Finite Group Databasebeta

← #11235714

Subgroups — #11235714

Order 768.

3346 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#379C211yesmaximal, characteristic, complemented
2#84128C212yesmaximal, complemented
3#82125C213yesmaximal, complemented
4#82116C213yesmaximal, complemented
5#82111C216yesmaximal, complemented
6#73860C211yesmaximal, characteristic, complemented
7#73859C216yesmaximal, complemented
8#73851C212yesmaximal, complemented
9#73844C211yesmaximal, characteristic, complemented
10#73843C213yesmaximal, complemented
11#73837C213yesmaximal, complemented
12#45458—31nomaximal, Sylow
13C26×C3C2211yescharacteristic
14C22×#235C2211yescharacteristic, complemented
15C22×C3×C22⋊C4C2213yes—
16C22×C3×C22⋊C4C2213yes—
17C2×#953C2212yes—
18C2×#953C2216yes—
19C2×#953C2216yes—
20C2×#924C2213yescomplemented
21C2×#885C2216yescomplemented
22C3×#379C2213yes—
23C3×#379C2211yescharacteristic
24C3×#379C2211yescharacteristic
25C3×#379C2213yes—
26#5247C2216yes—
27#5247C2212yes—
28#5247C2216yes—
29#5213C2216yescomplemented
30#5211C2213yescomplemented
31C4×#217C2213yescomplemented
32#4961C2216yescomplemented
33#4960C2213yescomplemented
34#4960C2213yescomplemented
35#4959C2216yescomplemented
36#4957C2216yescomplemented
37#4956C2216yescomplemented
38#4956C2216yescomplemented
39#4953C2213yescomplemented
40#4953C2213yescomplemented
41#4952C2216yescomplemented
42#4951C2216yescomplemented
43#4947C2213yescomplemented
44#4947C2213yescomplemented
45#4947C2213yescomplemented
46#4947C2216yescomplemented
47#4947C2213yescomplemented
48#4945C2216yescomplemented
49#4945C2216yescomplemented
50#4944C2216yescomplemented
51#2846—32no—
52#2840—31no—
53#2812—31no—
54#2736—33no—
55#2684—36no—
56#2670—33no—
57#2561—33no—
58#2472—36no—
59#2404—33no—
60#2375—32no—
61C2×#379D311yescharacteristic, complemented
62C25×C3—24no—
63C25×C3—46no—
64C25×C3—46no—
65C25×C3C2313yes—
66C25×C3C2313yes—
67C25×C3C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2313yes—
70C22×#67C2312yes—
71C22×#67C2316yes—
72C22×#67C2316yes—
73C2×#235—46no—
74C22×D12C2311yescharacteristic, complemented
75C22×C4×D3C2312yescomplemented
76C22×Dic6C2311yescharacteristic, complemented
77C22×Dic6C2313yescomplemented
78C2×C3×C22⋊C4—46no—
79C2×C3×C22⋊C4—46no—
80C2×C3×C22⋊C4C2316yes—
81C2×C3×C22⋊C4C2313yes—
82C2×C3×C22⋊C4C2313yes—
83C2×C3×C22⋊C4C2313yes—
84C2×C3×C22⋊C4C2313yes—
85C2×C3×C22⋊C4C2313yes—
86C2×C3×C22⋊C4C2313yes—
87#953—28no—
88#953—412no—
89#953—412no—
90C2×#217C2316yes—
91C2×#217C2316yes—
92C2×#217C2316yes—
93C2×#217C2316yes—
94C2×#217C2316yes—
95C2×#217C2316yes—
96C2×#217C2316yes—
97C2×(C2×D3)⋊C4C2316yescomplemented
98C2×(C2×D3)⋊C4C2316yescomplemented
99C2×C4⋊Dic3C2313yescomplemented
100#924—46no—
101C2×Dic3⋊C4C2316yescomplemented
102C2×C4×Dic3C2316yescomplemented
103#885—412no—
104C3⋊#95C2316yescomplemented
105C3⋊#95C2313yescomplemented
106C3⋊#95C2313yescomplemented
107C3⋊#95C2316yescomplemented
108C3⋊#95C2316yescomplemented
109C3⋊#95C2313yescomplemented
110C3⋊#95C2313yescomplemented
111C26C2×D311yescharacteristic
112C22×#41—31no—
113C2×#124—36no—
114C2×C22⋊Q8—33no—
115C2×#120—32no—
116C2×#120—36no—
117C2×#120—36no—
118C22×C22⋊C4C2×D313yes—
119C22×C22⋊C4C2×D313yes—
120#397—33no—
121#397—33no—
122#397—36no—
123#394—36no—
124#394—36no—
125#393—33no—
126#393—33no—
127#393—33no—
128#393—36no—
129#393—33no—
130#392—33no—
131#392—33no—
132#390—36no—
133#388—36no—
134#388—36no—
135#388—36no—
136#388—36no—
137#386—36no—
138#386—36no—
139#386—36no—
140#381—33no—
141#379—36no—
142#379—32no—
143#379—36no—
144#379C2×D311yescharacteristic
145#379C2×D311yescharacteristic
146#379C2×D313yes—
147#379C2×D313yes—
148C4×C22⋊C4—33no—
149C24×C3—26no—
150C24×C3—26no—
151C24×C3—26no—
152C24×C3—26no—
153C24×C3—212no—
154C24×C3—212no—
155C24×C3—46no—
156C24×C3—43no—
157C24×C3—43no—
158C24×C3—46no—
159C24×C3—43no—
160C24×C3—43no—
161C24×C3—448no—
162C24×C3—448no—
163C24×C3—416no—
164C24×C3#4114yescomplemented
165C24×C3C2413yes—
166C24×C3C2413yes—
167C24×C3C2411yescharacteristic
168C23×D3C2412yes—
169C22×C3×C4—26no—
170C22×C3×C4—26no—
171C22×C3×C4—46no—
172C22×C3×C4C2411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×C3×C4C2413yes—
175C22×C3×C4C2413yes—
176C2×#67—212no—
177C2×#67—212no—
178C2×#67—212no—
179C2×#67—46no—
180C2×#67—46no—
181C2×#67—46no—
182C2×#67—46no—
183C22×Dic3C2412yes—
184C22×Dic3C2416yes—
185C22×Dic3C2416yes—
186#235—416no—
187C2×D12—43no—
188C2×C4×D3—46no—
189C2×Dic6—43no—
190C2×Dic6—46no—
191C2×Dic6—43no—
192C3×C22⋊C4—212no—
193C3×C22⋊C4—43no—
194C3×C22⋊C4—43no—
195C3×C22⋊C4—43no—
196C3×C22⋊C4—43no—
197C3×C22⋊C4—43no—
198C3×C22⋊C4—43no—
199#217—212no—
200#217—212no—
201#217—212no—
202#217—46no—
203#217—46no—
204#217—46no—
205#217—46no—
206(C2×D3)⋊C4—46no—
207(C2×D3)⋊C4—46no—
208C4⋊Dic3—43no—
209Dic3⋊C4—46no—
210C4×Dic3—46no—
211C25—24no—
212C25—46no—
213C25—46no—
214C25C22×D313yes—
215C25C22×D313yes—
216C25C22×D311yescharacteristic
217C2×#41—126no—
218C22×Q8—33no—
219C22×Q8—31no—
220C22×D4—32no—
221C22×D4—36no—
222C22×D4—36no—
223C22×D4—31no—
224C23×C4—32no—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D313yes—
227#124—1212no—
228C22⋊Q8—126no—
229#120—68no—
230#120—1212no—
231#120—1212no—
232C2×C4⋊C4—36no—
233C2×C4⋊C4—33no—
234C2×C22⋊C4—36no—
235C2×C22⋊C4—36no—
236C2×C22⋊C4—36no—
237C2×C22⋊C4—36no—
238C2×C22⋊C4—36no—
239C2×C22⋊C4—36no—
240C2×C22⋊C4—36no—
241C2×C22⋊C4—36no—
242C2×C22⋊C4—36no—
243C2×C22⋊C4—46no—
244C2×C22⋊C4—46no—
245C2×C22⋊C4C22×D316yes—
246C2×C22⋊C4C22×D313yes—
247C2×C22⋊C4C22×D313yes—
248C2×C22⋊C4C22×D313yes—
249C2×C22⋊C4C22×D313yes—
250C2×C22⋊C4C22×D313yes—
251C2×C22⋊C4C22×D313yes—
252C2×C42—36no—
253#95—33no—
254#95—33no—
255#95—36no—
256#95—33no—
257#95—33no—
258#95—36no—
259#95—36no—
260C23×C3—21no—
261C23×C3—23no—
262C23×C3—224no—
263C23×C3—23no—
264C23×C3—26no—
265C23×C3—26no—
266C23×C3—26no—
267C23×C3—26no—
268C23×C3—23no—
269C23×C3—23no—
270C23×C3—23no—
271C23×C3—412no—
272C23×C3—424no—
273C23×C3—424no—
274C23×C3—424no—
275C23×C3—412no—
276C23×C3—412no—
277C23×C3—412no—
278C23×C3—424no—
279C23×C3—4128no—
280C23×C3—44no—
281C23×C3—412no—
282C23×C3—412no—
283C23×C3—412no—
284C23×C3C2×#4116yes—
285C23×C3H(5,𝔽2)16yes—
286C23×C3H(5,𝔽2)16yes—
287C23×C3C2511yescharacteristic
288C22×D3—22no—
289C22×D3—26no—
290C22×D3—26no—
291C2×C3×C4—26no—
292C2×C3×C4—23no—
293C2×C3×C4—23no—
294C2×C3×C4—26no—
295C2×C3×C4—23no—
296C2×C3×C4—23no—
297C2×C3×C4—44no—
298C2×C3×C4—86no—
299#67—48no—
300#67—812no—
301#67—812no—
302C2×Dic3—26no—
303C2×Dic3—26no—
304C2×Dic3—26no—
305C2×Dic3—26no—
306C2×Dic3—26no—
307C2×Dic3—26no—
308C2×Dic3—26no—
309D12—44no—
310C4×D3—48no—
311Dic6—44no—
312Dic6—86no—
313C24—26no—
314C24—26no—
315C24—26no—
316C24—26no—
317C24—212no—
318C24—212no—
319C24—32no—
320C24—46no—
321C24—43no—
322C24—43no—
323C24—43no—
324C24—46no—
325C24—43no—
326C24—448no—
327C24—448no—
328C24—416no—
329C24#23514yescomplemented
330C24C23×D313yes—
331C24C23×D313yes—
332C24C23×D311yescharacteristic
333#41—1216no—
334C2×Q8—123no—
335C2×Q8—123no—
336C2×Q8—126no—
337C2×D4—612no—
338C2×D4—612no—
339C2×D4—612no—
340C2×D4—123no—
341C2×D4—126no—
342C2×D4—126no—
343C2×D4—126no—
344C2×D4—126no—
345C22×C4—26no—
346C22×C4—26no—
347C22×C4—32no—
348C22×C4—36no—
349C22×C4—36no—
350C22×C4—46no—
351C22×C4—126no—
352C22×C4C23×D311yescharacteristic
353C22×C4C23×D311yescharacteristic
354C22×C4C23×D313yes—
355C22×C4C23×D313yes—
356C4⋊C4—126no—
357C4⋊C4—123no—
358C22⋊C4—212no—
359C22⋊C4—43no—
360C22⋊C4—43no—
361C22⋊C4—43no—
362C22⋊C4—43no—
363C22⋊C4—43no—
364C22⋊C4—43no—
365C22⋊C4—612no—
366C22⋊C4—612no—
367C22⋊C4—612no—
368C22⋊C4—126no—
369C22⋊C4—126no—
370C22⋊C4—126no—
371C22⋊C4—126no—
372C22⋊C4—126no—
373C22⋊C4—126no—
374C42—126no—
375C22×C3—26no—
376C22×C3—26no—
377C22×C3—212no—
378C22×C3—26no—
379C22×C3—26no—
380C22×C3—212no—
381C22×C3—448no—
382C22×C3—416no—
383C22×C3—448no—
384C22×C3—46no—
385C22×C3—43no—
386C22×C3—43no—
387C22×C3—46no—
388C22×C3—43no—
389C22×C3—43no—
390C22×C3H(3,𝔽4)14yes—
391C22×C3C22×#4111yescharacteristic
392C22×C3C2×H(5,𝔽2)13yes—
393C22×C3C2×H(5,𝔽2)13yes—
394C2×D3—46no—
395C2×D3—42no—
396C2×D3—46no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—43no—
400C3×C4—43no—
401Dic3—46no—
402Dic3—42no—
403Dic3—46no—
404C23—21no—
405C23—23no—
406C23—224no—
407C23—23no—
408C23—23no—
409C23—23no—
410C23—23no—
411C23—26no—
412C23—26no—
413C23—26no—
414C23—26no—
415C23—412no—
416C23—424no—
417C23—424no—
418C23—412no—
419C23—424no—
420C23—412no—
421C23—424no—
422C23—412no—
423C23—4128no—
424C23—44no—
425C23—412no—
426C23—412no—
427C23—412no—
428C23—62no—
429C23—66no—
430C23—66no—
431C23C2×#23516yes—
432C23#100416yes—
433C23#100416yes—
434C23C24×D311yescharacteristic
435Q8—124no—
436Q8—246no—
437D4—124no—
438D4—128no—
439D4—2412no—
440D4—2412no—
441C2×C4—26no—
442C2×C4—26no—
443C2×C4—23no—
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—44no—
448C2×C4—66no—
449C2×C4—66no—
450C2×C4—66no—
451C2×C4—66no—
452C2×C4—66no—
453C2×C4—66no—
454C2×C4—66no—
455C2×C4—86no—
456C2×C4—128no—
457C2×C3—24no—
458C2×C3—46no—
459C2×C3—46no—
460C2×C3#349013yes—
461C2×C3C2×H(3,𝔽4)11yescharacteristic
462C2×C3#353213yes—
463D3—82no—
464C22—26no—
465C22—26no—
466C22—212no—
467C22—212no—
468C22—26no—
469C22—26no—
470C22—448no—
471C22—416no—
472C22—448no—
473C22—43no—
474C22—43no—
475C22—43no—
476C22—46no—
477C22—43no—
478C22—46no—
479C22—122no—
480C22—126no—
481C22—126no—
482C22#559414yes—
483C22C22×#23511yescharacteristic
484C22C2×#100413yes—
485C22C2×#100413yes—
486C4—41no—
487C4—41no—
488C4—43no—
489C4—43no—
490C4—122no—
491C4—126no—
492C4—126no—
493C3#4545811yescharacteristic, complemented, Sylow
494C2—24no—
495C2—46no—
496C2—46no—
497C2—242no—
498C2#8569713yes—
499C2C2×#559411yescharacteristic
500C2#8592113yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#379C2
hypercenterC2×#379D3
hypocenterC3#45458
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C2×C3×#379C2semidirect
2. #84128C2semidirect
3. #82125C2semidirect
4. #82116C2semidirect
5. #82111C2semidirect
6. #73860C2semidirect
7. #73859C2semidirect
8. #73851C2semidirect
9. #73844C2semidirect
10. #73843C2semidirect
11. #73837C2semidirect
14. C22×#235C22semidirect
20. C2×#924C22semidirect
21. C2×#885C22semidirect
29. #5213C22semidirect
30. #5211C22semidirect
31. C4×#217C22semidirect
32. #4961C22semidirect
33. #4960C22semidirect
34. #4960C22semidirect
35. #4959C22semidirect
36. #4957C22semidirect
37. #4956C22semidirect
38. #4956C22semidirect
39. #4953C22semidirect
40. #4953C22semidirect
41. #4952C22semidirect
42. #4951C22semidirect
43. #4947C22semidirect
44. #4947C22semidirect
45. #4947C22semidirect
46. #4947C22semidirect
47. #4947C22semidirect
48. #4945C22semidirect
49. #4945C22semidirect
50. #4944C22semidirect
61. C2×#379D3semidirect
74. C22×D12C23semidirect
75. C22×C4×D3C23semidirect
76. C22×Dic6C23semidirect
77. C22×Dic6C23semidirect
97. C2×(C2×D3)⋊C4C23semidirect
98. C2×(C2×D3)⋊C4C23semidirect
99. C2×C4⋊Dic3C23semidirect
101. C2×Dic3⋊C4C23semidirect
102. C2×C4×Dic3C23semidirect
104. C3⋊#95C23semidirect
105. C3⋊#95C23semidirect
106. C3⋊#95C23semidirect
107. C3⋊#95C23semidirect
108. C3⋊#95C23semidirect
109. C3⋊#95C23semidirect
110. C3⋊#95C23semidirect
164. C24×C3#41semidirect
329. C24#235semidirect
493. C3#45458semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3792
#841282
#821252
#821162
#821112
#738602
#738592
#738512
#738442
#738432
#738372
#454583