Finite Group Databasebeta

← #10877402

Subgroups — #10877402

Order 768.

752 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73515C211yesmaximal, characteristic
2#73492C211yesmaximal, characteristic, complemented
3C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
4#71973C211yesmaximal, characteristic
5#71969C211yesmaximal, characteristic, complemented
6#69933C211yesmaximal, characteristic
7#69926C211yesmaximal, characteristic, complemented
8#69777C211yesmaximal, characteristic, complemented
9#69768C211yesmaximal, characteristic, complemented
10#69768C211yesmaximal, characteristic, complemented
11#69763C211yesmaximal, characteristic, complemented
12#68678C211yesmaximal, characteristic, complemented
13#68644C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68637C211yesmaximal, characteristic, complemented
16#27696—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31#4961—21no—
32#4961—21no—
33#4948—21no—
34#4947—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4936C2211yescharacteristic
38#4936C2211yescharacteristic
39#4935C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4670—21no—
43#4670—21no—
44#4655—21no—
45#4646—21no—
46C3×#336C2211yescharacteristic
47C3×#336C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2500—31no—
61#2439—31no—
62#2101—31no—
63#2100—31no—
64#2073—31no—
65#2023—31no—
66#1990—31no—
67#1934—31no—
68#1934—31no—
69#1893—31no—
70#1875—31no—
71#1835—31no—
72#1825—31no—
73C8×C22⋊C4D311yescharacteristic, complemented
74#1735—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×D12C2311yescharacteristic
78C22×Dic6—21no—
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#930—22no—
94#930—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100#926—22no—
101#926—22no—
102#926—22no—
103#926—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118#869—22no—
119#869—41no—
120C3×C22⋊C8—22no—
121C3×C4×C8—22no—
122C3×C4×C8—41no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#821—22no—
133#821—22no—
134#821—41no—
135#821—41no—
136C23×C8C2×D311yescharacteristic
137C2×C4⋊Q8—31no—
138C2×#121—31no—
139C2×#117—31no—
140C2×#117—31no—
141C2×#102—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×C22⋊C8C2×D311yescharacteristic
146C2×C4×C8C2×D311yescharacteristic
147C2×C4×C8C2×D311yescharacteristic
148#396—61no—
149#394—61no—
150#394—61no—
151#393—61no—
152#391—61no—
153#390—31no—
154#390—31no—
155#389—31no—
156#389—31no—
157#388—61no—
158#388—61no—
159#387—61no—
160#384—31no—
161#382—31no—
162#382—31no—
163#380—31no—
164C4×C22⋊C4C2×D311yescharacteristic
165#340—31no—
166#340—31no—
167#340—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#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—42no—
194C2×#67—81no—
195C2×#67—81no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×Dic6—42no—
210C2×Dic6—42no—
211C2×Dic6—81no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—44no—
224C2×C3×C8#4112yes—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231#217—42no—
232#217—42no—
233#217—81no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—42no—
253C4⋊Dic3—81no—
254C4⋊Dic3—81no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C22×Q8—61no—
261C22×D4—31no—
262C22×D4—61no—
263C23×C4C22×D311yescharacteristic
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—41no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C4⋊Q8—62no—
272C4⋊Q8—121no—
273#121—62no—
274#121—62no—
275#117—62no—
276#117—62no—
277#117—62no—
278#117—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4D1211yescharacteristic
300C2×C22⋊C4D1211yescharacteristic
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#102—62no—
306#102—62no—
307#102—121no—
308#102—121no—
309C22⋊C8—22no—
310C4×C8—22no—
311C4×C8—41no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95D1211yescharacteristic
320#95D1211yescharacteristic
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3#12412yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4#12412yes—
353C2×C3×C4#13512yes—
354C2×C3×C4#13512yes—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
361#67—84no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—41no—
384D12—41no—
385D12—41no—
386D12—41no—
387Dic6—84no—
388C3×C8—22no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C24—61no—
393C24C2×D1211yescharacteristic
394C2×Q8—122no—
395C2×Q8—122no—
396C2×Q8—241no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—122no—
404C2×D4—241no—
405C2×D4—241no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4#23512yes—
423C22×C4#23512yes—
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—44no—
437C2×C8#23712yes—
438C4⋊C4—32no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4—241no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—241no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—122no—
472C42—122no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3C2×#12411yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54711yescharacteristic
483C22×C3C2×#13511yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3#57611yescharacteristic
486C22×C3#57711yescharacteristic
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4#52911yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4#55111yescharacteristic
502C3×C4#55211yescharacteristic
503Dic3—42no—
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23#87512yes—
524C23C22×D1211yescharacteristic
525Q8—244no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—244no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—62no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4#87512yes—
569C2×C4#90412yes—
570C2×C4#90412yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×#23511yescharacteristic
573C2×C4C2×#23711yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#101011yescharacteristic
577C8—22no—
578C8—41no—
579C8—41no—
580C8—41no—
581C2×C3—42no—
582C2×C3#248811yescharacteristic
583C2×C3#313111yescharacteristic
584C2×C3#313611yescharacteristic
585C2×C3#329711yescharacteristic
586C2×C3#330111yescharacteristic
587C2×C3#334011yescharacteristic
588C2×C3#334111yescharacteristic
589D3—161no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—241no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22C2×#87511yescharacteristic
605C22#555411yescharacteristic
606C22#555511yescharacteristic
607C22C2×#90411yescharacteristic
608C22C2×#90411yescharacteristic
609C22#575111yescharacteristic
610C22#575211yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—122no—
617C4—241no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4—241no—
622C4—241no—
623C4#549511yescharacteristic
624C4#553111yescharacteristic
625C4#553811yescharacteristic
626C4#556411yescharacteristic
627C3#2769611yescharacteristic, complemented, Sylow
628C2—42no—
629C2—481no—
630C2#7350411yescharacteristic
631C2#7846311yescharacteristic
632C2#7856311yescharacteristic
633C2#7941711yescharacteristic
634C2#7941811yescharacteristic
635C2#7944111yescharacteristic
636C2#7945011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C8×C22⋊C4C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC8×C22⋊C4D3
hypocenterC3#27696
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #73492C2semidirect
3. C3×C8×C22⋊C4C2semidirect
5. #71969C2semidirect
7. #69926C2semidirect
8. #69777C2semidirect
9. #69768C2semidirect
10. #69768C2semidirect
11. #69763C2semidirect
12. #68678C2semidirect
13. #68644C2semidirect
14. #68643C2semidirect
15. #68637C2semidirect
73. C8×C22⋊C4D3semidirect
627. C3#27696semidirect

Maximal subgroups

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

subgroupindex
#735152
#734922
C3×C8×C22⋊C42
#719732
#719692
#699332
#699262
#697772
#697682
#697632
#686782
#686442
#686432
#686372
#276963