Finite Group Databasebeta

← #10957262

Subgroups — #10957262

Order 768.

2960 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic, complemented
2C2×#5817C211yesmaximal, characteristic
3C2×#5574C211yesmaximal, characteristic, complemented
4Q8×#217C211yesmaximal, characteristic, complemented
5#82490C213yesmaximal, complemented
6#82488C213yesmaximal, complemented
7#82466C213yesmaximal, complemented
8#82461C211yesmaximal, characteristic, complemented
9#82457C211yesmaximal, characteristic, complemented
10C4×#947C213yesmaximal, complemented
11C4×Q8×Dic3C211yesmaximal, characteristic, complemented
12#82272C213yesmaximal, complemented
13#82244C213yesmaximal, complemented
14#82236C213yesmaximal, complemented
15#82225C213yesmaximal, complemented
16#34312—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×#239C2211yescharacteristic
19C3×#519—28no—
20C2×C3×C4×Q8C2213yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2213yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×#117C2213yes—
25C2×C3×#117C2213yes—
26#5817C4116yescomplemented
27C2×#947C2213yes—
28C2×#946C2213yes—
29C2×Q8×Dic3C2211yescharacteristic
30C2×Q8×Dic3C2211yescharacteristic
31C2×Q8×Dic3C2211yescharacteristic
32C2×#930C2213yes—
33C2×C4×#67C2213yes—
34#5574—28no—
35C2×#893C2213yes—
36C2×D3×C4⋊C4C2213yes—
37C2×C4×D12C2213yes—
38C2×#872C2213yes—
39#5229C2213yes—
40#5229C2213yes—
41#5212C2213yes—
42C4×#217C2213yes—
43#4990C2213yes—
44#4990C2216yes—
45#4990C2213yes—
46#4989C2216yes—
47#4988C2213yes—
48#4987C2213yes—
49#4977C2213yes—
50#4976C2213yes—
51#4976C2213yes—
52Dic3×C4⋊C4C2213yes—
53Dic3×C4⋊C4C2213yes—
54#4972C2213yes—
55#4957C2213yes—
56#4957C2213yes—
57#4956C2216yes—
58#4945C2211yescharacteristic
59#4945C2213yes—
60#4940C2216yes—
61#4939C2213yes—
62C4×(C2×D3)⋊C4C2216yes—
63C4×(C2×D3)⋊C4C2216yes—
64C4×(C2×D3)⋊C4C2213yes—
65C4×C4⋊Dic3C2213yes—
66#4933C2216yes—
67C4×Dic3⋊C4C2213yes—
68C42×Dic3C2213yes—
69#4928C2213yes—
70C2×#548—31no—
71C2×#520—31no—
72C2×#519D311yescharacteristic, complemented
73#2367—33no—
74#2363—33no—
75#2360—33no—
76#2352—33no—
77#2351—31no—
78#2341—31no—
79Q8×C22⋊C4—31no—
80#2324—33no—
81#2323—33no—
82C4×#124—33no—
83#2293—33no—
84C42×Q8—31no—
85C22×C3×Q8—24no—
86C22×C3×Q8—24no—
87C22×C3×Q8C2×C418yescomplemented
88C22×C3×Q8C2313yes—
89C22×C3×Q8C2311yescharacteristic
90C23×C3×C4C2313yes—
91C22×#67C2313yes—
92C2×#239—24no—
93C2×#239—24no—
94C2×#239C2×C418yes—
95C22×D12C2313yes—
96C22×C4×D3C2313yes—
97C22×C4×D3C2316yes—
98C3×C4×Q8—24no—
99C3×C4×Q8—212no—
100C3×C4×Q8—212no—
101C3×C4×Q8D418yescomplemented
102C3×#117—212no—
103C3×#117—212no—
104C2×C3×C4⋊C4C2313yes—
105C2×C3×C4⋊C4C2313yes—
106C2×C3×C4⋊C4C2313yes—
107C2×C3×C4⋊C4C2313yes—
108C2×C3×C22⋊C4C2313yes—
109C2×C3×C22⋊C4C2311yescharacteristic
110C2×C3×C42C2313yes—
111C2×C3×C42C2316yes—
112C2×C3×C42C2313yes—
113C2×#217C2311yescharacteristic
114C2×#217C2313yes—
115C2×#217C2311yescharacteristic
116#947C2×C4124yescomplemented
117#946C2×C4124yescomplemented
118Q8×Dic3—24no—
119Q8×Dic3—24no—
120Q8×Dic3C2×C418yescomplemented
121#930C2×C4124yescomplemented
122C4×#67C2×C4124yescomplemented
123C2×(C2×D3)⋊C4C2313yes—
124C2×(C2×D3)⋊C4C2313yes—
125C2×(C2×D3)⋊C4C2313yes—
126C2×(C2×D3)⋊C4C2313yes—
127C2×(C2×D3)⋊C4C2316yes—
128C2×(C2×D3)⋊C4C2313yes—
129C2×(C2×D3)⋊C4C2316yes—
130C2×C4⋊Dic3C2313yes—
131C2×C4⋊Dic3C2313yes—
132C2×C4⋊Dic3C2313yes—
133C2×Dic3⋊C4C2313yes—
134C2×Dic3⋊C4C2311yescharacteristic
135C2×Dic3⋊C4C2313yes—
136C2×Dic3⋊C4C2313yes—
137C2×Dic3⋊C4C2313yes—
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2313yes—
140C2×C4×Dic3C2313yes—
141C2×C4×Dic3C2313yes—
142C2×C4×Dic3C2313yes—
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2313yes—
145#893—212no—
146D3×C4⋊C4—212no—
147C4×D12—212no—
148#872—212no—
149C3⋊#95C2316yes—
150C3⋊#95C2311yescharacteristic
151C3⋊#95C2313yes—
152C3⋊#95C2313yes—
153C3⋊#95C2316yes—
154C3⋊#95C2316yes—
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2313yes—
157C3⋊#95C2313yes—
158C22×#41—31no—
159C23×Q8C2×D311yescharacteristic
160#548—316no—
161C2×#124—33no—
162C2×C22⋊Q8—33no—
163C2×#121—33no—
164#520—68no—
165#519—28no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8C2×D313yes—
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×Q8C2×D313yes—
172C2×C4×Q8C2×D311yescharacteristic
173C2×C4×D4—33no—
174C2×C4×D4—33no—
175C2×C4×D4—33no—
176C2×#117—33no—
177C2×#117C2×D313yes—
178C2×#117C2×D313yes—
179C22×C4⋊C4—33no—
180#391—33no—
181#391—33no—
182#390—33no—
183#390—33no—
184#390—33no—
185#390—33no—
186#389—33no—
187#389—33no—
188#389—33no—
189#389—33no—
190#388—36no—
191#388—33no—
192#388—33no—
193#388—33no—
194#388—31no—
195#388—36no—
196#387—36no—
197#386—36no—
198#385—36no—
199#382—33no—
200#380—33no—
201#380—33no—
202C4×C4⋊C4—33no—
203C4×C4⋊C4—33no—
204C4×C4⋊C4—33no—
205C4×C4⋊C4—33no—
206C4×C22⋊C4—33no—
207C4×C22⋊C4—33no—
208C4×C22⋊C4—36no—
209C4×C22⋊C4—36no—
210C43—33no—
211C24×C3C2411yescharacteristic
212C23×D3C2413yes—
213C2×C3×Q8—232no—
214C2×C3×Q8—26no—
215C2×C3×Q8—26no—
216C2×C3×Q8C22×C4112yes—
217C2×C3×Q8C22×C414yes—
218C2×C3×Q8C2×D414yes—
219C2×C3×Q8#4114yes—
220C22×C3×C4—26no—
221C22×C3×C4—26no—
222C22×C3×C4C22×C4112yes—
223C22×C3×C4C2413yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2413yes—
226C22×C3×C4C2413yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2413yes—
229C2×#67—43no—
230C2×#67—43no—
231C2×#67C22×C4112yes—
232C22×Dic3C2413yes—
233C22×Dic3C2413yes—
234C22×Dic3C2411yescharacteristic
235C22×Dic3C2411yescharacteristic
236C22×Dic3C2413yes—
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2411yescharacteristic
239#239—232no—
240C2×D12—26no—
241C2×D12—26no—
242C2×D12C22×C4112yes—
243C2×C4×D3—26no—
244C2×C4×D3—26no—
245C2×C4×D3—212no—
246C2×C4×D3—46no—
247C2×C4×D3—46no—
248C2×C4×D3C22×C4112yes—
249C3×C4⋊C4—26no—
250C3×C4⋊C4—26no—
251C3×C4⋊C4—26no—
252C3×C4⋊C4C2×D4112yes—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—26no—
255C3×C42—212no—
256C3×C42—26no—
257C3×C42C2×D4112yes—
258#217—22no—
259#217—43no—
260#217C22×C414yes—
261(C2×D3)⋊C4—26no—
262(C2×D3)⋊C4—26no—
263(C2×D3)⋊C4—26no—
264(C2×D3)⋊C4—46no—
265(C2×D3)⋊C4—43no—
266(C2×D3)⋊C4C22×C4112yes—
267(C2×D3)⋊C4C22×C4124yes—
268C4⋊Dic3—26no—
269C4⋊Dic3—26no—
270C4⋊Dic3C22×C4112yes—
271Dic3⋊C4—22no—
272Dic3⋊C4—26no—
273Dic3⋊C4—43no—
274Dic3⋊C4—43no—
275Dic3⋊C4C22×C4112yes—
276C4×Dic3—26no—
277C4×Dic3—26no—
278C4×Dic3—22no—
279C4×Dic3—43no—
280C4×Dic3—43no—
281C4×Dic3—43no—
282C4×Dic3C22×C4112yes—
283C2×#41—38no—
284C2×#41—64no—
285C2×#41—64no—
286C22×Q8—24no—
287C22×Q8—24no—
288C22×Q8C4×D318yescomplemented
289C22×Q8C22×D313yes—
290C22×Q8C22×D311yescharacteristic
291C22×D4—33no—
292C22×D4—33no—
293C23×C4—33no—
294C23×C4—36no—
295C23×C4C22×D313yes—
296#124—324no—
297C22⋊Q8—324no—
298#121—324no—
299C4×Q8—24no—
300C4×Q8—212no—
301C4×Q8—212no—
302C4×Q8—38no—
303C4×Q8—64no—
304C4×Q8—64no—
305C4×Q8#6718yescomplemented
306C4×D4—324no—
307C4×D4—612no—
308C4×D4—612no—
309#117—212no—
310#117—212no—
311#117—612no—
312C2×C4⋊C4—33no—
313C2×C4⋊C4—33no—
314C2×C4⋊C4—33no—
315C2×C4⋊C4—33no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—33no—
318C2×C4⋊C4—33no—
319C2×C4⋊C4—33no—
320C2×C4⋊C4—612no—
321C2×C4⋊C4C22×D313yes—
322C2×C4⋊C4C22×D313yes—
323C2×C4⋊C4C22×D313yes—
324C2×C4⋊C4C22×D313yes—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—33no—
327C2×C22⋊C4—36no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—33no—
330C2×C22⋊C4—33no—
331C2×C22⋊C4—33no—
332C2×C22⋊C4—33no—
333C2×C22⋊C4—36no—
334C2×C22⋊C4—33no—
335C2×C22⋊C4C22×D313yes—
336C2×C22⋊C4C22×D311yescharacteristic
337C2×C42—33no—
338C2×C42—33no—
339C2×C42—33no—
340C2×C42—31no—
341C2×C42—33no—
342C2×C42—33no—
343C2×C42—33no—
344C2×C42C22×D316yes—
345C2×C42C22×D313yes—
346C2×C42C22×D313yes—
347#95—31no—
348#95—33no—
349#95—33no—
350#95—33no—
351#95—36no—
352#95—36no—
353#95—33no—
354#95—36no—
355#95—31no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3C23×C412yes—
363C23×C3C2511yescharacteristic
364C22×D3—23no—
365C22×D3—23no—
366C22×D3—23no—
367C22×D3—23no—
368C22×D3—23no—
369C22×D3—23no—
370C22×D3C23×C416yes—
371C3×Q8—224no—
372C3×Q8C4×D4116yes—
373C2×C3×C4—23no—
374C2×C3×C4—23no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—23no—
378C2×C3×C4—23no—
379C2×C3×C4—224no—
380C2×C3×C4—23no—
381C2×C3×C4—21no—
382C2×C3×C4—23no—
383C2×C3×C4—21no—
384C2×C3×C4—23no—
385C2×C3×C4—21no—
386C2×C3×C4C23×C416yes—
387C2×C3×C4C23×C416yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D416yes—
390C2×C3×C4C22×D416yes—
391C2×C3×C4C2×#4116yes—
392#67—412no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—23no—
400C2×Dic3—23no—
401C2×Dic3—21no—
402C2×Dic3—23no—
403C2×Dic3—23no—
404C2×Dic3—21no—
405C2×Dic3—23no—
406C2×Dic3—23no—
407C2×Dic3—21no—
408C2×Dic3—23no—
409C2×Dic3—23no—
410C2×Dic3C23×C412yes—
411C2×Dic3C23×C416yes—
412C2×Dic3C23×C412yes—
413C2×Dic32−1+412yes—
414D12—224no—
415C4×D3—224no—
416C4×D3—812no—
417C24—33no—
418C24C23×D311yescharacteristic
419#41—632no—
420C2×Q8—232no—
421C2×Q8—26no—
422C2×Q8—26no—
423C2×Q8C2×C4×D3112yes—
424C2×Q8C2×C4×D314yes—
425C2×Q8#23514yes—
426C2×Q8C2×#6714yes—
427C2×D4—312no—
428C2×D4—312no—
429C2×D4—66no—
430C2×D4—66no—
431C2×D4—123no—
432C2×D4—123no—
433C22×C4—26no—
434C22×C4—26no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—33no—
438C22×C4—31no—
439C22×C4—312no—
440C22×C4—33no—
441C22×C4—33no—
442C22×C4—31no—
443C22×C4—66no—
444C22×C4—66no—
445C22×C4—612no—
446C22×C4—126no—
447C22×C4—126no—
448C22×C4C2×C4×D3112yes—
449C22×C4C23×D313yes—
450C22×C4C23×D311yescharacteristic
451C22×C4C23×D313yes—
452C22×C4C23×D313yes—
453C22×C4C23×D311yescharacteristic
454C22×C4C23×D313yes—
455C4⋊C4—26no—
456C4⋊C4—26no—
457C4⋊C4—26no—
458C4⋊C4—312no—
459C4⋊C4—312no—
460C4⋊C4—66no—
461C4⋊C4—66no—
462C4⋊C4—66no—
463C4⋊C4—62no—
464C4⋊C4—123no—
465C4⋊C4—123no—
466C4⋊C4C2×#67112yes—
467C22⋊C4—26no—
468C22⋊C4—22no—
469C22⋊C4—34no—
470C22⋊C4—312no—
471C22⋊C4—324no—
472C22⋊C4—66no—
473C22⋊C4—62no—
474C22⋊C4—66no—
475C22⋊C4—66no—
476C22⋊C4—123no—
477C22⋊C4—126no—
478C22⋊C4—123no—
479C42—212no—
480C42—26no—
481C42—312no—
482C42—62no—
483C42—66no—
484C42—66no—
485C42—123no—
486C42—123no—
487C42—123no—
488C42C2×#67112yes—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3C24×C411yescharacteristic
497C22×C3C23×D411yescharacteristic
498C22×C3C22×#4111yescharacteristic
499C22×C3C2×2−1+411yescharacteristic
500C22×C3#58511yescharacteristic
501C22×C3#58511yescharacteristic
502C22×C3#58511yescharacteristic
503C2×D3—26no—
504C2×D3—26no—
505C2×D3—26no—
506C2×D3—43no—
507C2×D3—43no—
508C2×D3—43no—
509C2×D3—43no—
510C3×C4—26no—
511C3×C4—22no—
512C3×C4—26no—
513C3×C4—43no—
514C3×C4—41no—
515C3×C4C2×C4×D4112yes—
516Dic3—22no—
517Dic3—26no—
518Dic3—22no—
519Dic3—22no—
520Dic3—41no—
521Dic3—43no—
522Dic3—43no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—36no—
530C23—63no—
531C23—63no—
532C23—63no—
533C23—63no—
534C23—63no—
535C23—63no—
536C23C22×C4×D312yes—
537C23C24×D311yescharacteristic
538Q8—224no—
539Q8C4×#67116yes—
540D4—624no—
541D4—1212no—
542C2×C4—23no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—23no—
546C2×C4—23no—
547C2×C4—23no—
548C2×C4—224no—
549C2×C4—23no—
550C2×C4—21no—
551C2×C4—23no—
552C2×C4—21no—
553C2×C4—23no—
554C2×C4—21no—
555C2×C4—32no—
556C2×C4—36no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—63no—
565C2×C4—63no—
566C2×C4—61no—
567C2×C4—624no—
568C2×C4—63no—
569C2×C4—61no—
570C2×C4—63no—
571C2×C4—63no—
572C2×C4—63no—
573C2×C4—61no—
574C2×C4—63no—
575C2×C4—61no—
576C2×C4—63no—
577C2×C4—2412no—
578C2×C4C22×C4×D316yes—
579C2×C4C22×C4×D316yes—
580C2×C4C2×#23516yes—
581C2×C4C22×#6716yes—
582C2×C4C22×#6712yes—
583C2×C4C22×#6716yes—
584C2×C3—42no—
585C2×C3C22×C4×D411yescharacteristic
586C2×C3#342811yescharacteristic
587C2×C3C4×2−1+411yescharacteristic
588C2×C3#349711yescharacteristic
589C2×C3#350011yescharacteristic
590C2×C3#351611yescharacteristic
591C2×C3#352411yescharacteristic
592D3—46no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—66no—
601C22—66no—
602C22—66no—
603C22—123no—
604C22—123no—
605C22—123no—
606C22—123no—
607C22C23×C4×D311yescharacteristic
608C22C22×#23511yescharacteristic
609C22D3×2−1+411yescharacteristic
610C22#596811yescharacteristic
611C22#596811yescharacteristic
612C22#596811yescharacteristic
613C22C23×#6711yescharacteristic
614C4—26no—
615C4—22no—
616C4—26no—
617C4—43no—
618C4—41no—
619C4—66no—
620C4—62no—
621C4—62no—
622C4—62no—
623C4—123no—
624C4—121no—
625C4—123no—
626C4C2×C4×#67112yes—
627C3#3431211yescharacteristic, complemented, Sylow
628C2—42no—
629C2—126no—
630C2#8587211yescharacteristic
631C2#8587311yescharacteristic
632C2#8587911yescharacteristic
633C2#8588311yescharacteristic
634C2C22×C4×#6711yescharacteristic
635C2#8729111yescharacteristic
636C2#8729311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC2×C3×#519C2
socle, cosocleC23×C3C25
prime radicalC2×#5817C2
hypercenterC2×#519D3
hypocenterC3#34312
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
1. C2×C3×#519C2semidirect
3. C2×#5574C2semidirect
4. Q8×#217C2semidirect
5. #82490C2semidirect
6. #82488C2semidirect
7. #82466C2semidirect
8. #82461C2semidirect
9. #82457C2semidirect
10. C4×#947C2semidirect
11. C4×Q8×Dic3C2semidirect
12. #82272C2semidirect
13. #82244C2semidirect
14. #82236C2semidirect
15. #82225C2semidirect
26. #5817C4semidirect
72. C2×#519D3semidirect
87. C22×C3×Q8C2×C4semidirect
101. C3×C4×Q8D4semidirect
116. #947C2×C4semidirect
117. #946C2×C4semidirect
120. Q8×Dic3C2×C4semidirect
121. #930C2×C4semidirect
122. C4×#67C2×C4semidirect
288. C22×Q8C4×D3semidirect
305. C4×Q8#67semidirect
627. C3#34312semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×#58172
C2×#55742
Q8×#2172
#824902
#824882
#824662
#824612
#824572
C4×#9472
C4×Q8×Dic32
#822722
#822442
#822362
#822252
#343123