Finite Group Databasebeta

← #10830064

Subgroups — #10830064

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5568C211yesmaximal, characteristic
2C2×#4873C211yesmaximal, characteristic, complemented
3C2×#4798C211yesmaximal, characteristic
4#76076C211yesmaximal, characteristic, complemented
5C3×#1863C211yesmaximal, characteristic, complemented
6#71632C211yesmaximal, characteristic, complemented
7#70638C211yesmaximal, characteristic
8#70284C211yesmaximal, characteristic, complemented
9#70283C211yesmaximal, characteristic, complemented
10#70226C211yesmaximal, characteristic
11#69628C211yesmaximal, characteristic
12#68905C211yesmaximal, characteristic, complemented
13#68896C211yesmaximal, characteristic
14#68677C211yesmaximal, characteristic
15#68619C211yesmaximal, characteristic, complemented
16#24195—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22C22×#204C2211yescharacteristic
23#5568—24no—
24#5568—24no—
25C2×#898—21no—
26C2×#896—21no—
27C2×#893C2211yescharacteristic
28C2×#887C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×#876—21no—
31C2×#875C2211yescharacteristic
32C2×#872—21no—
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×#107C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4980—21no—
44#4979—21no—
45#4977C2211yescharacteristic
46#4973C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53#4929C2211yescharacteristic
54#4873—24no—
55#4798—24no—
56#4658—21no—
57#4652—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#464—31no—
66C2×C4×Q16—31no—
67#2581—31no—
68#2043—31no—
69#2042—31no—
70#2016—31no—
71#2015—31no—
72#1984—31no—
73#1958—31no—
74#1878—31no—
75#1877—31no—
76#1863D311yescharacteristic, complemented
77#1862—31no—
78#1805—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96#947—22no—
97#947D414yes—
98#944—22no—
99#944—22no—
100C2×#216—22no—
101C2×#216—22no—
102C2×#216—41no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C2×Dic12—22no—
124C2×Dic12—22no—
125C2×Dic12—22no—
126C2×#204—22no—
127C2×#204—22no—
128C2×#204—41no—
129#898—28no—
130#896—28no—
131#893—22no—
132#893—22no—
133#887—22no—
134#887—22no—
135#887—22no—
136#887D414yes—
137#876—28no—
138#875—22no—
139#875—22no—
140#872—28no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C3×#107—22no—
144C3×#103—22no—
145C3×#103—22no—
146C3×#95—21no—
147#835—41no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155#821—22no—
156#816—41no—
157Dic3⋊C8—22no—
158C8×Dic3—22no—
159#810—22no—
160#810—22no—
161#810—41no—
162#809—22no—
163#809—41no—
164C22×Q16—31no—
165C22×Q16—31no—
166C22×SD16—31no—
167#556—64no—
168#556—64no—
169C2×C4⋊Q8—31no—
170C2×#126—61no—
171C2×#126—61no—
172C2×#124—31no—
173C2×#124—31no—
174C2×C22⋊Q8—61no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4—31no—
180C2×#117—61no—
181#464—64no—
182C4×Q16—64no—
183C2×#107C2×D311yescharacteristic
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103C2×D311yescharacteristic
191C2×#103C2×D311yescharacteristic
192C2×#102—31no—
193C2×#102—31no—
194C2×#102—31no—
195C2×C4×C8—31no—
196#400—61no—
197#395—61no—
198#391—31no—
199#391—31no—
200#389—31no—
201#389C2×D311yescharacteristic
202#387—61no—
203#387—61no—
204#382—31no—
205#340—31no—
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C23×D3—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8C2×D412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12C2×D412yes—
238C2×C4×D3—24no—
239C2×C4×D3—42no—
240C2×C4×D3—42no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—22no—
252C2×Dic6—22no—
253C2×Dic6—41no—
254C2×Dic6C2×D412yes—
255C2×Dic6#4112yes—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8#4112yes—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—42no—
269C3×C4⋊C4C2×D412yes—
270C3×C42—24no—
271C3×C42—21no—
272C3×C42—21no—
273#216—44no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—24no—
279C4⋊Dic3—22no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282Dic3⋊C4—24no—
283Dic3⋊C4—24no—
284Dic3⋊C4—22no—
285Dic3⋊C4—22no—
286Dic3⋊C4—24no—
287Dic3⋊C4—42no—
288Dic3⋊C4—42no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—24no—
292C4×Dic3—42no—
293C4×Dic3C2×D412yes—
294C4×Dic3C2×D412yes—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301Dic12—28no—
302#204—44no—
303C22×Q8—31no—
304C22×Q8—31no—
305C22×Q8—31no—
306C22×Q8C22×D311yescharacteristic
307C22×D4—31no—
308C23×C4—61no—
309C2×Q16—62no—
310C2×Q16—62no—
311C2×Q16—62no—
312C2×Q16—62no—
313C2×Q16—62no—
314C2×Q16—121no—
315C2×SD16—62no—
316C2×SD16—62no—
317C2×SD16—121no—
318C22×C8—31no—
319C22×C8—31no—
320C22×C8C22×D311yescharacteristic
321C22×C8C22×D311yescharacteristic
322C4⋊Q8—62no—
323C4⋊Q8—62no—
324#126—68no—
325#126—68no—
326#124—34no—
327#124—62no—
328#124—62no—
329#124—62no—
330C22⋊Q8—68no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—34no—
334C4×Q8—62no—
335C4×Q8—62no—
336C4×Q8—62no—
337C4×Q8—62no—
338C4×Q8—62no—
339C4×D4—62no—
340C4×D4—62no—
341#117—68no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C22⋊C4—61no—
357C2×C22⋊C4—61no—
358C2×C22⋊C4—61no—
359C2×C22⋊C4—61no—
360C2×C22⋊C4—61no—
361C2×C42—21no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—61no—
365C2×C42—61no—
366C2×C42C22×D311yescharacteristic
367#107—22no—
368C4⋊C8—62no—
369#103—22no—
370#103—22no—
371#103—62no—
372#103—62no—
373#103—121no—
374#103—121no—
375#103—121no—
376#102—62no—
377#102—62no—
378#102—121no—
379C4×C8—62no—
380#95—21no—
381#95—61no—
382#95—61no—
383#95—61no—
384#95—61no—
385#95—61no—
386#95—61no—
387#95—61no—
388C23×C3C22×D411yescharacteristic
389C22×D3—22no—
390C22×D3—41no—
391C22×D3—41no—
392C22×D3—41no—
393C22×D3—41no—
394C22×D3—41no—
395C22×D3—41no—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—41no—
399C3×Q8—41no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4H(5,𝔽2)11yescharacteristic
423C2×C3×C42−1+411yescharacteristic
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3#12012yes—
443C2×Dic3#12012yes—
444C2×Dic3#13512yes—
445D12—22no—
446D12—22no—
447D12—41no—
448D12—41no—
449C4×D3—48no—
450Dic6—22no—
451Dic6—22no—
452Dic6—22no—
453Dic6—22no—
454Dic6—44no—
455Dic6—41no—
456Dic6—41no—
457Dic6—41no—
458Dic6—41no—
459C3×C8—22no—
460C3×C8—41no—
461C3⋊C8—41no—
462C3⋊C8—41no—
463C24—61no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—32no—
470C2×Q8—32no—
471C2×Q8—62no—
472C2×Q8—62no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—121no—
484C2×Q8D3×D412yes—
485C2×D4—32no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4—61no—
489C2×D4—61no—
490C2×D4—61no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—64no—
500C22×C4—61no—
501C22×C4—61no—
502C22×C4—61no—
503C22×C4—122no—
504C22×C4—122no—
505C22×C4D3×D411yescharacteristic
506C22×C4D3×D411yescharacteristic
507C22×C4C23×D311yescharacteristic
508Q16—68no—
509Q16—124no—
510SD16—124no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8#23512yes—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—64no—
536C4⋊C4—61no—
537C4⋊C4—61no—
538C4⋊C4—122no—
539C4⋊C4—122no—
540C4⋊C4D3×D412yes—
541C22⋊C4—64no—
542C22⋊C4—64no—
543C22⋊C4—64no—
544C22⋊C4—64no—
545C22⋊C4—64no—
546C42—24no—
547C42—21no—
548C42—21no—
549C42—32no—
550C42—32no—
551C42—64no—
552C42—61no—
553C42—61no—
554C42—122no—
555C22×C3C2×#12011yescharacteristic
556C22×C3#54811yescharacteristic
557C22×C3#55011yescharacteristic
558C22×C3C2×#13511yescharacteristic
559C22×C3#57511yescharacteristic
560C22×C3#57811yescharacteristic
561C22×C3#57811yescharacteristic
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—42no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C3×C4—42no—
570C3×C4—41no—
571C3×C4—42no—
572C3×C4—41no—
573C3×C4—81no—
574C3×C4—81no—
575C3×C4#52511yescharacteristic
576C3×C4#53711yescharacteristic
577C3×C4#54611yescharacteristic
578C3×C4#54711yescharacteristic
579Dic3—22no—
580Dic3—22no—
581Dic3—22no—
582Dic3—42no—
583Dic3—42no—
584Dic3—41no—
585Dic3—81no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23C2×D3×D411yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—41no—
597Q8—41no—
598Q8—62no—
599Q8—62no—
600Q8—62no—
601Q8—62no—
602Q8—124no—
603Q8—121no—
604Q8—121no—
605Q8—121no—
606Q8—121no—
607D4—62no—
608D4—62no—
609D4—121no—
610D4—121no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—32no—
620C2×C4—32no—
621C2×C4—32no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—62no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—62no—
643C2×C4—128no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4C2×#23511yescharacteristic
652C2×C4C2×D3×D411yescharacteristic
653C2×C4C2×D3×D411yescharacteristic
654C2×C4#100711yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4#100911yescharacteristic
657C8—22no—
658C8—41no—
659C8—121no—
660C8—121no—
661C2×C3#240711yescharacteristic
662C2×C3#300211yescharacteristic
663C2×C3#301111yescharacteristic
664C2×C3#336411yescharacteristic
665C2×C3#337711yescharacteristic
666C2×C3#339311yescharacteristic
667C2×C3#340211yescharacteristic
668D3—82no—
669C22—122no—
670C22—122no—
671C22—122no—
672C22—241no—
673C22—241no—
674C22—241no—
675C22—241no—
676C22#555311yescharacteristic
677C22D3×#12011yescharacteristic
678C22#563311yescharacteristic
679C22#576611yescharacteristic
680C22D3×#13511yescharacteristic
681C22#577311yescharacteristic
682C22#577911yescharacteristic
683C4—41no—
684C4—42no—
685C4—41no—
686C4—42no—
687C4—62no—
688C4—62no—
689C4—62no—
690C4—81no—
691C4—81no—
692C4—121no—
693C4—122no—
694C4—122no—
695C4—241no—
696C4#550711yescharacteristic
697C4#554711yescharacteristic
698C4#562911yescharacteristic
699C4#565411yescharacteristic
700C3#2419511yescharacteristic, complemented, Sylow
701C2—242no—
702C2#7440411yescharacteristic
703C2#7971011yescharacteristic
704C2#7976511yescharacteristic
705C2#8008211yescharacteristic
706C2#8018511yescharacteristic
707C2#8110111yescharacteristic
708C2#8121611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1863D3
hypocenterC3#24195
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
2. C2×#4873C2semidirect
4. #76076C2semidirect
5. C3×#1863C2semidirect
6. #71632C2semidirect
8. #70284C2semidirect
9. #70283C2semidirect
12. #68905C2semidirect
15. #68619C2semidirect
76. #1863D3semidirect
700. C3#24195semidirect

Maximal subgroups

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

subgroupindex
C2×#55682
C2×#48732
C2×#47982
#760762
C3×#18632
#716322
#706382
#702842
#702832
#702262
#696282
#689052
#688962
#686772
#686192
#241953