Finite Group Databasebeta

← #76790

Subgroups — #76790

Order 384.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#947C211yesmaximal, characteristic, complemented
2C2×#946C211yesmaximal, characteristic, complemented
3C2×#882C211yesmaximal, characteristic
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#393C211yesmaximal, characteristic, complemented
6#4961C211yesmaximal, characteristic, complemented
7#4957C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4945C211yesmaximal, characteristic, complemented
10#4675C211yesmaximal, characteristic, complemented
11#4672C211yesmaximal, characteristic, complemented
12#4670C211yesmaximal, characteristic, complemented
13#4666C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2472—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic, complemented
19C22×#67C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29#947—24no—
30#946—24no—
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×C4⋊Dic3C2211yescharacteristic, complemented
44C2×Dic3⋊C4C2211yescharacteristic, complemented
45C2×Dic3⋊C4C2211yescharacteristic, complemented
46C2×C4×Dic3C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48#882—24no—
49D3×C22⋊C4—24no—
50C3×#95C2211yescharacteristic, complemented
51C3×#95C2211yescharacteristic, complemented
52C3×#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#124—31no—
57C2×C22⋊Q8—31no—
58C2×#120—31no—
59C22×C22⋊C4—31no—
60#397—31no—
61#394—31no—
62#394—31no—
63#393D311yescharacteristic, complemented
64#390—31no—
65#388—31no—
66#388—31no—
67#386—31no—
68#386—31no—
69#379—31no—
70C4×C22⋊C4—31no—
71C24×C3C2311yescharacteristic
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3C2311yescharacteristic
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C2×C3×Q8—22no—
83C2×C3×Q8—22no—
84C2×C3×Q8—41no—
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—41no—
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C2×D12—22no—
100C2×D12—22no—
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—22no—
104C2×D12—41no—
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—41no—
114#217—22no—
115#217—22no—
116(C2×D3)⋊C4—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4D414yes—
127(C2×D3)⋊C4D414yescomplemented
128C4⋊Dic3—22no—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131C4×Dic3—22no—
132C4×Dic3—41no—
133C25—31no—
134C22×Q8C2×D311yescharacteristic, complemented
135C22×D4—31no—
136C22×D4—31no—
137C22×D4—31no—
138C23×C4—31no—
139C23×C4—31no—
140#124—64no—
141C22⋊Q8—64no—
142#120—64no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊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—64no—
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C22⋊C4C2×D311yescharacteristic
163C2×C22⋊C4C2×D311yescharacteristic
164C2×C42—31no—
165C2×C42—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95C2×D311yescharacteristic, complemented
170#95C2×D311yescharacteristic, complemented
171#95C2×D311yescharacteristic, complemented
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3#4112yes—
179C23×C3C2411yescharacteristic
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—28no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—44no—
199C22×D3—44no—
200C22×D3—44no—
201C22×D3—44no—
202C22×D3—44no—
203C22×D3—44no—
204C22×D3C2×D412yes—
205C22×D3C2×D412yes—
206C22×D3#4112yes—
207C22×D3#4112yes—
208C3×Q8—44no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4C2×D412yes—
226C2×C3×C4C2×D412yes—
227#67—44no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3C2×D412yes—
242C2×Dic3C2×D412yes—
243D12—44no—
244D12—44no—
245C4×D3—44no—
246C4×D3—44no—
247C24—31no—
248C24—31no—
249C24—31no—
250C24—62no—
251C24—62no—
252C24—62no—
253C24—62no—
254C24—62no—
255C24—121no—
256C24—121no—
257C24C22×D311yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—121no—
269C2×D4—121no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—34no—
293C22⋊C4—34no—
294C22⋊C4—41no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C42—62no—
308C42—121no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3C22×D411yescharacteristic
317C22×C3C22×D411yescharacteristic
318C22×C3C2×#4111yescharacteristic
319C22×C3C2×#4111yescharacteristic
320C22×C3C2×#4111yescharacteristic
321C22×C3H(5,𝔽2)11yescharacteristic
322C22×C3H(5,𝔽2)11yescharacteristic
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—22no—
329C2×D3—22no—
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—416no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3#12414yescomplemented
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—41no—
351Dic3—22no—
352Dic3—22no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—32no—
363C23—32no—
364C23—32no—
365C23—32no—
366C23—68no—
367C23—61no—
368C23—61no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—124no—
385C23—124no—
386C23—124no—
387C23—124no—
388C23—124no—
389C23—124no—
390C23#23912yes—
391C23C23×D311yescharacteristic
392Q8—44no—
393D4—124no—
394D4—124no—
395D4—124no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—32no—
413C2×C4—32no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—124no—
428C2×C4—124no—
429C2×C4D3×D412yes—
430C2×C4D3×D412yes—
431C2×C3—42no—
432C2×C3#52511yescharacteristic
433C2×C3C2×#12411yescharacteristic
434C2×C3#53811yescharacteristic
435C2×C3D4211yescharacteristic
436C2×C3#54611yescharacteristic
437C2×C3#54611yescharacteristic
438C2×C3#55111yescharacteristic
439D3—24no—
440D3—42no—
441D3—81no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—34no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22—62no—
458C22—62no—
459C22—1216no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22C2×D3×D411yescharacteristic
472C22C2×D3×D411yescharacteristic
473C22#100411yescharacteristic
474C22C2×#23911yescharacteristic
475C22D3×#4111yescharacteristic
476C22D3×#4111yescharacteristic
477C22#100911yescharacteristic
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—62no—
485C4—62no—
486C4—121no—
487C4—121no—
488C4—121no—
489C3#247211yescharacteristic, complemented, Sylow
490C2—42no—
491C2—64no—
492C2—122no—
493C2—241no—
494C2#559011yescharacteristic
495C2#562711yescharacteristic
496C2#563011yescharacteristic
497C2#563711yescharacteristic
498C2D3×#12411yescharacteristic
499C2#567611yescharacteristic
500C2#567811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC2×#882C2
hypercenter#393D3
hypocenterC3#2472
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#947C2semidirect
2. C2×#946C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#393C2semidirect
6. #4961C2semidirect
7. #4957C2semidirect
8. #4955C2semidirect
9. #4945C2semidirect
10. #4675C2semidirect
11. #4672C2semidirect
12. #4670C2semidirect
13. #4666C2semidirect
14. #4665C2semidirect
15. #4665C2semidirect
18. C22×C3×Q8C22semidirect
39. C2×(C2×D3)⋊C4C22semidirect
40. C2×(C2×D3)⋊C4C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×C4⋊Dic3C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×Dic3⋊C4C22semidirect
46. C2×C4×Dic3C22semidirect
47. C2×C4×Dic3C22semidirect
50. C3×#95C22semidirect
51. C3×#95C22semidirect
52. C3×#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
63. #393D3semidirect
127. (C2×D3)⋊C4D4semidirect
134. C22×Q8C2×D3semidirect
169. #95C2×D3semidirect
170. #95C2×D3semidirect
171. #95C2×D3semidirect
344. C2×D3#124semidirect
489. C3#2472semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#9462
C2×#8822
C2×D3×C22⋊C42
C3×#3932
#49612
#49572
#49552
#49452
#46752
#46722
#46702
#46662
#46652
#24723