Finite Group Databasebeta

← #35218002

Subgroups — #35218002

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×#19104717C211yesmaximal, characteristic, complemented
2C2×#19104704C211yesmaximal, characteristic
3C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic163C211yesmaximal, characteristic, complemented
5#23781601C211yesmaximal, characteristic, complemented
6#23781593C211yesmaximal, characteristic, complemented
7#23781581C211yesmaximal, characteristic, complemented
8#23781580C211yesmaximal, characteristic, complemented
9#23781559C211yesmaximal, characteristic, complemented
10#23781558C211yesmaximal, characteristic, complemented
11#23781542C211yesmaximal, characteristic, complemented
12#23781541C211yesmaximal, characteristic, complemented
13#23781540C211yesmaximal, characteristic, complemented
14#23781523C211yesmaximal, characteristic, complemented
15#379×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C23×Dic163C2211yescharacteristic
20C23×Dic163C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic, complemented
22#19104717—24no—
23#19104704—24no—
24(C22⋊D163)⋊C4—24no—
25C2×C4⋊Dic163C2211yescharacteristic, complemented
26C2×C4⋊Dic163C2211yescharacteristic, complemented
27C2×Dic163⋊C4C2211yescharacteristic, complemented
28C2×Dic163⋊C4C2211yescharacteristic, complemented
29C2×Dic163⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic163—24no—
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
41C2×C4×Dic163C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C25×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic
55C23×D163C2311yescharacteristic
56C2×C22⋊D163—22no—
57C2×C22⋊D163—22no—
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—22no—
62C2×C22⋊D163—41no—
63C2×C22⋊D163—41no—
64C2×C22⋊D163D414yes—
65C22×Dic163—22no—
66C22×Dic163—22no—
67C22×Dic163—22no—
68C22×Dic163—22no—
69C22×Dic163—41no—
70C22×Dic163—41no—
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C22×Dic163C2311yescharacteristic
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C22×Dic163C2311yescharacteristic
77C22×Dic163C2311yescharacteristic
78C2×C4×D163—22no—
79C2×C4×D163—22no—
80C2×C4×D163—41no—
81C4⋊Dic163—22no—
82C4⋊Dic163—41no—
83Dic163⋊C4—22no—
84Dic163⋊C4—22no—
85Dic163⋊C4—41no—
86C22⋊Dic163—22no—
87C22⋊Dic163—22no—
88C22⋊Dic163—22no—
89C22⋊Dic163—22no—
90C22⋊Dic163—22no—
91C22⋊Dic163—22no—
92C22⋊Dic163D414yescomplemented
93(C2×D163)⋊C4—22no—
94(C2×D163)⋊C4—22no—
95(C2×D163)⋊C4—41no—
96C4×Dic163—22no—
97C24×C163—22no—
98C24×C163—22no—
99C24×C163—22no—
100C24×C163—22no—
101C24×C163—22no—
102C24×C163—41no—
103C24×C163—41no—
104C24×C163C2311yescharacteristic
105C24×C163C2311yescharacteristic
106C24×C163C2311yescharacteristic
107C22×C4×C163C2311yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C22⋊C4×C163—22no—
112C22⋊C4×C163—22no—
113C22⋊C4×C163—22no—
114C22⋊C4×C163—22no—
115C22⋊C4×C163—41no—
116C22⋊C4×C163—41no—
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—21no—
122C22×D163—21no—
123C22×D163C2×D412yes—
124C22⋊D163—28no—
125C22⋊D163—44no—
126C22⋊D163—44no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
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—44no—
146C2×Dic163—44no—
147C2×Dic163C2×D412yes—
148C2×Dic163C2×D412yes—
149C2×Dic163C2×D412yes—
150C4×D163—44no—
151C23×C163—21no—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—28no—
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—21no—
159C23×C163—21no—
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—44no—
169C23×C163—44no—
170C23×C163—44no—
171C23×C163—44no—
172C23×C163—44no—
173C23×C163—44no—
174C23×C163C2×D412yes—
175C23×C163C2×D412yes—
176C23×C163#4112yes—
177C23×C163#4112yes—
178C23×C163#4112yes—
179C23×C163C2411yescharacteristic
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—21no—
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—22no—
196C22×C163—22no—
197C22×C163—22no—
198C22×C163—22no—
199C22×C163—22no—
200C22×C163—22no—
201C22×C163—22no—
202C22×C163—22no—
203C22×C163—22no—
204C22×C163—416no—
205C22×C163—41no—
206C22×C163—41no—
207C22×C163—41no—
208C22×C163—41no—
209C22×C163—41no—
210C22×C163—41no—
211C22×C163—41no—
212C22×C163—41no—
213C22×C163#12314yescomplemented
214C22×C163C22×D411yescharacteristic
215C22×C163C22×D411yescharacteristic
216C22×C163C2×#4111yescharacteristic
217C22×C163C2×#4111yescharacteristic
218C22×C163C2×#4111yescharacteristic
219C22×C163H(5,𝔽2)11yescharacteristic
220C22×C163H(5,𝔽2)11yescharacteristic
221C2×D163—22no—
222C2×D163—22no—
223C2×D163—22no—
224C2×D163—41no—
225C2×D163—41no—
226C2×D163—41no—
227C2×D163—41no—
228C4×C163—41no—
229C4×C163—41no—
230C4×C163—41no—
231C4×C163—41no—
232Dic163—22no—
233Dic163—22no—
234Dic163—22no—
235Dic163—41no—
236Dic163—41no—
237Dic163—41no—
238Dic163—41no—
239C2×C163—24no—
240C2×C163—42no—
241C2×C163—42no—
242C2×C163C2×#12311yescharacteristic
243C2×C163#52511yescharacteristic
244C2×C163#53811yescharacteristic
245C2×C163D4211yescharacteristic
246C2×C163#54611yescharacteristic
247C2×C163#54611yescharacteristic
248C2×C163#55111yescharacteristic
249D163—42no—
250C163#241811yescharacteristic, complemented, Sylow
251#2418—1631nomaximal, Sylow
252C2×#123—1631no—
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#379D16311yescharacteristic, complemented
267C25C2×D16311yescharacteristic
268C22×D4—1631no—
269C22×D4—1631no—
270C22×D4—1631no—
271C23×C4—1631no—
272C23×C4—1631no—
273C23×C4—1631no—
274#123—3264no—
275#120—3264no—
276C4×D4—3264no—
277C2×C4⋊C4—1631no—
278C2×C4⋊C4—1631no—
279C2×C4⋊C4—1631no—
280C2×C4⋊C4—1631no—
281C2×C4⋊C4—1631no—
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⋊C4C2×D16311yescharacteristic
294C2×C22⋊C4C2×D16311yescharacteristic
295C2×C22⋊C4C2×D16311yescharacteristic
296C2×C22⋊C4C2×D16311yescharacteristic
297C2×C22⋊C4C2×D16311yescharacteristic
298C2×C22⋊C4C2×D16311yescharacteristic
299C2×C42—1631no—
300#95—1631no—
301#95—1631no—
302#95—1631no—
303#95—1631no—
304#95—1631no—
305#95—1631no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24—1631no—
314C24C22×D16311yescharacteristic
315C24C22×D16311yescharacteristic
316C24C22×D16311yescharacteristic
317C2×D4—3262no—
318C2×D4—3262no—
319C2×D4—3262no—
320C2×D4—3262no—
321C2×D4—3262no—
322C2×D4—3262no—
323C2×D4—6521no—
324C2×D4—6521no—
325C2×D4—1634no—
326C22×C4—3262no—
327C22×C4—3262no—
328C22×C4—3262no—
329C22×C4—3262no—
330C22×C4—3262no—
331C22×C4—3262no—
332C22×C4—6521no—
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×C4—1631no—
342C22×C4C22×D16311yescharacteristic
343C22×C4C22×D16311yescharacteristic
344C22×C4C22×D16311yescharacteristic
345C22×C4C22×D16311yescharacteristic
346C4⋊C4—3262no—
347C4⋊C4—3262no—
348C4⋊C4—3262no—
349C4⋊C4—6521no—
350C4⋊C4—6521no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—3262no—
358C22⋊C4—3262no—
359C22⋊C4—3262no—
360C22⋊C4—3262no—
361C22⋊C4—3262no—
362C22⋊C4—3262no—
363C22⋊C4—3262no—
364C22⋊C4—3262no—
365C22⋊C4—6521no—
366C22⋊C4—1634no—
367C42—3262no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—3261no—
392C23—3261no—
393C23—3261no—
394C23—3261no—
395C23—3261no—
396C23—3261no—
397C23—1632no—
398C23D4×D16312yes—
399C23D4×D16312yes—
400C23#1457969812yes—
401C23#1457969812yes—
402C23#1457969912yes—
403C23C23×D16311yescharacteristic
404D4—3268no—
405D4—6524no—
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—3261no—
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—3261no—
434C2×C4—3261no—
435C2×C4—3261no—
436C2×C4—3261no—
437C2×C4—6524no—
438C2×C4—6524no—
439C2×C4—6524no—
440C2×C4—1632no—
441C2×C4—1632no—
442C2×C4—1632no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—3262no—
465C22—3262no—
466C22—3262no—
467C22—6521no—
468C22—6521no—
469C22—6521no—
470C22—6521no—
471C22#1910471714yescomplemented
472C22C2×D4×D16311yescharacteristic
473C22C2×D4×D16311yescharacteristic
474C22C2×#1457969811yescharacteristic
475C22C2×#1457969811yescharacteristic
476C22C2×#1457969911yescharacteristic
477C22#1910478011yescharacteristic
478C22#1910478011yescharacteristic
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—3262no—
484C4—3262no—
485C4—3262no—
486C4—6521no—
487C4—6521no—
488C4—6521no—
489C4—6521no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—6522no—
494C2C2×#1910471711yescharacteristic
495C2#2378217111yescharacteristic
496C2#2378225011yescharacteristic
497C2#2378229711yescharacteristic
498C2#2378230211yescharacteristic
499C2#2378230511yescharacteristic
500C2#2378234111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#379×C163C2
prime radicalC2×#19104704C2
hypercenter#379D163
hypocenterC163#2418
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#19104717C2semidirect
3. C2×(C22⋊D163)⋊C4C2semidirect
4. C22×C22⋊Dic163C2semidirect
5. #23781601C2semidirect
6. #23781593C2semidirect
7. #23781581C2semidirect
8. #23781580C2semidirect
9. #23781559C2semidirect
10. #23781558C2semidirect
11. #23781542C2semidirect
12. #23781541C2semidirect
13. #23781540C2semidirect
14. #23781523C2semidirect
15. #379×C163C2semidirect
21. C22×C4×D163C22semidirect
25. C2×C4⋊Dic163C22semidirect
26. C2×C4⋊Dic163C22semidirect
27. C2×Dic163⋊C4C22semidirect
28. C2×Dic163⋊C4C22semidirect
29. C2×Dic163⋊C4C22semidirect
38. C2×(C2×D163)⋊C4C22semidirect
39. C2×(C2×D163)⋊C4C22semidirect
40. C2×(C2×D163)⋊C4C22semidirect
41. C2×C4×Dic163C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
92. C22⋊Dic163D4semidirect
213. C22×C163#123semidirect
250. C163#2418semidirect
266. #379D163semidirect
471. C22#19104717semidirect

Maximal subgroups

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

subgroupindex
C2×#191047172
C2×#191047042
C2×(C22⋊D163)⋊C42
C22×C22⋊Dic1632
#237816012
#237815932
#237815812
#237815802
#237815592
#237815582
#237815422
#237815412
#237815402
#237815232
#379×C1632
#2418163