
177C2×C4×C163C22×C412yes—
178C2×C4×C163C2×D412yes—
179C22×C163—22no—
180C22×C163—22no—
181C22×C163—22no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
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—22no—
203C2×D163—22no—
204C2×D163—22no—
205C2×D163—416no—
206C2×D163#11714yescomplemented
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—42no—
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—24no—
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#388D16311yescharacteristic, complemented
239#385—1631no—
240#385—1631no—
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×C4—1631no—
252#117—3264no—
253C2×C4⋊C4—1631no—
254C2×C4⋊C4—1631no—
255C2×C4⋊C4—1631no—
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—1638no—
270C2×C22⋊C4C2×D16311yescharacteristic
271C2×C22⋊C4C2×D16311yescharacteristic
272C2×C22⋊C4C2×D16311yescharacteristic
273C2×C42—1631no—
274C2×C42—1631no—
275C2×C42—1631no—
276C2×C42C2×D16311yescharacteristic, complemented
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—1634no—
292C24—1631no—
293C24—1631no—
294C24C22×D16311yescharacteristic
295C22×C4—3262no—
296C22×C4—3262no—
297C22×C4—3262no—
298C22×C4—3262no—
299C22×C4—3262no—
300C22×C4—3262no—
301C22×C4—6521no—
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—3262no—
321C4⋊C4—6521no—
322C4⋊C4—6521no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—3262no—
326C22⋊C4—3262no—
327C22⋊C4—3262no—
328C22⋊C4—3262no—
329C22⋊C4—3262no—
330C22⋊C4—3262no—
331C22⋊C4—6521no—
332C22⋊C4—1634no—
333C22⋊C4—1634no—
334C22⋊C4—1634no—
335C22⋊C4—1634no—
336C22⋊C4C4×D16314yescomplemented
337C42—41no—
338C42—3262no—
339C42—3262no—
340C42—6521no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—3268no—
348C23—3268no—
349C23—3261no—
350C23—3261no—
351C23—3261no—
352C23—3261no—
353C23—3261no—
354C23—3261no—
355C23—3261no—
356C23—3261no—
357C23—6524no—
358C23—6524no—
359C23—6524no—
360C23—6524no—
361C23—6524no—
362C23—1632no—
363C23—1632no—
364C23—1632no—
365C23—1632no—
366C23—1632no—
367C23—1632no—
368C23C2×C4×D16312yes—
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—3268no—
386C2×C4—3261no—
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—6524no—
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—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—3262no—
418C22—3262no—
419C22—3262no—
420C22—3262no—
421C22—3262no—
422C22—3262no—
423C22—3262no—
424C22—3262no—
425C22—3262no—
426C22—3262no—
427C22—3262no—
428C22—3262no—
429C22—65216no—
430C22—1634no—
431C22—1634no—
432C22C22×C4×D16311yescharacteristic
433C22C2×D4×D16311yescharacteristic
434C22#41×D16311yescharacteristic
435C22#41×D16311yescharacteristic
436C22#41×D16311yescharacteristic
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—42no—
452C2—3264no—
453C2—3264no—
454C2#117×D16311yescharacteristic
455C2C4×D4×D16311yescharacteristic
456C2#2378213711yescharacteristic
457C2#2378230411yescharacteristic
458C2#2378230811yescharacteristic
459C2#2378234011yescharacteristic
460C2#2378234311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#388×C163C2
socle, cosocleC23×C163C24
prime radicalC24×D163C22
hypercenter#388D163
hypocenterC163#2342
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#19104693C2semidirect
4. #23781559C2semidirect
5. #23781558C2semidirect
6. #23781553C2semidirect
7. #23781539C2semidirect
8. #23781536C2semidirect
9. #23781527C2semidirect
10. #23781524C2semidirect
12. C4×(C2×D163)⋊C4C2semidirect
13. C4×(C2×D163)⋊C4C2semidirect
14. #23781512C2semidirect
15. #388×C163C2semidirect
22. C2×C4⋊Dic163C22semidirect
23. C2×Dic163⋊C4C22semidirect
24. C2×Dic163⋊C4C22semidirect
37. C22⋊C4×D163C4semidirect
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
47. C2×C4⋊C4×C163C22semidirect
51. C2×C42×C163C22semidirect
52. #95×C163C22semidirect
53. #95×C163C22semidirect
79. C2×C4×D163C2×C4semidirect
80. C2×C4×D163C2×C4semidirect
86. C22⋊Dic163C2×C4semidirect
108. C22⋊C4×C163C2×C4semidirect
206. C2×D163#117semidirect
230. C163#2342semidirect
238. #388D163semidirect
256. C2×C4⋊C4C2×D163semidirect
276. C2×C42C2×D163semidirect
283. #95C2×D163semidirect
284. #95C2×D163semidirect
336. C22⋊C4C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191046932
C2×C22⋊C4×D1632
#237815592
#237815582
#237815532
#237815392
#237815362
#237815272
#237815242
#237815202
C4×(C2×D163)⋊C42
#237815122
#388×C1632
#2342163