Finite Group Databasebeta

← #10849317

Subgroups — #10849317

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5559C211yesmaximal, characteristic
2C2×#4857C211yesmaximal, characteristic, complemented
3C2×#4764C211yesmaximal, characteristic
4#76078C211yesmaximal, characteristic, complemented
5C3×#1864C211yesmaximal, characteristic, complemented
6#71606C211yesmaximal, characteristic, complemented
7#70584C211yesmaximal, characteristic, complemented
8#70216C211yesmaximal, characteristic, complemented
9#70178C211yesmaximal, characteristic, complemented
10#70161C211yesmaximal, characteristic
11#69628C211yesmaximal, characteristic, complemented
12#68894C211yesmaximal, characteristic, complemented
13#68783C211yesmaximal, characteristic, complemented
14#68630C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#25658—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×#929—21no—
21C2×C4×#67—21no—
22#5559—24no—
23#5559—24no—
24C2×#893C2211yescharacteristic
25C2×#890C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#885—21no—
28C2×#883—21no—
29C2×#875C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#106C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#4984—21no—
44#4984C2211yescharacteristic
45#4979—21no—
46Dic3×C4⋊C4C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4857—24no—
55#4764—24no—
56#4657—21no—
57#4647—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#559—31no—
65C2×#490—31no—
66C2×#479—31no—
67#2584—31no—
68#2042—31no—
69#2042—31no—
70#2042—31no—
71#2042—31no—
72#1985—31no—
73#1982—31no—
74#1864—31no—
75#1864D311yescharacteristic, complemented
76#1862—31no—
77#1862—31no—
78#1763—31no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C22×#67—21no—
82C22×#67—21no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×Dic6D411yescharacteristic
86C22×Dic6C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99#940—22no—
100#940D414yes—
101#936—22no—
102#936—22no—
103#929—28no—
104C4×#67—28no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×Dic3⋊C4D411yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#893—22no—
126#893—22no—
127#890—22no—
128#890—22no—
129#887—22no—
130#887D414yes—
131#885—28no—
132#883—28no—
133#875—22no—
134#875—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#106—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#95—21no—
141#832—22no—
142#832—22no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#821—22no—
148#821—22no—
149#816—22no—
150#816—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#810—22no—
154#810—22no—
155#809—22no—
156#809—22no—
157#807—22no—
158#559—64no—
159#559—64no—
160C2×#127—31no—
161C2×#125—31no—
162C2×#124—31no—
163C2×#124—31no—
164C2×#124—61no—
165C2×#123—61no—
166C2×#123—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×D4—31no—
170C2×C4×D4—61no—
171C2×C4×D4C2×D311yescharacteristic
172#490—64no—
173#479—64no—
174C2×#107—31no—
175C2×#106C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#102—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102C2×D311yescharacteristic
188C2×#102C2×D311yescharacteristic
189C2×#97—31no—
190#398—61no—
191#395—61no—
192#389—31no—
193#389—31no—
194#389—31no—
195#389—61no—
196#389—61no—
197#389C2×D311yescharacteristic
198C4×C4⋊C4—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C24×C3—21no—
206C23×D3—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4C2×D412yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—81no—
218C22×C3×C4—81no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C2×#67—24no—
223C2×#67—24no—
224C2×#67—42no—
225C2×#67—42no—
226C2×#67—81no—
227C2×#67—81no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
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—81no—
243C2×C4×D3—81no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—22no—
250C2×Dic6—41no—
251C2×Dic6—41no—
252C2×Dic6C2×D412yes—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—42no—
264C3×C4⋊C4C2×D412yes—
265C3×C4⋊C4#4112yes—
266C3×C22⋊C4—24no—
267C3×C42—21no—
268C3×C42—21no—
269#217—24no—
270#217—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274C4⋊Dic3—21no—
275C4⋊Dic3—22no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—42no—
279C4⋊Dic3—81no—
280C4⋊Dic3#4112yes—
281Dic3⋊C4—22no—
282Dic3⋊C4—22no—
283Dic3⋊C4—24no—
284Dic3⋊C4—42no—
285Dic3⋊C4—42no—
286Dic3⋊C4—81no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—24no—
290C4×Dic3—81no—
291C4×Dic3C2×D412yes—
292C4×Dic3C2×D412yes—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C22×Q8—31no—
300C22×Q8—31no—
301C22×D4—31no—
302C22×D4—61no—
303C22×D4—61no—
304C22×D4C22×D311yescharacteristic
305C23×C4—21no—
306C23×C4—61no—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8C22×D311yescharacteristic
310C22×C8C22×D311yescharacteristic
311#127—34no—
312#127—62no—
313#125—62no—
314#125—62no—
315#124—62no—
316#124—62no—
317#124—68no—
318#124—62no—
319#124—62no—
320#123—68no—
321#123—68no—
322C4×Q8—34no—
323C4×Q8—62no—
324C4×Q8—62no—
325C4×Q8—62no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—62no—
329C4×D4—68no—
330C4×D4—62no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357#107—62no—
358#106—22no—
359C4⋊C8—62no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#102—22no—
365#102—22no—
366#102—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371#102—62no—
372#97—62no—
373#95—21no—
374#95—61no—
375#95—61no—
376#95—61no—
377#95—61no—
378C23×C3—22no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3C22×D411yescharacteristic
386C22×D3—22no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—41no—
392C22×D3—41no—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—41no—
396C3×D4—41no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—84no—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419#67—48no—
420#67—84no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3#12012yes—
441C2×Dic3#12012yes—
442D12—22no—
443D12—22no—
444D12—41no—
445D12—41no—
446C4×D3—84no—
447Dic6—22no—
448Dic6—22no—
449Dic6—44no—
450Dic6—41no—
451Dic6—41no—
452C3×C8—41no—
453C3×C8—41no—
454C3⋊C8—41no—
455C3⋊C8—41no—
456C24—21no—
457C24—61no—
458C2×Q8—32no—
459C2×Q8—62no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—121no—
466C2×Q8—121no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—32no—
473C2×D4—64no—
474C2×D4—64no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—122no—
481C2×D4—122no—
482C2×D4—241no—
483C2×D4—241no—
484C2×D4D3×D412yes—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—21no—
489C22×C4—31no—
490C22×C4—31no—
491C22×C4—31no—
492C22×C4—31no—
493C22×C4—64no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—241no—
500C22×C4—241no—
501C22×C4D3×D411yescharacteristic
502C22×C4D3×D411yescharacteristic
503C22×C4C23×D311yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—32no—
521C4⋊C4—42no—
522C4⋊C4—62no—
523C4⋊C4—62no—
524C4⋊C4—62no—
525C4⋊C4—64no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—122no—
530C4⋊C4—122no—
531C4⋊C4—122no—
532C4⋊C4—241no—
533C4⋊C4—241no—
534C4⋊C4#23512yes—
535C4⋊C4D3×D412yes—
536C22⋊C4—24no—
537C22⋊C4—64no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—64no—
541C22⋊C4—64no—
542C42—21no—
543C42—21no—
544C42—32no—
545C42—32no—
546C42—64no—
547C42—61no—
548C42—61no—
549C42—241no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—42no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3C2×#12011yescharacteristic
558C22×C3#54811yescharacteristic
559C22×C3#54811yescharacteristic
560C22×C3#57511yescharacteristic
561C22×C3#57511yescharacteristic
562C22×C3#57611yescharacteristic
563C22×C3#57711yescharacteristic
564C2×D3—42no—
565C2×D3—42no—
566C2×D3—42no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C3×C4—42no—
572C3×C4—41no—
573C3×C4—42no—
574C3×C4—41no—
575C3×C4—81no—
576C3×C4#53211yescharacteristic
577C3×C4#53611yescharacteristic
578C3×C4#54711yescharacteristic
579C3×C4#54711yescharacteristic
580Dic3—22no—
581Dic3—22no—
582Dic3—42no—
583Dic3—42no—
584Dic3—41no—
585Dic3—41no—
586Dic3—81no—
587C23—22no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—62no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23C2×D3×D411yescharacteristic
602Q8—62no—
603Q8—62no—
604Q8—124no—
605Q8—121no—
606Q8—121no—
607D4—22no—
608D4—22no—
609D4—41no—
610D4—41no—
611D4—62no—
612D4—62no—
613D4—128no—
614D4—121no—
615D4—121no—
616D4—244no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
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—61no—
643C2×C4—61no—
644C2×C4—62no—
645C2×C4—61no—
646C2×C4—84no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—244no—
655C2×C4C2×#23511yescharacteristic
656C2×C4C2×D3×D411yescharacteristic
657C2×C4C2×D3×D411yescharacteristic
658C2×C4#100711yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4#101011yescharacteristic
661C8—41no—
662C8—41no—
663C8—121no—
664C8—121no—
665C2×C3—82no—
666C2×C3#243211yescharacteristic
667C2×C3#300511yescharacteristic
668C2×C3#301211yescharacteristic
669C2×C3#337011yescharacteristic
670C2×C3#337011yescharacteristic
671C2×C3#337111yescharacteristic
672C2×C3#337111yescharacteristic
673D3—82no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—122no—
682C22—122no—
683C22—122no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22#555211yescharacteristic
689C22D3×#12011yescharacteristic
690C22#563311yescharacteristic
691C22#576211yescharacteristic
692C22#576811yescharacteristic
693C22#577311yescharacteristic
694C22#577411yescharacteristic
695C4—41no—
696C4—42no—
697C4—41no—
698C4—42no—
699C4—62no—
700C4—62no—
701C4—81no—
702C4—121no—
703C4—122no—
704C4—122no—
705C4—121no—
706C4—241no—
707C4#550711yescharacteristic
708C4#554811yescharacteristic
709C4#562811yescharacteristic
710C4#565511yescharacteristic
711C3#2565811yescharacteristic, complemented, Sylow
712C2—82no—
713C2—242no—
714C2#7441111yescharacteristic
715C2#7962711yescharacteristic
716C2#7985211yescharacteristic
717C2#8008311yescharacteristic
718C2#8014911yescharacteristic
719C2#8105111yescharacteristic
720C2#8120211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1864D3
hypocenterC3#25658
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×#4857C2semidirect
4. #76078C2semidirect
5. C3×#1864C2semidirect
6. #71606C2semidirect
7. #70584C2semidirect
8. #70216C2semidirect
9. #70178C2semidirect
11. #69628C2semidirect
12. #68894C2semidirect
13. #68783C2semidirect
14. #68630C2semidirect
15. #68620C2semidirect
75. #1864D3semidirect
711. C3#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#55592
C2×#48572
C2×#47642
#760782
C3×#18642
#716062
#705842
#702162
#701782
#701612
#696282
#688942
#687832
#686302
#686202
#256583