Finite Group Databasebeta

← #10877389

Subgroups — #10877389

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82177C211yesmaximal, characteristic
2#73899C211yesmaximal, characteristic, complemented
3C3×#1823C211yesmaximal, characteristic, complemented
4C2×#4529C211yesmaximal, characteristic, complemented
5#69947C211yesmaximal, characteristic, complemented
6#69932C211yesmaximal, characteristic
7#69515C211yesmaximal, characteristic
8#68902C211yesmaximal, characteristic, complemented
9#68898C211yesmaximal, characteristic, complemented
10#68853C211yesmaximal, characteristic, complemented
11#68852C211yesmaximal, characteristic, complemented
12#68847C211yesmaximal, characteristic, complemented
13#68842C211yesmaximal, characteristic, complemented
14#68810C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#27695—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#929—21no—
20C2×#929—21no—
21C22×C4⋊Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#878—21no—
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C4×#217C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4993—21no—
33#4993—21no—
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4984C2211yescharacteristic
37#4975C2211yescharacteristic
38#4975C2211yescharacteristic
39#4975C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4946—21no—
44#4657—21no—
45#4656—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2501—31no—
61#2440—31no—
62#2099—31no—
63#2099—31no—
64#2071—31no—
65#2023—31no—
66#1990—31no—
67#1934—31no—
68#1934—31no—
69#1893—31no—
70#1874—31no—
71#1834—31no—
72#1823D311yescharacteristic, complemented
73C8×C22⋊C4—31no—
74C2×#340—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×#67—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×Dic6—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×#217D411yescharacteristic
92C2×#217D411yescharacteristic
93#930—22no—
94#930—22no—
95#929—44no—
96#929—44no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—41no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8—22no—
120C22×C3⋊C8—22no—
121C22×C3⋊C8—41no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#888—22no—
125#888—41no—
126#878—44no—
127C3×C22⋊C8—22no—
128C3×#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#821—22no—
134#821—41no—
135C8×Dic3—22no—
136C8×Dic3—41no—
137#809—22no—
138#809—41no—
139C23×C8—31no—
140C2×C4⋊Q8—31no—
141C2×#123—61no—
142C2×#123—61no—
143C2×C22⋊Q8—61no—
144C2×#121—31no—
145C22×C4⋊C4—31no—
146C22×C4⋊C4C2×D311yescharacteristic
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×C22⋊C8C2×D311yescharacteristic
152C2×C4×C8—31no—
153C2×C4×C8—31no—
154#399—61no—
155#399—61no—
156#398—61no—
157#395—61no—
158#390—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#385—61no—
163#384—31no—
164#384—31no—
165#384—31no—
166#380C2×D311yescharacteristic
167C4×C22⋊C4—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#336—31no—
181#336—31no—
182C24×C3C2×D411yescharacteristic
183C23×D3—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—42no—
200C2×#67—42no—
201C2×#67—42no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—81no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×Dic6—42no—
219C2×Dic6—42no—
220C2×Dic6—81no—
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×C22⋊C4—42no—
237#217D814yes—
238#217SD1614yes—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—42no—
250C4⋊Dic3—42no—
251C4⋊Dic3#4112yes—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—81no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—44no—
270C2×C3⋊C8#4112yes—
271C22×Q8—61no—
272C22×D4—31no—
273C22×D4—61no—
274C23×C4—21no—
275C23×C4—61no—
276C23×C4C22×D311yescharacteristic
277C22×C8—31no—
278C22×C8—31no—
279C22×C8—62no—
280C22×C8—62no—
281C22×C8—121no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—62no—
285C4⋊Q8—121no—
286#123—124no—
287#123—124no—
288C22⋊Q8—124no—
289#121—62no—
290#121—62no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—62no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—121no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323#102—62no—
324#102—62no—
325#102—121no—
326#102—121no—
327C22⋊C8—22no—
328C4×C8—62no—
329C4×C8—121no—
330#95—21no—
331#95—31no—
332#95—31no—
333#95—61no—
334#95—61no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3#12412yes—
340C23×C3C2×D812yes—
341C23×C3C2×SD1612yes—
342C23×C3C22×D411yescharacteristic
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
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—41no—
368C2×C3×C4—84no—
369C2×C3×C4#12412yes—
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376#67—84no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—84no—
395C2×Dic3C2×D812yes—
396C2×Dic3C2×D812yes—
397C2×Dic3C2×SD1612yes—
398C2×Dic3C2×SD1612yes—
399D12—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403Dic6—84no—
404C3×C8—41no—
405C3×C8—41no—
406C3⋊C8—22no—
407C3⋊C8—41no—
408C24—61no—
409C24D3×D411yescharacteristic
410C2×Q8—122no—
411C2×Q8—122no—
412C2×Q8—241no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—122no—
420C2×D4—122no—
421C2×D4—122no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—241no—
443C22×C4#23512yes—
444C22×C4D3×D411yescharacteristic
445C22×C4#23712yes—
446C22×C4C23×D311yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—32no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—124no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—241no—
485C22⋊C4—34no—
486C22⋊C4—34no—
487C22⋊C4—42no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—61no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3#48614yes—
504C22×C3C2×#12411yescharacteristic
505C22×C3#54611yescharacteristic
506C22×C3#54711yescharacteristic
507C22×C3C22×D811yescharacteristic
508C22×C3C22×SD1611yescharacteristic
509C22×C3#57711yescharacteristic
510C22×C3#57811yescharacteristic
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C3×C4—22no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#52911yescharacteristic
524C3×C4#53911yescharacteristic
525C3×C4#55111yescharacteristic
526C3×C4#55211yescharacteristic
527Dic3—22no—
528Dic3—22no—
529Dic3—22no—
530Dic3—22no—
531Dic3—42no—
532Dic3—81no—
533Dic3—81no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23#88512yes—
546C23D3×D812yes—
547C23D3×SD1612yes—
548C23C2×D3×D411yescharacteristic
549Q8—244no—
550D4—121no—
551D4—121no—
552D4—121no—
553D4—121no—
554D4—244no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—32no—
564C2×C4—32no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—84no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—244no—
596C2×C4#88512yes—
597C2×C4C2×#23511yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4#100911yescharacteristic
602C2×C4#101011yescharacteristic
603C8—41no—
604C8—41no—
605C8—62no—
606C8—121no—
607C2×C3—24no—
608C2×C3#248811yescharacteristic
609C2×C3C2×#48611yescharacteristic
610C2×C3#313711yescharacteristic
611C2×C3#329411yescharacteristic
612C2×C3#329811yescharacteristic
613C2×C3#333411yescharacteristic
614C2×C3#333511yescharacteristic
615D3—161no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—241no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22#476014yes—
630C22C2×#88511yescharacteristic
631C22#565411yescharacteristic
632C22#565511yescharacteristic
633C22C2×D3×D811yescharacteristic
634C22C2×D3×SD1611yescharacteristic
635C22#577811yescharacteristic
636C22#577911yescharacteristic
637C4—22no—
638C4—62no—
639C4—62no—
640C4—62no—
641C4—62no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—122no—
647C4—241no—
648C4—241no—
649C4#552711yescharacteristic
650C4#553111yescharacteristic
651C4#564911yescharacteristic
652C4#566511yescharacteristic
653C3#2769511yescharacteristic, complemented, Sylow
654C2—24no—
655C2—481no—
656C2#7407611yescharacteristic
657C2C2×#476011yescharacteristic
658C2#7895211yescharacteristic
659C2#8129111yescharacteristic
660C2#8130211yescharacteristic
661C2#8143511yescharacteristic
662C2#8145011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1823D3
hypocenterC3#27695
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. #73899C2semidirect
3. C3×#1823C2semidirect
4. C2×#4529C2semidirect
5. #69947C2semidirect
8. #68902C2semidirect
9. #68898C2semidirect
10. #68853C2semidirect
11. #68852C2semidirect
12. #68847C2semidirect
13. #68842C2semidirect
14. #68810C2semidirect
15. #68803C2semidirect
72. #1823D3semidirect
653. C3#27695semidirect

Maximal subgroups

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

subgroupindex
#821772
#738992
C3×#18232
C2×#45292
#699472
#699322
#695152
#689022
#688982
#688532
#688522
#688472
#688422
#688102
#688032
#276953