Finite Group Databasebeta

← #82608

Subgroups — #82608

Order 384.

816 conjugacy classes of subgroups.

The subgroup lattice has 542 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×#939C211yesmaximal, characteristic, complemented
4C2×#938C211yesmaximal, characteristic, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5222C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#5220C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4953C211yesmaximal, characteristic, complemented
14#4952C211yesmaximal, characteristic, complemented
15#4949C211yesmaximal, characteristic, complemented
16#2668—31nomaximal, Sylow
17C25×C3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic, complemented
25C23×Dic3C2211yescharacteristic, complemented
26C22×C4×D3C2211yescharacteristic, complemented
27C3×#120—24no—
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31#953—24no—
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C2×#217C2211yescharacteristic
41C2×#217C2211yescharacteristic, complemented
42C2×#217C2211yescharacteristic, complemented
43#939—24no—
44#938—24no—
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×C4⋊Dic3C2211yescharacteristic, complemented
48C2×Dic3⋊C4C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#121—31no—
57C2×#121—31no—
58C2×#120—31no—
59C2×#120D311yescharacteristic, complemented
60#399—31no—
61#397—31no—
62#397—31no—
63#394—31no—
64#394—31no—
65#392—31no—
66#386—31no—
67#386—31no—
68#386—31no—
69#386—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—41no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4D414yescomplemented
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—22no—
101C2×#67—41no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67D414yescomplemented
107C22×Dic3—22no—
108C22×Dic3—41no—
109C22×Dic3—41no—
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C2×C4×D3—22no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123#217—22no—
124#217—22no—
125#217—22no—
126#217—22no—
127#217—22no—
128#217—22no—
129#217—41no—
130#217—41no—
131#217—41no—
132#217—41no—
133#217—41no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4—41no—
139C25C2×D311yescharacteristic
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4C2×D311yescharacteristic
145C22×D4C2×D311yescharacteristic
146C22×D4C2×D311yescharacteristic
147C23×C4—31no—
148C23×C4—31no—
149#121—64no—
150#121—64no—
151#120—24no—
152#120—64no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊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
172#95—31no—
173#95—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—44no—
203C23×C3—44no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3—44no—
208C23×C3—44no—
209C23×C3C2×D412yes—
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×D3C2×D412yes—
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×C4—21no—
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×Dic3—21no—
257C2×Dic3—44no—
258C2×Dic3C2×D412yes—
259C4×D3—44no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—31no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24C22×D311yescharacteristic
269C24C22×D311yescharacteristic
270C24C22×D311yescharacteristic
271C24C22×D311yescharacteristic
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—34no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—62no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4#6714yescomplemented
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4C22×D311yescharacteristic
307C22×C4C22×D311yescharacteristic
308C22×C4C22×D311yescharacteristic
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—121no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22⋊C4—121no—
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—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—416no—
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×C3—41no—
357C22×C3C22×D411yescharacteristic
358C22×C3C22×D411yescharacteristic
359C22×C3C2×#4111yescharacteristic
360C22×C3H(5,𝔽2)11yescharacteristic
361C22×C3H(5,𝔽2)11yescharacteristic
362C22×C3H(5,𝔽2)11yescharacteristic
363C22×C3H(5,𝔽2)11yescharacteristic
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—41no—
371C3×C4—22no—
372C3×C4—41no—
373C3×C4—41no—
374Dic3—22no—
375Dic3—41no—
376Dic3—41no—
377Dic3—41no—
378Dic3—41no—
379Dic3—41no—
380Dic3—41no—
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—32no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23D3×D412yes—
420C23#23712yes—
421C23C2×#6712yes—
422C23C2×#6712yes—
423C23C23×D311yescharacteristic
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—21no—
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×C4—61no—
460C2×C4—124no—
461C2×C4—124no—
462C2×C4C2×#6712yes—
463C2×C3—42no—
464C2×C3—42no—
465C2×C3—42no—
466C2×C3—42no—
467C2×C3#53411yescharacteristic
468C2×C3#53411yescharacteristic
469C2×C3#53811yescharacteristic
470C2×C3D4211yescharacteristic
471C2×C3#54611yescharacteristic
472C2×C3#54611yescharacteristic
473C2×C3#56011yescharacteristic
474D3—42no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—416no—
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—41no—
503C22—41no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—121no—
508C22—121no—
509C22—121no—
510C22—121no—
511C22C2×D3×D411yescharacteristic
512C22C2×#23711yescharacteristic
513C22#100411yescharacteristic
514C22#100411yescharacteristic
515C22#100411yescharacteristic
516C22#100411yescharacteristic
517C22C22×#6711yescharacteristic
518C4—22no—
519C4—41no—
520C4—41no—
521C4—62no—
522C4—121no—
523C4—121no—
524C4—121no—
525C4—121no—
526C4—121no—
527C4—121no—
528C3#266811yescharacteristic, complemented, Sylow
529C2—42no—
530C2—42no—
531C2—42no—
532C2—42no—
533C2—122no—
534C2#558911yescharacteristic
535C2#559111yescharacteristic
536C2#559311yescharacteristic
537C2#559511yescharacteristic
538C2D4×#6711yescharacteristic
539C2#580411yescharacteristic
540C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2668
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×#939C2semidirect
4. C2×#938C2semidirect
5. #5247C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #5220C2semidirect
9. #5220C2semidirect
10. #4959C2semidirect
11. #4956C2semidirect
12. #4956C2semidirect
13. #4953C2semidirect
14. #4952C2semidirect
15. #4949C2semidirect
24. C22×#67C22semidirect
25. C23×Dic3C22semidirect
26. C22×C4×D3C22semidirect
41. C2×#217C22semidirect
42. C2×#217C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×C4⋊Dic3C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
59. C2×#120D3semidirect
91. C2×C3×D4D4semidirect
106. C2×#67D4semidirect
292. C2×D4#67semidirect
528. C3#2668semidirect

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×#9392
C2×#9382
#52472
#52232
#52222
#52202
#49592
#49562
#49532
#49522
#49492
#26683