Finite Group Databasebeta

← #10868321

Subgroups — #10868321

Order 768.

902 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76047C211yesmaximal, characteristic, complemented
2#73653C211yesmaximal, characteristic, complemented
3C3×#1918C211yesmaximal, characteristic, complemented
4#70641C211yesmaximal, characteristic, complemented
5#70628C211yesmaximal, characteristic, complemented
6#69627C211yesmaximal, characteristic, complemented
7#69462C211yesmaximal, characteristic, complemented
8#68897C211yesmaximal, characteristic, complemented
9#68897C211yesmaximal, characteristic, complemented
10#68892C211yesmaximal, characteristic, complemented
11#68862C211yesmaximal, characteristic, complemented
12#68791C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic, complemented
14#68674C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#27062—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C2×#898—21no—
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C3×#107C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4994—21no—
32#4988C2211yescharacteristic
33#4988C2211yescharacteristic
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4939—21no—
46#4939C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48#4675—21no—
49#4669—21no—
50#4665—21no—
51#4665—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2366—31no—
62#2093—31no—
63#2054—31no—
64#2041—31no—
65#2011—31no—
66#1970—31no—
67#1925—31no—
68#1925—31no—
69#1918D311yescharacteristic, complemented
70#1916—31no—
71#1879—31no—
72#1879—31no—
73#1802—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D24—22no—
114C2×D24—22no—
115C2×D24—22no—
116C2×D24—41no—
117C2×D24—41no—
118C2×D24—41no—
119#898—44no—
120#893—22no—
121#893—22no—
122#893—41no—
123#893—41no—
124#874—22no—
125#874—22no—
126C4×D12—44no—
127C3×#107—22no—
128C3×#95—21no—
129C3×#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#821—22no—
134#821—41no—
135Dic3⋊C8—41no—
136C8×Dic3—41no—
137#809—22no—
138#809—22no—
139#809—22no—
140#809—41no—
141#809—41no—
142#809—41no—
143#807—41no—
144C22×D8—31no—
145C22×D8—31no—
146C2×#127—31no—
147C2×#126—61no—
148C2×C4×D4—31no—
149C2×C4×D4—31no—
150C2×C4×D4—61no—
151C2×#107—31no—
152C2×#107C2×D311yescharacteristic
153C2×C4⋊C8—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×#102—31no—
161C2×#102—31no—
162C2×C4×C8—31no—
163#397—61no—
164#396—61no—
165#390—31no—
166#390—31no—
167#390—31no—
168#390—31no—
169#390—31no—
170#390—61no—
171#389C2×D311yescharacteristic
172#388—61no—
173#386—61no—
174#386—61no—
175#384—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4C2×D311yescharacteristic
179#340—31no—
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#336—31no—
184#336—31no—
185#336—31no—
186#336C2×D311yescharacteristic
187C23×D3—21no—
188C23×D3—21no—
189C23×D3—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—24no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—24no—
220C2×D12—42no—
221C2×D12—42no—
222C2×D12—81no—
223C2×D12—81no—
224C2×D12#4112yes—
225C2×D12#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8#4112yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4#4112yes—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—41no—
265C4⋊Dic3D814yescomplemented
266Dic3⋊C4—42no—
267Dic3⋊C4—81no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279D24—44no—
280D24—44no—
281C22×D4—31no—
282C22×D4—31no—
283C22×D4—31no—
284C22×D4—61no—
285C22×D4—61no—
286C23×C4—61no—
287C23×C4—61no—
288C2×D8—62no—
289C2×D8—62no—
290C2×D8—62no—
291C2×D8—121no—
292C2×D8—121no—
293C2×D8—121no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#127—62no—
299#127—62no—
300#126—124no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—124no—
304C4×D4—121no—
305C4×D4—121no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C42—21no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—22no—
334#107—121no—
335C4⋊C8—121no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#102—121no—
343#102—121no—
344C4×C8—121no—
345#95—21no—
346#95—21no—
347#95—31no—
348#95—31no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4C2×D812yes—
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3C2×D812yes—
410C2×Dic3C2×D812yes—
411D12—22no—
412D12—22no—
413D12—22no—
414D12—22no—
415D12—48no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—41no—
424D12—84no—
425C4×D3—84no—
426C4×D3—84no—
427C3×C8—22no—
428C3×C8—41no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C24—61no—
432C24—61no—
433C24—61no—
434C2×D4—32no—
435C2×D4—32no—
436C2×D4—64no—
437C2×D4—64no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—241no—
457C2×D4—241no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—61no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—241no—
472C22×C4—241no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476D8—124no—
477D8—124no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8#23512yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—34no—
498C4⋊C4—42no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—121no—
502C4⋊C4—122no—
503C4⋊C4—241no—
504C4⋊C4#23912yes—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—122no—
525C22×C3#52511yescharacteristic
526C22×C3#54611yescharacteristic
527C22×C3#55011yescharacteristic
528C22×C3C22×D811yescharacteristic
529C22×C3#57511yescharacteristic
530C22×C3#57611yescharacteristic
531C22×C3#57711yescharacteristic
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C3×C4—22no—
554C3×C4—41no—
555C3×C4—42no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4#52911yescharacteristic
560C3×C4#53811yescharacteristic
561C3×C4#55111yescharacteristic
562C3×C4#55311yescharacteristic
563Dic3—22no—
564Dic3—22no—
565Dic3—41no—
566Dic3—41no—
567Dic3—81no—
568C23—62no—
569C23—62no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23C2×D3×D411yescharacteristic
590D4—62no—
591D4—62no—
592D4—62no—
593D4—62no—
594D4—128no—
595D4—121no—
596D4—121no—
597D4—121no—
598D4—121no—
599D4—121no—
600D4—121no—
601D4—121no—
602D4—121no—
603D4—244no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—32no—
611C2×C4—32no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—244no—
637C2×C4—244no—
638C2×C4D3×D812yes—
639C2×C4C2×#23511yescharacteristic
640C2×C4C2×#23911yescharacteristic
641C2×C4D3×#4111yescharacteristic
642C2×C4D3×#4111yescharacteristic
643C2×C4#100911yescharacteristic
644C2×C4#100911yescharacteristic
645C8—22no—
646C8—41no—
647C8—121no—
648C8—121no—
649C2×C3#246711yescharacteristic
650C2×C3#310011yescharacteristic
651C2×C3#311811yescharacteristic
652C2×C3#329411yescharacteristic
653C2×C3#330911yescharacteristic
654C2×C3#339111yescharacteristic
655C2×C3#340111yescharacteristic
656D3—82no—
657D3—82no—
658D3—161no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—122no—
663C22—122no—
664C22—122no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22—241no—
669C22—241no—
670C22—241no—
671C22—241no—
672C22—241no—
673C22—241no—
674C22—241no—
675C22—241no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22#555311yescharacteristic
681C22#563011yescharacteristic
682C22#565611yescharacteristic
683C22C2×D3×D811yescharacteristic
684C22#576911yescharacteristic
685C22#577711yescharacteristic
686C22#577811yescharacteristic
687C4—22no—
688C4—42no—
689C4—41no—
690C4—62no—
691C4—62no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—121no—
696C4—121no—
697C4—241no—
698C4#552711yescharacteristic
699C4#558011yescharacteristic
700C4#569511yescharacteristic
701C4#570411yescharacteristic
702C3#2706211yescharacteristic, complemented, Sylow
703C2—242no—
704C2—242no—
705C2—481no—
706C2#7500211yescharacteristic
707C2#7957811yescharacteristic
708C2#7982211yescharacteristic
709C2#8098011yescharacteristic
710C2#8124711yescharacteristic
711C2#8126811yescharacteristic
712C2#8149611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1918C2
socleC23×C3C22×D4
prime radical#68674C2
hypercenter#1918D3
hypocenterC3#27062
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
1. #76047C2semidirect
2. #73653C2semidirect
3. C3×#1918C2semidirect
4. #70641C2semidirect
5. #70628C2semidirect
6. #69627C2semidirect
7. #69462C2semidirect
8. #68897C2semidirect
9. #68897C2semidirect
10. #68892C2semidirect
11. #68862C2semidirect
12. #68791C2semidirect
13. #68780C2semidirect
15. #68610C2semidirect
69. #1918D3semidirect
265. C4⋊Dic3D8semidirect
702. C3#27062semidirect

Maximal subgroups

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

subgroupindex
#760472
#736532
C3×#19182
#706412
#706282
#696272
#694622
#688972
#688922
#688622
#687912
#687802
#686742
#686102
#270623