Finite Group Databasebeta

← #10840206

Subgroups — #10840206

Order 768.

756 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74324C211yesmaximal, characteristic
2#74240C211yesmaximal, characteristic
3C3×#1836C211yesmaximal, characteristic, complemented
4#69837C211yesmaximal, characteristic
5#69813C211yesmaximal, characteristic, complemented
6#69588C211yesmaximal, characteristic, complemented
7#69540C211yesmaximal, characteristic, complemented
8#68890C211yesmaximal, characteristic, complemented
9#68879C211yesmaximal, characteristic, complemented
10#68879C211yesmaximal, characteristic, complemented
11#68873C211yesmaximal, characteristic, complemented
12#68833C211yesmaximal, characteristic, complemented
13#68825C211yesmaximal, characteristic, complemented
14#68824C211yesmaximal, characteristic, complemented
15#68813C211yesmaximal, characteristic, complemented
16#24980—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×#384C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#4990—21no—
33#4990—21no—
34#4988—21no—
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39Dic3×C4⋊C4—21no—
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4672—21no—
47#4670—21no—
48#4670—21no—
49#4661—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2536—31no—
61#2354—31no—
62#2077—31no—
63#2076—31no—
64#2036—31no—
65#2035—31no—
66#1955—31no—
67#1949—31no—
68#1873—31no—
69#1873—31no—
70#1836D311yescharacteristic, complemented
71#1825—31no—
72#1825—31no—
73#1803—31no—
74#1763—31no—
75C22×D12D411yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
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)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#897—22no—
112#897—22no—
113#897—41no—
114#897—41no—
115#892—22no—
116#892—22no—
117#892—41no—
118#892—41no—
119C4×D12—22no—
120C4×D12—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#97—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131(C2×D3)⋊C8—41no—
132(C2×D3)⋊C8—41no—
133(C2×D3)⋊C8—41no—
134(C2×D3)⋊C8—41no—
135#816—22no—
136#816—22no—
137#810—41no—
138#810—41no—
139#808—41no—
140#807—41no—
141C4⋊(C3⋊C8)—22no—
142C2×C22⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×D4—31no—
146C2×#117—31no—
147C2×#117—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×#97C2×D311yescharacteristic
160#398—61no—
161#394—61no—
162#390—61no—
163#388—61no—
164#388—61no—
165#388—61no—
166#388—61no—
167#384C2×D311yescharacteristic
168#382C2×D311yescharacteristic
169#380—31no—
170#380—31no—
171#380—31no—
172#380—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#340C2×D311yescharacteristic
185C23×D3C2×D411yescharacteristic
186C23×D3C2×D411yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—22no—
201C2×D12—41no—
202C2×D12—41no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—81no—
237C3×C42#4112yes—
238C3×C42#4112yes—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—81no—
248(C2×D3)⋊C4—81no—
249C4⋊Dic3—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—42no—
256C4⋊Dic3—42no—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—42no—
259C4×Dic3—42no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C22×Q8—31no—
270C22×D4—31no—
271C23×C4—31no—
272C23×C4—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—121no—
280C22⋊Q8—121no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×D4—62no—
284C4×D4—62no—
285#117—62no—
286#117—62no—
287#117—121no—
288#117—121no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—62no—
320#103—62no—
321#103—62no—
322#103—121no—
323#103—121no—
324C22⋊C8—121no—
325C22⋊C8—121no—
326C22⋊C8—121no—
327C22⋊C8—121no—
328#97—22no—
329#95—21no—
330#95—21no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3C22×D411yescharacteristic
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4#12312yes—
368C2×C3×C4#12312yes—
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393D12—44no—
394C4×D3—41no—
395C4×D3—41no—
396C4×D3—41no—
397C4×D3—41no—
398C4×D3—41no—
399C4×D3—41no—
400C4×D3—41no—
401C4×D3—41no—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406C3×C8—41no—
407C3×C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C24—31no—
411C24—31no—
412C2×Q8—32no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×D4—62no—
419C2×D4—121no—
420C2×D4—121no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—42no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—81no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—241no—
487C22⋊C4—241no—
488C42—122no—
489C42—122no—
490C42—122no—
491C42—122no—
492C42#23512yes—
493C42#23912yes—
494C22×C3C2×#12311yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3#54611yescharacteristic
497C22×C3#57511yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—41no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—41no—
514C2×D3—41no—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4#52911yescharacteristic
522C3×C4#54011yescharacteristic
523C3×C4#55311yescharacteristic
524C3×C4#55511yescharacteristic
525Dic3—42no—
526Dic3—41no—
527Dic3—41no—
528Dic3—42no—
529Dic3—81no—
530Dic3—81no—
531C23—61no—
532C23—61no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23—61no—
545C23C2×D3×D411yescharacteristic
546Q8—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
550D4—124no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4#89412yes—
594C2×C4#89412yes—
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23911yescharacteristic
597C2×C4D3×#4111yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4#100911yescharacteristic
600C2×C4#100911yescharacteristic
601C8—41no—
602C8—41no—
603C8—121no—
604C8—121no—
605C2×C3#242011yescharacteristic
606C2×C3#309311yescharacteristic
607C2×C3#309711yescharacteristic
608C2×C3#331311yescharacteristic
609C2×C3#331511yescharacteristic
610C2×C3#331711yescharacteristic
611C2×C3#332111yescharacteristic
612D3—81no—
613D3—81no—
614C22—121no—
615C22—121no—
616C22—121no—
617C22—121no—
618C22—121no—
619C22—121no—
620C22—121no—
621C22—121no—
622C22—121no—
623C22—121no—
624C22—121no—
625C22—121no—
626C22—121no—
627C22—121no—
628C22C2×#89411yescharacteristic
629C22#567611yescharacteristic
630C22#567611yescharacteristic
631C22#577311yescharacteristic
632C22#577511yescharacteristic
633C22#577711yescharacteristic
634C22#577911yescharacteristic
635C4—22no—
636C4—22no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—122no—
642C4—121no—
643C4—121no—
644C4—122no—
645C4—241no—
646C4—241no—
647C4#553011yescharacteristic
648C4#553211yescharacteristic
649C4#569111yescharacteristic
650C4#569711yescharacteristic
651C3#2498011yescharacteristic, complemented, Sylow
652C2—241no—
653C2—241no—
654C2#7423111yescharacteristic
655C2#7921611yescharacteristic
656C2#7922011yescharacteristic
657C2#8169911yescharacteristic
658C2#8170311yescharacteristic
659C2#8172611yescharacteristic
660C2#8175011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1836C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1836D3
hypocenterC3#24980
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
3. C3×#1836C2semidirect
5. #69813C2semidirect
6. #69588C2semidirect
7. #69540C2semidirect
8. #68890C2semidirect
9. #68879C2semidirect
10. #68879C2semidirect
11. #68873C2semidirect
12. #68833C2semidirect
13. #68825C2semidirect
14. #68824C2semidirect
15. #68813C2semidirect
70. #1836D3semidirect
651. C3#24980semidirect

Maximal subgroups

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

subgroupindex
#743242
#742402
C3×#18362
#698372
#698132
#695882
#695402
#688902
#688792
#688732
#688332
#688252
#688242
#688132
#249803