Finite Group Databasebeta

← #82398

Subgroups — #82398

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic
3C2×C4×#67C211yesmaximal, characteristic, complemented
4C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
6#5220C211yesmaximal, characteristic, complemented
7#5213C211yesmaximal, characteristic, complemented
8#5210C211yesmaximal, characteristic, complemented
9#4990C211yesmaximal, characteristic, complemented
10#4989C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
14#4940C211yesmaximal, characteristic, complemented
15C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
16#2358—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic, complemented
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C3×C4×D4—24no—
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic, complemented
30C2×C3×C42C2211yescharacteristic, complemented
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic, complemented
34C2×#217C2211yescharacteristic, complemented
35#937C418yescomplemented
36C4×#67—24no—
37C2×(C2×D3)⋊C4—24no—
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47C2×Dic3⋊C4C2211yescharacteristic, complemented
48C2×C4×Dic3C2211yescharacteristic, complemented
49C2×C4×Dic3C2211yescharacteristic, complemented
50D3×C22⋊C4—24no—
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C3⋊#95C2211yescharacteristic, complemented
57C2×#120—31no—
58C2×C4×D4—31no—
59C2×C4×D4D311yescharacteristic, complemented
60C22×C22⋊C4—31no—
61C22×C22⋊C4—31no—
62#390—31no—
63#388—31no—
64#388—31no—
65#386—31no—
66#386—31no—
67#386—31no—
68#385—31no—
69#385—31no—
70C4×C22⋊C4—31no—
71C4×C22⋊C4—31no—
72C24×C3C2311yescharacteristic
73C24×C3C2311yescharacteristic
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—22no—
78C23×D3—41no—
79C23×D3—41no—
80C23×D3C2×C414yescomplemented
81C23×D3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4C2×C414yescomplemented
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—41no—
100C2×#67—41no—
101C2×#67C2×C414yescomplemented
102C2×#67C2×C414yescomplemented
103C22×Dic3—22no—
104C22×Dic3—22no—
105C22×Dic3—41no—
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C2×C4×D3—22no—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C3×C4⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4D414yescomplemented
121C3×C42—22no—
122#217—22no—
123#217—22no—
124#217—41no—
125#217C2×C414yescomplemented
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4C2×C414yescomplemented
132(C2×D3)⋊C4C2×C414yescomplemented
133(C2×D3)⋊C4D414yescomplemented
134Dic3⋊C4—22no—
135Dic3⋊C4—41no—
136C4×Dic3—22no—
137C4×Dic3—41no—
138C25—31no—
139C22×D4—31no—
140C22×D4—31no—
141C22×D4C2×D311yescharacteristic
142C23×C4—31no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4C2×D311yescharacteristic
146C23×C4C2×D311yescharacteristic, complemented
147#120—38no—
148C4×D4—24no—
149C4×D4—64no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4C2×D311yescharacteristic, complemented
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—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—64no—
166C2×C22⋊C4—64no—
167C2×C22⋊C4C2×D311yescharacteristic
168C2×C22⋊C4C2×D311yescharacteristic, complemented
169C2×C42—31no—
170C2×C42—31no—
171C2×C42C2×D311yescharacteristic, complemented
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95—31no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3C22×C412yes—
189C23×C3C22×C412yes—
190C23×C3C2×D412yes—
191C23×C3#4112yes—
192C23×C3C2411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—28no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—28no—
205C22×D3—44no—
206C22×D3—44no—
207C22×D3—44no—
208C22×D3—44no—
209C22×D3—44no—
210C22×D3C22×C412yes—
211C22×D3C22×C412yes—
212C22×D3C2×D412yes—
213C22×D3#4112yes—
214C3×D4—28no—
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×C4—21no—
226C2×C3×C4—44no—
227C2×C3×C4—44no—
228C2×C3×C4C22×C412yes—
229C2×C3×C4C2×D412yes—
230C2×C3×C4C2×D412yes—
231C2×C3×C4C2×D412yes—
232#67—28no—
233#67—44no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—44no—
250C2×Dic3C22×C412yes—
251C2×Dic3C22×C412yes—
252C2×Dic3C2×D412yes—
253C4×D3—44no—
254C4×D3—44no—
255C24—31no—
256C24—31no—
257C24—34no—
258C24—62no—
259C24—62no—
260C24—62no—
261C24—62no—
262C24—121no—
263C24—121no—
264C24C22×D311yescharacteristic
265C24C22×D311yescharacteristic
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—34no—
269C2×D4—34no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4C4×D314yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4C22×D311yescharacteristic
297C22×C4C22×D311yescharacteristic
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C22×C4C22×D311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—62no—
303C4⋊C4—121no—
304C22⋊C4—22no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—34no—
308C22⋊C4—34no—
309C22⋊C4—62no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—121no—
317C22⋊C4#6714yescomplemented
318C42—22no—
319C42—62no—
320C42—121no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3C4×D414yescomplemented
335C22×C3C23×C411yescharacteristic
336C22×C3C22×D411yescharacteristic
337C22×C3C22×D411yescharacteristic
338C22×C3C2×#4111yescharacteristic
339C22×C3C2×#4111yescharacteristic
340C22×C3H(5,𝔽2)11yescharacteristic
341C22×C3H(5,𝔽2)11yescharacteristic
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—22no—
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—416no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3C4×D414yescomplemented
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—41no—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—41no—
366Dic3—41no—
367Dic3—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—32no—
379C23—32no—
380C23—32no—
381C23—32no—
382C23—68no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—68no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—124no—
395C23—124no—
396C23—124no—
397C23—124no—
398C23—124no—
399C23C2×C4×D312yes—
400C23C2×C4×D312yes—
401C23#23512yes—
402C23C2×#6712yes—
403C23C23×D311yescharacteristic
404D4—28no—
405D4—68no—
406D4—124no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4C2×C4×D312yes—
442C2×C4D3×D412yes—
443C2×C4C2×#6712yes—
444C2×C4C2×#6712yes—
445C2×C3—24no—
446C2×C3—42no—
447C2×C3C2×C4×D411yescharacteristic
448C2×C3C2×C4×D411yescharacteristic
449C2×C3#51811yescharacteristic
450C2×C3D4211yescharacteristic
451C2×C3#54611yescharacteristic
452C2×C3#54611yescharacteristic
453C2×C3#55111yescharacteristic
454D3—24no—
455D3—42no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—34no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—62no—
471C22—62no—
472C22—62no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—1216no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22—121no—
484C22C4×#6714yescomplemented
485C22C22×C4×D311yescharacteristic
486C22C2×#23511yescharacteristic
487C22C2×D3×D411yescharacteristic
488C22#100411yescharacteristic
489C22D3×#4111yescharacteristic
490C22#100911yescharacteristic
491C22C22×#6711yescharacteristic
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—62no—
498C4—62no—
499C4—62no—
500C4—121no—
501C4—121no—
502C4—121no—
503C3#235811yescharacteristic, complemented, Sylow
504C2—24no—
505C2—42no—
506C2—64no—
507C2—122no—
508C2C4×D3×D411yescharacteristic
509C2#554511yescharacteristic
510C2#555011yescharacteristic
511C2#555611yescharacteristic
512C2C2×C4×#6711yescharacteristic
513C2D4×#6711yescharacteristic
514C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC2×C3×C4×D4C2
socle, cosocleC23×C3C24
prime radicalC2×#937C2
hypercenterC2×C4×D4D3
hypocenterC3#2358
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
3. C2×C4×#67C2semidirect
4. C22×(C2×D3)⋊C4C2semidirect
5. C2×D3×C22⋊C4C2semidirect
6. #5220C2semidirect
7. #5213C2semidirect
8. #5210C2semidirect
9. #4990C2semidirect
10. #4989C2semidirect
11. #4957C2semidirect
12. #4956C2semidirect
13. Dic3×C22⋊C4C2semidirect
14. #4940C2semidirect
15. C4×(C2×D3)⋊C4C2semidirect
20. C23×C3×C4C22semidirect
27. C2×C3×C4⋊C4C22semidirect
29. C2×C3×C22⋊C4C22semidirect
30. C2×C3×C42C22semidirect
33. C2×#217C22semidirect
34. C2×#217C22semidirect
35. #937C4semidirect
46. C2×Dic3⋊C4C22semidirect
47. C2×Dic3⋊C4C22semidirect
48. C2×C4×Dic3C22semidirect
49. C2×C4×Dic3C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
59. C2×C4×D4D3semidirect
80. C23×D3C2×C4semidirect
85. C2×C3×D4C2×C4semidirect
101. C2×#67C2×C4semidirect
102. C2×#67C2×C4semidirect
120. C3×C22⋊C4D4semidirect
125. #217C2×C4semidirect
131. (C2×D3)⋊C4C2×C4semidirect
132. (C2×D3)⋊C4C2×C4semidirect
133. (C2×D3)⋊C4D4semidirect
146. C23×C4C2×D3semidirect
152. C2×C4⋊C4C2×D3semidirect
168. C2×C22⋊C4C2×D3semidirect
171. C2×C42C2×D3semidirect
274. C2×D4C4×D3semidirect
317. C22⋊C4#67semidirect
334. C22×C3C4×D4semidirect
356. C2×D3C4×D4semidirect
484. C22C4×#67semidirect
503. C3#2358semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9372
C2×C4×#672
C22×(C2×D3)⋊C42
C2×D3×C22⋊C42
#52202
#52132
#52102
#49902
#49892
#49572
#49562
Dic3×C22⋊C42
#49402
C4×(C2×D3)⋊C42
#23583