Finite Group Databasebeta

← #77123

Subgroups — #77123

Order 384.

824 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic, complemented
2C2×#884C211yesmaximal, characteristic, complemented
3C2×#882C211yesmaximal, characteristic, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#4991C211yesmaximal, characteristic, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4961C211yesmaximal, characteristic, complemented
10#4961C211yesmaximal, characteristic, complemented
11#4960C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4952C211yesmaximal, characteristic, complemented
14#4675C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2617—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic, complemented
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic, complemented
31C2×C3×C22⋊C4C2211yescharacteristic, complemented
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic, complemented
35#937—24no—
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×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×C4⋊Dic3C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47#884—24no—
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×#121—31no—
57C2×#120—31no—
58C2×#120—31no—
59C22×C22⋊C4—31no—
60#397—31no—
61#397—31no—
62#394—31no—
63#394—31no—
64#394—31no—
65#394D311yescharacteristic, complemented
66#392—31no—
67#392—31no—
68#386—31no—
69#386—31no—
70#380—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3D414yescomplemented
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C2×#67—41no—
101C2×#67—41no—
102C2×#67—41no—
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—41no—
113C2×D12—41no—
114C2×C4×D3—22no—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3D414yescomplemented
120C3×C4⋊C4—41no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—41no—
125#217—22no—
126#217—22no—
127#217—41no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137C4⋊Dic3—41no—
138Dic3⋊C4—22no—
139C25—31no—
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4C2×D311yescharacteristic
147C23×C4—31no—
148C23×C4—31no—
149#121—64no—
150#120—64no—
151#120—64no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4C2×D311yescharacteristic, complemented
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—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—64no—
168C2×C22⋊C4C2×D311yescharacteristic
169C2×C22⋊C4C2×D311yescharacteristic
170C2×C22⋊C4C2×D311yescharacteristic, complemented
171C2×C22⋊C4C2×D311yescharacteristic, complemented
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95C2×D311yescharacteristic, complemented
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×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3C2×D412yes—
192C23×C3C2411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—28no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—44no—
212C22×D3—44no—
213C22×D3—44no—
214C22×D3—44no—
215C22×D3—44no—
216C22×D3—44no—
217C22×D3C2×D412yes—
218C22×D3C2×D412yes—
219C22×D3C2×D412yes—
220C22×D3#4112yes—
221C3×D4—44no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4C2×D412yes—
237#67—44no—
238#67—44no—
239#67—44no—
240#67—44no—
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—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3C2×D412yes—
256D12—44no—
257D12—44no—
258C4×D3—44no—
259C4×D3—44no—
260C24—31no—
261C24—31no—
262C24—34no—
263C24—31no—
264C24—62no—
265C24—62no—
266C24—121no—
267C24—121no—
268C24—121no—
269C24—121no—
270C24C22×D311yescharacteristic
271C24C22×D311yescharacteristic
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—121no—
293C22×C4—31no—
294C22×C4—34no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—62no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C22×C4C22×D311yescharacteristic
308C22×C4C22×D311yescharacteristic
309C4⋊C4—41no—
310C4⋊C4—62no—
311C4⋊C4—121no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—62no—
323C22⋊C4—62no—
324C22⋊C4—62no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3C22×D411yescharacteristic
343C22×C3C22×D411yescharacteristic
344C22×C3C2×#4111yescharacteristic
345C22×C3H(5,𝔽2)11yescharacteristic
346C22×C3H(5,𝔽2)11yescharacteristic
347C22×C3H(5,𝔽2)11yescharacteristic
348C22×C3H(5,𝔽2)11yescharacteristic
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—416no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—41no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3#12114yes—
371C3×C4—22no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—41no—
376Dic3—22no—
377Dic3—41no—
378Dic3—41no—
379Dic3—41no—
380Dic3—41no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—32no—
395C23—32no—
396C23—32no—
397C23—32no—
398C23—68no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—61no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—124no—
417C23—124no—
418C23—124no—
419C23—124no—
420C23—124no—
421C23—124no—
422C23D3×D412yes—
423C23C23×D311yescharacteristic
424D4—44no—
425D4—124no—
426D4—124no—
427D4—124no—
428D4—124no—
429D4—124no—
430D4—124no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—32no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—124no—
461C2×C4—124no—
462C2×C4D3×D412yes—
463C2×C3—42no—
464C2×C3—81no—
465C2×C3C2×#12111yescharacteristic
466C2×C3#53411yescharacteristic
467C2×C3#53411yescharacteristic
468C2×C3#53411yescharacteristic
469C2×C3#53511yescharacteristic
470C2×C3#53811yescharacteristic
471C2×C3#53811yescharacteristic
472D3—24no—
473D3—42no—
474D3—81no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—34no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—62no—
498C22—62no—
499C22—1216no—
500C22—121no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22—121no—
509C22—121no—
510C22—121no—
511C22C2×D3×D411yescharacteristic
512C22C2×D3×D411yescharacteristic
513C22#100411yescharacteristic
514C22#100411yescharacteristic
515C22D3×#4111yescharacteristic
516C22#100911yescharacteristic
517C22#100911yescharacteristic
518C4—22no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—62no—
524C4—121no—
525C4—121no—
526C4—121no—
527C4—121no—
528C3#261711yescharacteristic, complemented, Sylow
529C2—42no—
530C2—64no—
531C2—81no—
532C2—122no—
533C2—241no—
534C2#558911yescharacteristic
535C2D3×#12111yescharacteristic
536C2#560511yescharacteristic
537C2#560711yescharacteristic
538C2#565311yescharacteristic
539C2#565711yescharacteristic
540C2#567811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2617
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#937C2semidirect
2. C2×#884C2semidirect
3. C2×#882C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#394C2semidirect
6. #5223C2semidirect
7. #4991C2semidirect
8. #4987C2semidirect
9. #4961C2semidirect
10. #4961C2semidirect
11. #4960C2semidirect
12. #4956C2semidirect
13. #4952C2semidirect
14. #4675C2semidirect
15. #4665C2semidirect
22. C22×#67C22semidirect
27. C2×C3×C4⋊C4C22semidirect
30. C2×C3×C22⋊C4C22semidirect
31. C2×C3×C22⋊C4C22semidirect
34. C2×#217C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×C4⋊Dic3C22semidirect
46. C2×Dic3⋊C4C22semidirect
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
65. #394D3semidirect
79. C23×D3D4semidirect
119. C2×C4×D3D4semidirect
154. C2×C4⋊C4C2×D3semidirect
170. C2×C22⋊C4C2×D3semidirect
171. C2×C22⋊C4C2×D3semidirect
177. #95C2×D3semidirect
528. C3#2617semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8842
C2×#8822
C2×D3×C22⋊C42
C3×#3942
#52232
#49912
#49872
#49612
#49602
#49562
#49522
#46752
#46652
#26173