Finite Group Databasebeta

← #10848760

Subgroups — #10848760

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5479C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic, complemented
3C2×#5086C211yesmaximal, characteristic, complemented
4C2×#5082C211yesmaximal, characteristic, complemented
5#71700C212yesmaximal, complemented
6#71699C212yesmaximal
7#71529C211yesmaximal, characteristic, complemented
8#69690C212yesmaximal, complemented
9#69687C212yesmaximal, complemented
10#69583C211yesmaximal, characteristic, complemented
11#69581C211yesmaximal, characteristic, complemented
12#25604—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×#216C2212yes—
17C22×#215C2212yes—
18C2×#930C2211yescharacteristic
19C2×#929—21no—
20C2×C4×#67—21no—
21C2×#924C2211yescharacteristic
22#5479—24no—
23#5479—24no—
24C2×#876—21no—
25C2×#875C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#872—21no—
28C2×#870C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#391C2212yes—
31C3×#389—21no—
32C3×#389—21no—
33C3×#384C2211yescharacteristic
34C3×#380—21no—
35C3×#380—21no—
36C2×#835C2212yes—
37C2×#835C2212yes—
38C2×#830C2212yes—
39#5086—24no—
40#5082—24no—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#529—31no—
49C2×#497—31no—
50C2×#491—31no—
51#2857D311yescharacteristic, complemented
52#2001—32no—
53#2000—32no—
54#1992—31no—
55#1959—31no—
56#1878—32no—
57#1877—32no—
58#1749—31no—
59C22×C3×Q8C2312yes—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×Dic6C2311yescharacteristic
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C22⋊Q8—42no—
70C3×C22⋊Q8—42no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—42no—
88C2×#215—24no—
89C2×#215—24no—
90C2×#215—42no—
91#930—22no—
92#930—22no—
93#929—28no—
94C4×#67—28no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101#924—22no—
102#924—22no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2312yes—
110#876—28no—
111#875—22no—
112#875D414yes—
113C4×D12—22no—
114C4×D12—22no—
115#872—28no—
116#870—22no—
117#870D414yes—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×#95—22no—
121C3×#95—22no—
122#835—24no—
123#835—24no—
124#830—24no—
125#810—42no—
126#809—42no—
127#803—22no—
128#803—22no—
129C22×Q16—32no—
130C22×SD16—32no—
131C2×C4⋊Q8C2×D311yescharacteristic
132#529—64no—
133#529—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#124—31no—
137C2×#123—61no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8C2×D312yes—
140C2×#121—31no—
141C2×C4×Q8—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—61no—
145C22×C42C2×D311yescharacteristic
146#497—64no—
147#491—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×#97—31no—
155#391C2×D312yes—
156#389—21no—
157#389—21no—
158#384C2×D311yescharacteristic
159#380—21no—
160#380—21no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—41no—
183C22×C3×C4—41no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×#67—24no—
191C2×#67—42no—
192C2×#67—42no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12C2×D412yes—
200C2×C4×D3—24no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6C2×D412yes—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4—44no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4—42no—
214C3×C22⋊C4—44no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42C2×D412yes—
220C3×C42C2×D412yes—
221#217—24no—
222#216—48no—
223#215—48no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3C2×D412yes—
231C4⋊Dic3C2×D412yes—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234Dic3⋊C4—24no—
235Dic3⋊C4—24no—
236C4×Dic3—24no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8#4114yes—
243C22×Q8—31no—
244C22×Q8C22×D312yes—
245C22×D4—31no—
246C22×D4—61no—
247C23×C4—61no—
248C23×C4#6711yescharacteristic
249C23×C4#6711yescharacteristic
250C23×C4C22×D311yescharacteristic
251C2×Q16—64no—
252C2×Q16—64no—
253C2×Q16—122no—
254C2×SD16—64no—
255C2×SD16—64no—
256C2×SD16—122no—
257C22×C8—32no—
258C22×C8—32no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#126—68no—
262#125—34no—
263#125—62no—
264#124—34no—
265#124—62no—
266#123—68no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271#121—62no—
272#121—62no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—68no—
278#117—68no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D312yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—61no—
300C2×C42#6711yescharacteristic
301C2×C42#6711yescharacteristic
302C2×C42C22×D311yescharacteristic
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—122no—
307#102—122no—
308C22⋊C8—64no—
309#97—62no—
310#97—62no—
311#95—22no—
312#95—22no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3#13712yes—
320C23×C3C22×D411yescharacteristic
321C22×D3—22no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4#12012yes—
355C2×C3×C4#12012yes—
356C2×C3×C4#13712yes—
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4112yes—
360C2×C3×C4H(5,𝔽2)12yes—
361#67—48no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371D12—24no—
372D12—42no—
373C4×D3—48no—
374Dic6—24no—
375Dic6—42no—
376C3⋊C8—24no—
377C3⋊C8—42no—
378C24—61no—
379C24C2×#6711yescharacteristic
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×D4—32no—
391C2×D4—64no—
392C2×D4—61no—
393C2×D4—62no—
394C2×D4—62no—
395C2×D4—122no—
396C2×D4—122no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—64no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421Q16—128no—
422SD16—128no—
423C2×C8—34no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—32no—
432C4⋊C4—32no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—64no—
439C4⋊C4—64no—
440C4⋊C4—64no—
441C4⋊C4—64no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C22⋊C4—44no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C22⋊C4—64no—
450C42—22no—
451C42—22no—
452C42—21no—
453C42—21no—
454C42—64no—
455C42D3×D412yes—
456C42D3×D412yes—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#53211yescharacteristic
466C22×C3#53511yescharacteristic
467C22×C3C2×#13711yescharacteristic
468C22×C3C2×#13711yescharacteristic
469C22×C3#57511yescharacteristic
470C22×C3#57511yescharacteristic
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—82no—
484C3×C4—82no—
485C3×C4#54612yes—
486C3×C4#55012yes—
487Dic3—42no—
488Dic3—42no—
489Dic3—42no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—62no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23#94212yes—
504C23C22×#6711yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—64no—
510Q8—122no—
511D4—64no—
512D4—128no—
513D4—122no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—128no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4#93712yes—
547C2×C4#93712yes—
548C2×C4#94212yes—
549C2×C4C2×D3×D411yescharacteristic
550C2×C4C2×D3×D411yescharacteristic
551C2×C4#100412yes—
552C2×C4D3×#4112yes—
553C8—64no—
554C8—122no—
555C2×C3—42no—
556C2×C3#243111yescharacteristic
557C2×C3#300711yescharacteristic
558C2×C3#300711yescharacteristic
559C2×C3#317211yescharacteristic
560C2×C3#318111yescharacteristic
561C2×C3#321811yescharacteristic
562C2×C3#321911yescharacteristic
563D3—82no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22—241no—
578C22C2×#93711yescharacteristic
579C22#580511yescharacteristic
580C22C2×#94211yescharacteristic
581C22C2×#94211yescharacteristic
582C22#582211yescharacteristic
583C22#582211yescharacteristic
584C22#583211yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—82no—
591C4—82no—
592C4—122no—
593C4—122no—
594C4—122no—
595C4#559912yes—
596C4#561212yes—
597C3#2560411yescharacteristic, complemented, Sylow
598C2—42no—
599C2—242no—
600C2#8265211yescharacteristic
601C2#8273511yescharacteristic
602C2#8273511yescharacteristic
603C2#8321411yescharacteristic
604C2#8330911yescharacteristic
605C2#8334511yescharacteristic
606C2#8334611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2857D3
hypocenterC3#25604
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C3×#2857C2semidirect
3. C2×#5086C2semidirect
4. C2×#5082C2semidirect
5. #71700C2semidirect
7. #71529C2semidirect
8. #69690C2semidirect
9. #69687C2semidirect
10. #69583C2semidirect
11. #69581C2semidirect
51. #2857D3semidirect
597. C3#25604semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C3×#28572
C2×#50862
C2×#50822
#717002
#716992
#715292
#696902
#696872
#695832
#695812
#256043