Finite Group Databasebeta

← #82586

Subgroups — #82586

Order 384.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 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
3C22×#217C211yesmaximal, characteristic, complemented
4C2×#939C211yesmaximal, characteristic, complemented
5C2×#938C211yesmaximal, characteristic, complemented
6C2×#67⋊C4C211yesmaximal, characteristic, complemented
7#5223C211yesmaximal, characteristic, complemented
8#5222C211yesmaximal, characteristic, complemented
9#5221C211yesmaximal, characteristic, complemented
10#5220C211yesmaximal, characteristic, complemented
11#4961C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4950C211yesmaximal, characteristic, complemented
14#4948C211yesmaximal, characteristic, complemented
15#4944C211yesmaximal, characteristic, complemented
16#2424—31nomaximal, Sylow
17C25×C3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic, complemented
25C23×Dic3C2211yescharacteristic
26C23×Dic3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic, complemented
28C3×#120—24no—
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32#953—24no—
33C2×#217—24no—
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C2×#217C2211yescharacteristic
41C2×#217C2211yescharacteristic, complemented
42C2×#217C2211yescharacteristic, complemented
43C2×#217C2211yescharacteristic, complemented
44#939—24no—
45#938—24no—
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×C4⋊Dic3C2211yescharacteristic, complemented
49C2×C4⋊Dic3C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51C2×Dic3⋊C4C2211yescharacteristic, complemented
52C2×C4×Dic3C2211yescharacteristic, complemented
53#67⋊C4—24no—
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C3⋊#95C2211yescharacteristic, complemented
57C3⋊#95C2211yescharacteristic, complemented
58C2×#121—31no—
59C2×#121—31no—
60C2×#120—31no—
61C2×#120D311yescharacteristic, complemented
62C2×C4×D4—31no—
63C22×C22⋊C4—31no—
64#396—31no—
65#394—31no—
66#394—31no—
67#392—31no—
68#390—31no—
69#388—31no—
70#386—31no—
71#386—31no—
72#385—31no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—22no—
78C24×C3—41no—
79C24×C3—41no—
80C24×C3C2311yescharacteristic
81C24×C3C2311yescharacteristic
82C24×C3C2311yescharacteristic
83C24×C3C2311yescharacteristic
84C23×D3C2311yescharacteristic
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—22no—
88C2×C3×D4—22no—
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4—41no—
93C2×C3×D4—41no—
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—22no—
103C2×#67—22no—
104C2×#67—22no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—41no—
108C2×#67D414yescomplemented
109C22×Dic3—22no—
110C22×Dic3—22no—
111C22×Dic3—22no—
112C22×Dic3—22no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119C22×Dic3C2311yescharacteristic
120C22×Dic3C2311yescharacteristic
121C22×Dic3C2311yescharacteristic
122C2×C4×D3—22no—
123C2×C4×D3—22no—
124C2×C4×D3—41no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4D414yescomplemented
128#217—22no—
129#217—22no—
130#217—22no—
131#217—22no—
132#217—22no—
133#217—22no—
134#217—22no—
135#217—22no—
136#217—41no—
137#217D414yescomplemented
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140C4⋊Dic3—22no—
141C4⋊Dic3—41no—
142Dic3⋊C4—22no—
143Dic3⋊C4—22no—
144C4×Dic3—22no—
145C25C2×D311yescharacteristic
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4C2×D311yescharacteristic
151C22×D4C2×D311yescharacteristic
152C22×D4C2×D311yescharacteristic
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156#121—64no—
157#121—64no—
158#120—24no—
159#120—64no—
160C4×D4—64no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—64no—
178C2×C22⋊C4C2×D311yescharacteristic
179C2×C22⋊C4C2×D311yescharacteristic
180C2×C22⋊C4C2×D311yescharacteristic
181C2×C42—31no—
182#95—31no—
183#95—31no—
184#95—31no—
185#95—31no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—28no—
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—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—44no—
213C23×C3—44no—
214C23×C3—44no—
215C23×C3—44no—
216C23×C3C2×D412yes—
217C23×C3C2×D412yes—
218C23×C3C2×D412yes—
219C23×C3#4112yes—
220C23×C3#4112yes—
221C23×C3C2411yescharacteristic
222C22×D3—21no—
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—21no—
228C22×D3C2×D412yes—
229C3×D4—44no—
230C3×D4—44no—
231C3×D4—44no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4C2×D412yes—
240C2×C3×C4C2×D412yes—
241#67—28no—
242#67—44no—
243#67—44no—
244#67—44no—
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—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—44no—
264C2×Dic3—44no—
265C2×Dic3C2×D412yes—
266C2×Dic3C2×D412yes—
267C2×Dic3C2×D412yes—
268C4×D3—44no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—31no—
275C24—41no—
276C24—41no—
277C24C22×D311yescharacteristic
278C24C22×D311yescharacteristic
279C24C22×D311yescharacteristic
280C24C22×D311yescharacteristic
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—34no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4C22×D311yescharacteristic
319C22×C4C22×D311yescharacteristic
320C22×C4C22×D311yescharacteristic
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—121no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—34no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—62no—
335C22⋊C4—62no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—121no—
339C22⋊C4#6714yescomplemented
340C42—62no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—416no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3#12114yescomplemented
370C22×C3C22×D411yescharacteristic
371C22×C3C22×D411yescharacteristic
372C22×C3C22×D411yescharacteristic
373C22×C3C2×#4111yescharacteristic
374C22×C3C2×#4111yescharacteristic
375C22×C3H(5,𝔽2)11yescharacteristic
376C22×C3H(5,𝔽2)11yescharacteristic
377C2×D3—22no—
378C2×D3—22no—
379C2×D3—22no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3—41no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—41no—
387Dic3—22no—
388Dic3—22no—
389Dic3—22no—
390Dic3—41no—
391Dic3—41no—
392Dic3—41no—
393Dic3—41no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—28no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—32no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23D3×D412yes—
432C23D3×D412yes—
433C23#23712yes—
434C23#23712yes—
435C23C2×#6712yes—
436C23C23×D311yescharacteristic
437D4—44no—
438D4—44no—
439D4—44no—
440D4—68no—
441D4—124no—
442D4—124no—
443D4—124no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—124no—
473C2×C4—124no—
474C2×C4—124no—
475C2×C4C2×#6712yes—
476C2×C4C2×#6712yes—
477C2×C3—24no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—81no—
481C2×C3C2×#12111yescharacteristic
482C2×C3#52511yescharacteristic
483C2×C3#53811yescharacteristic
484C2×C3D4211yescharacteristic
485C2×C3D4211yescharacteristic
486C2×C3#54611yescharacteristic
487C2×C3#54611yescharacteristic
488D3—42no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—416no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22#93814yescomplemented
525C22C2×D3×D411yescharacteristic
526C22C2×D3×D411yescharacteristic
527C22C2×#23711yescharacteristic
528C22C2×#23711yescharacteristic
529C22#100411yescharacteristic
530C22#100411yescharacteristic
531C22C22×#6711yescharacteristic
532C4—22no—
533C4—22no—
534C4—41no—
535C4—62no—
536C4—62no—
537C4—62no—
538C4—121no—
539C4—121no—
540C4—121no—
541C4—121no—
542C3#242411yescharacteristic, complemented, Sylow
543C2—24no—
544C2—42no—
545C2—42no—
546C2—81no—
547C2—122no—
548C2#558611yescharacteristic
549C2#559011yescharacteristic
550C2#559111yescharacteristic
551C2#559311yescharacteristic
552C2C2×#93811yescharacteristic
553C2D4×#6711yescharacteristic
554C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2424
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. C22×#217C2semidirect
4. C2×#939C2semidirect
5. C2×#938C2semidirect
6. C2×#67⋊C4C2semidirect
7. #5223C2semidirect
8. #5222C2semidirect
9. #5221C2semidirect
10. #5220C2semidirect
11. #4961C2semidirect
12. #4956C2semidirect
13. #4950C2semidirect
14. #4948C2semidirect
15. #4944C2semidirect
24. C22×#67C22semidirect
27. C22×C4×D3C22semidirect
41. C2×#217C22semidirect
42. C2×#217C22semidirect
43. C2×#217C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×C4⋊Dic3C22semidirect
49. C2×C4⋊Dic3C22semidirect
50. C2×Dic3⋊C4C22semidirect
51. C2×Dic3⋊C4C22semidirect
52. C2×C4×Dic3C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
57. C3⋊#95C22semidirect
61. C2×#120D3semidirect
108. C2×#67D4semidirect
127. C3×C22⋊C4D4semidirect
137. #217D4semidirect
339. C22⋊C4#67semidirect
369. C22×C3#121semidirect
524. C22#938semidirect
542. C3#2424semidirect

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
C22×#2172
C2×#9392
C2×#9382
C2×#67⋊C42
#52232
#52222
#52212
#52202
#49612
#49562
#49502
#49482
#49442
#24243