Finite Group Databasebeta

← #10949539

Subgroups — #10949539

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4916C211yesmaximal, characteristic, complemented
2C2×#4911C211yesmaximal, characteristic, complemented
3#81823C211yesmaximal, characteristic, complemented
4#79589C211yesmaximal, characteristic, complemented
5C3×#2210C211yesmaximal, characteristic, complemented
6#71477C211yesmaximal, characteristic, complemented
7#71462C211yesmaximal, characteristic, complemented
8#71324C211yesmaximal, characteristic, complemented
9#70889C211yesmaximal, characteristic, complemented
10#69292C211yesmaximal, characteristic, complemented
11#69285C211yesmaximal, characteristic, complemented
12#69237C211yesmaximal, characteristic, complemented
13#69221C211yesmaximal, characteristic, complemented
14#69204C211yesmaximal, characteristic, complemented
15D3×#357C211yesmaximal, characteristic, complemented
16#33589—31nomaximal, Sylow
17C2×#919C2211yescharacteristic
18C2×#914—21no—
19C2×#912C2211yescharacteristic
20C2×D3×D8C2211yescharacteristic
21#5674—21no—
22#5546—21no—
23C2×C3×SD32C2211yescharacteristic, complemented
24C2×C3×D16C2211yescharacteristic
25C3×#495C2211yescharacteristic
26C3×C4×D8C2211yescharacteristic, complemented
27#5191C2211yescharacteristic
28C2×#829C2211yescharacteristic, complemented
29#5176—21no—
30#5151C2211yescharacteristic
31Dic3×D8C2211yescharacteristic, complemented
32C2×#827C2211yescharacteristic
33C2×#826C2211yescharacteristic
34C2×#826C2211yescharacteristic, complemented
35#5058—21no—
36#5018—21no—
37#4916—24no—
38#4911—24no—
39C2×#800C2211yescharacteristic
40C2×#800C2211yescharacteristic, complemented
41C2×#798C2211yescharacteristic
42C2×D3×C16C2211yescharacteristic
43D3×#107C2211yescharacteristic
44#4873C2211yescharacteristic, complemented
45#4771—21no—
46#4757—21no—
47C4×Dic12C2211yescharacteristic, complemented
48#4695C2211yescharacteristic
49#4689C2211yescharacteristic
50C3×C4⋊C16C2211yescharacteristic, complemented
51C3×#358C2211yescharacteristic, complemented
52C3×#357C2211yescharacteristic
53#4564C2211yescharacteristic, complemented
54#4562C2211yescharacteristic
55#4509C2211yescharacteristic
56#4509C2211yescharacteristic, complemented
57Dic3⋊C16C2211yescharacteristic, complemented
58Dic3⋊C16C2211yescharacteristic, complemented
59#4491C2211yescharacteristic
60#4491C2211yescharacteristic, complemented
61C4⋊(C3⋊C16)C2211yescharacteristic, complemented
62C2×#509—31no—
63C2×#508—31no—
64#3155—31no—
65#2951—31no—
66#2211—31no—
67#2211—31no—
68#2210—31no—
69#2210D311yescharacteristic, complemented
70#2207—31no—
71#2207—31no—
72#2206—31no—
73#2206—31no—
74#2151—31no—
75#2139—31no—
76C2×#357—31no—
77C2×#239—21no—
78C2×#237—21no—
79C2×D3×D4—21no—
80C2×C3×Q16C2311yescharacteristic
81C2×C3×SD16—21no—
82C2×C3×D8C2311yescharacteristic
83C2×C3×D8C2311yescharacteristic
84C3×#124—21no—
85C3×C4×D4—21no—
86#947—21no—
87C2×#215—21no—
88C2×#215—21no—
89#940—21no—
90D4×Dic3—21no—
91C2×#214—21no—
92C2×#213—21no—
93C2×#213—21no—
94C4×#67—41no—
95C2×Dic3⋊C4—41no—
96#919—22no—
97#919—22no—
98#914—44no—
99#912—22no—
100#912D414yescomplemented
101D3×D8—22no—
102D3×D8—22no—
103C2×Dic12C2311yescharacteristic
104C2×D24—21no—
105C2×D24C2311yescharacteristic
106C2×D24C2311yescharacteristic
107C2×ΓL(1,𝔽25)D411yescharacteristic
108C2×D3×C8C2311yescharacteristic
109D3×C4⋊C4—21no—
110#887—21no—
111#884—41no—
112#884—41no—
113#882—41no—
114#882—41no—
115#67⋊C4—41no—
116#877—41no—
117#874—21no—
118#872D411yescharacteristic
119C4×Dic6—21no—
120C3×SD32—22no—
121C3×D16—22no—
122C2×C3×C16C2311yescharacteristic
123C2×C3×C16C2311yescharacteristic
124C3×#107C2311yescharacteristic
125C3×#102—21no—
126C3×C4×C8C2311yescharacteristic
127#832—21no—
128#829—22no—
129#827—22no—
130#826—22no—
131#826—22no—
132#818C2311yescharacteristic
133#816—21no—
134#815D411yescharacteristic
135C8×Dic3C2311yescharacteristic
136C2×C3⋊C16C2311yescharacteristic
137C2×C3⋊C16C2311yescharacteristic
138#810—21no—
139#807—21no—
140#803D411yescharacteristic
141#800—22no—
142#800—22no—
143#798—22no—
144D3×C16—22no—
145C2×AGL(1,ℤ/8)—61no—
146C2×#135—31no—
147C2×#135—31no—
148C22×D8—31no—
149#535—61no—
150#520—61no—
151#509—64no—
152#508—64no—
153C2×Q32—31no—
154C2×SD32—31no—
155C2×SD32—31no—
156C2×SD32—31no—
157C2×SD32C2×D311yescharacteristic, complemented
158C2×D16—31no—
159C2×D16—31no—
160C2×D16C2×D311yescharacteristic
161C2×M32—31no—
162C22×C16—31no—
163#496—61no—
164#496—61no—
165#495—31no—
166#495C2×D311yescharacteristic
167#493—31no—
168#493—31no—
169#440—61no—
170#440—61no—
171C4×Q16—31no—
172C4×Q16—31no—
173C4×D8—31no—
174C4×D8C2×D311yescharacteristic, complemented
175C2×#107—31no—
176#417—61no—
177#405—31no—
178C4⋊C16—31no—
179C4⋊C16—31no—
180C4⋊C16—31no—
181C4⋊C16C2×D311yescharacteristic, complemented
182#358—31no—
183#358C2×D311yescharacteristic, complemented
184#357—31no—
185#357—31no—
186#357—31no—
187#357—31no—
188#357—31no—
189#357C2×D311yescharacteristic
190C23×D3—41no—
191C2×C3×Q8—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C22×C3×C4—41no—
195C2×#67—41no—
196C2×#67—41no—
197C2×#67—41no—
198C2×#67—41no—
199C22×Dic3—41no—
200C22×Dic3—41no—
201#239—42no—
202#239—42no—
203#237—24no—
204#237—42no—
205D3×D4—42no—
206D3×D4—42no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—41no—
210C2×D12—41no—
211C2×D12—41no—
212C2×C4×D3—41no—
213C2×C4×D3—41no—
214C2×C4×D3C2×D411yescharacteristic
215C2×Dic6—21no—
216C3×Q16—21no—
217C3×Q16—21no—
218C3×SD16—42no—
219C3×D8—21no—
220C3×D8—21no—
221C3×D8—21no—
222C3×D8C2×D412yes—
223C2×C3×C8C2×D411yescharacteristic
224C2×C3×C8C2411yescharacteristic
225C3×C4⋊C4—21no—
226C3×C22⋊C4—41no—
227C3×C22⋊C4—41no—
228C3×C22⋊C4—41no—
229C3×C42C2×D411yescharacteristic
230#217—41no—
231#215—42no—
232#215—42no—
233#214—24no—
234#213—42no—
235#213—42no—
236(C2×D3)⋊C4—41no—
237(C2×D3)⋊C4—41no—
238(C2×D3)⋊C4D811yescharacteristic
239(C2×D3)⋊C4D811yescharacteristic
240C4⋊Dic3—21no—
241Dic3⋊C4—41no—
242Dic3⋊C4—41no—
243Dic3⋊C4—41no—
244Dic3⋊C4D811yescharacteristic
245Dic3⋊C4D811yescharacteristic
246C4×Dic3—41no—
247C4×Dic3—41no—
248C4×Dic3C2×D411yescharacteristic
249C2×C3⋊C8C2×D411yescharacteristic
250C2×C3⋊C8C2×D411yescharacteristic
251Dic12—21no—
252Dic12C2×D412yes—
253D24—21no—
254D24—21no—
255D24—21no—
256D24—21no—
257D24—42no—
258ΓL(1,𝔽25)—22no—
259D3×C8C2×D412yes—
260D3×C8#4112yes—
261C3×C16—21no—
262C3×C16—21no—
263C3⋊C16—21no—
264C3⋊C16—21no—
265C2×#41—61no—
266C2×#41—61no—
267C22×D4—61no—
268AGL(1,ℤ/8)—124no—
269#135—34no—
270#135—62no—
271#135—62no—
272#135—62no—
273C2×Q16—31no—
274C2×Q16C22×D311yescharacteristic
275C2×SD16—21no—
276C2×SD16—61no—
277C2×SD16—61no—
278C2×SD16—61no—
279C2×D8—31no—
280C2×D8—31no—
281C2×D8—62no—
282C2×D8—62no—
283C2×D8—61no—
284C2×D8—61no—
285C2×D8—61no—
286C2×D8C22×D311yescharacteristic
287C2×D8C22×D311yescharacteristic
288C2×M16—31no—
289C22×C8—31no—
290#127—61no—
291#127—61no—
292#124—21no—
293#124—61no—
294#121—121no—
295#121—121no—
296#120—121no—
297#120—121no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×D4—21no—
301C4×D4—61no—
302C4×D4—121no—
303C4×D4—121no—
304#117—31no—
305#117—121no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—121no—
308Q32—62no—
309SD32—22no—
310SD32—62no—
311SD32—62no—
312SD32—62no—
313D16—22no—
314D16—62no—
315D16—62no—
316M32—62no—
317C2×C16—31no—
318C2×C16—31no—
319C2×C16—62no—
320C2×C16C22×D311yescharacteristic
321C2×C16C22×D311yescharacteristic
322#107—31no—
323#107—61no—
324#107C22×D311yescharacteristic
325#103—61no—
326#103—61no—
327#102—21no—
328#102—61no—
329#97—31no—
330#97—31no—
331C4×C8—31no—
332C4×C8C22×D311yescharacteristic
333C23×C3—41no—
334C23×C3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—82no—
338C22×D3—161no—
339C22×D3—161no—
340C22×D3C2×D811yescharacteristic
341C3×Q8—41no—
342C3×Q8—41no—
343C3×D4—22no—
344C3×D4—41no—
345C3×D4—41no—
346C3×D4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—161no—
350C2×C3×C4C2×D811yescharacteristic
351C2×C3×C4C2×D811yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353#67—44no—
354#67—82no—
355#67—161no—
356#67—161no—
357C2×Dic3—41no—
358C2×Dic3—44no—
359C2×Dic3—41no—
360C2×Dic3—161no—
361C2×Dic3C2×D811yescharacteristic
362C2×Dic3C2×D811yescharacteristic
363C2×Dic3C2×D811yescharacteristic
364D12—41no—
365D12—41no—
366D12—41no—
367D12—41no—
368D12—82no—
369D12—161no—
370D12—161no—
371C4×D3—82no—
372C4×D3—161no—
373C4×D3#12112yes—
374C4×D3AGL(1,ℤ/8)12yes—
375Dic6—22no—
376Dic6—41no—
377C3×C8—21no—
378C3×C8C22×D411yescharacteristic
379C3×C8C2×#4111yescharacteristic
380C3⋊C8—21no—
381C3⋊C8#12112yes—
382C24—121no—
383#41—64no—
384#41—122no—
385#41—122no—
386#41—122no—
387C2×Q8—21no—
388C2×Q8—61no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—122no—
394C2×D4—122no—
395C2×D4—121no—
396C2×D4—121no—
397C2×D4—121no—
398C2×D4—121no—
399C2×D4—121no—
400C2×D4—121no—
401C2×D4—121no—
402C22×C4—31no—
403C22×C4—41no—
404C22×C4—121no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4—121no—
408Q16—21no—
409Q16—21no—
410Q16—32no—
411Q16—61no—
412SD16—42no—
413SD16—64no—
414SD16—122no—
415SD16—122no—
416D8—21no—
417D8—21no—
418D8—21no—
419D8—61no—
420D8—61no—
421D8—61no—
422D8—61no—
423D8—122no—
424D8—122no—
425D8—122no—
426D8D3×D412yes—
427M16—62no—
428C2×C8—31no—
429C2×C8—32no—
430C2×C8—32no—
431C2×C8—31no—
432C2×C8D3×D411yescharacteristic
433C2×C8C23×D311yescharacteristic
434C4⋊C4—21no—
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—61no—
438C4⋊C4—121no—
439C4⋊C4—121no—
440C4⋊C4—121no—
441C22⋊C4—31no—
442C22⋊C4—31no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—121no—
447C22⋊C4—121no—
448C22⋊C4—121no—
449C42—31no—
450C42—121no—
451C42—121no—
452C42D3×D411yescharacteristic
453C16—21no—
454C16—21no—
455C16—61no—
456C16—61no—
457C22×C3—42no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3C22×D811yescharacteristic
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—164no—
473C2×D3#45912yes—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4C2×#12111yescharacteristic
479C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
480Dic3—21no—
481Dic3—21no—
482Dic3—42no—
483Dic3—81no—
484Dic3#45912yes—
485C23—31no—
486C23—41no—
487C23—41no—
488C23—121no—
489C23—121no—
490C23—242no—
491C23—481no—
492C23—481no—
493Q8—41no—
494Q8—41no—
495Q8—62no—
496Q8—121no—
497D4—22no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—124no—
502D4—121no—
503D4—121no—
504D4—121no—
505D4—121no—
506D4—242no—
507D4—242no—
508D4—481no—
509D4—481no—
510D4—481no—
511D4—481no—
512C2×C4—31no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—31no—
516C2×C4—31no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—124no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—161no—
523C2×C4—242no—
524C2×C4—481no—
525C2×C4—481no—
526C2×C4D3×D811yescharacteristic
527C2×C4D3×D811yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C8—21no—
530C8—32no—
531C8—61no—
532C8C2×D3×D411yescharacteristic
533C8D3×#4111yescharacteristic
534C2×C3—82no—
535C2×C3—161no—
536C2×C3C2×#45911yescharacteristic
537C2×C3#341511yescharacteristic
538C2×C3#341611yescharacteristic
539D3—22no—
540D3—161no—
541D3—161no—
542C22—32no—
543C22—42no—
544C22—61no—
545C22—61no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—241no—
552C22—241no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22—484no—
558C22C2×D3×D811yescharacteristic
559C4—21no—
560C4—21no—
561C4—32no—
562C4—61no—
563C4—61no—
564C4—81no—
565C4—81no—
566C4—122no—
567C4—241no—
568C4D3×#12111yescharacteristic
569C4D3×AGL(1,ℤ/8)11yescharacteristic
570C3#3358911yescharacteristic, complemented, Sylow
571C2—62no—
572C2—82no—
573C2—161no—
574C2—481no—
575C2—481no—
576C2D3×#45911yescharacteristic
577C2#8203211yescharacteristic
578C2#8203611yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2210C2
socleC22×C3C22×D8
hypercenter#2210D3
hypocenterC3#33589
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#4916C2semidirect
2. C2×#4911C2semidirect
3. #81823C2semidirect
4. #79589C2semidirect
5. C3×#2210C2semidirect
6. #71477C2semidirect
7. #71462C2semidirect
8. #71324C2semidirect
9. #70889C2semidirect
10. #69292C2semidirect
11. #69285C2semidirect
12. #69237C2semidirect
13. #69221C2semidirect
14. #69204C2semidirect
15. D3×#357C2semidirect
23. C2×C3×SD32C22semidirect
26. C3×C4×D8C22semidirect
28. C2×#829C22semidirect
31. Dic3×D8C22semidirect
34. C2×#826C22semidirect
40. C2×#800C22semidirect
44. #4873C22semidirect
47. C4×Dic12C22semidirect
50. C3×C4⋊C16C22semidirect
51. C3×#358C22semidirect
53. #4564C22semidirect
56. #4509C22semidirect
57. Dic3⋊C16C22semidirect
58. Dic3⋊C16C22semidirect
60. #4491C22semidirect
61. C4⋊(C3⋊C16)C22semidirect
69. #2210D3semidirect
100. #912D4semidirect
157. C2×SD32C2×D3semidirect
174. C4×D8C2×D3semidirect
181. C4⋊C16C2×D3semidirect
183. #358C2×D3semidirect
570. C3#33589semidirect

Maximal subgroups

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

subgroupindex
C2×#49162
C2×#49112
#818232
#795892
C3×#22102
#714772
#714622
#713242
#708892
#692922
#692852
#692372
#692212
#692042
D3×#3572
#335893