Finite Group Databasebeta

← #10957271

Subgroups — #10957271

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×#548C211yesmaximal, characteristic
2C2×#5827C211yesmaximal, characteristic, complemented
3C2×#5812C211yesmaximal, characteristic, complemented
4#84089C211yesmaximal, characteristic, complemented
5Q8×#217C211yesmaximal, characteristic, complemented
6#82955C213yesmaximal, complemented
7Dic3×#124C213yesmaximal, complemented
8#82747C213yesmaximal, complemented
9#82744C213yesmaximal, complemented
10#82637C213yesmaximal, complemented
11#82633C213yesmaximal, complemented
12#82460C211yesmaximal, characteristic, complemented
13C4×Q8×Dic3C211yesmaximal, characteristic, complemented
14#82335C213yesmaximal, complemented
15#82319C213yesmaximal, complemented
16#34312—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C23×C3×Q8C2211yescharacteristic
19C3×#548C4116yescomplemented
20C2×C3×#124C2213yes—
21C2×C3×C22⋊Q8C2213yes—
22C2×C3×#121C2213yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2213yes—
25#5827—28no—
26#5812—28no—
27C2×Q8×Dic3C2213yes—
28C2×Q8×Dic3C2213yes—
29C2×Q8×Dic3C2211yescharacteristic
30C2×Q8×Dic3C2211yescharacteristic
31C2×Q8×Dic3C2211yescharacteristic
32C2×Q8×Dic3C2211yescharacteristic
33C2×D4×Dic3C2213yes—
34C2×D4×Dic3C2213yes—
35C2×#926C2213yes—
36C2×#926C2213yes—
37C2×#926C2213yes—
38C22×C4⋊Dic3C2213yes—
39#5229C2213yes—
40#5229C2213yes—
41#5221C2213yes—
42#5221C2213yes—
43#5221C2213yes—
44#5221C2213yes—
45#5220C2216yes—
46#5212C2213yes—
47#5212C2213yes—
48C4×#217C2213yes—
49C4×#217C2213yes—
50#4984C2213yes—
51#4984C2213yes—
52#4984C2213yes—
53#4984C2213yes—
54#4980C2216yes—
55Dic3×C4⋊C4C2213yes—
56Dic3×C4⋊C4C2213yes—
57#4951C2213yes—
58#4951C2216yes—
59#4951C2213yes—
60#4951C2216yes—
61#4951C2211yescharacteristic
62#4951C2213yes—
63#4950C2216yes—
64Dic3×C22⋊C4C2216yes—
65Dic3×C22⋊C4C2216yes—
66#4936C2213yes—
67C4×C4⋊Dic3C2213yes—
68C4×C4⋊Dic3C2213yes—
69C42×Dic3C2213yes—
70C2×#548D311yescharacteristic
71C2×#520—31no—
72C2×#519—31no—
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—
85C2×C3×#41—24no—
86C2×C3×#41—24no—
87C2×C3×#41C2×C418yes—
88C22×C3×Q8—24no—
89C22×C3×Q8—24no—
90C22×C3×Q8C2×C418yescomplemented
91C22×C3×Q8C2313yes—
92C22×C3×Q8C2311yescharacteristic
93C22×C3×D4C2313yes—
94C22×C3×D4C2313yes—
95C23×C3×C4C2313yes—
96C23×C3×C4C2313yes—
97C23×Dic3C2316yes—
98C3×#124C2×C4124yescomplemented
99C3×C22⋊Q8C2×C4124yescomplemented
100C3×#121C2×C4124yescomplemented
101C3×C4×Q8C2×C418yescomplemented
102C3×C4×D4C2×C4124yescomplemented
103C2×C3×C4⋊C4C2313yes—
104C2×C3×C4⋊C4C2313yes—
105C2×C3×C22⋊C4C2313yes—
106C2×C3×C22⋊C4C2316yes—
107C2×C3×C22⋊C4C2311yescharacteristic
108C2×C3×C42C2313yes—
109C2×#217C2313yes—
110C2×#217C2313yes—
111C2×#217C2313yes—
112C2×#217C2311yescharacteristic
113C2×#217C2313yes—
114C2×#217C2311yescharacteristic
115C2×#217C2316yes—
116C2×#217C2313yes—
117C2×#217C2313yes—
118Q8×Dic3—24no—
119Q8×Dic3—24no—
120Q8×Dic3—24no—
121Q8×Dic3—212no—
122Q8×Dic3—212no—
123Q8×Dic3D418yescomplemented
124D4×Dic3—212no—
125D4×Dic3—212no—
126#926—212no—
127#926—212no—
128#926—212no—
129C2×C4⋊Dic3—212no—
130C2×C4⋊Dic3C2313yes—
131C2×C4⋊Dic3C2313yes—
132C2×C4⋊Dic3C2313yes—
133C2×C4⋊Dic3C2313yes—
134C2×C4⋊Dic3C2313yes—
135C2×C4⋊Dic3C2313yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2313yes—
138C2×C4⋊Dic3C2313yes—
139C2×C4⋊Dic3C2313yes—
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2313yes—
142C2×C4×Dic3C2313yes—
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2316yes—
145C2×C4×Dic3C2313yes—
146C2×C4×Dic3C2313yes—
147C2×C4×Dic3C2313yes—
148C2×C4×Dic3C2313yes—
149C3⋊#95C2311yescharacteristic
150C3⋊#95C2313yes—
151C3⋊#95C2316yes—
152C3⋊#95C2316yes—
153C3⋊#95C2313yes—
154C3⋊#95C2316yes—
155C3⋊#95C2313yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2313yes—
158C22×#41C2×D311yescharacteristic
159C23×Q8C2×D311yescharacteristic
160#548Dic3116yescomplemented
161C2×#124C2×D313yes—
162C2×C22⋊Q8C2×D313yes—
163C2×#121C2×D313yes—
164#520—68no—
165#519—68no—
166C2×C4×Q8—31no—
167C2×C4×Q8—33no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8—33no—
172C2×C4×Q8C2×D311yescharacteristic
173C2×C4×D4—33no—
174C2×C4×D4—33no—
175C2×C4×D4C2×D313yes—
176C2×#117—33no—
177C2×#117—33no—
178C2×#117—33no—
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—36no—
192#388—33no—
193#388—33no—
194#388—31no—
195#388—33no—
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
212C24×C3C2413yes—
213C3×#41—232no—
214C2×C3×Q8—232no—
215C2×C3×Q8—26no—
216C2×C3×Q8—26no—
217C2×C3×Q8C22×C4112yes—
218C2×C3×Q8C22×C414yes—
219C2×C3×Q8C2×D414yes—
220C2×C3×Q8#4114yes—
221C2×C3×D4—26no—
222C2×C3×D4—26no—
223C2×C3×D4—43no—
224C2×C3×D4—43no—
225C2×C3×D4C22×C4112yes—
226C2×C3×D4C22×C4112yes—
227C22×C3×C4—26no—
228C22×C3×C4—26no—
229C22×C3×C4—26no—
230C22×C3×C4—26no—
231C22×C3×C4C22×C4112yes—
232C22×C3×C4C22×C4112yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2413yes—
235C22×C3×C4C2413yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2413yes—
238C22×Dic3—212no—
239C22×Dic3—46no—
240C22×Dic3—46no—
241C22×Dic3C2413yes—
242C22×Dic3C2413yes—
243C22×Dic3C2413yes—
244C22×Dic3C2411yescharacteristic
245C22×Dic3C2411yescharacteristic
246C22×Dic3C2413yes—
247C22×Dic3C2411yescharacteristic
248C22×Dic3C2411yescharacteristic
249C3×C4⋊C4C22×C4112yes—
250C3×C4⋊C4C22×C4112yes—
251C3×C22⋊C4C22×C414yes—
252C3×C22⋊C4C22×C4112yes—
253C3×C22⋊C4C22×C4124yes—
254C3×C42C22×C4112yes—
255#217—26no—
256#217—22no—
257#217—26no—
258#217—26no—
259#217—26no—
260#217—22no—
261#217—46no—
262#217—43no—
263#217—43no—
264C4⋊Dic3—26no—
265C4⋊Dic3—26no—
266C4⋊Dic3—26no—
267C4⋊Dic3—26no—
268C4⋊Dic3—22no—
269C4⋊Dic3—26no—
270C4⋊Dic3—26no—
271C4⋊Dic3—43no—
272C4⋊Dic3—43no—
273C4⋊Dic3C2×D4112yes—
274C4×Dic3—26no—
275C4×Dic3—26no—
276C4×Dic3—26no—
277C4×Dic3—212no—
278C4×Dic3—22no—
279C4×Dic3—43no—
280C4×Dic3—43no—
281C4×Dic3—43no—
282C4×Dic3C2×D4112yes—
283C2×#41—24no—
284C2×#41—24no—
285C2×#41C2×Dic318yes—
286C22×Q8—24no—
287C22×Q8—24no—
288C22×Q8C2×Dic318yescomplemented
289C22×Q8C22×D313yes—
290C22×Q8C22×D311yescharacteristic
291C22×D4C22×D313yes—
292C22×D4C22×D313yes—
293C23×C4—36no—
294C23×C4C22×D313yes—
295C23×C4C22×D313yes—
296#124C2×Dic3124yescomplemented
297C22⋊Q8C2×Dic3124yescomplemented
298#121C2×Dic3124yescomplemented
299C4×Q8—38no—
300C4×Q8—612no—
301C4×Q8—612no—
302C4×Q8—64no—
303C4×Q8—64no—
304C4×Q8—64no—
305C4×Q8C2×Dic318yescomplemented
306C4×D4—612no—
307C4×D4—612no—
308C4×D4C2×Dic3124yescomplemented
309#117—612no—
310#117—612no—
311#117—612no—
312C2×C4⋊C4—33no—
313C2×C4⋊C4—33no—
314C2×C4⋊C4—33no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—33no—
317C2×C4⋊C4—33no—
318C2×C4⋊C4—33no—
319C2×C4⋊C4—33no—
320C2×C4⋊C4—33no—
321C2×C4⋊C4—33no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4C22×D313yes—
324C2×C4⋊C4C22×D313yes—
325C2×C22⋊C4—33no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—33no—
328C2×C22⋊C4—33no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—33no—
331C2×C22⋊C4—33no—
332C2×C22⋊C4—36no—
333C2×C22⋊C4—33no—
334C2×C22⋊C4C22×D311yescharacteristic
335C2×C22⋊C4C22×D313yes—
336C2×C22⋊C4C22×D316yes—
337C2×C42—33no—
338C2×C42—33no—
339C2×C42—36no—
340C2×C42—33no—
341C2×C42—33no—
342C2×C42—31no—
343C2×C42—33no—
344C2×C42—33no—
345C2×C42—33no—
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—23no—
358C23×C3—21no—
359C23×C3—23no—
360C23×C3—21no—
361C23×C3—23no—
362C23×C3—21no—
363C23×C3—23no—
364C23×C3—23no—
365C23×C3—23no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3C23×C416yes—
369C23×C3C23×C412yes—
370C23×C3C2511yescharacteristic
371C3×Q8—224no—
372C3×Q8C4×D4116yes—
373C3×D4—224no—
374C3×D4—412no—
375C2×C3×C4—23no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—23no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—23no—
382C2×C3×C4—23no—
383C2×C3×C4—224no—
384C2×C3×C4—224no—
385C2×C3×C4C23×C412yes—
386C2×C3×C4C23×C416yes—
387C2×C3×C4C23×C416yes—
388C2×C3×C4C23×C412yes—
389C2×C3×C4C23×C416yes—
390C2×C3×C4C22×D416yes—
391C2×C3×C4C2×#4116yes—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—23no—
397C2×Dic3—23no—
398C2×Dic3—23no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—23no—
402C2×Dic3—23no—
403C2×Dic3—23no—
404C2×Dic3—21no—
405C2×Dic3—23no—
406C2×Dic3—23no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—23no—
410C2×Dic3—23no—
411C2×Dic3—21no—
412C2×Dic3—23no—
413C2×Dic3—812no—
414C2×Dic3C22×D416yes—
415C2×Dic3C22×D412yes—
416C2×Dic32−1+412yes—
417C24C23×D311yescharacteristic
418C24C23×D313yes—
419#41—232no—
420C2×Q8—232no—
421C2×Q8—26no—
422C2×Q8—26no—
423C2×Q8D3×D414yes—
424C2×Q8#23714yes—
425C2×Q8C22×Dic3112yes—
426C2×Q8C22×Dic314yes—
427C2×D4—26no—
428C2×D4—26no—
429C2×D4—43no—
430C2×D4—43no—
431C2×D4C22×Dic3112yes—
432C2×D4C22×Dic3112yes—
433C22×C4—26no—
434C22×C4—26no—
435C22×C4—26no—
436C22×C4—26no—
437C22×C4—31no—
438C22×C4—33no—
439C22×C4—31no—
440C22×C4—33no—
441C22×C4—33no—
442C22×C4—31no—
443C22×C4—33no—
444C22×C4—31no—
445C22×C4—612no—
446C22×C4—126no—
447C22×C4—126no—
448C22×C4C22×Dic3112yes—
449C22×C4C22×Dic3112yes—
450C22×C4C23×D311yescharacteristic
451C22×C4C23×D313yes—
452C22×C4C23×D313yes—
453C22×C4C23×D311yescharacteristic
454C22×C4C23×D313yes—
455C4⋊C4—312no—
456C4⋊C4—66no—
457C4⋊C4—62no—
458C4⋊C4—66no—
459C4⋊C4—66no—
460C4⋊C4—66no—
461C4⋊C4—66no—
462C4⋊C4—66no—
463C4⋊C4—123no—
464C4⋊C4—123no—
465C4⋊C4C22×Dic3112yes—
466C4⋊C4C22×Dic3112yes—
467C22⋊C4—62no—
468C22⋊C4—66no—
469C22⋊C4—66no—
470C22⋊C4—66no—
471C22⋊C4—62no—
472C22⋊C4—66no—
473C22⋊C4—126no—
474C22⋊C4—123no—
475C22⋊C4—123no—
476C22⋊C4C22×Dic314yes—
477C22⋊C4C22×Dic3112yes—
478C22⋊C4C22×Dic3124yes—
479C42—312no—
480C42—612no—
481C42—62no—
482C42—66no—
483C42—66no—
484C42—66no—
485C42—123no—
486C42—123no—
487C42—123no—
488C42C22×Dic3112yes—
489C22×C3—22no—
490C22×C3—26no—
491C22×C3—22no—
492C22×C3—26no—
493C22×C3—22no—
494C22×C3—26no—
495C22×C3—41no—
496C22×C3—43no—
497C22×C3—43no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—43no—
501C22×C3—41no—
502C22×C3—43no—
503C22×C3C24×C411yescharacteristic
504C22×C3C23×D411yescharacteristic
505C22×C3C22×#4111yescharacteristic
506C22×C3C2×2−1+411yescharacteristic
507C22×C3#58511yescharacteristic
508C22×C3#58511yescharacteristic
509C22×C3#58511yescharacteristic
510C3×C4—22no—
511C3×C4—26no—
512C3×C4—26no—
513C3×C4—22no—
514C3×C4C2×C4×D4112yes—
515Dic3—22no—
516Dic3—26no—
517Dic3—22no—
518Dic3—43no—
519Dic3—41no—
520Dic3—41no—
521Dic3—43no—
522Dic3—43no—
523C23—21no—
524C23—23no—
525C23—21no—
526C23—23no—
527C23—23no—
528C23—23no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—23no—
533C23—21no—
534C23—23no—
535C23C23×Dic316yes—
536C23C23×Dic312yes—
537C23C24×D311yescharacteristic
538Q8—224no—
539Q8D4×Dic3116yes—
540D4—224no—
541D4—412no—
542C2×C4—23no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—23no—
546C2×C4—23no—
547C2×C4—23no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—224no—
551C2×C4—224no—
552C2×C4—32no—
553C2×C4—36no—
554C2×C4—32no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—63no—
559C2×C4—61no—
560C2×C4—63no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—63no—
564C2×C4—63no—
565C2×C4—63no—
566C2×C4—63no—
567C2×C4—63no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—63no—
571C2×C4—61no—
572C2×C4—63no—
573C2×C4—63no—
574C2×C4—61no—
575C2×C4—63no—
576C2×C4—2412no—
577C2×C4C2×D3×D416yes—
578C2×C4C2×#23716yes—
579C2×C4C23×Dic312yes—
580C2×C4C23×Dic316yes—
581C2×C4C23×Dic316yes—
582C2×C4C23×Dic312yes—
583C2×C4C23×Dic316yes—
584C2×C3—42no—
585C2×C3—46no—
586C2×C3C22×C4×D411yescharacteristic
587C2×C3#342811yescharacteristic
588C2×C3C4×2−1+411yescharacteristic
589C2×C3#349711yescharacteristic
590C2×C3#350011yescharacteristic
591C2×C3#351611yescharacteristic
592C2×C3#352411yescharacteristic
593C22—22no—
594C22—26no—
595C22—22no—
596C22—26no—
597C22—22no—
598C22—26no—
599C22—43no—
600C22—41no—
601C22—41no—
602C22—43no—
603C22—41no—
604C22—43no—
605C22—41no—
606C22—43no—
607C22C22×D3×D411yescharacteristic
608C22C22×#23711yescharacteristic
609C22#596411yescharacteristic
610C22#596411yescharacteristic
611C22D3×2−1+411yescharacteristic
612C22#596811yescharacteristic
613C22C24×Dic311yescharacteristic
614C4—22no—
615C4—26no—
616C4—26no—
617C4—22no—
618C4—66no—
619C4—62no—
620C4—62no—
621C4—123no—
622C4—121no—
623C4—123no—
624C4—121no—
625C4—123no—
626C4C2×D4×Dic3112yes—
627C3#3431211yescharacteristic, complemented, Sylow
628C2—42no—
629C2—46no—
630C2#8671511yescharacteristic
631C2#8671911yescharacteristic
632C2#8672411yescharacteristic
633C2#8673711yescharacteristic
634C2C22×D4×Dic311yescharacteristic
635C2#8726011yescharacteristic
636C2Dic3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
Fitting, prime radicalC2×C3×#548C2
socle, cosocleC23×C3C25
hypercenterC2×#548D3
hypocenterC3#34312
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
2. C2×#5827C2semidirect
3. C2×#5812C2semidirect
4. #84089C2semidirect
5. Q8×#217C2semidirect
6. #82955C2semidirect
7. Dic3×#124C2semidirect
8. #82747C2semidirect
9. #82744C2semidirect
10. #82637C2semidirect
11. #82633C2semidirect
12. #82460C2semidirect
13. C4×Q8×Dic3C2semidirect
14. #82335C2semidirect
15. #82319C2semidirect
19. C3×#548C4semidirect
90. C22×C3×Q8C2×C4semidirect
98. C3×#124C2×C4semidirect
99. C3×C22⋊Q8C2×C4semidirect
100. C3×#121C2×C4semidirect
101. C3×C4×Q8C2×C4semidirect
102. C3×C4×D4C2×C4semidirect
123. Q8×Dic3D4semidirect
160. #548Dic3semidirect
288. C22×Q8C2×Dic3semidirect
296. #124C2×Dic3semidirect
297. C22⋊Q8C2×Dic3semidirect
298. #121C2×Dic3semidirect
305. C4×Q8C2×Dic3semidirect
308. C4×D4C2×Dic3semidirect
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×#5482
C2×#58272
C2×#58122
#840892
Q8×#2172
#829552
Dic3×#1242
#827472
#827442
#826372
#826332
#824602
C4×Q8×Dic32
#823352
#823192
#343123