Finite Group Databasebeta

← #35217676

Subgroups — #35217676

Order 20864.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104705C211yesmaximal, characteristic
2#23781567C211yesmaximal, characteristic, complemented
3#23781566C211yesmaximal, characteristic, complemented
4#23781565C211yesmaximal, characteristic, complemented
5#23781558C211yesmaximal, characteristic, complemented
6#23781557C211yesmaximal, characteristic, complemented
7#23781556C211yesmaximal, characteristic, complemented
8#23781555C211yesmaximal, characteristic, complemented
9#23781554C211yesmaximal, characteristic, complemented
10#23781543C211yesmaximal, characteristic, complemented
11#23781543C211yesmaximal, characteristic, complemented
12#23781541C211yesmaximal, characteristic, complemented
13#23781520C211yesmaximal, characteristic, complemented
14#23781520C211yesmaximal, characteristic, complemented
15#388×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic, complemented
21#19104705C418yescomplemented
22C2×C4⋊Dic163C2211yescharacteristic, complemented
23C2×Dic163⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic163C2211yescharacteristic
25C2×C22⋊Dic163C2211yescharacteristic
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
36C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
37C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
39C2×C4×Dic163C2211yescharacteristic, complemented
40C2×C4×Dic163C2211yescharacteristic, complemented
41C163⋊#95C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C2×C4⋊C4×C163C2211yescharacteristic, complemented
46C2×C22⋊C4×C163C2211yescharacteristic
47C2×C22⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C42×C163C2211yescharacteristic, complemented
50#95×C163C2211yescharacteristic, complemented
51#95×C163C2211yescharacteristic, complemented
52C23×D163—41no—
53C23×D163—41no—
54C23×D163—41no—
55C23×D163—41no—
56C23×D163—41no—
57C23×D163—41no—
58C23×D163C2×C414yescomplemented
59C23×D163C2311yescharacteristic
60C23×D163C2311yescharacteristic
61C23×D163C2311yescharacteristic
62C2×D652—41no—
63C2×D652—41no—
64C2×D652—41no—
65C2×D652—41no—
66C2×D652C2×C414yes—
67C2×D652C2×C414yes—
68C2×C22⋊D163—41no—
69C2×C22⋊D163—41no—
70C2×C22⋊D163C2×C414yes—
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C22×Dic163C2311yescharacteristic
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C2×C4×D163—41no—
77C2×C4×D163—41no—
78C2×C4×D163—41no—
79C4⋊Dic163—41no—
80Dic163⋊C4—41no—
81C22⋊Dic163—41no—
82C22⋊Dic163—41no—
83(C2×D163)⋊C4—41no—
84(C2×D163)⋊C4—41no—
85(C2×D163)⋊C4—41no—
86(C2×D163)⋊C4—41no—
87(C2×D163)⋊C4—41no—
88(C2×D163)⋊C4—41no—
89(C2×D163)⋊C4—41no—
90(C2×D163)⋊C4—41no—
91(C2×D163)⋊C4—41no—
92(C2×D163)⋊C4—41no—
93(C2×D163)⋊C4—41no—
94(C2×D163)⋊C4C2×C414yescomplemented
95(C2×D163)⋊C4C2×C414yescomplemented
96C4×Dic163—41no—
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—41no—
106C22⋊C4×C163—41no—
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—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—44no—
129C22×D163—44no—
130C22×D163—44no—
131C22×D163—44no—
132C22×D163—44no—
133C22×D163—44no—
134C22×D163—44no—
135C22×D163C22×C412yes—
136C22×D163C22×C412yes—
137C22×D163C22×C412yes—
138D652—44no—
139D652—44no—
140C22⋊D163—44no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—21no—
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×Dic163C22×C412yes—
156C4×D163—44no—
157C23×C163—21no—
158C23×C163—21no—
159C23×C163—21no—
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—21no—
163C23×C163C22×C412yes—
164C23×C163C2411yescharacteristic
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×C163—21no—
177C2×C4×C163—21no—
178C2×C4×C163—21no—
179C2×C4×C163—21no—
180C2×C4×C163—21no—
181C2×C4×C163C22×C412yes—
182C2×C4×C163C22×C412yes—
183C22×C163—22no—
184C22×C163—22no—
185C22×C163—22no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163C23×C411yescharacteristic
191C22×C163H(5,𝔽2)11yescharacteristic
192C22×C163H(5,𝔽2)11yescharacteristic
193C22×C163H(5,𝔽2)11yescharacteristic
194C22×C163H(5,𝔽2)11yescharacteristic
195C22×C163H(5,𝔽2)11yescharacteristic
196C22×C163H(5,𝔽2)11yescharacteristic
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—22no—
206C2×D163—416no—
207C2×D163—41no—
208C2×D163—41no—
209C2×D163—41no—
210C2×D163—41no—
211C2×D163—41no—
212C2×D163—41no—
213C2×D163—41no—
214C2×D163—41no—
215C2×D163—41no—
216C2×D163—41no—
217C2×D163—41no—
218C2×D163—41no—
219C4×C163—22no—
220C4×C163—22no—
221C4×C163—41no—
222C4×C163—41no—
223C4×C163—41no—
224C4×C163—41no—
225Dic163—22no—
226Dic163—41no—
227Dic163—41no—
228Dic163—41no—
229Dic163—41no—
230C2×C163—42no—
231C2×C163#51811yescharacteristic
232C2×C163#51811yescharacteristic
233C2×C163#51811yescharacteristic
234C2×C163#56011yescharacteristic
235C2×C163#56011yescharacteristic
236C2×C163#56011yescharacteristic
237C2×C163H(3,𝔽4)11yescharacteristic
238D163—42no—
239D163—42no—
240D163—42no—
241C163#237511yescharacteristic, complemented, Sylow
242#2375—1631nomaximal, Sylow
243C2×#120—1631no—
244#390—1631no—
245#390—1631no—
246#390—1631no—
247#388—1631no—
248#388—1631no—
249#388—1631no—
250#388—1631no—
251#388—1631no—
252#388D16311yescharacteristic, complemented
253#386—1631no—
254#386—1631no—
255#386—1631no—
256#379—1631no—
257#379—1631no—
258C25—1631no—
259C22×D4—1631no—
260C22×D4—1631no—
261C22×D4—1631no—
262C23×C4—1631no—
263#120—1638no—
264C2×C4⋊C4—1631no—
265C2×C4⋊C4—1631no—
266C2×C4⋊C4C2×D16311yescharacteristic, complemented
267C2×C22⋊C4—1631no—
268C2×C22⋊C4—1631no—
269C2×C22⋊C4—1631no—
270C2×C22⋊C4—1631no—
271C2×C22⋊C4—1631no—
272C2×C22⋊C4—1631no—
273C2×C22⋊C4—1631no—
274C2×C22⋊C4—1631no—
275C2×C22⋊C4—1631no—
276C2×C22⋊C4—1631no—
277C2×C22⋊C4—1631no—
278C2×C22⋊C4—1631no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4—1631no—
282C2×C22⋊C4C2×D16311yescharacteristic
283C2×C22⋊C4C2×D16311yescharacteristic
284C2×C22⋊C4C2×D16311yescharacteristic
285C2×C42—1631no—
286C2×C42—1631no—
287C2×C42C2×D16311yescharacteristic, complemented
288#95—1631no—
289#95—1631no—
290#95—1631no—
291#95—1631no—
292#95C2×D16311yescharacteristic, complemented
293#95C2×D16311yescharacteristic, complemented
294C24—6521no—
295C24—6521no—
296C24—6521no—
297C24—6521no—
298C24—6521no—
299C24—6521no—
300C24—1631no—
301C24—1631no—
302C24—1631no—
303C24—1634no—
304C24C22×D16311yescharacteristic
305C2×D4—6521no—
306C2×D4—6521no—
307C2×D4—6521no—
308C2×D4—6521no—
309C2×D4—6521no—
310C2×D4—6521no—
311C2×D4—1634no—
312C2×D4—1634no—
313C2×D4—1634no—
314C22×C4—6521no—
315C22×C4—6521no—
316C22×C4—6521no—
317C22×C4—1631no—
318C22×C4—1631no—
319C22×C4—1631no—
320C22×C4—1631no—
321C22×C4—1631no—
322C22×C4C22×D16311yescharacteristic
323C22×C4C22×D16311yescharacteristic
324C22×C4C22×D16311yescharacteristic
325C22×C4C22×D16311yescharacteristic
326C22×C4C22×D16311yescharacteristic
327C22×C4C22×D16311yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—6521no—
330C4⋊C4—6521no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—6521no—
334C22⋊C4—6521no—
335C22⋊C4—6521no—
336C22⋊C4—6521no—
337C22⋊C4—6521no—
338C22⋊C4—6521no—
339C22⋊C4—6521no—
340C22⋊C4—6521no—
341C22⋊C4—6521no—
342C22⋊C4—6521no—
343C22⋊C4—6521no—
344C22⋊C4—6521no—
345C22⋊C4—6521no—
346C22⋊C4—1634no—
347C22⋊C4—1634no—
348C22⋊C4C4×D16314yescomplemented
349C42—41no—
350C42—6521no—
351C42—6521no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—3261no—
359C23—3261no—
360C23—3261no—
361C23—3261no—
362C23—3261no—
363C23—3261no—
364C23—3261no—
365C23—3261no—
366C23—3261no—
367C23—3261no—
368C23—3261no—
369C23—3261no—
370C23—3261no—
371C23—3261no—
372C23—3261no—
373C23—3261no—
374C23—3261no—
375C23—3261no—
376C23—6524no—
377C23—6524no—
378C23—6524no—
379C23—6524no—
380C23—6524no—
381C23—6524no—
382C23—6524no—
383C23—1632no—
384C23—1632no—
385C23—1632no—
386C23C2×C4×D16312yes—
387C23C23×D16311yescharacteristic
388D4—6524no—
389D4—6524no—
390D4—6524no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—3261no—
411C2×C4—3261no—
412C2×C4—3261no—
413C2×C4—3261no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—3261no—
417C2×C4—3261no—
418C2×C4—3261no—
419C2×C4—3261no—
420C2×C4—3261no—
421C2×C4—6524no—
422C2×C4—1632no—
423C2×C4C2×C4×D16312yes—
424C2×C4C2×C4×D16312yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—3262no—
433C22—3262no—
434C22—3262no—
435C22—3262no—
436C22—3262no—
437C22—3262no—
438C22—3262no—
439C22—3262no—
440C22—3262no—
441C22—65216no—
442C22—6521no—
443C22—6521no—
444C22—6521no—
445C22—6521no—
446C22—6521no—
447C22—6521no—
448C22—6521no—
449C22—6521no—
450C22—6521no—
451C22—6521no—
452C22—6521no—
453C22—6521no—
454C22C22×C4×D16311yescharacteristic
455C22#1910478011yescharacteristic
456C22#1910478011yescharacteristic
457C22#1910478111yescharacteristic
458C22#1910478111yescharacteristic
459C22#1910478111yescharacteristic
460C22#1910478111yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—3262no—
468C4—6521no—
469C4—6521no—
470C4—6521no—
471C4—6521no—
472C2—42no—
473C2—6522no—
474C2—6522no—
475C2—6522no—
476C2#2378213711yescharacteristic
477C2#2378213711yescharacteristic
478C2#2378213911yescharacteristic
479C2#2378240311yescharacteristic
480C2#2378240511yescharacteristic
481C2#2378240611yescharacteristic
482C2#2378240711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#388×C163C2
socle, cosocleC23×C163C24
prime radicalC2×#19104705C2
hypercenter#388D163
hypocenterC163#2375
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #23781567C2semidirect
3. #23781566C2semidirect
4. #23781565C2semidirect
5. #23781558C2semidirect
6. #23781557C2semidirect
7. #23781556C2semidirect
8. #23781555C2semidirect
9. #23781554C2semidirect
10. #23781543C2semidirect
11. #23781543C2semidirect
12. #23781541C2semidirect
13. #23781520C2semidirect
14. #23781520C2semidirect
15. #388×C163C2semidirect
20. C22×C4×D163C22semidirect
21. #19104705C4semidirect
22. C2×C4⋊Dic163C22semidirect
23. C2×Dic163⋊C4C22semidirect
36. C2×(C2×D163)⋊C4C22semidirect
37. C2×(C2×D163)⋊C4C22semidirect
38. C2×(C2×D163)⋊C4C22semidirect
39. C2×C4×Dic163C22semidirect
40. C2×C4×Dic163C22semidirect
41. C163⋊#95C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C2×C4⋊C4×C163C22semidirect
49. C2×C42×C163C22semidirect
50. #95×C163C22semidirect
51. #95×C163C22semidirect
58. C23×D163C2×C4semidirect
94. (C2×D163)⋊C4C2×C4semidirect
95. (C2×D163)⋊C4C2×C4semidirect
108. C22⋊C4×C163C2×C4semidirect
241. C163#2375semidirect
252. #388D163semidirect
266. C2×C4⋊C4C2×D163semidirect
287. C2×C42C2×D163semidirect
292. #95C2×D163semidirect
293. #95C2×D163semidirect
348. 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
#237815672
#237815662
#237815652
#237815582
#237815572
#237815562
#237815552
#237815542
#237815432
#237815412
#237815202
#388×C1632
#2375163