Finite Group Databasebeta

← #35217995

Subgroups — #35217995

Order 20864.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104703C211yesmaximal, characteristic
2C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
4#23781600C211yesmaximal, characteristic, complemented
5#23781593C211yesmaximal, characteristic, complemented
6#23781581C211yesmaximal, characteristic, complemented
7#23781579C211yesmaximal, characteristic, complemented
8#23781558C211yesmaximal, characteristic, complemented
9#23781555C211yesmaximal, characteristic, complemented
10#23781544C211yesmaximal, characteristic, complemented
11#23781541C211yesmaximal, characteristic, complemented
12#23781536C211yesmaximal, characteristic, complemented
13#23781522C211yesmaximal, characteristic, complemented
14#23781520C211yesmaximal, characteristic, complemented
15C2×#123×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104703—24no—
22C4×C22⋊D163—24no—
23C2×C4⋊Dic163C2211yescharacteristic, complemented
24C2×Dic163⋊C4C2211yescharacteristic, complemented
25C2×Dic163⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic163C2211yescharacteristic
27C2×C22⋊Dic163C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×C22⋊Dic163C2211yescharacteristic, complemented
30C2×C22⋊Dic163C2211yescharacteristic, complemented
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
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C22⋊C4×D163—24no—
40C2×C4×Dic163C2211yescharacteristic, complemented
41C163⋊#95C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C22×D4×C163C2211yescharacteristic
48C23×C4×C163C2211yescharacteristic, complemented
49#123×C163—24no—
50C2×C4⋊C4×C163C2211yescharacteristic, complemented
51C2×C4⋊C4×C163C2211yescharacteristic, complemented
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic, complemented
55C23×D163—22no—
56C23×D163—22no—
57C23×D163—22no—
58C23×D163—22no—
59C23×D163—22no—
60C23×D163—41no—
61C23×D163—41no—
62C23×D163C2311yescharacteristic
63C23×D163C2311yescharacteristic
64C2×C22⋊D163—22no—
65C2×C22⋊D163—22no—
66C2×C22⋊D163—22no—
67C2×C22⋊D163—22no—
68C2×C22⋊D163—41no—
69C2×C22⋊D163D414yes—
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C22×Dic163C2311yescharacteristic
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C2×C4×D163—22no—
77C2×C4×D163—22no—
78C2×C4×D163—22no—
79C2×C4×D163—22no—
80C2×C4×D163—41no—
81C2×C4×D163—41no—
82C4⋊Dic163—41no—
83Dic163⋊C4—22no—
84Dic163⋊C4—41no—
85C22⋊Dic163—22no—
86C22⋊Dic163—22no—
87C22⋊Dic163—22no—
88C22⋊Dic163—41no—
89C22⋊Dic163—41no—
90(C2×D163)⋊C4—22no—
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—22no—
97(C2×D163)⋊C4—41no—
98C4×Dic163—22no—
99C24×C163C2311yescharacteristic
100C24×C163C2311yescharacteristic
101C2×D4×C163—22no—
102C2×D4×C163—22no—
103C2×D4×C163—41no—
104C22×C4×C163—22no—
105C22×C4×C163—22no—
106C22×C4×C163—41no—
107C22×C4×C163C2311yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C4⋊C4×C163—22no—
113C4⋊C4×C163—22no—
114C22⋊C4×C163—22no—
115C22⋊C4×C163—22no—
116C22⋊C4×C163D414yescomplemented
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—28no—
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×D163C2×D412yes—
135C22×D163#4112yes—
136C22×D163#4112yes—
137C22×D163#4112yes—
138C22⋊D163—28no—
139C22⋊D163—44no—
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×Dic163—21no—
154C2×Dic163—21no—
155C2×Dic163—21no—
156C2×Dic163—21no—
157C2×Dic163C2×D412yes—
158C4×D163—44no—
159C4×D163—44no—
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—21no—
163C23×C163—21no—
164C23×C163—21no—
165C23×C163—21no—
166C23×C163—21no—
167C23×C163—21no—
168C23×C163—21no—
169C23×C163—21no—
170C23×C163—21no—
171C23×C163—21no—
172C23×C163C2×D412yes—
173C23×C163C2×D412yes—
174C23×C163C2411yescharacteristic
175D4×C163—44no—
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×C163—21no—
189C2×C4×C163—44no—
190C2×C4×C163C2×D412yes—
191C2×C4×C163C2×D412yes—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—22no—
196C22×C163—22no—
197C22×C163—22no—
198C22×C163—41no—
199C22×C163—41no—
200C22×C163—41no—
201C22×C163—41no—
202C22×C163—41no—
203C22×C163—41no—
204C22×C163—41no—
205C22×C163—41no—
206C22×C163C22×D411yescharacteristic
207C22×C163C22×D411yescharacteristic
208C22×C163C2×#4111yescharacteristic
209C22×C163C2×#4111yescharacteristic
210C22×C163C2×#4111yescharacteristic
211C22×C163H(5,𝔽2)11yescharacteristic
212C22×C163H(5,𝔽2)11yescharacteristic
213C2×D163—22no—
214C2×D163—22no—
215C2×D163—22no—
216C2×D163—22no—
217C2×D163—22no—
218C2×D163—22no—
219C2×D163—22no—
220C2×D163—22no—
221C2×D163—22no—
222C2×D163—416no—
223C2×D163—41no—
224C2×D163—41no—
225C2×D163—41no—
226C2×D163—41no—
227C2×D163#12314yescomplemented
228C4×C163—22no—
229C4×C163—22no—
230C4×C163—41no—
231C4×C163—41no—
232C4×C163—41no—
233Dic163—22no—
234Dic163—41no—
235Dic163—41no—
236Dic163—41no—
237Dic163—41no—
238Dic163—41no—
239C2×C163—42no—
240C2×C163—42no—
241C2×C163C2×#12311yescharacteristic
242C2×C163#52511yescharacteristic
243C2×C163#53811yescharacteristic
244C2×C163D4211yescharacteristic
245C2×C163#54611yescharacteristic
246C2×C163#54611yescharacteristic
247C2×C163#55111yescharacteristic
248D163—24no—
249D163—42no—
250C163#241811yescharacteristic, complemented, Sylow
251#2418—1631nomaximal, Sylow
252C2×#123D16311yescharacteristic, complemented
253C2×#120—1631no—
254C2×C4×D4—1631no—
255C22×C22⋊C4—1631no—
256#399—1631no—
257#397—1631no—
258#394—1631no—
259#394—1631no—
260#388—1631no—
261#388—1631no—
262#386—1631no—
263#386—1631no—
264#385—1631no—
265#380—1631no—
266#379—1631no—
267C25—1631no—
268C22×D4—1631no—
269C22×D4—1631no—
270C22×D4C2×D16311yescharacteristic
271C23×C4—1631no—
272C23×C4—1631no—
273C23×C4C2×D16311yescharacteristic, complemented
274#123—24no—
275#120—3264no—
276C4×D4—3264no—
277C2×C4⋊C4—1631no—
278C2×C4⋊C4—1631no—
279C2×C4⋊C4—1631no—
280C2×C4⋊C4C2×D16311yescharacteristic, complemented
281C2×C4⋊C4C2×D16311yescharacteristic, complemented
282C2×C22⋊C4—3264no—
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⋊C4—1631no—
293C2×C22⋊C4—1631no—
294C2×C22⋊C4—1631no—
295C2×C22⋊C4—1631no—
296C2×C22⋊C4C2×D16311yescharacteristic
297C2×C22⋊C4C2×D16311yescharacteristic
298C2×C22⋊C4C2×D16311yescharacteristic, complemented
299C2×C42—1631no—
300#95—1631no—
301#95—1631no—
302#95—1631no—
303#95—1631no—
304#95—1631no—
305#95—1631no—
306C24—3262no—
307C24—3262no—
308C24—3262no—
309C24—3262no—
310C24—3262no—
311C24—6521no—
312C24—6521no—
313C24—1631no—
314C24—1631no—
315C24C22×D16311yescharacteristic
316C24C22×D16311yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—3262no—
321C2×D4—3262no—
322C2×D4—3262no—
323C2×D4—3262no—
324C2×D4—6521no—
325C2×D4—1634no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—3262no—
330C22×C4—3262no—
331C22×C4—3262no—
332C22×C4—3262no—
333C22×C4—6521no—
334C22×C4—6521no—
335C22×C4—1631no—
336C22×C4—1631no—
337C22×C4—1631no—
338C22×C4—1631no—
339C22×C4—1631no—
340C22×C4—1631no—
341C22×C4C22×D16311yescharacteristic
342C22×C4C22×D16311yescharacteristic
343C22×C4C22×D16311yescharacteristic
344C22×C4C22×D16311yescharacteristic
345C22×C4C22×D16311yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—3262no—
349C4⋊C4—6521no—
350C4⋊C4—6521no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—3262no—
354C22⋊C4—3262no—
355C22⋊C4—3262no—
356C22⋊C4—3262no—
357C22⋊C4—3262no—
358C22⋊C4—3262no—
359C22⋊C4—3262no—
360C22⋊C4—3262no—
361C22⋊C4—3262no—
362C22⋊C4—3262no—
363C22⋊C4—6521no—
364C22⋊C4—6521no—
365C22⋊C4—6521no—
366C22⋊C4C22⋊D16314yescomplemented
367C42—3262no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—3268no—
381C23—3261no—
382C23—3261no—
383C23—3261no—
384C23—3261no—
385C23—3261no—
386C23—3261no—
387C23—3261no—
388C23—3261no—
389C23—3261no—
390C23—3261no—
391C23—6524no—
392C23—6524no—
393C23—6524no—
394C23—6524no—
395C23—6524no—
396C23—6524no—
397C23—1632no—
398C23—1632no—
399C23—1632no—
400C23—1632no—
401C23D4×D16312yes—
402C23C2×C22⋊D16312yes—
403C23C23×D16311yescharacteristic
404D4—44no—
405D4—3268no—
406D4—6524no—
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—44no—
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—3261no—
434C2×C4—3261no—
435C2×C4—3261no—
436C2×C4—3261no—
437C2×C4—3261no—
438C2×C4—6524no—
439C2×C4—6524no—
440C2×C4—1632no—
441C2×C4C2×C22⋊D16312yes—
442C2×C4C2×C22⋊D16312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—3262no—
458C22—3262no—
459C22—3262no—
460C22—3262no—
461C22—3262no—
462C22—3262no—
463C22—3262no—
464C22—3262no—
465C22—3262no—
466C22—65216no—
467C22—6521no—
468C22—6521no—
469C22—6521no—
470C22—6521no—
471C22—1634no—
472C22C2×D4×D16311yescharacteristic
473C22C22×C22⋊D16311yescharacteristic
474C22#41×D16311yescharacteristic
475C22#41×D16311yescharacteristic
476C22#41×D16311yescharacteristic
477C22#1910478011yescharacteristic
478C22#1910478111yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—3262no—
485C4—6521no—
486C4—6521no—
487C4—6521no—
488C4—6521no—
489C4—6521no—
490C2—42no—
491C2—42no—
492C2—3264no—
493C2—6522no—
494C2#123×D16311yescharacteristic
495C2#2378217211yescharacteristic
496C2#2378225111yescharacteristic
497C2D4×C22⋊D16311yescharacteristic
498C2#2378230311yescharacteristic
499C2#2378230711yescharacteristic
500C2#2378234311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC2×#123×C163C2
prime radicalC2×#19104703C2
hypercenterC2×#123D163
hypocenterC163#2418
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D163C2semidirect
3. C2×C22⋊C4×D163C2semidirect
4. #23781600C2semidirect
5. #23781593C2semidirect
6. #23781581C2semidirect
7. #23781579C2semidirect
8. #23781558C2semidirect
9. #23781555C2semidirect
10. #23781544C2semidirect
11. #23781541C2semidirect
12. #23781536C2semidirect
13. #23781522C2semidirect
14. #23781520C2semidirect
15. C2×#123×C163C2semidirect
23. C2×C4⋊Dic163C22semidirect
24. C2×Dic163⋊C4C22semidirect
25. C2×Dic163⋊C4C22semidirect
29. C2×C22⋊Dic163C22semidirect
30. C2×C22⋊Dic163C22semidirect
40. C2×C4×Dic163C22semidirect
41. C163⋊#95C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
48. C23×C4×C163C22semidirect
50. C2×C4⋊C4×C163C22semidirect
51. C2×C4⋊C4×C163C22semidirect
54. C2×C22⋊C4×C163C22semidirect
116. C22⋊C4×C163D4semidirect
227. C2×D163#123semidirect
250. C163#2418semidirect
252. C2×#123D163semidirect
273. C23×C4C2×D163semidirect
280. C2×C4⋊C4C2×D163semidirect
281. C2×C4⋊C4C2×D163semidirect
298. C2×C22⋊C4C2×D163semidirect
366. C22⋊C4C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047032
C2×C4×C22⋊D1632
C2×C22⋊C4×D1632
#237816002
#237815932
#237815812
#237815792
#237815582
#237815552
#237815442
#237815412
#237815362
#237815222
#237815202
C2×#123×C1632
#2418163