Finite Group Databasebeta

← #10830061

Subgroups — #10830061

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×#5584C211yesmaximal, characteristic
2C3×#2581C211yesmaximal, characteristic, complemented
3C2×#5028C211yesmaximal, characteristic
4C2×C4×#216C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6#69691C211yesmaximal, characteristic, complemented
7#69690C211yesmaximal, characteristic, complemented
8#69689C211yesmaximal, characteristic
9#69687C211yesmaximal, characteristic
10#69633C211yesmaximal, characteristic, complemented
11#69631C211yesmaximal, characteristic, complemented
12#69628C211yesmaximal, characteristic
13#69625C211yesmaximal, characteristic
14#69559C211yesmaximal, characteristic, complemented
15#69551C211yesmaximal, characteristic
16#24195—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#946—21no—
20C22×#216C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C22×#215C2211yescharacteristic
23#5584—24no—
24#5584—24no—
25C2×#898—21no—
26C2×#898—21no—
27C2×#892—21no—
28C2×#875C2211yescharacteristic
29C2×#875C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×#869C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#400—21no—
35C3×#395—21no—
36C3×#391C2211yescharacteristic
37C3×#391C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#387—21no—
41C3×#387—21no—
42C3×#382C2211yescharacteristic
43C2×#835C2211yescharacteristic
44C4×#216—24no—
45#5028—24no—
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#809C2211yescharacteristic
54C2×#809C2211yescharacteristic
55C2×#807C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#464—31no—
66C2×C4×Q16—31no—
67#2581D311yescharacteristic, complemented
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#1863—31no—
77#1862—31no—
78#1805—31no—
79C22×C3×Q8D411yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102#946—28no—
103C2×#216—22no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—22no—
108C2×#216—41no—
109C2×#215—22no—
110C2×#215—22no—
111C2×#215—41no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#898—28no—
129#898—28no—
130#892—28no—
131#875—22no—
132#875—22no—
133#875—22no—
134#875D414yes—
135C4×D12—22no—
136C4×D12—22no—
137#869—22no—
138#869—22no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C4×Dic6—22no—
142C4×Dic6D414yes—
143C3×#95—21no—
144C3×#95—21no—
145C3×#95—21no—
146C3×#95—21no—
147C3×#95—21no—
148C3×#95—21no—
149C3×#95—21no—
150C3×#95—21no—
151#835—22no—
152#810—22no—
153#810—22no—
154#810—22no—
155#810—41no—
156#810—41no—
157#810—41no—
158#809—22no—
159#809—22no—
160#809—41no—
161#807—22no—
162C4⋊(C3⋊C8)—22no—
163C4×C3⋊C8—22no—
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×Q8C2×D311yescharacteristic
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4—31no—
180C2×#117—61no—
181#464—64no—
182C4×Q16—64no—
183C2×#107—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103—31no—
192C2×#102—31no—
193C2×#102—31no—
194C2×#102—31no—
195C2×C4×C8—31no—
196#400—21no—
197#395—21no—
198#391C2×D311yescharacteristic
199#391C2×D311yescharacteristic
200#389C2×D311yescharacteristic
201#389C2×D311yescharacteristic
202#387—21no—
203#387—21no—
204#382C2×D311yescharacteristic
205#340—31no—
206#340—31no—
207#336—31no—
208#336—31no—
209#336—31no—
210#336—31no—
211C23×D3—21no—
212C2×C3×Q8—22no—
213C2×C3×Q8—22no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—41no—
220C2×C3×Q8#4112yes—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12C2×D412yes—
241C2×C4×D3—24no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6C2×D412yes—
255C2×Dic6C2×D412yes—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—42no—
266C3×C4⋊C4—42no—
267C3×C4⋊C4—42no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—42no—
273C3×C42C2×D412yes—
274C3×C42C2×D412yes—
275#216—28no—
276#216—44no—
277#215—44no—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—24no—
282(C2×D3)⋊C4—24no—
283C4⋊Dic3—24no—
284C4⋊Dic3—24no—
285C4⋊Dic3—21no—
286C4⋊Dic3C2×D412yes—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8#4112yes—
303C22×Q8—31no—
304C22×Q8—31no—
305C22×Q8#6711yescharacteristic
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×C8—31no—
321C22×C8—31no—
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—22no—
334C4×Q8—22no—
335C4×Q8—34no—
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—21no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4#6711yescharacteristic
351C2×C4⋊C4#6711yescharacteristic
352C2×C4⋊C4#6711yescharacteristic
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—61no—
363C2×C42—61no—
364C2×C42C22×D311yescharacteristic
365C2×C42C22×D311yescharacteristic
366C2×C42C22×D311yescharacteristic
367#107—62no—
368C4⋊C8—62no—
369#103—62no—
370#103—62no—
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—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
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—44no—
399C3×Q8—41no—
400C3×Q8—41no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4#12012yes—
429C2×C3×C4#12012yes—
430C2×C3×C4#13512yes—
431C2×C3×C4C22×D411yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C2×#4111yescharacteristic
434C2×C3×C4C2×#4111yescharacteristic
435C2×C3×C4H(5,𝔽2)11yescharacteristic
436C2×C3×C42−1+411yescharacteristic
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446D12—22no—
447D12—22no—
448D12—41no—
449D12—41no—
450C4×D3—48no—
451Dic6—22no—
452Dic6—22no—
453Dic6—22no—
454Dic6—22no—
455Dic6—41no—
456Dic6—41no—
457Dic6—41no—
458Dic6—41no—
459C3⋊C8—22no—
460C3⋊C8—41no—
461C3⋊C8—41no—
462C3⋊C8—41no—
463C24—61no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—32no—
472C2×Q8—32no—
473C2×Q8—41no—
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—61no—
484C2×Q8#23912yes—
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—64no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—122no—
500C22×C4—122no—
501C22×C4C2×#6711yescharacteristic
502C22×C4C2×#6711yescharacteristic
503C22×C4C2×#6711yescharacteristic
504C22×C4C2×#6711yescharacteristic
505C22×C4C2×#6711yescharacteristic
506C22×C4C2×#6711yescharacteristic
507C22×C4C23×D311yescharacteristic
508Q16—68no—
509Q16—124no—
510SD16—124no—
511C2×C8—32no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—32no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—64no—
537C4⋊C4—64no—
538C4⋊C4—64no—
539C4⋊C4—64no—
540C4⋊C4—61no—
541C22⋊C4—64no—
542C22⋊C4—64no—
543C22⋊C4—64no—
544C22⋊C4—64no—
545C22⋊C4—64no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—64no—
552C42—64no—
553C42D3×D412yes—
554C42D3×D412yes—
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—22no—
570C3×C4—22no—
571C3×C4—22no—
572C3×C4—42no—
573C3×C4—41no—
574C3×C4—41no—
575C3×C4—41no—
576C3×C4—81no—
577C3×C4—81no—
578C3×C4—81no—
579C3×C4#52511yescharacteristic
580C3×C4#53711yescharacteristic
581C3×C4#54611yescharacteristic
582C3×C4#54711yescharacteristic
583Dic3—42no—
584Dic3—42no—
585Dic3—42no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23C22×#6711yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—44no—
597Q8—41no—
598Q8—41no—
599Q8—62no—
600Q8—62no—
601Q8—62no—
602Q8—62no—
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—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—128no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4#93712yes—
649C2×C4#93712yes—
650C2×C4#95012yes—
651C2×C4C2×D3×D411yescharacteristic
652C2×C4C2×D3×D411yescharacteristic
653C2×C4C2×#23911yescharacteristic
654C2×C4#100711yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4#100911yescharacteristic
657C8—62no—
658C8—121no—
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—
676C22C2×#93711yescharacteristic
677C22#581711yescharacteristic
678C22C2×#95011yescharacteristic
679C22#582211yescharacteristic
680C22#583411yescharacteristic
681C22#583811yescharacteristic
682C22#583811yescharacteristic
683C4—22no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—122no—
694C4—122no—
695C4—122no—
696C4#562711yescharacteristic
697C4#562911yescharacteristic
698C4#563011yescharacteristic
699C4#563111yescharacteristic
700C3#2419511yescharacteristic, complemented, Sylow
701C2—242no—
702C2#8281211yescharacteristic
703C2#8292611yescharacteristic
704C2#8341111yescharacteristic
705C2#8367011yescharacteristic
706C2#8368111yescharacteristic
707C2#8368911yescharacteristic
708C2#8370011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2581C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2581D3
hypocenterC3#24195
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C3×#2581C2semidirect
4. C2×C4×#216C2semidirect
5. #71634C2semidirect
6. #69691C2semidirect
7. #69690C2semidirect
10. #69633C2semidirect
11. #69631C2semidirect
14. #69559C2semidirect
67. #2581D3semidirect
700. C3#24195semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25812
C2×#50282
C2×C4×#2162
#716342
#696912
#696902
#696892
#696872
#696332
#696312
#696282
#696252
#695592
#695512
#241953