Finite Group Databasebeta

← #16849033

Subgroups — #16849033

Order 3968.

1116 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D31C211yesmaximal, characteristic
2C2×#11758471C211yesmaximal, characteristic, complemented
3C2×#11758460C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D31C211yesmaximal, characteristic, complemented
5#14520965C211yesmaximal, characteristic, complemented
6#14520964C211yesmaximal, characteristic, complemented
7#14520963C211yesmaximal, characteristic, complemented
8#14520927C211yesmaximal, characteristic, complemented
9#14520925C211yesmaximal, characteristic, complemented
10#14520923C211yesmaximal, characteristic, complemented
11#14520920C211yesmaximal, characteristic, complemented
12#14520904C211yesmaximal, characteristic, complemented
13#14520904C211yesmaximal, characteristic, complemented
14#14520904C211yesmaximal, characteristic, complemented
15C31×#394C211yesmaximal, characteristic, complemented
16C24×D31C2211yescharacteristic
17C24×D31C2211yescharacteristic
18C22×D4×C31C2211yescharacteristic
19C22×C22⋊D31C2211yescharacteristic
20C22×C22⋊D31C2211yescharacteristic
21C2×D4×D31—24no—
22C2×D4×D31—24no—
23C2×D4×D31—24no—
24C22×D124C2211yescharacteristic
25C22×C4×D31C2211yescharacteristic
26C22×C4×D31C2211yescharacteristic, complemented
27C22×C4×D31C2211yescharacteristic, complemented
28C2×C4⋊C4×C31C2211yescharacteristic, complemented
29C2×C22⋊C4×C31C2211yescharacteristic, complemented
30C2×C22⋊C4×C31C2211yescharacteristic, complemented
31C2×C22⋊C4×C31C2211yescharacteristic, complemented
32C2×C22⋊C4×C31C2211yescharacteristic, complemented
33C2×C22⋊Dic31C2211yescharacteristic, complemented
34C2×C22⋊Dic31C2211yescharacteristic, complemented
35C2×C22⋊Dic31C2211yescharacteristic, complemented
36C2×C22⋊Dic31C2211yescharacteristic, complemented
37C2×(C2×D31)⋊C4C2211yescharacteristic
38C2×(C2×D31)⋊C4C2211yescharacteristic
39C2×(C2×D31)⋊C4C2211yescharacteristic
40C2×(C2×D31)⋊C4C2211yescharacteristic
41C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
47C2×C4⋊Dic31C2211yescharacteristic, complemented
48C2×Dic31⋊C4C2211yescharacteristic, complemented
49C2×Dic31⋊C4C2211yescharacteristic, complemented
50#11758471—24no—
51#11758460—24no—
52C22⋊C4×D31—24no—
53C31×#95C2211yescharacteristic, complemented
54C31⋊#95C2211yescharacteristic, complemented
55C31⋊#95C2211yescharacteristic, complemented
56C31⋊#95C2211yescharacteristic, complemented
57C24×C31C2311yescharacteristic
58C24×C31C2311yescharacteristic
59C23×D31—22no—
60C23×D31—22no—
61C23×D31—22no—
62C23×D31—22no—
63C23×D31—22no—
64C23×D31—22no—
65C23×D31—22no—
66C23×D31—22no—
67C23×D31—22no—
68C23×D31—41no—
69C23×D31—41no—
70C23×D31—41no—
71C23×D31—41no—
72C23×D31D414yes—
73C23×D31C2311yescharacteristic
74C23×D31C2311yescharacteristic
75C23×D31C2311yescharacteristic
76C2×D4×C31—22no—
77C2×D4×C31—22no—
78C2×D4×C31—22no—
79C22×C4×C31C2311yescharacteristic
80C22×C4×C31C2311yescharacteristic
81C22×C4×C31C2311yescharacteristic
82C22×C4×C31C2311yescharacteristic
83C22×C4×C31C2311yescharacteristic
84C2×C22⋊D31—22no—
85C2×C22⋊D31—22no—
86C2×C22⋊D31—22no—
87C2×C22⋊D31—22no—
88C2×C22⋊D31—22no—
89C2×C22⋊D31D414yes—
90C22×Dic31C2311yescharacteristic
91C22×Dic31C2311yescharacteristic
92C22×Dic31C2311yescharacteristic
93C22×Dic31C2311yescharacteristic
94C22×Dic31C2311yescharacteristic
95D4×D31—232no—
96C2×D124—22no—
97C2×D124—22no—
98C2×D124D414yes—
99C2×C4×D31—22no—
100C2×C4×D31—22no—
101C2×C4×D31—22no—
102C2×C4×D31—22no—
103C2×C4×D31—22no—
104C2×C4×D31—22no—
105C2×C4×D31—22no—
106C2×C4×D31—41no—
107C2×C4×D31—41no—
108C4⋊C4×C31—22no—
109C22⋊C4×C31—22no—
110C22⋊C4×C31—22no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊Dic31—22no—
114C22⋊Dic31—22no—
115C22⋊Dic31—41no—
116C22⋊Dic31—41no—
117(C2×D31)⋊C4—22no—
118(C2×D31)⋊C4—22no—
119(C2×D31)⋊C4—22no—
120(C2×D31)⋊C4—22no—
121(C2×D31)⋊C4—22no—
122(C2×D31)⋊C4—22no—
123(C2×D31)⋊C4—22no—
124(C2×D31)⋊C4—22no—
125(C2×D31)⋊C4—41no—
126(C2×D31)⋊C4—41no—
127C4⋊Dic31—22no—
128Dic31⋊C4—22no—
129Dic31⋊C4—22no—
130C23×C31—21no—
131C23×C31—21no—
132C23×C31—21no—
133C23×C31—21no—
134C23×C31—21no—
135C23×C31—21no—
136C23×C31—21no—
137C23×C31—21no—
138C23×C31—21no—
139C23×C31—21no—
140C23×C31—21no—
141C23×C31—21no—
142C23×C31C2×D412yes—
143C23×C31C2×D412yes—
144C23×C31C2411yescharacteristic
145C22×D31—21no—
146C22×D31—21no—
147C22×D31—21no—
148C22×D31—21no—
149C22×D31—21no—
150C22×D31—21no—
151C22×D31—21no—
152C22×D31—21no—
153C22×D31—28no—
154C22×D31—21no—
155C22×D31—21no—
156C22×D31—21no—
157C22×D31—21no—
158C22×D31—21no—
159C22×D31—21no—
160C22×D31—28no—
161C22×D31—28no—
162C22×D31—44no—
163C22×D31—44no—
164C22×D31—44no—
165C22×D31—44no—
166C22×D31—44no—
167C22×D31—44no—
168C22×D31—44no—
169C22×D31—44no—
170C22×D31—44no—
171C22×D31—44no—
172C22×D31—44no—
173C22×D31C2×D412yes—
174C22×D31C2×D412yes—
175C22×D31C2×D412yes—
176C22×D31C2×D412yes—
177C22×D31C2×D412yes—
178C22×D31#4112yes—
179C22×D31#4112yes—
180D4×C31—28no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—21no—
186C2×C4×C31—21no—
187C2×C4×C31—21no—
188C2×C4×C31—21no—
189C2×C4×C31—21no—
190C2×C4×C31—21no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31C2×D412yes—
196C22⋊D31—28no—
197C22⋊D31—28no—
198C2×Dic31—21no—
199C2×Dic31—21no—
200C2×Dic31—21no—
201C2×Dic31—21no—
202C2×Dic31—21no—
203C2×Dic31—21no—
204C2×Dic31—21no—
205C2×Dic31—21no—
206C2×Dic31—21no—
207C2×Dic31—21no—
208C2×Dic31—21no—
209C2×Dic31—21no—
210C2×Dic31—21no—
211C2×Dic31—21no—
212C2×Dic31C2×D412yes—
213D124—28no—
214C4×D31—28no—
215C4×D31—44no—
216C4×D31—44no—
217#2405—311nomaximal, Sylow
218C22×C31—22no—
219C22×C31—22no—
220C22×C31—22no—
221C22×C31—22no—
222C22×C31—22no—
223C22×C31—22no—
224C22×C31—41no—
225C22×C31—41no—
226C22×C31—41no—
227C22×C31—41no—
228C22×C31—41no—
229C22×C31—41no—
230C22×C31—41no—
231C22×C31—41no—
232C22×C31C22×D411yescharacteristic
233C22×C31C22×D411yescharacteristic
234C22×C31C22×D411yescharacteristic
235C22×C31C2×#4111yescharacteristic
236C22×C31C2×#4111yescharacteristic
237C22×C31H(5,𝔽2)11yescharacteristic
238C22×C31H(5,𝔽2)11yescharacteristic
239C2×D31—22no—
240C2×D31—22no—
241C2×D31—22no—
242C2×D31—22no—
243C2×D31—22no—
244C2×D31—22no—
245C2×D31—22no—
246C2×D31—22no—
247C2×D31—22no—
248C2×D31—22no—
249C2×D31—22no—
250C2×D31—22no—
251C2×D31—22no—
252C2×D31—22no—
253C2×D31—22no—
254C2×D31—416no—
255C2×D31—416no—
256C2×D31—41no—
257C2×D31—41no—
258C2×D31—41no—
259C2×D31—41no—
260C2×D31#12014yes—
261C2×D31#12314yescomplemented
262C4×C31—22no—
263C4×C31—41no—
264C4×C31—41no—
265C4×C31—41no—
266C4×C31—41no—
267Dic31—22no—
268Dic31—41no—
269Dic31—41no—
270Dic31—41no—
271Dic31—41no—
272C23×D4—311no—
273C2×#123—311no—
274C2×#123—311no—
275C22×C22⋊C4—311no—
276#394—311no—
277#394—311no—
278#394—311no—
279#394D3111yescharacteristic, complemented
280#386—311no—
281#386—311no—
282#385—311no—
283#385—311no—
284#379—311no—
285#379—311no—
286#379—311no—
287C2×C31—42no—
288C2×C31—42no—
289C2×C31C2×#12011yescharacteristic
290C2×C31C2×#12311yescharacteristic
291C2×C31#53411yescharacteristic
292C2×C31#54611yescharacteristic
293C2×C31#54611yescharacteristic
294C2×C31#54611yescharacteristic
295C2×C31#54611yescharacteristic
296D31—24no—
297D31—24no—
298D31—42no—
299C25—311no—
300C25—311no—
301C22×D4—311no—
302C22×D4—311no—
303C22×D4—311no—
304C22×D4—624no—
305C22×D4—624no—
306C22×D4—624no—
307C22×D4C2×D3111yescharacteristic
308C23×C4—311no—
309C23×C4—311no—
310C23×C4—311no—
311#123—624no—
312#123—624no—
313C2×C4⋊C4—311no—
314C2×C4⋊C4—311no—
315C2×C4⋊C4—311no—
316C2×C4⋊C4C2×D3111yescharacteristic, complemented
317C2×C22⋊C4—311no—
318C2×C22⋊C4—311no—
319C2×C22⋊C4—311no—
320C2×C22⋊C4—311no—
321C2×C22⋊C4—311no—
322C2×C22⋊C4—311no—
323C2×C22⋊C4—311no—
324C2×C22⋊C4—311no—
325C2×C22⋊C4—311no—
326C2×C22⋊C4—311no—
327C2×C22⋊C4—311no—
328C2×C22⋊C4—311no—
329C2×C22⋊C4—311no—
330C2×C22⋊C4—311no—
331C2×C22⋊C4—624no—
332C2×C22⋊C4C2×D3111yescharacteristic, complemented
333C2×C22⋊C4C2×D3111yescharacteristic, complemented
334C2×C22⋊C4C2×D3111yescharacteristic, complemented
335C2×C22⋊C4C2×D3111yescharacteristic, complemented
336#95—311no—
337#95—311no—
338#95—311no—
339#95C2×D3111yescharacteristic, complemented
340C31#240511yescharacteristic, complemented, Sylow
341C24—311no—
342C24—311no—
343C24—311no—
344C24—314no—
345C24—622no—
346C24—622no—
347C24—622no—
348C24—622no—
349C24—622no—
350C24—622no—
351C24—622no—
352C24—622no—
353C24—622no—
354C24—1241no—
355C24—1241no—
356C24—1241no—
357C24—1241no—
358C24C22×D3111yescharacteristic
359C24C22×D3111yescharacteristic
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—314no—
364C2×D4—314no—
365C2×D4—622no—
366C2×D4—622no—
367C2×D4—622no—
368C2×D4—6232no—
369C2×D4—622no—
370C2×D4—622no—
371C2×D4—622no—
372C2×D4—622no—
373C22×C4—311no—
374C22×C4—311no—
375C22×C4—311no—
376C22×C4—311no—
377C22×C4—311no—
378C22×C4—622no—
379C22×C4—622no—
380C22×C4—622no—
381C22×C4—622no—
382C22×C4—622no—
383C22×C4—622no—
384C22×C4—622no—
385C22×C4—1241no—
386C22×C4—1241no—
387C22×C4C22×D3111yescharacteristic
388C22×C4C22×D3111yescharacteristic
389C22×C4C22×D3111yescharacteristic
390C22×C4C22×D3111yescharacteristic
391C22×C4C22×D3111yescharacteristic
392C4⋊C4—22no—
393C4⋊C4—622no—
394C4⋊C4—622no—
395C4⋊C4—622no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—622no—
401C22⋊C4—622no—
402C22⋊C4—622no—
403C22⋊C4—622no—
404C22⋊C4—622no—
405C22⋊C4—622no—
406C22⋊C4—622no—
407C22⋊C4—622no—
408C22⋊C4—622no—
409C22⋊C4—622no—
410C22⋊C4—1241no—
411C22⋊C4—1241no—
412C22⋊C4—1241no—
413C22⋊C4—1241no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—312no—
427C23—312no—
428C23—312no—
429C23—312no—
430C23—312no—
431C23—312no—
432C23—312no—
433C23—628no—
434C23—628no—
435C23—628no—
436C23—621no—
437C23—621no—
438C23—621no—
439C23—621no—
440C23—621no—
441C23—621no—
442C23—621no—
443C23—621no—
444C23—621no—
445C23—621no—
446C23—621no—
447C23—621no—
448C23—621no—
449C23—621no—
450C23—1244no—
451C23—1244no—
452C23—1244no—
453C23—1244no—
454C23—1244no—
455C23—1244no—
456C23—1244no—
457C23—1244no—
458C23—1244no—
459C23—1244no—
460C23—1244no—
461C23D4×D3112yes—
462C23D4×D3112yes—
463C23C23×D3111yescharacteristic
464D4—28no—
465D4—628no—
466D4—628no—
467D4—628no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—312no—
483C2×C4—628no—
484C2×C4—621no—
485C2×C4—621no—
486C2×C4—621no—
487C2×C4—621no—
488C2×C4—621no—
489C2×C4—621no—
490C2×C4—621no—
491C2×C4—621no—
492C2×C4—621no—
493C2×C4—621no—
494C2×C4—621no—
495C2×C4—621no—
496C2×C4—621no—
497C2×C4—621no—
498C2×C4—1244no—
499C2×C4—1244no—
500C2×C4D4×D3112yes—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—314no—
516C22—314no—
517C22—622no—
518C22—622no—
519C22—622no—
520C22—622no—
521C22—622no—
522C22—622no—
523C22—622no—
524C22—622no—
525C22—622no—
526C22—622no—
527C22—622no—
528C22—622no—
529C22—622no—
530C22—622no—
531C22—622no—
532C22—12416no—
533C22—12416no—
534C22—1241no—
535C22—1241no—
536C22—1241no—
537C22—1241no—
538C22C2×D4×D3111yescharacteristic
539C22C2×D4×D3111yescharacteristic
540C22C2×D4×D3111yescharacteristic
541C22#1175856111yescharacteristic
542C22#41×D3111yescharacteristic
543C22#41×D3111yescharacteristic
544C22#1175856611yescharacteristic
545C4—22no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—622no—
551C4—1241no—
552C4—1241no—
553C4—1241no—
554C4—1241no—
555C2—42no—
556C2—42no—
557C2—624no—
558C2—624no—
559C2—1242no—
560C2#120×D3111yescharacteristic
561C2#123×D3111yescharacteristic
562C2#1452160711yescharacteristic
563C2#1452168811yescharacteristic
564C2#1452169111yescharacteristic
565C2#1452169211yescharacteristic
566C2#1452169611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#394C2
prime radicalC22×D4×D31C2
hypercenter#394D31
hypocenterC31#2405
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#11758471C2semidirect
3. C2×#11758460C2semidirect
4. C2×C22⋊C4×D31C2semidirect
5. #14520965C2semidirect
6. #14520964C2semidirect
7. #14520963C2semidirect
8. #14520927C2semidirect
9. #14520925C2semidirect
10. #14520923C2semidirect
11. #14520920C2semidirect
12. #14520904C2semidirect
13. #14520904C2semidirect
14. #14520904C2semidirect
15. C31×#394C2semidirect
26. C22×C4×D31C22semidirect
27. C22×C4×D31C22semidirect
28. C2×C4⋊C4×C31C22semidirect
29. C2×C22⋊C4×C31C22semidirect
30. C2×C22⋊C4×C31C22semidirect
31. C2×C22⋊C4×C31C22semidirect
32. C2×C22⋊C4×C31C22semidirect
33. C2×C22⋊Dic31C22semidirect
34. C2×C22⋊Dic31C22semidirect
35. C2×C22⋊Dic31C22semidirect
36. C2×C22⋊Dic31C22semidirect
41. C2×(C2×D31)⋊C4C22semidirect
42. C2×(C2×D31)⋊C4C22semidirect
43. C2×(C2×D31)⋊C4C22semidirect
44. C2×(C2×D31)⋊C4C22semidirect
45. C2×(C2×D31)⋊C4C22semidirect
46. C2×(C2×D31)⋊C4C22semidirect
47. C2×C4⋊Dic31C22semidirect
48. C2×Dic31⋊C4C22semidirect
49. C2×Dic31⋊C4C22semidirect
53. C31×#95C22semidirect
54. C31⋊#95C22semidirect
55. C31⋊#95C22semidirect
56. C31⋊#95C22semidirect
261. C2×D31#123semidirect
279. #394D31semidirect
316. C2×C4⋊C4C2×D31semidirect
332. C2×C22⋊C4C2×D31semidirect
333. C2×C22⋊C4C2×D31semidirect
334. C2×C22⋊C4C2×D31semidirect
335. C2×C22⋊C4C2×D31semidirect
339. #95C2×D31semidirect
340. C31#2405semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117584712
C2×#117584602
C2×C22⋊C4×D312
#145209652
#145209642
#145209632
#145209272
#145209252
#145209232
#145209202
#145209042
C31×#3942
#240531