Finite Group Databasebeta

← #10876212

Subgroups — #10876212

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#82269C211yesmaximal, characteristic
2#73868C211yesmaximal, characteristic, complemented
3C3×#1854C211yesmaximal, characteristic, complemented
4#71576C211yesmaximal, characteristic, complemented
5#70640C211yesmaximal, characteristic, complemented
6#70623C211yesmaximal, characteristic, complemented
7#70586C211yesmaximal, characteristic, complemented
8#70569C211yesmaximal, characteristic, complemented
9#69959C211yesmaximal, characteristic
10#69911C211yesmaximal, characteristic, complemented
11#69521C211yesmaximal, characteristic
12#68902C211yesmaximal, characteristic, complemented
13#68902C211yesmaximal, characteristic, complemented
14#68792C211yesmaximal, characteristic, complemented
15#68791C211yesmaximal, characteristic, complemented
16#27591—31nomaximal, Sylow
17C23×D12C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic, complemented
21C2×C3×#106C2211yescharacteristic, complemented
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24#5212C2211yescharacteristic
25C2×#830C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4988C2211yescharacteristic
33#4988C2211yescharacteristic
34#4977C2211yescharacteristic, complemented
35#4975C2211yescharacteristic, complemented
36Dic3×C4⋊C4C2211yescharacteristic, complemented
37Dic3×C4⋊C4C2211yescharacteristic, complemented
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic, complemented
43C2×#807C2211yescharacteristic, complemented
44#4955—21no—
45#4955—21no—
46#4945—21no—
47#4944—21no—
48#4939—21no—
49#4939—21no—
50#4654—21no—
51#4653—21no—
52C3×#340C2211yescharacteristic, complemented
53C3×#340C2211yescharacteristic, complemented
54#4550C2211yescharacteristic, complemented
55#4550C2211yescharacteristic, complemented
56#4550C2211yescharacteristic, complemented
57#4550C2211yescharacteristic, complemented
58#4529C2211yescharacteristic, complemented
59#4529C2211yescharacteristic, complemented
60#2314—31no—
61#2309—31no—
62#2092—31no—
63#2092—31no—
64#2071—31no—
65#2071—31no—
66#1926—31no—
67#1926—31no—
68#1925—31no—
69#1925—31no—
70#1854—31no—
71#1854D311yescharacteristic, complemented
72#1818—31no—
73#1786—31no—
74#1786—31no—
75C24×D3—21no—
76C23×C3×C4C2311yescharacteristic
77C22×D12—21no—
78C22×D12—22no—
79C22×D12—22no—
80C22×D12—22no—
81C22×D12—22no—
82C22×D12—41no—
83C22×D12—41no—
84C22×D12C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#117—41no—
89C3×#117—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
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)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108#926—22no—
109#926—22no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C3×#107—41no—
123C3×#106—41no—
124C3×C22⋊C8—22no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#830—22no—
132#821—22no—
133#821—22no—
134#821—41no—
135#821—41no—
136#809—22no—
137#809—22no—
138#809—41no—
139#809—41no—
140#808—41no—
141#807—41no—
142C23×D4—31no—
143C2×#117—31no—
144C2×#117C2×D311yescharacteristic
145C2×#107—31no—
146C2×#107C2×D311yescharacteristic, complemented
147C2×#106—31no—
148C2×#106C2×D311yescharacteristic, complemented
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#390—31no—
160#390—31no—
161#390—31no—
162#390—31no—
163#390—61no—
164#390—61no—
165#388—61no—
166#388—61no—
167#387—61no—
168#387—61no—
169#384—31no—
170#382—31no—
171#380—31no—
172#380C2×D311yescharacteristic
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#336—31no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C23×D3—21no—
188C23×D3—24no—
189C23×D3—42no—
190C23×D3—42no—
191C23×D3—81no—
192C23×D3—81no—
193C23×D3—81no—
194C23×D3—81no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4#4112yes—
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—24no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—44no—
225C2×D12—44no—
226C2×D12—44no—
227C2×D12—44no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12#4112yes—
231C2×D12#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—21no—
245C3×C4⋊C4—21no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C42—42no—
249#217—22no—
250#217—22no—
251#217—81no—
252(C2×D3)⋊C4—81no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4—81no—
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—
260C4⋊Dic3—22no—
261C4⋊Dic3—22no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266Dic3⋊C4—81no—
267Dic3⋊C4—81no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
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—31no—
283C22×D4—62no—
284C22×D4—62no—
285C22×D4—62no—
286C22×D4—61no—
287C22×D4—62no—
288C22×D4—121no—
289C22×D4—121no—
290C23×C4C22×D311yescharacteristic
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#117—41no—
296#117—41no—
297#117—62no—
298#117—62no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
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—21no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42—61no—
327#107—41no—
328#107—121no—
329#106—41no—
330#106—121no—
331#102—62no—
332#102—62no—
333#102—62no—
334#102—62no—
335#102—121no—
336#102—121no—
337#102—121no—
338#102—121no—
339C22⋊C8—22no—
340C22⋊C8—62no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3#12312yes—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—48no—
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—84no—
371C22×D3—84no—
372C22×D3—84no—
373C22×D3—84no—
374C22×D3—84no—
375C22×D3—84no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
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—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4#12312yes—
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
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—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419D12—22no—
420D12—22no—
421D12—22no—
422D12—22no—
423D12—48no—
424D12—41no—
425D12—41no—
426D12—41no—
427D12—41no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—64no—
433C24—61no—
434C24—61no—
435C24—122no—
436C24—122no—
437C24—241no—
438C24—241no—
439C24—241no—
440C24—241no—
441C24D3×D411yescharacteristic
442C2×D4—32no—
443C2×D4—32no—
444C2×D4—64no—
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—61no—
455C2×D4—124no—
456C2×D4—124no—
457C2×D4—124no—
458C2×D4—124no—
459C2×D4—122no—
460C2×D4—122no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4#23512yes—
476C22×C4D3×D411yescharacteristic
477C22×C4#23912yes—
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—21no—
498C4⋊C4—21no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—241no—
506C4⋊C4—241no—
507C22⋊C4—42no—
508C22⋊C4—62no—
509C22⋊C4—62no—
510C22⋊C4—81no—
511C22⋊C4—241no—
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—42no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—61no—
525C42—241no—
526C42—241no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3C2×#12311yescharacteristic
535C22×C3#52511yescharacteristic
536C22×C3#53511yescharacteristic
537C22×C3#57611yescharacteristic
538C22×C3#57611yescharacteristic
539C22×C3#57711yescharacteristic
540C22×C3#57711yescharacteristic
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—816no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C3×C4—22no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4#55111yescharacteristic
562C3×C4#55111yescharacteristic
563C3×C4#55111yescharacteristic
564C3×C4#55111yescharacteristic
565Dic3—41no—
566Dic3—41no—
567Dic3—41no—
568Dic3—41no—
569Dic3—81no—
570Dic3—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—62no—
578C23—62no—
579C23—128no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—244no—
593C23—244no—
594C23—244no—
595C23—244no—
596C23—244no—
597C23—244no—
598C23#89412yes—
599C23C2×D3×D411yescharacteristic
600D4—62no—
601D4—62no—
602D4—62no—
603D4—62no—
604D4—128no—
605D4—121no—
606D4—121no—
607D4—121no—
608D4—121no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
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—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4#89412yes—
646C2×C4C2×#23511yescharacteristic
647C2×C4#100411yescharacteristic
648C2×C4#100411yescharacteristic
649C2×C4C2×#23911yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C8—41no—
653C8—41no—
654C8—121no—
655C8—121no—
656C2×C3—42no—
657C2×C3#248211yescharacteristic
658C2×C3#309611yescharacteristic
659C2×C3#309611yescharacteristic
660C2×C3#312211yescharacteristic
661C2×C3#312311yescharacteristic
662C2×C3#322411yescharacteristic
663C2×C3#322411yescharacteristic
664D3—82no—
665D3—82no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—122no—
677C22—122no—
678C22—122no—
679C22—2416no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22C2×#89411yescharacteristic
689C22#559211yescharacteristic
690C22#565611yescharacteristic
691C22#576911yescharacteristic
692C22#577011yescharacteristic
693C22#577411yescharacteristic
694C22#577811yescharacteristic
695C4—22no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—121no—
701C4—121no—
702C4—121no—
703C4—121no—
704C4—241no—
705C4—241no—
706C4#555611yescharacteristic
707C4#555611yescharacteristic
708C4#563711yescharacteristic
709C4#563711yescharacteristic
710C3#2759111yescharacteristic, complemented, Sylow
711C2—42no—
712C2—242no—
713C2—242no—
714C2#7461411yescharacteristic
715C2#7917411yescharacteristic
716C2#7921911yescharacteristic
717C2#8011911yescharacteristic
718C2#8026311yescharacteristic
719C2#8135411yescharacteristic
720C2#8149811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1854D3
hypocenterC3#27591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. #73868C2semidirect
3. C3×#1854C2semidirect
4. #71576C2semidirect
5. #70640C2semidirect
6. #70623C2semidirect
7. #70586C2semidirect
8. #70569C2semidirect
10. #69911C2semidirect
12. #68902C2semidirect
13. #68902C2semidirect
14. #68792C2semidirect
15. #68791C2semidirect
20. C2×C3×#107C22semidirect
21. C2×C3×#106C22semidirect
34. #4977C22semidirect
35. #4975C22semidirect
36. Dic3×C4⋊C4C22semidirect
37. Dic3×C4⋊C4C22semidirect
42. C2×#808C22semidirect
43. C2×#807C22semidirect
52. C3×#340C22semidirect
53. C3×#340C22semidirect
54. #4550C22semidirect
55. #4550C22semidirect
56. #4550C22semidirect
57. #4550C22semidirect
58. #4529C22semidirect
59. #4529C22semidirect
71. #1854D3semidirect
146. C2×#107C2×D3semidirect
148. C2×#106C2×D3semidirect
179. #340C2×D3semidirect
180. #340C2×D3semidirect
710. C3#27591semidirect

Maximal subgroups

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

subgroupindex
#822692
#738682
C3×#18542
#715762
#706402
#706232
#705862
#705692
#699592
#699112
#695212
#689022
#687922
#687912
#275913