Finite Group Databasebeta

← #74876

Subgroups — #74876

Order 384.

808 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×#937C211yesmaximal, characteristic
2C2×C4×#67C211yesmaximal, characteristic, complemented
3C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#386C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4957C211yesmaximal, characteristic, complemented
8#4956C211yesmaximal, characteristic, complemented
9#4955C211yesmaximal, characteristic, complemented
10#4944C211yesmaximal, characteristic, complemented
11#4944C211yesmaximal, characteristic, complemented
12#4670C211yesmaximal, characteristic, complemented
13#4669C211yesmaximal, characteristic, complemented
14#4668C211yesmaximal, characteristic, complemented
15#4666C211yesmaximal, characteristic, complemented
16#2321—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic, complemented
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic, complemented
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic, complemented
31C2×#217C2211yescharacteristic, complemented
32#937C418yescomplemented
33C4×#67—24no—
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
42C2×C4⋊Dic3C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44C2×Dic3⋊C4C2211yescharacteristic, complemented
45C2×C4×Dic3C2211yescharacteristic, complemented
46C2×C4×Dic3C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48D3×C22⋊C4—24no—
49C3×#95C2211yescharacteristic, complemented
50C3×#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C2×#120—31no—
56C2×C4×D4—31no—
57C22×C22⋊C4—31no—
58#390—31no—
59#390—31no—
60#388—31no—
61#388—31no—
62#388—31no—
63#388—31no—
64#386—31no—
65#386—31no—
66#386D311yescharacteristic, complemented
67#385—31no—
68#379—31no—
69C4×C22⋊C4—31no—
70C24×C3C2311yescharacteristic
71C24×C3C2311yescharacteristic
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3C2×C414yescomplemented
79C23×D3C2311yescharacteristic
80C23×D3C2311yescharacteristic
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×C3×D4C2×C414yescomplemented
84C22×C3×C4—22no—
85C22×C3×C4—41no—
86C22×C3×C4—41no—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67C2×C414yes—
97C2×#67C2×C414yescomplemented
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C2×C4×D3—22no—
105C2×C4×D3—41no—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C2×C4×D3—41no—
109C2×C4×D3D414yescomplemented
110C3×C22⋊C4—22no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113#217—22no—
114#217—22no—
115#217—41no—
116#217—41no—
117#217C2×C414yescomplemented
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4C2×C414yescomplemented
125(C2×D3)⋊C4C2×C414yescomplemented
126C4⋊Dic3—41no—
127Dic3⋊C4—22no—
128Dic3⋊C4—41no—
129C4×Dic3—22no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132C25—31no—
133C22×D4—31no—
134C22×D4—31no—
135C22×D4C2×D311yescharacteristic
136C23×C4—31no—
137C23×C4—31no—
138C23×C4C2×D311yescharacteristic, complemented
139#120—38no—
140C4×D4—64no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊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⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—64no—
158C2×C22⋊C4C2×D311yescharacteristic
159C2×C22⋊C4C2×D311yescharacteristic
160C2×C22⋊C4C2×D311yescharacteristic, complemented
161C2×C42—31no—
162C2×C42—31no—
163C2×C42—31no—
164#95—31no—
165#95—31no—
166#95—31no—
167#95—31no—
168#95C2×D311yescharacteristic, complemented
169#95C2×D311yescharacteristic, complemented
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3C22×C412yes—
183C23×C3C22×C412yes—
184C23×C3C2411yescharacteristic
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—28no—
196C22×D3—44no—
197C22×D3—44no—
198C22×D3—44no—
199C22×D3—44no—
200C22×D3—44no—
201C22×D3—44no—
202C22×D3C22×C412yes—
203C22×D3C22×C412yes—
204C22×D3C2×D412yes—
205C22×D3#4112yes—
206C3×D4—44no—
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×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—44no—
221C2×C3×C4C22×C412yes—
222C2×C3×C4C2×D412yes—
223#67—28no—
224#67—44no—
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—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3C22×C412yes—
241C2×Dic3C22×C412yes—
242C2×Dic3C2×D412yes—
243C4×D3—44no—
244C4×D3—44no—
245C24—31no—
246C24—31no—
247C24—34no—
248C24—62no—
249C24—62no—
250C24—121no—
251C24—121no—
252C24—121no—
253C24—121no—
254C24C22×D311yescharacteristic
255C24C22×D311yescharacteristic
256C2×D4—34no—
257C2×D4—34no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4C4×D314yescomplemented
265C22×C4—22no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—34no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—62no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C4⋊C4—62no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—22no—
289C22⋊C4—34no—
290C22⋊C4—34no—
291C22⋊C4—34no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—62no—
305C42—121no—
306C42—121no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3C23×C411yescharacteristic
322C22×C3C22×D411yescharacteristic
323C22×C3C2×#4111yescharacteristic
324C22×C3H(5,𝔽2)11yescharacteristic
325C22×C3H(5,𝔽2)11yescharacteristic
326C22×C3H(5,𝔽2)11yescharacteristic
327C22×C3H(5,𝔽2)11yescharacteristic
328C2×D3—22no—
329C2×D3—22no—
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—416no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3C4×D414yes—
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—41no—
348Dic3—22no—
349Dic3—22no—
350Dic3—22no—
351Dic3—41no—
352Dic3—41no—
353Dic3—41no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—32no—
367C23—32no—
368C23—32no—
369C23—32no—
370C23—68no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—124no—
382C23—124no—
383C23—124no—
384C23—124no—
385C23—124no—
386C23—124no—
387C23C2×C4×D312yes—
388C23C2×C4×D312yes—
389C23C23×D311yescharacteristic
390D4—44no—
391D4—68no—
392D4—124no—
393C2×C4—21no—
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—32no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—44no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
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—124no—
426C2×C4—124no—
427C2×C4C2×C4×D312yes—
428C2×C4D3×D412yes—
429C2×C3—42no—
430C2×C3—42no—
431C2×C3C2×C4×D411yescharacteristic
432C2×C3#51811yescharacteristic
433C2×C3#51811yescharacteristic
434C2×C3#53411yescharacteristic
435C2×C3#53511yescharacteristic
436C2×C3#53811yescharacteristic
437C2×C3#53811yescharacteristic
438D3—24no—
439D3—42no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—34no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—62no—
456C22—62no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—1216no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22C22×C4×D311yescharacteristic
470C22C2×D3×D411yescharacteristic
471C22#100411yescharacteristic
472C22#100411yescharacteristic
473C22#100411yescharacteristic
474C22D3×#4111yescharacteristic
475C22#100911yescharacteristic
476C4—22no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—62no—
482C4—62no—
483C4—62no—
484C4—121no—
485C4—121no—
486C4—121no—
487C3#232111yescharacteristic, complemented, Sylow
488C2—42no—
489C2—42no—
490C2—64no—
491C2—122no—
492C2#548611yescharacteristic
493C2C4×D3×D411yescharacteristic
494C2#554511yescharacteristic
495C2#558911yescharacteristic
496C2#560711yescharacteristic
497C2#561311yescharacteristic
498C2#565711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#386C2
socle, cosocleC23×C3C24
prime radicalC2×#937C2
hypercenter#386D3
hypocenterC3#2321
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×#67C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C3×#386C2semidirect
5. #5221C2semidirect
6. #5213C2semidirect
7. #4957C2semidirect
8. #4956C2semidirect
9. #4955C2semidirect
10. #4944C2semidirect
11. #4944C2semidirect
12. #4670C2semidirect
13. #4669C2semidirect
14. #4668C2semidirect
15. #4666C2semidirect
19. C23×C3×C4C22semidirect
26. C2×C3×C22⋊C4C22semidirect
30. C2×#217C22semidirect
31. C2×#217C22semidirect
32. #937C4semidirect
42. C2×C4⋊Dic3C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×C4×Dic3C22semidirect
46. C2×C4×Dic3C22semidirect
47. C2×C4×Dic3C22semidirect
49. C3×#95C22semidirect
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
66. #386D3semidirect
78. C23×D3C2×C4semidirect
83. C2×C3×D4C2×C4semidirect
97. C2×#67C2×C4semidirect
109. C2×C4×D3D4semidirect
117. #217C2×C4semidirect
124. (C2×D3)⋊C4C2×C4semidirect
125. (C2×D3)⋊C4C2×C4semidirect
138. C23×C4C2×D3semidirect
160. C2×C22⋊C4C2×D3semidirect
168. #95C2×D3semidirect
169. #95C2×D3semidirect
264. C2×D4C4×D3semidirect
487. C3#2321semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×C4×#672
C2×D3×C22⋊C42
C3×#3862
#52212
#52132
#49572
#49562
#49552
#49442
#46702
#46692
#46682
#46662
#23213