Finite Group Databasebeta

← #35218198

Subgroups — #35218198

Order 20864.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104715C211yesmaximal, characteristic, complemented
2C2×#19104713C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D163C211yesmaximal, characteristic
4C2×C22⋊C4×D163C211yesmaximal, characteristic
5#23781601C211yesmaximal, characteristic, complemented
6#23781588C211yesmaximal, characteristic, complemented
7#23781577C211yesmaximal, characteristic, complemented
8#23781539C211yesmaximal, characteristic, complemented
9#23781536C211yesmaximal, characteristic, complemented
10#23781536C211yesmaximal, characteristic, complemented
11#23781536C211yesmaximal, characteristic, complemented
12#23781524C211yesmaximal, characteristic, complemented
13#23781522C211yesmaximal, characteristic, complemented
14#23781520C211yesmaximal, characteristic
15#396×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×Dic326C2211yescharacteristic, complemented
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic
22#19104715—24no—
23#19104713—24no—
24C2×C4⋊Dic163C2211yescharacteristic, complemented
25C2×C4⋊Dic163C2211yescharacteristic, complemented
26C2×Dic163⋊C4C2211yescharacteristic, complemented
27C2×Dic163⋊C4C2211yescharacteristic, complemented
28C2×Dic163⋊C4C2211yescharacteristic, complemented
29C2×Dic163⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C22⋊C4×D163—24no—
42C22⋊C4×D163—24no—
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C2×C4⋊C4×C163C2211yescharacteristic, complemented
49C2×C4⋊C4×C163C2211yescharacteristic, complemented
50C2×C4⋊C4×C163C2211yescharacteristic, complemented
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54#95×C163C2211yescharacteristic, complemented
55C23×D163—22no—
56C23×D163—22no—
57C23×D163—22no—
58C23×D163—22no—
59C23×D163—41no—
60C23×D163—41no—
61C23×D163Q814yescomplemented
62C23×D163C2311yescharacteristic
63C23×D163C2311yescharacteristic
64C2×Dic326—22no—
65C2×Dic326—22no—
66C2×Dic326—41no—
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C2×C4×D163—22no—
76C2×C4×D163—22no—
77C2×C4×D163—22no—
78C2×C4×D163—22no—
79C2×C4×D163—22no—
80C2×C4×D163—22no—
81C2×C4×D163—41no—
82C2×C4×D163—41no—
83C2×C4×D163—41no—
84C2×C4×D163—41no—
85C4⋊Dic163—22no—
86C4⋊Dic163—22no—
87Dic163⋊C4—22no—
88Dic163⋊C4—22no—
89Dic163⋊C4—41no—
90Dic163⋊C4—41no—
91C22⋊Dic163—22no—
92C22⋊Dic163—22no—
93C22⋊Dic163—41no—
94(C2×D163)⋊C4—22no—
95(C2×D163)⋊C4—22no—
96(C2×D163)⋊C4—22no—
97(C2×D163)⋊C4—22no—
98(C2×D163)⋊C4—22no—
99(C2×D163)⋊C4—22no—
100(C2×D163)⋊C4D414yes—
101(C2×D163)⋊C4D414yes—
102C24×C163C2311yescharacteristic
103C22×C4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105C22×C4×C163C2311yescharacteristic
106C22×C4×C163C2311yescharacteristic
107C22×C4×C163C2311yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C4⋊C4×C163—22no—
110C4⋊C4×C163—22no—
111C4⋊C4×C163—41no—
112C22⋊C4×C163—22no—
113C22⋊C4×C163—22no—
114C22⋊C4×C163—41no—
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—28no—
124C22×D163—28no—
125C22×D163—44no—
126C22×D163—44no—
127C22×D163—44no—
128C22×D163—44no—
129C22×D163—44no—
130C22×D163C2×D412yes—
131C22×D163C2×D412yes—
132C22×D163C2×Q812yes—
133C22×D163C2×Q812yes—
134C22×D163#4112yes—
135C22×D163#4112yes—
136Dic326—44no—
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—21no—
147C2×Dic163—21no—
148C2×Dic163—21no—
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163—21no—
152C2×Dic163—21no—
153C2×Dic163C2×D412yes—
154C2×Dic163C2×D412yes—
155C4×D163—44no—
156C4×D163—44no—
157C4×D163—44no—
158C4×D163—44no—
159C23×C163—21no—
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—21no—
163C23×C163—21no—
164C23×C163—21no—
165C23×C163C2×Q812yes—
166C23×C163C2411yescharacteristic
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×C163—21no—
182C2×C4×C163—21no—
183C2×C4×C163C2×D412yes—
184C2×C4×C163C2×D412yes—
185C22×C163—22no—
186C22×C163—22no—
187C22×C163—22no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163—41no—
191C22×C163—41no—
192C22×C163C22×D411yescharacteristic
193C22×C163C22×D411yescharacteristic
194C22×C163C22×Q811yescharacteristic
195C22×C163C2×#4111yescharacteristic
196C22×C163C2×#4111yescharacteristic
197C22×C163H(5,𝔽2)11yescharacteristic
198C22×C163H(5,𝔽2)11yescharacteristic
199C2×D163—22no—
200C2×D163—22no—
201C2×D163—22no—
202C2×D163—22no—
203C2×D163—22no—
204C2×D163—22no—
205C2×D163—22no—
206C2×D163—22no—
207C2×D163—22no—
208C2×D163—22no—
209C2×D163—22no—
210C2×D163—22no—
211C2×D163—416no—
212C2×D163C22⋊Q814yescomplemented
213C2×D163C22⋊Q814yescomplemented
214C4×C163—22no—
215C4×C163—22no—
216C4×C163—41no—
217C4×C163—41no—
218C4×C163—41no—
219C4×C163—41no—
220Dic163—22no—
221Dic163—22no—
222Dic163—41no—
223Dic163—41no—
224Dic163—41no—
225Dic163—41no—
226C2×C163—42no—
227C2×C163C2×C22⋊Q811yescharacteristic
228C2×C163C2×C22⋊Q811yescharacteristic
229C2×C163#54311yescharacteristic
230C2×C163D4211yescharacteristic
231C2×C163#54611yescharacteristic
232C2×C163#54611yescharacteristic
233C2×C163#55111yescharacteristic
234D163—24no—
235D163—24no—
236C163#243711yescharacteristic, complemented, Sylow
237#2437—1631nomaximal, Sylow
238C2×C22⋊Q8—1631no—
239C2×C22⋊Q8—1631no—
240C22×C22⋊C4—1631no—
241C22×C22⋊C4—1631no—
242#399—1631no—
243#396—1631no—
244#396D16311yescharacteristic, complemented
245#393—1631no—
246#385—1631no—
247#385—1631no—
248#385—1631no—
249#385—1631no—
250#380—1631no—
251#380—1631no—
252#379—1631no—
253C25—1631no—
254C22×Q8—1631no—
255C23×C4—1631no—
256C23×C4—1631no—
257C23×C4—1631no—
258C23×C4—1631no—
259C22⋊Q8—3264no—
260C22⋊Q8—3264no—
261C2×C4⋊C4—1631no—
262C2×C4⋊C4—1631no—
263C2×C4⋊C4—1631no—
264C2×C4⋊C4—1631no—
265C2×C4⋊C4—1631no—
266C2×C4⋊C4—1631no—
267C2×C4⋊C4C2×D16311yescharacteristic, complemented
268C2×C4⋊C4C2×D16311yescharacteristic, complemented
269C2×C4⋊C4C2×D16311yescharacteristic, complemented
270C2×C22⋊C4—3264no—
271C2×C22⋊C4—3264no—
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⋊C4—1631no—
283C2×C22⋊C4C2×D16311yescharacteristic
284C2×C22⋊C4C2×D16311yescharacteristic
285C2×C22⋊C4C2×D16311yescharacteristic
286#95—1631no—
287#95—1631no—
288#95—1631no—
289#95—1631no—
290#95—1631no—
291#95C2×D16311yescharacteristic, complemented
292C24—3262no—
293C24—3262no—
294C24—3262no—
295C24—3262no—
296C24—6521no—
297C24—6521no—
298C24—1634no—
299C24—1631no—
300C24—1631no—
301C24C22×D16311yescharacteristic
302C2×Q8—3262no—
303C2×Q8—3262no—
304C2×Q8—6521no—
305C22×C4—3262no—
306C22×C4—3262no—
307C22×C4—3262no—
308C22×C4—3262no—
309C22×C4—3262no—
310C22×C4—3262no—
311C22×C4—3262no—
312C22×C4—3262no—
313C22×C4—6521no—
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×C4—1631no—
323C22×C4C22×D16311yescharacteristic
324C22×C4C22×D16311yescharacteristic
325C22×C4C22×D16311yescharacteristic
326C22×C4C22×D16311yescharacteristic
327C22×C4C22×D16311yescharacteristic
328C22×C4C22×D16311yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—3262no—
333C4⋊C4—3262no—
334C4⋊C4—3262no—
335C4⋊C4—3262no—
336C4⋊C4—6521no—
337C4⋊C4—6521no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—3262no—
342C22⋊C4—3262no—
343C22⋊C4—3262no—
344C22⋊C4—3262no—
345C22⋊C4—3262no—
346C22⋊C4—3262no—
347C22⋊C4—3262no—
348C22⋊C4—3262no—
349C22⋊C4—6521no—
350C22⋊C4—1634no—
351C22⋊C4—1634no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—3268no—
359C23—3268no—
360C23—3261no—
361C23—3261no—
362C23—3261no—
363C23—3261no—
364C23—3261no—
365C23—3261no—
366C23—3261no—
367C23—3261no—
368C23—6524no—
369C23—6524no—
370C23—6524no—
371C23—6524no—
372C23—6524no—
373C23—1632no—
374C23—1632no—
375C23—1632no—
376C23—1632no—
377C23—1632no—
378C23—1632no—
379C23Q8×D16312yes—
380C23C23×D16311yescharacteristic
381Q8—6524no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—3261no—
399C2×C4—3261no—
400C2×C4—3261no—
401C2×C4—3261no—
402C2×C4—3261no—
403C2×C4—3261no—
404C2×C4—3261no—
405C2×C4—3261no—
406C2×C4—3261no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—3261no—
411C2×C4—3261no—
412C2×C4—3261no—
413C2×C4—3261no—
414C2×C4—6524no—
415C2×C4—6524no—
416C2×C4—6524no—
417C2×C4—6524no—
418C2×C4—1632no—
419C2×C4—1632no—
420C2×C4D4×D16312yes—
421C2×C4D4×D16312yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—3262no—
430C22—3262no—
431C22—3262no—
432C22—3262no—
433C22—3262no—
434C22—3262no—
435C22—3262no—
436C22—3262no—
437C22—3262no—
438C22—3262no—
439C22—3262no—
440C22—3262no—
441C22—65216no—
442C22—1634no—
443C22—1634no—
444C22C2×D4×D16311yescharacteristic
445C22C2×D4×D16311yescharacteristic
446C22C2×Q8×D16311yescharacteristic
447C22#41×D16311yescharacteristic
448C22#41×D16311yescharacteristic
449C22#1910478011yescharacteristic
450C22#1910478111yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—3262no—
458C4—3262no—
459C4—6521no—
460C4—6521no—
461C4—6521no—
462C4—6521no—
463C2—42no—
464C2—3264no—
465C2—3264no—
466C2C22⋊Q8×D16311yescharacteristic
467C2C22⋊Q8×D16311yescharacteristic
468C2#2378229011yescharacteristic
469C2#2378229911yescharacteristic
470C2#2378230411yescharacteristic
471C2#2378231211yescharacteristic
472C2#2378234011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#396×C163C2
prime radicalC24×D163C22
hypercenter#396D163
hypocenterC163#2437
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#19104715C2semidirect
2. C2×#19104713C2semidirect
5. #23781601C2semidirect
6. #23781588C2semidirect
7. #23781577C2semidirect
8. #23781539C2semidirect
9. #23781536C2semidirect
10. #23781536C2semidirect
11. #23781536C2semidirect
12. #23781524C2semidirect
13. #23781522C2semidirect
15. #396×C163C2semidirect
17. C22×Dic326C22semidirect
24. C2×C4⋊Dic163C22semidirect
25. C2×C4⋊Dic163C22semidirect
26. C2×Dic163⋊C4C22semidirect
27. C2×Dic163⋊C4C22semidirect
28. C2×Dic163⋊C4C22semidirect
29. C2×Dic163⋊C4C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
48. C2×C4⋊C4×C163C22semidirect
49. C2×C4⋊C4×C163C22semidirect
50. C2×C4⋊C4×C163C22semidirect
54. #95×C163C22semidirect
61. C23×D163Q8semidirect
212. C2×D163C22⋊Q8semidirect
213. C2×D163C22⋊Q8semidirect
236. C163#2437semidirect
244. #396D163semidirect
267. C2×C4⋊C4C2×D163semidirect
268. C2×C4⋊C4C2×D163semidirect
269. C2×C4⋊C4C2×D163semidirect
291. #95C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047152
C2×#191047132
C2×C22⋊C4×D1632
#237816012
#237815882
#237815772
#237815392
#237815362
#237815242
#237815222
#237815202
#396×C1632
#2437163