Finite Group Databasebeta

← #35218003

Subgroups — #35218003

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×#19104705C211yesmaximal, characteristic
3C2×C4×D652C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
5#23781602C211yesmaximal, characteristic, complemented
6#23781590C211yesmaximal, characteristic, complemented
7#23781582C211yesmaximal, characteristic, complemented
8#23781581C211yesmaximal, characteristic, complemented
9#23781556C211yesmaximal, characteristic, complemented
10#23781543C211yesmaximal, characteristic, complemented
11#23781543C211yesmaximal, characteristic, complemented
12#23781539C211yesmaximal, characteristic, complemented
13#23781522C211yesmaximal, characteristic, complemented
14#23781520C211yesmaximal, characteristic, complemented
15#388×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21C22×C4×D163C2211yescharacteristic
22C22×C4×D163C2211yescharacteristic, complemented
23#19104717—24no—
24#19104705—24no—
25C4×D652—24no—
26C2×C4⋊Dic163C2211yescharacteristic, complemented
27C2×C4⋊Dic163C2211yescharacteristic, complemented
28C2×C4⋊Dic163C2211yescharacteristic, complemented
29C2×Dic163⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
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
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D163—24no—
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C2×C4⋊C4×C163C2211yescharacteristic, complemented
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C42×C163C2211yescharacteristic, complemented
53#95×C163C2211yescharacteristic, complemented
54#95×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
64C23×D163C2311yescharacteristic
65C2×D652—22no—
66C2×D652—22no—
67C2×D652—22no—
68C2×D652—22no—
69C2×D652—41no—
70C2×D652D414yes—
71C2×C22⋊D163—22no—
72C2×C22⋊D163—22no—
73C2×C22⋊D163—41no—
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C22×Dic163C2311yescharacteristic
77C22×Dic163C2311yescharacteristic
78C22×Dic163C2311yescharacteristic
79C2×C4×D163—22no—
80C2×C4×D163—22no—
81C2×C4×D163—22no—
82C2×C4×D163—22no—
83C2×C4×D163—22no—
84C2×C4×D163—22no—
85C2×C4×D163—41no—
86C2×C4×D163—41no—
87C2×C4×D163—41no—
88C4⋊Dic163—22no—
89C4⋊Dic163—22no—
90C4⋊Dic163—41no—
91Dic163⋊C4—22no—
92C22⋊Dic163—22no—
93C22⋊Dic163—22no—
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)⋊C4—22no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4—41no—
104(C2×D163)⋊C4—41no—
105C24×C163C2311yescharacteristic
106C22×C4×C163C2311yescharacteristic
107C22×C4×C163C2311yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C4⋊C4×C163—41no—
113C22⋊C4×C163—22no—
114C22⋊C4×C163—41no—
115C22⋊C4×C163D414yescomplemented
116C42×C163—22no—
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—21no—
129C22×D163—21no—
130C22×D163—21no—
131C22×D163—21no—
132C22×D163—21no—
133C22×D163—21no—
134C22×D163—44no—
135C22×D163—44no—
136C22×D163—44no—
137C22×D163—44no—
138C22×D163—44no—
139C22×D163—44no—
140C22×D163C2×D412yes—
141C22×D163C2×D412yes—
142C22×D163#4112yes—
143C22×D163#4112yes—
144C22×D163#4112yes—
145D652—28no—
146D652—44no—
147C22⋊D163—44no—
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×Dic163—21no—
158C2×Dic163—21no—
159C2×Dic163—21no—
160C2×Dic163—21no—
161C2×Dic163—21no—
162C2×Dic163—21no—
163C4×D163—44no—
164C4×D163—44no—
165C4×D163—44no—
166C23×C163—21no—
167C23×C163—21no—
168C23×C163—21no—
169C23×C163—21no—
170C23×C163—21no—
171C23×C163—21no—
172C23×C163C2×D412yes—
173C23×C163C2411yescharacteristic
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×C163—21no—
184C2×C4×C163—21no—
185C2×C4×C163—21no—
186C2×C4×C163—21no—
187C2×C4×C163—21no—
188C2×C4×C163—21no—
189C2×C4×C163C2×D412yes—
190C2×C4×C163C2×D412yes—
191C2×C4×C163C2×D412yes—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—41no—
196C22×C163—41no—
197C22×C163—41no—
198C22×C163—41no—
199C22×C163C22×D411yescharacteristic
200C22×C163C22×D411yescharacteristic
201C22×C163C2×#4111yescharacteristic
202C22×C163C2×#4111yescharacteristic
203C22×C163C2×#4111yescharacteristic
204C22×C163H(5,𝔽2)11yescharacteristic
205C22×C163H(5,𝔽2)11yescharacteristic
206C2×D163—22no—
207C2×D163—22no—
208C2×D163—22no—
209C2×D163—22no—
210C2×D163—22no—
211C2×D163—22no—
212C2×D163—22no—
213C2×D163—22no—
214C2×D163—22no—
215C2×D163—22no—
216C2×D163—22no—
217C2×D163—22no—
218C2×D163—416no—
219C2×D163—41no—
220C2×D163—41no—
221C2×D163—41no—
222C2×D163—41no—
223C2×D163—41no—
224C2×D163—41no—
225C2×D163—41no—
226C2×D163—41no—
227C2×D163#12314yescomplemented
228C4×C163—22no—
229C4×C163—22no—
230C4×C163—22no—
231C4×C163—41no—
232C4×C163—41no—
233C4×C163—41no—
234Dic163—41no—
235Dic163—41no—
236Dic163—41no—
237Dic163—41no—
238Dic163—41no—
239C2×C163—42no—
240C2×C163C2×#12311yescharacteristic
241C2×C163#52511yescharacteristic
242C2×C163#53811yescharacteristic
243C2×C163D4211yescharacteristic
244C2×C163#54611yescharacteristic
245C2×C163#54611yescharacteristic
246C2×C163#55111yescharacteristic
247D163—24no—
248D163—42no—
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#388D16311yescharacteristic, complemented
262#386—1631no—
263#386—1631no—
264#385—1631no—
265#380—1631no—
266#379—1631no—
267C25—1631no—
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⋊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
299C2×C42C2×D16311yescharacteristic, complemented
300#95—1631no—
301#95—1631no—
302#95—1631no—
303#95—1631no—
304#95C2×D16311yescharacteristic, complemented
305#95C2×D16311yescharacteristic, complemented
306C24—3262no—
307C24—3262no—
308C24—3262no—
309C24—3262no—
310C24—3262no—
311C24—6521no—
312C24—6521no—
313C24—1631no—
314C24—1631no—
315C24—1631no—
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×C4C22×D16311yescharacteristic
341C22×C4C22×D16311yescharacteristic
342C22×C4C22×D16311yescharacteristic
343C22×C4C22×D16311yescharacteristic
344C22×C4C22×D16311yescharacteristic
345C22×C4C22×D16311yescharacteristic
346C4⋊C4—41no—
347C4⋊C4—3262no—
348C4⋊C4—3262no—
349C4⋊C4—3262no—
350C4⋊C4—6521no—
351C22⋊C4—22no—
352C22⋊C4—41no—
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—3262no—
364C22⋊C4—6521no—
365C22⋊C4—6521no—
366C22⋊C4D65214yescomplemented
367C42—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—3268no—
375C23—3261no—
376C23—3261no—
377C23—3261no—
378C23—3261no—
379C23—3261no—
380C23—3261no—
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—
401C23—1632no—
402C23C2×D65212yes—
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—21no—
420C2×C4—21no—
421C2×C4—21no—
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×C4D4×D16312yes—
441C2×C4C2×D65212yes—
442C2×C4C2×D65212yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—3262no—
451C22—3262no—
452C22—3262no—
453C22—3262no—
454C22—3262no—
455C22—3262no—
456C22—3262no—
457C22—3262no—
458C22—3262no—
459C22—3262no—
460C22—3262no—
461C22—3262no—
462C22—65216no—
463C22—6521no—
464C22—6521no—
465C22—6521no—
466C22—6521no—
467C22—6521no—
468C22—6521no—
469C22—6521no—
470C22—6521no—
471C22—1634no—
472C22C2×D4×D16311yescharacteristic
473C22C22×D65211yescharacteristic
474C22#41×D16311yescharacteristic
475C22#41×D16311yescharacteristic
476C22#41×D16311yescharacteristic
477C22#1910478011yescharacteristic
478C22#1910478111yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—6521no—
486C4—6521no—
487C4—6521no—
488C4—6521no—
489C4—6521no—
490C2—42no—
491C2—3264no—
492C2—6522no—
493C2—6522no—
494C2#123×D16311yescharacteristic
495C2#2378217711yescharacteristic
496C2#2378224911yescharacteristic
497C2D4×D65211yescharacteristic
498C2#2378231011yescharacteristic
499C2#2378231211yescharacteristic
500C2#2378234011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#388×C163C2
prime radicalC2×#19104705C2
hypercenter#388D163
hypocenterC163#2418
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#19104717C2semidirect
3. C2×C4×D652C2semidirect
4. C2×C22⋊C4×D163C2semidirect
5. #23781602C2semidirect
6. #23781590C2semidirect
7. #23781582C2semidirect
8. #23781581C2semidirect
9. #23781556C2semidirect
10. #23781543C2semidirect
11. #23781543C2semidirect
12. #23781539C2semidirect
13. #23781522C2semidirect
14. #23781520C2semidirect
15. #388×C163C2semidirect
22. C22×C4×D163C22semidirect
26. C2×C4⋊Dic163C22semidirect
27. C2×C4⋊Dic163C22semidirect
28. C2×C4⋊Dic163C22semidirect
29. C2×Dic163⋊C4C22semidirect
40. C2×(C2×D163)⋊C4C22semidirect
41. C2×(C2×D163)⋊C4C22semidirect
42. C2×(C2×D163)⋊C4C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
48. C2×C4⋊C4×C163C22semidirect
52. C2×C42×C163C22semidirect
53. #95×C163C22semidirect
54. #95×C163C22semidirect
115. C22⋊C4×C163D4semidirect
227. C2×D163#123semidirect
250. C163#2418semidirect
261. #388D163semidirect
281. C2×C4⋊C4C2×D163semidirect
299. C2×C42C2×D163semidirect
304. #95C2×D163semidirect
305. #95C2×D163semidirect
366. C22⋊C4D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047172
C2×#191047052
C2×C4×D6522
C2×C22⋊C4×D1632
#237816022
#237815902
#237815822
#237815812
#237815562
#237815432
#237815392
#237815222
#237815202
#388×C1632
#2418163