Finite Group Databasebeta

← #10868362

Subgroups — #10868362

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83173C211yesmaximal, characteristic, complemented
2#76048C211yesmaximal, characteristic, complemented
3C3×#2041C211yesmaximal, characteristic, complemented
4#71767C211yesmaximal, characteristic
5#70212C211yesmaximal, characteristic, complemented
6#70190C211yesmaximal, characteristic, complemented
7#70182C211yesmaximal, characteristic, complemented
8#70172C211yesmaximal, characteristic, complemented
9#70172C211yesmaximal, characteristic, complemented
10#70156C211yesmaximal, characteristic, complemented
11#70154C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69603C211yesmaximal, characteristic, complemented
14#68626C211yesmaximal, characteristic, complemented
15#68609C211yesmaximal, characteristic, complemented
16#27065—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C22×#213C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5221—21no—
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31#5220—21no—
32#5220—21no—
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#4992—21no—
41#4990—21no—
42#4988C2211yescharacteristic
43#4988C2211yescharacteristic
44#4977C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4674—21no—
51#4670—21no—
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2366—31no—
62#2092—31no—
63#2054—31no—
64#2041D311yescharacteristic, complemented
65#2011—31no—
66#1970—31no—
67#1926—31no—
68#1926—31no—
69#1921—31no—
70#1916—31no—
71#1879—31no—
72#1879—31no—
73#1804—31no—
74C4×#102—31no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×D12C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#127—22no—
85C3×#127—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96D4×Dic3—22no—
97D4×Dic3—22no—
98D4×Dic3—41no—
99D4×Dic3—41no—
100D4×Dic3—44no—
101C2×#213—22no—
102C2×#213—22no—
103C2×#213—22no—
104C2×#213—41no—
105C2×#213—41no—
106C2×#213—41no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C3×#102—22no—
125C3×#102—22no—
126C3×#102—41no—
127C3×#102—41no—
128C3×#95—21no—
129#832—22no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#821—41no—
139#817—41no—
140Dic3⋊C8—41no—
141C8×Dic3—41no—
142#807—22no—
143C22×D8—31no—
144C22×D8—31no—
145C2×#127C2×D311yescharacteristic
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4—61no—
149C2×#107—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158C2×#102C2×D311yescharacteristic
159C2×#102C2×D311yescharacteristic
160C2×C4×C8—31no—
161#397—61no—
162#397—61no—
163#390—31no—
164#390—31no—
165#390—31no—
166#390—31no—
167#390—31no—
168#390—61no—
169#389C2×D311yescharacteristic
170#388—61no—
171#388—61no—
172#386—61no—
173#386—61no—
174#382—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C24×C3—21no—
187C24×C3—21no—
188C23×D3—21no—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—42no—
202C2×C3×D4—42no—
203C2×C3×D4—81no—
204C2×C3×D4—81no—
205C2×C3×D4#4112yes—
206C2×C3×D4#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—81no—
220C22×Dic3—81no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—81no—
242C3×C42—21no—
243C3×C42—21no—
244#217—42no—
245#217—42no—
246#217—42no—
247#217—42no—
248#217—81no—
249#217—81no—
250#213—44no—
251#213—44no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258C4⋊Dic3—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3—22no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3#4112yes—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—42no—
271C4×Dic3—81no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8#4112yes—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—31no—
281C22×D4C22×D311yescharacteristic
282C22×D4C22×D311yescharacteristic
283C23×C4—61no—
284C23×C4—61no—
285C2×D8—62no—
286C2×D8—62no—
287C2×D8—62no—
288C2×D8—121no—
289C2×D8—121no—
290C2×D8—121no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#127—22no—
296#127—22no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—121no—
300C4×D4—121no—
301C4×D4—124no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
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×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#106—121no—
329C4⋊C8—121no—
330#102—22no—
331#102—22no—
332#102—41no—
333#102—41no—
334#102—62no—
335#102—62no—
336#102—121no—
337#102—121no—
338C4×C8—121no—
339#95—21no—
340#95—31no—
341#95—31no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3C22×D411yescharacteristic
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—48no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—84no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
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#13512yes—
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—22no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—84no—
417C2×Dic3—84no—
418C2×Dic3AGL(1,ℤ/8)12yes—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—41no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—22no—
426C3⋊C8—41no—
427C24—21no—
428C24—21no—
429C24—61no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—81no—
451C2×D4—81no—
452C2×D4#23712yes—
453C2×D4#23712yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472D8—124no—
473D8—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—32no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—81no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C22⋊C4—241no—
511C42—21no—
512C42—21no—
513C42—61no—
514C42—61no—
515C42—61no—
516C42—61no—
517C42—122no—
518C42—241no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3#52511yescharacteristic
534C22×C3#54611yescharacteristic
535C22×C3#55011yescharacteristic
536C22×C3C2×#13511yescharacteristic
537C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
538C22×C3#57611yescharacteristic
539C22×C3#57611yescharacteristic
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C3×C4—22no—
548C3×C4—41no—
549C3×C4—81no—
550C3×C4—81no—
551C3×C4#52911yescharacteristic
552C3×C4#53811yescharacteristic
553C3×C4#55111yescharacteristic
554C3×C4#55311yescharacteristic
555Dic3—22no—
556Dic3—41no—
557Dic3—41no—
558Dic3—41no—
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562C23—22no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23C2×D3×D411yescharacteristic
584D4—22no—
585D4—22no—
586D4—22no—
587D4—22no—
588D4—48no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—84no—
594D4—121no—
595D4—121no—
596D4—121no—
597D4—121no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—32no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—244no—
631C2×C4—244no—
632C2×C4#91612yes—
633C2×C4C2×#23711yescharacteristic
634C2×C4C2×#23711yescharacteristic
635C2×C4#100411yescharacteristic
636C2×C4D3×#4111yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4#100911yescharacteristic
639C8—41no—
640C8—41no—
641C8—62no—
642C8—121no—
643C2×C3—82no—
644C2×C3—82no—
645C2×C3#246711yescharacteristic
646C2×C3#310511yescharacteristic
647C2×C3#312211yescharacteristic
648C2×C3#330511yescharacteristic
649C2×C3#331811yescharacteristic
650C2×C3#339611yescharacteristic
651C2×C3#339911yescharacteristic
652D3—161no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—241no—
668C22—241no—
669C22—241no—
670C22—241no—
671C22—241no—
672C22—241no—
673C22—241no—
674C22#558611yescharacteristic
675C22#559911yescharacteristic
676C22#567611yescharacteristic
677C22#575711yescharacteristic
678C22C2×#91611yescharacteristic
679C22D3×AGL(1,ℤ/8)11yescharacteristic
680C22#577411yescharacteristic
681C4—22no—
682C4—41no—
683C4—62no—
684C4—81no—
685C4—81no—
686C4—121no—
687C4—122no—
688C4—121no—
689C4—121no—
690C4—241no—
691C4—241no—
692C4#560111yescharacteristic
693C4#561711yescharacteristic
694C4#564911yescharacteristic
695C4#565811yescharacteristic
696C3#2706511yescharacteristic, complemented, Sylow
697C2—82no—
698C2—82no—
699C2—481no—
700C2#7692011yescharacteristic
701C2#7999311yescharacteristic
702C2#8013311yescharacteristic
703C2#8032811yescharacteristic
704C2#8037511yescharacteristic
705C2#8157911yescharacteristic
706C2#8172711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radical#71767C2
hypercenter#2041D3
hypocenterC3#27065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
1. #83173C2semidirect
2. #76048C2semidirect
3. C3×#2041C2semidirect
5. #70212C2semidirect
6. #70190C2semidirect
7. #70182C2semidirect
8. #70172C2semidirect
9. #70172C2semidirect
10. #70156C2semidirect
11. #70154C2semidirect
12. Dic3×#102C2semidirect
13. #69603C2semidirect
14. #68626C2semidirect
15. #68609C2semidirect
64. #2041D3semidirect
696. C3#27065semidirect

Maximal subgroups

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

subgroupindex
#831732
#760482
C3×#20412
#717672
#702122
#701902
#701822
#701722
#701562
#701542
Dic3×#1022
#696032
#686262
#686092
#270653