Finite Group Databasebeta

← #10877404

Subgroups — #10877404

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#82257C211yesmaximal, characteristic
2#73898C211yesmaximal, characteristic, complemented
3C3×#1825C211yesmaximal, characteristic, complemented
4#71580C211yesmaximal, characteristic, complemented
5#69947C211yesmaximal, characteristic, complemented
6#69933C211yesmaximal, characteristic
7#69515C211yesmaximal, characteristic
8#68903C211yesmaximal, characteristic, complemented
9#68898C211yesmaximal, characteristic, complemented
10#68853C211yesmaximal, characteristic, complemented
11#68852C211yesmaximal, characteristic, complemented
12#68848C211yesmaximal, characteristic, complemented
13#68843C211yesmaximal, characteristic, complemented
14#68812C211yesmaximal, characteristic, complemented
15#68803C211yesmaximal, characteristic, complemented
16#27696—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24C4×#217C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#4988C2211yescharacteristic
30#4988C2211yescharacteristic
31#4984C2211yescharacteristic
32#4977C2211yescharacteristic
33#4977C2211yescharacteristic
34#4975C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38#4959—21no—
39#4959—21no—
40#4948—21no—
41#4947—21no—
42#4940—21no—
43#4940—21no—
44#4654—21no—
45#4646—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#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#1825D311yescharacteristic, complemented
73C8×C22⋊C4—31no—
74#1735—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×D12C2311yescharacteristic
78C22×Dic6—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217—21no—
91C2×#217D411yescharacteristic
92C2×#217D411yescharacteristic
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—41no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118#888—22no—
119#888—41no—
120C3×C22⋊C8—22no—
121C3×#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#821—22no—
131#821—41no—
132C8×Dic3—22no—
133C8×Dic3—41no—
134#809—22no—
135#809—41no—
136C23×C8—31no—
137C2×C4⋊Q8—31no—
138C2×#121—31no—
139C2×#117—31no—
140C2×#117C2×D311yescharacteristic
141C2×#102—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×C22⋊C8C2×D311yescharacteristic
146C2×C4×C8—31no—
147C2×C4×C8—31no—
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#380C2×D311yescharacteristic
164C4×C22⋊C4—31no—
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#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
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×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
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—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—81no—
228C3×C42—42no—
229#217—22no—
230#217—22no—
231#217—42no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—81no—
236(C2×D3)⋊C4—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—42no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—44no—
259C2×C3⋊C8#4112yes—
260C22×Q8—61no—
261C22×D4—31no—
262C22×D4—61no—
263C23×C4C22×D311yescharacteristic
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—62no—
267C22×C8—62no—
268C22×C8—121no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C4⋊Q8—62no—
272C4⋊Q8—121no—
273#121—62no—
274#121—62no—
275#117—22no—
276#117—22no—
277#117—62no—
278#117—121no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—21no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305#102—62no—
306#102—62no—
307#102—121no—
308#102—121no—
309C22⋊C8—22no—
310C4×C8—62no—
311C4×C8—121no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
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—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4#12412yes—
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—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
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#13512yes—
382C2×Dic3#13512yes—
383D12—41no—
384D12—41no—
385D12—41no—
386D12—41no—
387Dic6—84no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—22no—
391C3⋊C8—41no—
392C24—61no—
393C24D3×D411yescharacteristic
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—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4#23512yes—
422C22×C4D3×D411yescharacteristic
423C22×C4#23712yes—
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—32no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—124no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—32no—
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—42no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—81no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C42—42no—
468C42—61no—
469C42—61no—
470C42—61no—
471C42—61no—
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—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#52911yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4#55111yescharacteristic
502C3×C4#55211yescharacteristic
503Dic3—22no—
504Dic3—22no—
505Dic3—41no—
506Dic3—41no—
507Dic3—42no—
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#88512yes—
524C23C2×D3×D411yescharacteristic
525Q8—244no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—244no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4#88512yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4#100911yescharacteristic
576C2×C4#101011yescharacteristic
577C8—41no—
578C8—41no—
579C8—62no—
580C8—121no—
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×#88511yescharacteristic
605C22#565411yescharacteristic
606C22#565511yescharacteristic
607C22D3×#13511yescharacteristic
608C22D3×#13511yescharacteristic
609C22#577411yescharacteristic
610C22#577811yescharacteristic
611C4—22no—
612C4—62no—
613C4—62no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—122no—
620C4—121no—
621C4—241no—
622C4—241no—
623C4#552711yescharacteristic
624C4#553111yescharacteristic
625C4#564911yescharacteristic
626C4#566511yescharacteristic
627C3#2769611yescharacteristic, complemented, Sylow
628C2—42no—
629C2—481no—
630C2#7407611yescharacteristic
631C2#7882511yescharacteristic
632C2#7887611yescharacteristic
633C2#8138211yescharacteristic
634C2#8138911yescharacteristic
635C2#8152511yescharacteristic
636C2#8153211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1825C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1825D3
hypocenterC3#27696
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. #73898C2semidirect
3. C3×#1825C2semidirect
4. #71580C2semidirect
5. #69947C2semidirect
8. #68903C2semidirect
9. #68898C2semidirect
10. #68853C2semidirect
11. #68852C2semidirect
12. #68848C2semidirect
13. #68843C2semidirect
14. #68812C2semidirect
15. #68803C2semidirect
72. #1825D3semidirect
627. C3#27696semidirect

Maximal subgroups

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

subgroupindex
#822572
#738982
C3×#18252
#715802
#699472
#699332
#695152
#689032
#688982
#688532
#688522
#688482
#688432
#688122
#688032
#276963