Finite Group Databasebeta

← #77106

Subgroups — #77106

Order 384.

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×D3×D4C211yesmaximal, characteristic
2C2×#894C211yesmaximal, characteristic, complemented
3C2×#883C211yesmaximal, characteristic, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4955C211yesmaximal, characteristic, complemented
12#4955C211yesmaximal, characteristic, complemented
13#4955C211yesmaximal, characteristic, complemented
14#4668C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2405—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C24×D3C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C2×D3×D4—24no—
23C2×D3×D4—24no—
24C2×D3×D4—24no—
25C22×D12C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic, complemented
28C22×C4×D3C2211yescharacteristic, complemented
29C2×C3×C4⋊C4C2211yescharacteristic, complemented
30C2×C3×C22⋊C4C2211yescharacteristic, complemented
31C2×C3×C22⋊C4C2211yescharacteristic, complemented
32C2×C3×C22⋊C4C2211yescharacteristic, complemented
33C2×C3×C22⋊C4C2211yescharacteristic, complemented
34C2×#217C2211yescharacteristic, complemented
35C2×#217C2211yescharacteristic, complemented
36C2×#217C2211yescharacteristic, complemented
37C2×#217C2211yescharacteristic, complemented
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×C4⋊Dic3C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51#894—24no—
52#883—24no—
53D3×C22⋊C4—24no—
54C3×#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C3⋊#95C2211yescharacteristic, complemented
57C3⋊#95C2211yescharacteristic, complemented
58C23×D4—31no—
59C2×#123—31no—
60C2×#123—31no—
61C22×C22⋊C4—31no—
62#394—31no—
63#394—31no—
64#394—31no—
65#394D311yescharacteristic, complemented
66#386—31no—
67#386—31no—
68#385—31no—
69#385—31no—
70#379—31no—
71#379—31no—
72#379—31no—
73C24×C3C2311yescharacteristic
74C24×C3C2311yescharacteristic
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—22no—
78C23×D3—22no—
79C23×D3—22no—
80C23×D3—22no—
81C23×D3—22no—
82C23×D3—22no—
83C23×D3—22no—
84C23×D3—41no—
85C23×D3—41no—
86C23×D3—41no—
87C23×D3—41no—
88C23×D3D414yes—
89C23×D3C2311yescharacteristic
90C23×D3C2311yescharacteristic
91C23×D3C2311yescharacteristic
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—22no—
103C2×#67—22no—
104C2×#67—22no—
105C2×#67D414yes—
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111D3×D4—232no—
112C2×D12—22no—
113C2×D12—22no—
114C2×D12D414yes—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—22no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C3×C4⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129#217—22no—
130#217—22no—
131#217—41no—
132#217—41no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—22no—
144Dic3⋊C4—22no—
145Dic3⋊C4—22no—
146C25—31no—
147C25—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—31no—
151C22×D4—64no—
152C22×D4—64no—
153C22×D4—64no—
154C22×D4C2×D311yescharacteristic
155C23×C4—31no—
156C23×C4—31no—
157C23×C4—31no—
158#123—64no—
159#123—64no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4C2×D311yescharacteristic, complemented
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⋊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—31no—
178C2×C22⋊C4—64no—
179C2×C22⋊C4C2×D311yescharacteristic, complemented
180C2×C22⋊C4C2×D311yescharacteristic, complemented
181C2×C22⋊C4C2×D311yescharacteristic, complemented
182C2×C22⋊C4C2×D311yescharacteristic, complemented
183#95—31no—
184#95—31no—
185#95—31no—
186#95C2×D311yescharacteristic, complemented
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×C3C2×D412yes—
200C23×C3C2×D412yes—
201C23×C3C2411yescharacteristic
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—28no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—28no—
218C22×D3—28no—
219C22×D3—44no—
220C22×D3—44no—
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—44no—
224C22×D3—44no—
225C22×D3—44no—
226C22×D3—44no—
227C22×D3—44no—
228C22×D3—44no—
229C22×D3—44no—
230C22×D3C2×D412yes—
231C22×D3C2×D412yes—
232C22×D3C2×D412yes—
233C22×D3C2×D412yes—
234C22×D3C2×D412yes—
235C22×D3#4112yes—
236C22×D3#4112yes—
237C3×D4—28no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4C2×D412yes—
253#67—28no—
254#67—28no—
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—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3C2×D412yes—
270D12—28no—
271C4×D3—28no—
272C4×D3—44no—
273C4×D3—44no—
274C24—31no—
275C24—31no—
276C24—31no—
277C24—34no—
278C24—62no—
279C24—62no—
280C24—62no—
281C24—62no—
282C24—62no—
283C24—62no—
284C24—62no—
285C24—62no—
286C24—62no—
287C24—121no—
288C24—121no—
289C24—121no—
290C24—121no—
291C24C22×D311yescharacteristic
292C24C22×D311yescharacteristic
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—34no—
297C2×D4—34no—
298C2×D4—62no—
299C2×D4—62no—
300C2×D4—62no—
301C2×D4—632no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—62no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—31no—
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×C4C22×D311yescharacteristic
321C22×C4C22×D311yescharacteristic
322C22×C4C22×D311yescharacteristic
323C22×C4C22×D311yescharacteristic
324C22×C4C22×D311yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—62no—
327C4⋊C4—62no—
328C4⋊C4—62no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—62no—
334C22⋊C4—62no—
335C22⋊C4—62no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—62no—
339C22⋊C4—62no—
340C22⋊C4—62no—
341C22⋊C4—62no—
342C22⋊C4—62no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3C22×D411yescharacteristic
362C22×C3C22×D411yescharacteristic
363C22×C3C22×D411yescharacteristic
364C22×C3C2×#4111yescharacteristic
365C22×C3C2×#4111yescharacteristic
366C22×C3H(5,𝔽2)11yescharacteristic
367C22×C3H(5,𝔽2)11yescharacteristic
368C2×D3—22no—
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—22no—
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—22no—
379C2×D3—22no—
380C2×D3—22no—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—416no—
384C2×D3—416no—
385C2×D3—41no—
386C2×D3—41no—
387C2×D3—41no—
388C2×D3—41no—
389C2×D3#12014yes—
390C2×D3#12314yescomplemented
391C3×C4—22no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4—41no—
396Dic3—22no—
397Dic3—41no—
398Dic3—41no—
399Dic3—41no—
400Dic3—41no—
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—32no—
414C23—32no—
415C23—32no—
416C23—32no—
417C23—32no—
418C23—32no—
419C23—32no—
420C23—68no—
421C23—68no—
422C23—68no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—61no—
434C23—61no—
435C23—61no—
436C23—61no—
437C23—124no—
438C23—124no—
439C23—124no—
440C23—124no—
441C23—124no—
442C23—124no—
443C23—124no—
444C23—124no—
445C23—124no—
446C23—124no—
447C23—124no—
448C23D3×D412yes—
449C23D3×D412yes—
450C23C23×D311yescharacteristic
451D4—28no—
452D4—68no—
453D4—68no—
454D4—68no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—32no—
470C2×C4—68no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—124no—
486C2×C4—124no—
487C2×C4D3×D412yes—
488C2×C3—42no—
489C2×C3—42no—
490C2×C3C2×#12011yescharacteristic
491C2×C3C2×#12311yescharacteristic
492C2×C3#53411yescharacteristic
493C2×C3#54611yescharacteristic
494C2×C3#54611yescharacteristic
495C2×C3#54611yescharacteristic
496C2×C3#54611yescharacteristic
497D3—24no—
498D3—24no—
499D3—42no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—34no—
507C22—34no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—62no—
522C22—62no—
523C22—62no—
524C22—62no—
525C22—62no—
526C22—62no—
527C22—62no—
528C22—62no—
529C22—62no—
530C22—62no—
531C22—1216no—
532C22—1216no—
533C22—121no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22C2×D3×D411yescharacteristic
538C22C2×D3×D411yescharacteristic
539C22C2×D3×D411yescharacteristic
540C22#100411yescharacteristic
541C22D3×#4111yescharacteristic
542C22D3×#4111yescharacteristic
543C22#100911yescharacteristic
544C4—22no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—62no—
550C4—121no—
551C4—121no—
552C4—121no—
553C4—121no—
554C3#240511yescharacteristic, complemented, Sylow
555C2—42no—
556C2—42no—
557C2—64no—
558C2—64no—
559C2—122no—
560C2D3×#12011yescharacteristic
561C2#560511yescharacteristic
562C2#561011yescharacteristic
563C2#561211yescharacteristic
564C2D3×#12311yescharacteristic
565C2#565411yescharacteristic
566C2#567611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×D3×D4C2
hypercenter#394D3
hypocenterC3#2405
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#894C2semidirect
3. C2×#883C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#394C2semidirect
6. #5222C2semidirect
7. #4989C2semidirect
8. #4961C2semidirect
9. #4959C2semidirect
10. #4956C2semidirect
11. #4955C2semidirect
12. #4955C2semidirect
13. #4955C2semidirect
14. #4668C2semidirect
15. #4665C2semidirect
27. C22×C4×D3C22semidirect
28. C22×C4×D3C22semidirect
29. C2×C3×C4⋊C4C22semidirect
30. C2×C3×C22⋊C4C22semidirect
31. C2×C3×C22⋊C4C22semidirect
32. C2×C3×C22⋊C4C22semidirect
33. C2×C3×C22⋊C4C22semidirect
34. C2×#217C22semidirect
35. C2×#217C22semidirect
36. C2×#217C22semidirect
37. C2×#217C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
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×Dic3⋊C4C22semidirect
54. C3×#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
57. C3⋊#95C22semidirect
65. #394D3semidirect
163. C2×C4⋊C4C2×D3semidirect
179. C2×C22⋊C4C2×D3semidirect
180. C2×C22⋊C4C2×D3semidirect
181. C2×C22⋊C4C2×D3semidirect
182. C2×C22⋊C4C2×D3semidirect
186. #95C2×D3semidirect
390. C2×D3#123semidirect
554. C3#2405semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#8942
C2×#8832
C2×D3×C22⋊C42
C3×#3942
#52222
#49892
#49612
#49592
#49562
#49552
#46682
#46652
#24053