Finite Group Databasebeta

← #10876208

Subgroups — #10876208

Order 768.

954 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
1#76428C211yesmaximal, characteristic
2#74358C211yesmaximal, characteristic, complemented
3C3×#2071C211yesmaximal, characteristic, complemented
4#70198C211yesmaximal, characteristic
5#70198C211yesmaximal, characteristic
6#70156C211yesmaximal, characteristic, complemented
7#70155C211yesmaximal, characteristic, complemented
8#69654C211yesmaximal, characteristic, complemented
9#69585C211yesmaximal, characteristic, complemented
10#68877C211yesmaximal, characteristic, complemented
11#68876C211yesmaximal, characteristic, complemented
12#68792C211yesmaximal, characteristic, complemented
13#68791C211yesmaximal, characteristic, complemented
14#68680C211yesmaximal, characteristic, complemented
15#68665C211yesmaximal, characteristic, complemented
16#27591—31nomaximal, Sylow
17C22×D3×D4C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×#872C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C3×#390C2211yescharacteristic
23C3×#384C2211yescharacteristic, complemented
24#5221C2211yescharacteristic
25C2×#832C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic, complemented
32C2×#817C2211yescharacteristic, complemented
33#4990—21no—
34#4990—21no—
35#4988C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4978—21no—
39#4978—21no—
40Dic3×C4⋊C4C2211yescharacteristic, complemented
41Dic3×C4⋊C4C2211yescharacteristic, complemented
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic, complemented
45C2×#807C2211yescharacteristic, complemented
46#4957—21no—
47#4957—21no—
48#4955—21no—
49#4955—21no—
50#4939C2211yescharacteristic
51#4936C2211yescharacteristic, complemented
52C3×#340C2211yescharacteristic, complemented
53C3×#340C2211yescharacteristic, complemented
54C3×#336C2211yescharacteristic, complemented
55#4550C2211yescharacteristic, complemented
56#4550C2211yescharacteristic, complemented
57#4529C2211yescharacteristic, complemented
58#4529C2211yescharacteristic, complemented
59#4524C2211yescharacteristic, complemented
60#2314—31no—
61#2309—31no—
62#2092—31no—
63#2092—31no—
64#2071—31no—
65#2071D311yescharacteristic, complemented
66#1926—31no—
67#1926—31no—
68#1925—31no—
69#1925—31no—
70#1854—31no—
71#1854—31no—
72#1818—31no—
73#1786—31no—
74#1786—31no—
75C24×D3—21no—
76C22×C3×D4C2311yescharacteristic
77C22×#67—21no—
78C2×D3×D4—22no—
79C2×D3×D4—22no—
80C2×D3×D4—22no—
81C2×D3×D4—22no—
82C2×D3×D4—41no—
83C2×D3×D4—41no—
84C22×D12C2311yescharacteristic
85C22×C4×D3C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—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×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#892—41no—
119#892—41no—
120#872—22no—
121#872—22no—
122C3×#102—22no—
123C3×#102—22no—
124#832—41no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#821—41no—
133#821—41no—
134(C2×D3)⋊C8—22no—
135(C2×D3)⋊C8—22no—
136#818—41no—
137#817—41no—
138#809—22no—
139#809—22no—
140#808—41no—
141#807—41no—
142C23×D4—31no—
143C2×#117—31no—
144C2×#117—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#390—31no—
160#390—31no—
161#390—31no—
162#390—61no—
163#390—61no—
164#390C2×D311yescharacteristic
165#388—61no—
166#388—61no—
167#387—61no—
168#387—61no—
169#384C2×D311yescharacteristic, complemented
170#382—31no—
171#380—31no—
172#380—31no—
173#379—61no—
174#379—61no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic, complemented
180#340C2×D311yescharacteristic, complemented
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic, complemented
185C24×C3—21no—
186C23×D3—21no—
187C23×D3—24no—
188C23×D3—42no—
189C23×D3—42no—
190C23×D3—81no—
191C23×D3—81no—
192C23×D3—81no—
193C23×D3—81no—
194C23×D3C2×D411yescharacteristic
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—24no—
208C2×#67—42no—
209C2×#67—42no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216D3×D4—44no—
217D3×D4—44no—
218D3×D4—44no—
219D3×D4—44no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12#4112yes—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3#4112yes—
231C2×C4×D3#4112yes—
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—21no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250#217—81no—
251#217—81no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
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—21no—
266C4⋊Dic3—21no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C4×Dic3—81no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C25—61no—
281C22×D4—31no—
282C22×D4—62no—
283C22×D4—62no—
284C22×D4—61no—
285C22×D4—62no—
286C22×D4—62no—
287C22×D4—121no—
288C22×D4—121no—
289C22×D4C22×D311yescharacteristic
290C23×C4—31no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#117—62no—
296#117—62no—
297#117—121no—
298#117—121no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—31no—
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—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—121no—
328#107—121no—
329#106—121no—
330#106—121no—
331#102—22no—
332#102—22no—
333#102—62no—
334#102—62no—
335#102—121no—
336#102—121no—
337#102—121no—
338#102—121no—
339C22⋊C8—62no—
340C22⋊C8—62no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3C22×D411yescharacteristic
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—22no—
362C22×D3—48no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—84no—
370C22×D3—84no—
371C22×D3—84no—
372C22×D3—84no—
373C22×D3—84no—
374C22×D3—84no—
375C22×D3#12312yes—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401#67—48no—
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×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3#12312yes—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—22no—
427C4×D3—22no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—21no—
433C24—31no—
434C24—64no—
435C24—61no—
436C24—122no—
437C24—122no—
438C24—241no—
439C24—241no—
440C24—241no—
441C24—241no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—32no—
448C2×D4—64no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—122no—
455C2×D4—124no—
456C2×D4—124no—
457C2×D4—124no—
458C2×D4—124no—
459C2×D4—122no—
460C2×D4#23712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—31no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4D3×D411yescharacteristic
477C22×C4D3×D411yescharacteristic
478C22×C4C23×D311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C22⋊C4—62no—
508C22⋊C4—62no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—122no—
512C22⋊C4—241no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C42—21no—
521C42—21no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—241no—
526C42—241no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3C2×#12311yescharacteristic
535C22×C3#52511yescharacteristic
536C22×C3#53511yescharacteristic
537C22×C3#57611yescharacteristic
538C22×C3#57611yescharacteristic
539C22×C3#57711yescharacteristic
540C22×C3#57711yescharacteristic
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—22no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—41no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—816no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4#55111yescharacteristic
562C3×C4#55111yescharacteristic
563C3×C4#55111yescharacteristic
564C3×C4#55111yescharacteristic
565Dic3—22no—
566Dic3—41no—
567Dic3—41no—
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571C23—22no—
572C23—32no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—62no—
586C23—128no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—244no—
594C23—244no—
595C23—244no—
596C23—244no—
597C23—244no—
598C23—244no—
599C23C2×D3×D411yescharacteristic
600D4—22no—
601D4—22no—
602D4—41no—
603D4—41no—
604D4—62no—
605D4—62no—
606D4—128no—
607D4—121no—
608D4—121no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—32no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4C2×#23711yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4#100911yescharacteristic
651C2×C4#100911yescharacteristic
652C8—41no—
653C8—41no—
654C8—121no—
655C8—121no—
656C2×C3—82no—
657C2×C3#248211yescharacteristic
658C2×C3#309611yescharacteristic
659C2×C3#309611yescharacteristic
660C2×C3#312211yescharacteristic
661C2×C3#312311yescharacteristic
662C2×C3#322411yescharacteristic
663C2×C3#322411yescharacteristic
664D3—42no—
665D3—82no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—62no—
670C22—62no—
671C22—62no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22—122no—
681C22—122no—
682C22—122no—
683C22—2416no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22#560611yescharacteristic
689C22D3×#12311yescharacteristic
690C22#567411yescharacteristic
691C22#575711yescharacteristic
692C22#576211yescharacteristic
693C22#577411yescharacteristic
694C22#577811yescharacteristic
695C4—41no—
696C4—41no—
697C4—62no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—121no—
702C4—121no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#565811yescharacteristic
707C4#565811yescharacteristic
708C4#570511yescharacteristic
709C4#570511yescharacteristic
710C3#2759111yescharacteristic, complemented, Sylow
711C2—82no—
712C2—122no—
713C2—242no—
714C2#7767611yescharacteristic
715C2#8031711yescharacteristic
716C2#8041311yescharacteristic
717C2#8128411yescharacteristic
718C2#8132711yescharacteristic
719C2#8157011yescharacteristic
720C2#8168811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenter#2071D3
hypocenterC3#27591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #74358C2semidirect
3. C3×#2071C2semidirect
6. #70156C2semidirect
7. #70155C2semidirect
8. #69654C2semidirect
9. #69585C2semidirect
10. #68877C2semidirect
11. #68876C2semidirect
12. #68792C2semidirect
13. #68791C2semidirect
14. #68680C2semidirect
15. #68665C2semidirect
23. C3×#384C22semidirect
31. C2×#818C22semidirect
32. C2×#817C22semidirect
40. Dic3×C4⋊C4C22semidirect
41. Dic3×C4⋊C4C22semidirect
44. C2×#808C22semidirect
45. C2×#807C22semidirect
51. #4936C22semidirect
52. C3×#340C22semidirect
53. C3×#340C22semidirect
54. C3×#336C22semidirect
55. #4550C22semidirect
56. #4550C22semidirect
57. #4529C22semidirect
58. #4529C22semidirect
59. #4524C22semidirect
65. #2071D3semidirect
169. #384C2×D3semidirect
179. #340C2×D3semidirect
180. #340C2×D3semidirect
184. #336C2×D3semidirect
710. C3#27591semidirect

Maximal subgroups

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

subgroupindex
#764282
#743582
C3×#20712
#701982
#701562
#701552
#696542
#695852
#688772
#688762
#687922
#687912
#686802
#686652
#275913