Finite Group Databasebeta

← #10876321

Subgroups — #10876321

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2280C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4#71576C212yesmaximal, complemented
5#71575C212yesmaximal, complemented
6#69520C212yesmaximal, complemented
7#69519C212yesmaximal, complemented
8#69505C211yesmaximal, characteristic, complemented
9#69504C211yesmaximal, characteristic, complemented
10#69501C211yesmaximal, characteristic, complemented
11#69501C211yesmaximal, characteristic, complemented
12#27602—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#217—21no—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C3×#381—21no—
20C3×#380C2211yescharacteristic
21C3×#380C2211yescharacteristic
22C3×#380C2211yescharacteristic
23C3×#380C2211yescharacteristic
24C3×#379—21no—
25C3×C4×C22⋊C4—21no—
26C3×C4×C22⋊C4—21no—
27#5212—21no—
28#5212—21no—
29#5212C2212yes—
30#5212C2212yes—
31C2×C3⋊#95—21no—
32C2×#830C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35C2×#830C2212yes—
36C2×#808C2212yescomplemented
37C2×#807C2212yescomplemented
38#4529C2212yescomplemented
39#4529C2212yescomplemented
40#4529C2212yescomplemented
41#4529C2212yescomplemented
42#4529C2212yescomplemented
43#4529C2212yescomplemented
44#2280D311yescharacteristic
45C2×#380—31no—
46#2077—32no—
47#2076—32no—
48#1854—32no—
49#1853—31no—
50#1852—31no—
51#1825—32no—
52#1823—31no—
53#1823—31no—
54#1779—31no—
55C25×C3D411yescharacteristic
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—24no—
59C23×C3×C4—42no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C23×Dic3—21no—
68C3×#117—24no—
69C3×#117—42no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—21no—
81C2×C3×C42—21no—
82C2×#217—22no—
83C2×#217—22no—
84C2×#217—44no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—42no—
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2312yes—
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2312yes—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#830—22no—
101#830—22no—
102#830—24no—
103#830—42no—
104#808—42no—
105#807—42no—
106C24×C4C2×D311yescharacteristic
107C2×#117C2×D311yescharacteristic
108C2×#117C2×D311yescharacteristic
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C22⋊C4—61no—
112C2×#107—32no—
113C2×#106—32no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—32no—
117C2×C22⋊C8—32no—
118#381—21no—
119#380—32no—
120#380—32no—
121#380—61no—
122#380—61no—
123#380C2×D311yescharacteristic
124#380C2×D311yescharacteristic
125#380C2×D311yescharacteristic
126#380C2×D311yescharacteristic
127#379—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C2×#95—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340—32no—
136#340—32no—
137C24×C3—22no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—41no—
143C24×C3—41no—
144C24×C3C2×D411yescharacteristic
145C24×C3C2×D411yescharacteristic
146C24×C3C2×D411yescharacteristic
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—48no—
170C22×C3×C4—48no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4114yes—
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—42no—
181C22×Dic3—42no—
182C22×Dic3—42no—
183C22×Dic3—42no—
184C22×Dic3—42no—
185C22×Dic3—42no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C22⋊C4—42no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C42—42no—
197C3×C42—42no—
198#217—44no—
199#217—44no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—42no—
205C4⋊Dic3—42no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C25#6711yescharacteristic
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—61no—
218C23×C4—61no—
219C23×C4#6711yescharacteristic
220C23×C4#6711yescharacteristic
221C23×C4#6711yescharacteristic
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C22×C8—32no—
226C22×C8—32no—
227#117—24no—
228#117—42no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—122no—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—124no—
249C2×C42—21no—
250C2×C42—21no—
251#107—122no—
252#106—122no—
253C22⋊C8—62no—
254C22⋊C8—62no—
255C22⋊C8—64no—
256C22⋊C8—122no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—62no—
262#95—62no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—28no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—44no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3—44no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3#12312yes—
287C23×C3#12312yes—
288C23×C3#13512yes—
289C23×C3AGL(1,ℤ/8)12yes—
290C23×C3#13712yes—
291C23×C3C22×D411yescharacteristic
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—44no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4#12312yes—
321C2×C3×C4#12312yes—
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4112yes—
325C2×C3×C4H(5,𝔽2)12yes—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—84no—
333C2×Dic3—84no—
334C3⋊C8—42no—
335C3⋊C8—42no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—41no—
342C24—41no—
343C24C2×#6711yescharacteristic
344C24C2×#6711yescharacteristic
345C24C2×#6711yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4—122no—
375C22×C4—122no—
376C22×C4—122no—
377C22×C4—122no—
378C22×C4#23512yes—
379C22×C4#23512yes—
380C22×C4#23714yes—
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C23×D311yescharacteristic
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—122no—
400C4⋊C4—122no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C42—42no—
410C42—42no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—416no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3#48214yes—
433C22×C3C2×#12311yescharacteristic
434C22×C3C2×#12311yescharacteristic
435C22×C3#53411yescharacteristic
436C22×C3C2×#13511yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3C2×#13711yescharacteristic
439C22×C3#57511yescharacteristic
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—41no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—81no—
446C3×C4—81no—
447C3×C4#55112yes—
448C3×C4#55112yes—
449Dic3—82no—
450Dic3—82no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—28no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23#92912yes—
475C23#93512yes—
476C23#94912yes—
477C23#95012yes—
478C23#95112yes—
479C23C22×#6711yescharacteristic
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—244no—
515C2×C4—244no—
516C2×C4#92912yes—
517C2×C4#93512yes—
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×#23511yescharacteristic
520C2×C4C2×#23712yes—
521C2×C4#100412yes—
522C8—122no—
523C8—122no—
524C2×C3—24no—
525C2×C3—42no—
526C2×C3—42no—
527C2×C3#248311yescharacteristic
528C2×C3C2×#48211yescharacteristic
529C2×C3#309111yescharacteristic
530C2×C3#309211yescharacteristic
531C2×C3#309511yescharacteristic
532C2×C3#319011yescharacteristic
533C2×C3#319411yescharacteristic
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—416no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22#500314yes—
556C22C2×#92911yescharacteristic
557C22C2×#93511yescharacteristic
558C22#580411yescharacteristic
559C22C2×#94911yescharacteristic
560C22C2×#95011yescharacteristic
561C22#582211yescharacteristic
562C22C2×#95111yescharacteristic
563C4—22no—
564C4—22no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—242no—
571C4—242no—
572C4#549512yes—
573C4#549512yes—
574C3#2760211yescharacteristic, complemented, Sylow
575C2—24no—
576C2—42no—
577C2—42no—
578C2#8211411yescharacteristic
579C2C2×#500311yescharacteristic
580C2#8222011yescharacteristic
581C2#8291311yescharacteristic
582C2#8291411yescharacteristic
583C2#8293111yescharacteristic
584C2#8293411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2280C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#2280D3
hypocenterC3#27602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
4. #71576C2semidirect
5. #71575C2semidirect
6. #69520C2semidirect
7. #69519C2semidirect
8. #69505C2semidirect
9. #69504C2semidirect
10. #69501C2semidirect
11. #69501C2semidirect
36. C2×#808C22semidirect
37. C2×#807C22semidirect
38. #4529C22semidirect
39. #4529C22semidirect
40. #4529C22semidirect
41. #4529C22semidirect
42. #4529C22semidirect
43. #4529C22semidirect
574. C3#27602semidirect

Maximal subgroups

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

subgroupindex
C3×#22802
C2×#52122
#721262
#715762
#715752
#695202
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