Finite Group Databasebeta

← #83114

Subgroups — #83114

Order 384.

666 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic, complemented
2C2×#882C211yesmaximal, characteristic
3#4994C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4990C211yesmaximal, characteristic, complemented
8#4988C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4955C211yesmaximal, characteristic, complemented
12#4955C211yesmaximal, characteristic, complemented
13#4951C211yesmaximal, characteristic, complemented
14#4940C211yesmaximal, characteristic, complemented
15#4939C211yesmaximal, characteristic, complemented
16#2685—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C3×#126—24no—
22C2×C3×C4⋊C4C2211yescharacteristic, complemented
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C4⋊C4C2211yescharacteristic, complemented
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C42C2211yescharacteristic, complemented
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
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
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×C4⋊Dic3C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47C2×Dic3⋊C4C2211yescharacteristic, complemented
48C2×C4×Dic3C2211yescharacteristic, complemented
49#882—24no—
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#126D311yescharacteristic, complemented
55C2×#120—31no—
56#399—31no—
57#396—31no—
58#394—31no—
59#394—31no—
60#394—31no—
61#394—31no—
62#390—31no—
63#390—31no—
64#388—31no—
65#388—31no—
66#388—31no—
67#379—31no—
68#379—31no—
69C24×C3C2311yescharacteristic
70C23×D3—22no—
71C23×D3—22no—
72C23×D3—22no—
73C23×D3—41no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3C2311yescharacteristic
78C23×D3C2311yescharacteristic
79C23×D3C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—22no—
87C2×#67—41no—
88C2×#67—41no—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C2×D12—22no—
95C2×D12—22no—
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—41no—
99C2×D12—41no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4D414yes—
106C3×C42—22no—
107#217—41no—
108#217—41no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123C4⋊Dic3—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126C4×Dic3—41no—
127C25—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131#126—24no—
132#120—64no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4C2×D311yescharacteristic, complemented
137C2×C4⋊C4C2×D311yescharacteristic, complemented
138C2×C4⋊C4C2×D311yescharacteristic, complemented
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊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⋊C4C2×D311yescharacteristic
156C2×C22⋊C4C2×D311yescharacteristic
157C2×C22⋊C4C2×D311yescharacteristic
158C2×C42—31no—
159C2×C42C2×D311yescharacteristic, complemented
160#95—31no—
161#95—31no—
162#95—31no—
163#95—31no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3C2×D412yes—
171C23×C3C2411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—44no—
192C22×D3—44no—
193C22×D3—44no—
194C22×D3—44no—
195C22×D3—44no—
196C22×D3—44no—
197C22×D3—44no—
198C22×D3#4112yes—
199C22×D3#4112yes—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
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×C4C2×D412yes—
217C2×C3×C4C2×D412yes—
218#67—44no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234D12—44no—
235D12—44no—
236C24—31no—
237C24—31no—
238C24—31no—
239C24—62no—
240C24—62no—
241C24—62no—
242C24—121no—
243C24—121no—
244C24—121no—
245C24—121no—
246C24C22×D311yescharacteristic
247C2×D4—62no—
248C2×D4—62no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4C22×D311yescharacteristic
262C22×C4C22×D311yescharacteristic
263C22×C4C22×D311yescharacteristic
264C22×C4C22×D311yescharacteristic
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4#6714yes—
292C42—22no—
293C42—121no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3C22×D411yescharacteristic
302C22×C3C2×#4111yescharacteristic
303C22×C3C2×#4111yescharacteristic
304C22×C3H(5,𝔽2)11yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C22×C3H(5,𝔽2)11yescharacteristic
307C22×C3H(5,𝔽2)11yescharacteristic
308C2×D3—22no—
309C2×D3—22no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—22no—
313C2×D3—22no—
314C2×D3—416no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336Dic3—41no—
337Dic3—41no—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—32no—
348C23—32no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—61no—
368C23—124no—
369C23—124no—
370C23—124no—
371C23—124no—
372C23—124no—
373C23—124no—
374C23—124no—
375C23C2×#6712yes—
376C23C23×D311yescharacteristic
377D4—124no—
378D4—124no—
379D4—124no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4C2×#6712yes—
412C2×C4C2×#6712yes—
413C2×C3—42no—
414C2×C3#53511yescharacteristic
415C2×C3#53811yescharacteristic
416C2×C3#53811yescharacteristic
417C2×C3#54611yescharacteristic
418C2×C3#54611yescharacteristic
419C2×C3#55111yescharacteristic
420C2×C3#56011yescharacteristic
421D3—42no—
422D3—42no—
423D3—81no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—1216no—
438C22—121no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22D3×#4111yescharacteristic
454C22D3×#4111yescharacteristic
455C22#100911yescharacteristic
456C22#100911yescharacteristic
457C22#100911yescharacteristic
458C22#100911yescharacteristic
459C22C22×#6711yescharacteristic
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—121no—
467C4—121no—
468C4—121no—
469C4—121no—
470C4—121no—
471C3#268511yescharacteristic, complemented, Sylow
472C2—42no—
473C2—122no—
474C2—122no—
475C2—241no—
476C2#570411yescharacteristic
477C2#570411yescharacteristic
478C2#570511yescharacteristic
479C2#571111yescharacteristic
480C2#582911yescharacteristic
481C2#582911yescharacteristic
482C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#126C2
prime radicalC2×#882C2
hypercenterC2×#126D3
hypocenterC3#2685
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
3. #4994C2semidirect
4. #4993C2semidirect
5. #4991C2semidirect
6. #4991C2semidirect
7. #4990C2semidirect
8. #4988C2semidirect
9. #4959C2semidirect
10. #4959C2semidirect
11. #4955C2semidirect
12. #4955C2semidirect
13. #4951C2semidirect
14. #4940C2semidirect
15. #4939C2semidirect
22. C2×C3×C4⋊C4C22semidirect
23. C2×C3×C4⋊C4C22semidirect
24. C2×C3×C4⋊C4C22semidirect
28. C2×C3×C42C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×C4⋊Dic3C22semidirect
46. C2×Dic3⋊C4C22semidirect
47. C2×Dic3⋊C4C22semidirect
48. C2×C4×Dic3C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C2×#126D3semidirect
136. C2×C4⋊C4C2×D3semidirect
137. C2×C4⋊C4C2×D3semidirect
138. C2×C4⋊C4C2×D3semidirect
159. C2×C42C2×D3semidirect
471. C3#2685semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1262
C2×#8822
#49942
#49932
#49912
#49902
#49882
#49592
#49552
#49512
#49402
#49392
#26853