Finite Group Databasebeta

← #82907

Subgroups — #82907

Order 384.

802 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×#123C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4992C211yesmaximal, characteristic, complemented
9#4991C211yesmaximal, characteristic, complemented
10#4988C211yesmaximal, characteristic, complemented
11#4960C211yesmaximal, characteristic, complemented
12#4959C211yesmaximal, characteristic, complemented
13#4957C211yesmaximal, characteristic, complemented
14#4955C211yesmaximal, characteristic, complemented
15#4951C211yesmaximal, characteristic, complemented
16#2678—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic, complemented
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic, complemented
24C22×D12C2211yescharacteristic
25C22×D12C2211yescharacteristic
26C3×#123—24no—
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C4⋊C4C2211yescharacteristic, complemented
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic, complemented
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic, complemented
35#937—24no—
36#930—24no—
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×C4⋊Dic3C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×C4×Dic3C2211yescharacteristic, complemented
51#882—24no—
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#123D311yescharacteristic, complemented
57C2×#121—31no—
58C2×#120—31no—
59C2×#120—31no—
60#397—31no—
61#396—31no—
62#394—31no—
63#394—31no—
64#392—31no—
65#392—31no—
66#390—31no—
67#390—31no—
68#388—31no—
69#386—31no—
70#379—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3—41no—
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—41no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—41no—
99C2×#67—41no—
100C2×#67—41no—
101C2×#67—41no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67D414yescomplemented
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—22no—
114C2×D12—22no—
115C2×D12—41no—
116C2×D12—41no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4D414yescomplemented
122#217—22no—
123#217—41no—
124#217—41no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—22no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139C25—31no—
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4C2×D311yescharacteristic
147C23×C4C2×D311yescharacteristic, complemented
148#123—24no—
149#121—64no—
150#120—64no—
151#120—64no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4C2×D311yescharacteristic, complemented
155C2×C4⋊C4C2×D311yescharacteristic, complemented
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⋊C4—31no—
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C22⋊C4C2×D311yescharacteristic
172C2×C22⋊C4C2×D311yescharacteristic, complemented
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×C3C2×D412yes—
191C23×C3C2×D412yes—
192C23×C3C2411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—44no—
213C22×D3—44no—
214C22×D3—44no—
215C22×D3—44no—
216C22×D3—44no—
217C22×D3—44no—
218C22×D3—44no—
219C22×D3C2×D412yes—
220C22×D3#4112yes—
221C3×D4—44no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
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×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—44no—
236C2×C3×C4C2×D412yes—
237C2×C3×C4C2×D412yes—
238#67—44no—
239#67—44no—
240#67—44no—
241#67—44no—
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—
257D12—44no—
258D12—44no—
259C24—31no—
260C24—31no—
261C24—31no—
262C24—62no—
263C24—62no—
264C24—62no—
265C24—121no—
266C24—121no—
267C24—121no—
268C24—121no—
269C24C22×D311yescharacteristic
270C24C22×D311yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—34no—
274C2×D4—41no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—41no—
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C22⋊C4—22no—
310C22⋊C4—22no—
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—121no—
325C22⋊C4#6714yescomplemented
326C42—121no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3C22×D411yescharacteristic
342C22×C3C22×D411yescharacteristic
343C22×C3C2×#4111yescharacteristic
344C22×C3H(5,𝔽2)11yescharacteristic
345C22×C3H(5,𝔽2)11yescharacteristic
346C22×C3H(5,𝔽2)11yescharacteristic
347C22×C3H(5,𝔽2)11yescharacteristic
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—416no—
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
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—
373C3×C4—41no—
374C3×C4—41no—
375Dic3—22no—
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—32no—
393C23—32no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—61no—
413C23—124no—
414C23—124no—
415C23—124no—
416C23—124no—
417C23—124no—
418C23—124no—
419C23—124no—
420C23D3×D412yes—
421C23C2×#6712yes—
422C23C23×D311yescharacteristic
423D4—44no—
424D4—124no—
425D4—124no—
426D4—124no—
427D4—124no—
428D4—124no—
429D4—124no—
430C2×C4—21no—
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—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—32no—
444C2×C4—44no—
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#53411yescharacteristic
464C2×C3#53511yescharacteristic
465C2×C3#53811yescharacteristic
466C2×C3D4211yescharacteristic
467C2×C3#54611yescharacteristic
468C2×C3#54611yescharacteristic
469C2×C3#56011yescharacteristic
470D3—42no—
471D3—42no—
472D3—81no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—1216no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22—121no—
509C22C2×D3×D411yescharacteristic
510C22#100411yescharacteristic
511C22D3×#4111yescharacteristic
512C22#100911yescharacteristic
513C22#100911yescharacteristic
514C22#100911yescharacteristic
515C22C22×#6711yescharacteristic
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—62no—
522C4—121no—
523C4—121no—
524C4—121no—
525C4—121no—
526C3#267811yescharacteristic, complemented, Sylow
527C2—42no—
528C2—42no—
529C2—122no—
530C2—122no—
531C2—241no—
532C2#565311yescharacteristic
533C2#565411yescharacteristic
534C2#565711yescharacteristic
535C2#566611yescharacteristic
536C2D4×#6711yescharacteristic
537C2#582911yescharacteristic
538C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
hypercenterC2×#123D3
hypocenterC3#2678
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×#937C2semidirect
3. C2×#930C2semidirect
4. C2×#882C2semidirect
5. #5223C2semidirect
6. #5213C2semidirect
7. #4994C2semidirect
8. #4992C2semidirect
9. #4991C2semidirect
10. #4988C2semidirect
11. #4960C2semidirect
12. #4959C2semidirect
13. #4957C2semidirect
14. #4955C2semidirect
15. #4951C2semidirect
19. C23×C3×C4C22semidirect
23. C22×#67C22semidirect
27. C2×C3×C4⋊C4C22semidirect
28. C2×C3×C4⋊C4C22semidirect
31. C2×C3×C22⋊C4C22semidirect
34. C2×#217C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×C4⋊Dic3C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×C4×Dic3C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
56. C2×#123D3semidirect
105. C2×#67D4semidirect
121. C3×C22⋊C4D4semidirect
147. C23×C4C2×D3semidirect
154. C2×C4⋊C4C2×D3semidirect
155. C2×C4⋊C4C2×D3semidirect
172. C2×C22⋊C4C2×D3semidirect
325. C22⋊C4#67semidirect
526. C3#2678semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9372
C2×#9302
C2×#8822
#52232
#52132
#49942
#49922
#49912
#49882
#49602
#49592
#49572
#49552
#49512
#26783