Finite Group Databasebeta

← #82595

Subgroups — #82595

Order 384.

796 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#120C211yesmaximal, characteristic, complemented
2C2×#953C211yesmaximal, characteristic, complemented
3C2×#936C211yesmaximal, characteristic, complemented
4C2×#885C211yesmaximal, characteristic, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5223C211yesmaximal, characteristic, complemented
8#5223C211yesmaximal, characteristic, complemented
9#5222C211yesmaximal, characteristic, complemented
10#5221C211yesmaximal, characteristic, complemented
11#4961C211yesmaximal, characteristic, complemented
12#4957C211yesmaximal, characteristic, complemented
13#4951C211yesmaximal, characteristic, complemented
14#4947C211yesmaximal, characteristic, complemented
15#4944C211yesmaximal, characteristic, complemented
16#2670—31nomaximal, Sylow
17C25×C3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic, complemented
25C22×Dic6C2211yescharacteristic, complemented
26C3×#120—24no—
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30#953—24no—
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C2×#217C2211yescharacteristic, complemented
41C2×#217C2211yescharacteristic, complemented
42C2×#217C2211yescharacteristic, complemented
43#936—24no—
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×C4⋊Dic3C2211yescharacteristic, complemented
48C2×Dic3⋊C4C2211yescharacteristic, complemented
49C2×C4×Dic3C2211yescharacteristic, complemented
50C2×C4×Dic3C2211yescharacteristic, complemented
51#885—24no—
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#124—31no—
57C2×#124—31no—
58C2×#120—31no—
59C2×#120D311yescharacteristic, complemented
60#394—31no—
61#394—31no—
62#393—31no—
63#392—31no—
64#392—31no—
65#392—31no—
66#390—31no—
67#390—31no—
68#388—31no—
69#388—31no—
70#379—31no—
71C24×C3—22no—
72C24×C3—22no—
73C24×C3—22no—
74C24×C3—41no—
75C24×C3—41no—
76C24×C3—41no—
77C24×C3—41no—
78C24×C3C2311yescharacteristic
79C24×C3C2311yescharacteristic
80C24×C3C2311yescharacteristic
81C24×C3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—22no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4—41no—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—41no—
101C2×#67—41no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—41no—
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C2×Dic6—22no—
115C2×Dic6—22no—
116C2×Dic6—41no—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4D414yescomplemented
120#217—22no—
121#217—22no—
122#217—22no—
123#217—22no—
124#217—22no—
125#217—22no—
126#217—41no—
127#217—41no—
128#217—41no—
129#217—41no—
130#217—41no—
131#217D414yescomplemented
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—41no—
135C4⋊Dic3—41no—
136Dic3⋊C4—41no—
137C4×Dic3—22no—
138C4×Dic3—22no—
139C25C2×D311yescharacteristic
140C22×Q8—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4C2×D311yescharacteristic
146C22×D4C2×D311yescharacteristic
147C22×D4C2×D311yescharacteristic
148#124—64no—
149#124—64no—
150#120—24no—
151#120—64no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4C2×D311yescharacteristic
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C22⋊C4C2×D311yescharacteristic
172C2×C42—31no—
173C2×C42—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95—31no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—44no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3—44no—
208C23×C3—44no—
209C23×C3—44no—
210C23×C3C2×D412yes—
211C23×C3C2×D412yes—
212C23×C3#4112yes—
213C23×C3C2411yescharacteristic
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—21no—
220C22×D3—21no—
221C3×D4—44no—
222C3×D4—44no—
223C3×D4—44no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4C2×D412yes—
232C2×C3×C4C2×D412yes—
233#67—44no—
234#67—44no—
235#67—44no—
236#67—44no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3C2×D412yes—
257C2×Dic3C2×D412yes—
258Dic6—44no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—31no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24C22×D311yescharacteristic
268C24C22×D311yescharacteristic
269C24C22×D311yescharacteristic
270C24C22×D311yescharacteristic
271C2×Q8—62no—
272C2×Q8—62no—
273C2×Q8—121no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C4⋊C4—121no—
306C4⋊C4—121no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—34no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4#6714yescomplemented
325C42—62no—
326C42—62no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—416no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3C22×D411yescharacteristic
357C22×C3C22×D411yescharacteristic
358C22×C3C2×#4111yescharacteristic
359C22×C3H(5,𝔽2)11yescharacteristic
360C22×C3H(5,𝔽2)11yescharacteristic
361C22×C3H(5,𝔽2)11yescharacteristic
362C22×C3H(5,𝔽2)11yescharacteristic
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—41no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—41no—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—41no—
373Dic3—22no—
374Dic3—22no—
375Dic3—41no—
376Dic3—41no—
377Dic3—41no—
378Dic3—41no—
379Dic3—41no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—61no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23D3×D412yes—
420C23#23712yes—
421C23C2×#6712yes—
422C23C23×D311yescharacteristic
423Q8—124no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—124no—
428D4—124no—
429D4—124no—
430D4—124no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—32no—
439C2×C4—32no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4C2×#6712yes—
460C2×C4C2×#6712yes—
461C2×C3—42no—
462C2×C3—42no—
463C2×C3—42no—
464C2×C3—81no—
465C2×C3#53511yescharacteristic
466C2×C3#53511yescharacteristic
467C2×C3#53811yescharacteristic
468C2×C3D4211yescharacteristic
469C2×C3#54611yescharacteristic
470C2×C3#54611yescharacteristic
471C2×C3H(3,𝔽4)11yescharacteristic
472D3—81no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—416no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22—121no—
509C22C2×D3×D411yescharacteristic
510C22C2×#23711yescharacteristic
511C22#100411yescharacteristic
512C22#100411yescharacteristic
513C22#100411yescharacteristic
514C22#100411yescharacteristic
515C22C22×#6711yescharacteristic
516C4—22no—
517C4—22no—
518C4—41no—
519C4—62no—
520C4—62no—
521C4—121no—
522C4—121no—
523C4—121no—
524C4—121no—
525C4—121no—
526C3#267011yescharacteristic, complemented, Sylow
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—241no—
532C2#559111yescharacteristic
533C2#559211yescharacteristic
534C2#559311yescharacteristic
535C2#559411yescharacteristic
536C2D4×#6711yescharacteristic
537C2#580511yescharacteristic
538C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2670
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×C3×#120C2semidirect
2. C2×#953C2semidirect
3. C2×#936C2semidirect
4. C2×#885C2semidirect
5. #5247C2semidirect
6. #5223C2semidirect
7. #5223C2semidirect
8. #5223C2semidirect
9. #5222C2semidirect
10. #5221C2semidirect
11. #4961C2semidirect
12. #4957C2semidirect
13. #4951C2semidirect
14. #4947C2semidirect
15. #4944C2semidirect
24. C22×#67C22semidirect
25. C22×Dic6C22semidirect
40. C2×#217C22semidirect
41. C2×#217C22semidirect
42. C2×#217C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×C4⋊Dic3C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×C4×Dic3C22semidirect
50. C2×C4×Dic3C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
59. C2×#120D3semidirect
119. C3×C22⋊C4D4semidirect
131. #217D4semidirect
324. C22⋊C4#67semidirect
526. C3#2670semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C2×#9532
C2×#9362
C2×#8852
#52472
#52232
#52222
#52212
#49612
#49572
#49512
#49472
#49442
#26703