Finite Group Databasebeta

← #77414

Subgroups — #77414

Order 384.

786 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#894C211yesmaximal, characteristic, complemented
2C2×#892C211yesmaximal, characteristic, complemented
3C2×#882C211yesmaximal, characteristic
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#396C211yesmaximal, characteristic, complemented
6#4994C211yesmaximal, characteristic, complemented
7#4991C211yesmaximal, characteristic, complemented
8#4989C211yesmaximal, characteristic, complemented
9#4988C211yesmaximal, characteristic, complemented
10#4961C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4955C211yesmaximal, characteristic, complemented
13#4944C211yesmaximal, characteristic, complemented
14#4669C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2411—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C4⋊C4C2211yescharacteristic, complemented
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
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, 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#894—24no—
49#892—24no—
50#882—24no—
51D3×C22⋊C4—24no—
52C3×#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#123—31no—
57C2×#120—31no—
58C2×#117—31no—
59C22×C22⋊C4—31no—
60#396—31no—
61#396D311yescharacteristic, complemented
62#394—31no—
63#394—31no—
64#390—31no—
65#390—31no—
66#388—31no—
67#388—31no—
68#386—31no—
69#385—31no—
70#379—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
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—41no—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—22no—
101C2×D12—41no—
102C2×C4×D3—22no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—41no—
106C2×C4×D3—41no—
107C2×C4×D3D414yescomplemented
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—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—22no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4D414yes—
128C4⋊Dic3—22no—
129Dic3⋊C4—22no—
130Dic3⋊C4—41no—
131C4×Dic3—22no—
132C4×Dic3—22no—
133C25—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4—31no—
137C23×C4—31no—
138C23×C4—31no—
139#123—64no—
140#120—64no—
141#117—64no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4C2×D311yescharacteristic, complemented
146C2×C4⋊C4C2×D311yescharacteristic, complemented
147C2×C4⋊C4C2×D311yescharacteristic, complemented
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—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—64no—
163C2×C22⋊C4C2×D311yescharacteristic
164C2×C22⋊C4C2×D311yescharacteristic
165C2×C22⋊C4C2×D311yescharacteristic
166C2×C42—31no—
167C2×C42—31no—
168#95—31no—
169#95—31no—
170#95—31no—
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—
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×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×C4C2×D412yes—
225C2×C3×C4C2×D412yes—
226#67—44no—
227C2×Dic3—21no—
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×Dic3C2×D412yes—
241C2×Dic3C2×D412yes—
242D12—44no—
243D12—44no—
244C4×D3—28no—
245C4×D3—44no—
246C24—31no—
247C24—31no—
248C24—31no—
249C24—62no—
250C24—62no—
251C24—62no—
252C24—62no—
253C24—62no—
254C24—121no—
255C24—121no—
256C24C22×D311yescharacteristic
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—121no—
265C2×D4—121no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—34no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4C22×D311yescharacteristic
278C22×C4C22×D311yescharacteristic
279C22×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—121no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—34no—
292C22⋊C4—41no—
293C22⋊C4—62no—
294C22⋊C4—62no—
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—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42—62no—
307C42—62no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3C22×D411yescharacteristic
316C22×C3C22×D411yescharacteristic
317C22×C3C2×#4111yescharacteristic
318C22×C3C2×#4111yescharacteristic
319C22×C3C2×#4111yescharacteristic
320C22×C3H(5,𝔽2)11yescharacteristic
321C22×C3H(5,𝔽2)11yescharacteristic
322C2×D3—22no—
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—416no—
332C2×D3—41no—
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#12114yes—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350Dic3—22no—
351Dic3—22no—
352Dic3—41no—
353Dic3—41no—
354Dic3—41no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—32no—
362C23—32no—
363C23—32no—
364C23—32no—
365C23—68no—
366C23—61no—
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—124no—
384C23—124no—
385C23—124no—
386C23—124no—
387C23—124no—
388C23—124no—
389C23#23912yes—
390C23C23×D311yescharacteristic
391D4—124no—
392D4—124no—
393D4—124no—
394C2×C4—21no—
395C2×C4—21no—
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—32no—
411C2×C4—32no—
412C2×C4—68no—
413C2×C4—61no—
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—124no—
427C2×C4D3×D412yes—
428C2×C4D3×D412yes—
429C2×C3—42no—
430C2×C3C2×#12111yescharacteristic
431C2×C3#52511yescharacteristic
432C2×C3#53511yescharacteristic
433C2×C3#54611yescharacteristic
434C2×C3#54611yescharacteristic
435C2×C3#55111yescharacteristic
436C2×C3#55111yescharacteristic
437D3—24no—
438D3—42no—
439D3—81no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—34no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—62no—
449C22—62no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22—1216no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22C2×D3×D411yescharacteristic
470C22C2×D3×D411yescharacteristic
471C22#100411yescharacteristic
472C22C2×#23911yescharacteristic
473C22D3×#4111yescharacteristic
474C22D3×#4111yescharacteristic
475C22#100911yescharacteristic
476C4—22no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—62no—
483C4—62no—
484C4—121no—
485C4—121no—
486C4—121no—
487C3#241111yescharacteristic, complemented, Sylow
488C2—42no—
489C2—64no—
490C2—122no—
491C2—241no—
492C2D3×#12111yescharacteristic
493C2#560711yescharacteristic
494C2#561011yescharacteristic
495C2#562711yescharacteristic
496C2#563011yescharacteristic
497C2#563711yescharacteristic
498C2#570511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#396C2
prime radicalC2×#882C2
hypercenter#396D3
hypocenterC3#2411
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#894C2semidirect
2. C2×#892C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#396C2semidirect
6. #4994C2semidirect
7. #4991C2semidirect
8. #4989C2semidirect
9. #4988C2semidirect
10. #4961C2semidirect
11. #4957C2semidirect
12. #4955C2semidirect
13. #4944C2semidirect
14. #4669C2semidirect
15. #4665C2semidirect
23. C2×C3×C4⋊C4C22semidirect
24. C2×C3×C4⋊C4C22semidirect
25. C2×C3×C4⋊C4C22semidirect
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
52. C3×#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
61. #396D3semidirect
107. C2×C4×D3D4semidirect
145. C2×C4⋊C4C2×D3semidirect
146. C2×C4⋊C4C2×D3semidirect
147. C2×C4⋊C4C2×D3semidirect
171. #95C2×D3semidirect
487. C3#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#8942
C2×#8922
C2×#8822
C2×D3×C22⋊C42
C3×#3962
#49942
#49912
#49892
#49882
#49612
#49572
#49552
#49442
#46692
#46652
#24113