Finite Group Databasebeta

← #10868331

Subgroups — #10868331

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#82343C211yesmaximal, characteristic, complemented
2#74333C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic, complemented
4#71613C211yesmaximal, characteristic, complemented
5#70481C211yesmaximal, characteristic, complemented
6#70447C211yesmaximal, characteristic, complemented
7#70209C211yesmaximal, characteristic
8#70190C211yesmaximal, characteristic, complemented
9#70177C211yesmaximal, characteristic, complemented
10#70173C211yesmaximal, characteristic, complemented
11#69586C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic, complemented
13#68815C211yesmaximal, characteristic, complemented
14#68791C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#27062—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×C4×#67—21no—
21C22×D24C2211yescharacteristic
22C2×#898—21no—
23C2×#893C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#384C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5213—21no—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4994—21no—
35#4992—21no—
36#4990—21no—
37#4988C2211yescharacteristic
38#4988C2211yescharacteristic
39#4984C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4957—21no—
48#4956—21no—
49#4939C2211yescharacteristic
50#4939C2211yescharacteristic
51C4×Dic3⋊C4C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
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#1918—31no—
70#1916—31no—
71#1879—31no—
72#1879—31no—
73#1802D311yescharacteristic, complemented
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×#67—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#940—22no—
96#940—22no—
97C2×#213—22no—
98C2×#213—22no—
99C2×#213—41no—
100C4×#67—44no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C2×D24—22no—
122C2×D24—41no—
123C2×D24—41no—
124#898—44no—
125#893—22no—
126#893—41no—
127C3×C4⋊C8—41no—
128C3×#102—41no—
129C3×#102—41no—
130#832—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#821—22no—
137#821—41no—
138C8×Dic3—41no—
139#809—22no—
140#809—22no—
141#809—41no—
142#807—22no—
143#807—41no—
144C22×D8—31no—
145C22×D8—31no—
146C2×#127—31no—
147C2×#126—61no—
148C2×C4×D4—31no—
149C2×C4×D4—61no—
150C2×C4×D4C2×D311yescharacteristic
151C2×#107—31no—
152C2×#107—31no—
153C2×C4⋊C8C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×#102C2×D311yescharacteristic
161C2×#102C2×D311yescharacteristic
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#389—31no—
172#388—61no—
173#386—61no—
174#386—61no—
175#384C2×D311yescharacteristic
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#336—31no—
184#336—31no—
185#336—31no—
186#336—31no—
187C24×C3—21no—
188C23×D3—21no—
189C23×D3—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4#4112yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—24no—
207C2×#67—42no—
208C2×#67—42no—
209C2×#67—81no—
210C2×#67—81no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12#4112yes—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
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—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—42no—
247C3×C42—21no—
248C3×C42—21no—
249#217—42no—
250#217—81no—
251#213—44no—
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—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266Dic3⋊C4—41no—
267Dic3⋊C4D814yescomplemented
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8#4112yes—
280D24—44no—
281C22×D4—31no—
282C22×D4—31no—
283C22×D4—61no—
284C22×D4—61no—
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
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—22no—
302C4×D4—41no—
303C4×D4—62no—
304C4×D4—121no—
305C4×D4—124no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
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—31no—
329C2×C42—31no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—121no—
335C4⋊C8—41no—
336#102—41no—
337#102—41no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—121no—
343#102—121no—
344C4×C8—121no—
345#95—31no—
346#95—31no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
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—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—41no—
375C3×D4—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
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×C4—41no—
390C2×C3×C4—84no—
391C2×C3×C4C2×D812yes—
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398#67—48no—
399#67—84no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3C2×D812yes—
417C2×Dic3C2×D812yes—
418D12—22no—
419D12—22no—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426C4×D3—84no—
427C3×C8—41no—
428C3×C8—41no—
429C3⋊C8—22no—
430C3⋊C8—41no—
431C24—21no—
432C24—61no—
433C24—61no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—32no—
440C2×D4—64no—
441C2×D4—64no—
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—122no—
454C2×D4—122no—
455C2×D4—241no—
456C2×D4—241no—
457C2×D4#23912yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—81no—
470C22×C4—122no—
471C22×C4—122no—
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—21no—
484C2×C8—32no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—34no—
496C4⋊C4—42no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—81no—
502C4⋊C4—121no—
503C4⋊C4—122no—
504C4⋊C4#23512yes—
505C22⋊C4—42no—
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—61no—
520C42—61no—
521C42—61no—
522C42—61no—
523C42—122no—
524C42—122no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3#52511yescharacteristic
533C22×C3#54611yescharacteristic
534C22×C3#55011yescharacteristic
535C22×C3C22×D811yescharacteristic
536C22×C3#57511yescharacteristic
537C22×C3#57611yescharacteristic
538C22×C3#57711yescharacteristic
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
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—42no—
555C3×C4—41no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4#52911yescharacteristic
559C3×C4#53811yescharacteristic
560C3×C4#55111yescharacteristic
561C3×C4#55311yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—41no—
565Dic3—41no—
566Dic3—81no—
567Dic3—81no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—62no—
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—22no—
591D4—22no—
592D4—41no—
593D4—41no—
594D4—62no—
595D4—62no—
596D4—128no—
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—22no—
608C2×C4—21no—
609C2×C4—21no—
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—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—84no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—244no—
638C2×C4D3×D812yes—
639C2×C4C2×#23511yescharacteristic
640C2×C4#100411yescharacteristic
641C2×C4#100411yescharacteristic
642C2×C4C2×#23911yescharacteristic
643C2×C4D3×#4111yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C8—41no—
646C8—41no—
647C8—62no—
648C8—121no—
649C2×C3—82no—
650C2×C3#246711yescharacteristic
651C2×C3#310011yescharacteristic
652C2×C3#311811yescharacteristic
653C2×C3#329411yescharacteristic
654C2×C3#330911yescharacteristic
655C2×C3#339111yescharacteristic
656C2×C3#340111yescharacteristic
657D3—82no—
658D3—161no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—122no—
667C22—122no—
668C22—122no—
669C22—241no—
670C22—241no—
671C22—241no—
672C22—241no—
673C22—241no—
674C22—241no—
675C22—241no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22#549011yescharacteristic
681C22#559911yescharacteristic
682C22#562711yescharacteristic
683C22C2×D3×D811yescharacteristic
684C22#576211yescharacteristic
685C22#577411yescharacteristic
686C22#577711yescharacteristic
687C4—22no—
688C4—41no—
689C4—42no—
690C4—62no—
691C4—62no—
692C4—81no—
693C4—81no—
694C4—121no—
695C4—121no—
696C4—241no—
697C4—241no—
698C4#551411yescharacteristic
699C4#552811yescharacteristic
700C4#561311yescharacteristic
701C4#563711yescharacteristic
702C3#2706211yescharacteristic, complemented, Sylow
703C2—82no—
704C2—242no—
705C2—481no—
706C2#7390611yescharacteristic
707C2#7875711yescharacteristic
708C2#7889711yescharacteristic
709C2#8032411yescharacteristic
710C2#8042411yescharacteristic
711C2#8094811yescharacteristic
712C2#8104811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radical#70209C2
hypercenter#1802D3
hypocenterC3#27062
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. #82343C2semidirect
2. #74333C2semidirect
3. C3×#1802C2semidirect
4. #71613C2semidirect
5. #70481C2semidirect
6. #70447C2semidirect
8. #70190C2semidirect
9. #70177C2semidirect
10. #70173C2semidirect
11. #69586C2semidirect
12. #68897C2semidirect
13. #68815C2semidirect
14. #68791C2semidirect
15. #68780C2semidirect
73. #1802D3semidirect
267. Dic3⋊C4D8semidirect
702. C3#27062semidirect

Maximal subgroups

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

subgroupindex
#823432
#743332
C3×#18022
#716132
#704812
#704472
#702092
#701902
#701772
#701732
#695862
#688972
#688152
#687912
#687802
#270623