Finite Group Databasebeta

← #10800221

Subgroups — #10800221

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C211yesmaximal, characteristic
2C4×D3×M16C211yesmaximal, characteristic, complemented
3#73752C211yesmaximal, characteristic, complemented
4C3×#1986C211yesmaximal, characteristic, complemented
5#71841C211yesmaximal, characteristic, complemented
6#69867C211yesmaximal, characteristic, complemented
7#69492C211yesmaximal, characteristic, complemented
8#68755C214yesmaximal, complemented
9#68729C214yesmaximal, complemented
10#21383—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×D3×M16C2212yes—
13#5728C411yescharacteristic, complemented
14#5728C411yescharacteristic, complemented
15#5725C411yescharacteristic, complemented
16#5725C411yescharacteristic, complemented
17#5691C412yescomplemented
18#5691C412yescomplemented
19#5627—21no—
20#5627—21no—
21#5624—21no—
22#5624—21no—
23#5580—21no—
24#5580—21no—
25C4×#239—21no—
26C4×#239—21no—
27#5524C2212yes—
28#5522C2212yes—
29#5477C2211yescharacteristic
30C2×C42×D3C2211yescharacteristic
31C3×C4×M16C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#825C2214yes—
34Dic3×M16C2212yes—
35#4987—21no—
36#4977—21no—
37C4×(C2×D3)⋊C4C2212yes—
38C4×C4⋊Dic3C2211yescharacteristic
39#4932C2212yes—
40C2×#805C2214yes—
41C4×#208C2211yescharacteristic
42#4707—21no—
43D3×#97—21no—
44C4×ΓL(1,𝔽25)—21no—
45C4×D3×C8—21no—
46#4664—21no—
47#4651—21no—
48C3×#339C2214yes—
49#4532C2214yes—
50#3444—31no—
51C2×C4×M16—31no—
52#2283—31no—
53#1986—31no—
54#1986D311yescharacteristic, complemented
55#1980—31no—
56#1980—31no—
57#1871—34no—
58#1759—34no—
59C2×#239—21no—
60C2×#239—21no—
61C2×#235C2312yes—
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3C2311yescharacteristic
65C2×C3×M16C2312yes—
66C3×C4⋊Q8C2×C411yescharacteristic
67C3×C4⋊Q8C2×C411yescharacteristic
68C3×#125C2×C412yes—
69C3×C22⋊Q8—21no—
70C3×C22⋊Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×#117C2312yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#947—21no—
78#947—21no—
79#946—21no—
80#946—21no—
81Q8×Dic3—21no—
82Q8×Dic3—21no—
83#930—21no—
84#930—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—21no—
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3C2312yes—
90C2×C4×Dic3C2311yescharacteristic
91C2×#208C2312yes—
92D3×M16—24no—
93D3×M16—24no—
94C2×ΓL(1,𝔽25)—22no—
95C2×D3×C8—22no—
96#898—22no—
97#898—22no—
98#897C2×C412yes—
99#897C2×C412yes—
100#895—21no—
101#895—21no—
102#895C2×C412yes—
103#895C2×C412yes—
104#894—22no—
105#894—22no—
106#893—22no—
107#893—22no—
108#893—21no—
109#893—21no—
110#893C2×C412yes—
111#893C2×C412yes—
112#892—21no—
113#892—21no—
114#892—22no—
115#892—22no—
116#890—21no—
117#890—21no—
118#887C2×C412yes—
119#887C2×C412yes—
120#886—22no—
121#882—22no—
122#67⋊C4—22no—
123#67⋊C4—22no—
124#875C2×C412yes—
125#874C2×C411yescharacteristic
126C4×D12—22no—
127C4×D12—22no—
128C4×D12—21no—
129C4×D12—21no—
130#872—22no—
131C42×D3—22no—
132C42×D3C2×C411yescharacteristic
133C42×D3C2×C411yescharacteristic
134C42×D3C2×C411yescharacteristic
135C42×D3C2×C411yescharacteristic
136C42×D3D411yescharacteristic
137C42×D3D411yescharacteristic
138C42×D3D411yescharacteristic
139C42×D3D411yescharacteristic
140#869C2×C411yescharacteristic
141C3×#97—21no—
142C3×C4×C8—21no—
143C3×#95—21no—
144C3⋊#95—21no—
145C3⋊#95—22no—
146C3⋊#95—22no—
147C3⋊#95—22no—
148#825—28no—
149#815—22no—
150C8×Dic3—22no—
151#805—28no—
152#803—21no—
153C4×C3⋊C8—21no—
154C22×M16—32no—
155#533—31no—
156#533—31no—
157#533C2×D311yescharacteristic
158#532—31no—
159#532—31no—
160#532—31no—
161#529—32no—
162#529—32no—
163#529—61no—
164#529—61no—
165#529—61no—
166#529—61no—
167#525—32no—
168#525—61no—
169#525—61no—
170C4×#41—32no—
171C4×#41—61no—
172C4×#41—61no—
173C22×C42—31no—
174C2×#104—34no—
175C2×#104—34no—
176C4×M16—32no—
177C4×M16—31no—
178C4×M16—61no—
179C4×M16—61no—
180C4×M16C2×D311yescharacteristic
181C2×#97—61no—
182C2×C4×C8—61no—
183#383—61no—
184#382—61no—
185#381—61no—
186#380—61no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4C2×D311yescharacteristic
189C4×C22⋊C4—32no—
190#376—32no—
191#339—34no—
192#339C2×D314yes—
193C23×D3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C22×C3×C4—22no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—22no—
201C22×Dic3—22no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205#239—44no—
206#239—44no—
207#235—24no—
208#235—24no—
209C2×D12—22no—
210C2×D12—22no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12C22×C412yes—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3C22⋊C411yescharacteristic
222C2×C4×D3C22⋊C411yescharacteristic
223C2×C4×D3C22⋊C411yescharacteristic
224C2×C4×D3C22⋊C411yescharacteristic
225C2×C4×D3C22⋊C411yescharacteristic
226C2×C4×D3C22⋊C411yescharacteristic
227C2×C4×D3C22⋊C411yescharacteristic
228C2×C4×D3C22⋊C411yescharacteristic
229C2×C4×D3C22×C411yescharacteristic
230C2×C4×D3C22×C411yescharacteristic
231C2×C4×D3C2×D411yescharacteristic
232C2×C4×D3C2×D411yescharacteristic
233C2×Dic6C22×C412yes—
234C3×M16—22no—
235C3×M16#4114yescomplemented
236C2×C3×C8—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4C22×C412yes—
243C3×C4⋊C4C22×C412yes—
244C3×C22⋊C4—22no—
245C3×C42—22no—
246C3×C42—24no—
247C3×C42C22×C411yescharacteristic
248C3×C42C22×C411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D411yescharacteristic
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—44no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—22no—
263C4⋊Dic3—22no—
264C4⋊Dic3—42no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267C4×Dic3—22no—
268C4×Dic3—22no—
269C4×Dic3—24no—
270C4×Dic3C22⋊C411yescharacteristic
271C4×Dic3C22⋊C411yescharacteristic
272C4×Dic3C22⋊C411yescharacteristic
273C4×Dic3C22⋊C411yescharacteristic
274C4×Dic3C22⋊C411yescharacteristic
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22⋊C411yescharacteristic
278C4×Dic3C22×C411yescharacteristic
279C4×Dic3C22×C411yescharacteristic
280C4×Dic3C2×D411yescharacteristic
281C4×Dic3C2×D411yescharacteristic
282#208—22no—
283#208#4114yescomplemented
284C2×C3⋊C8—22no—
285ΓL(1,𝔽25)—28no—
286D3×C8—28no—
287C2×#41—32no—
288C2×#41—61no—
289C2×#41—61no—
290C23×C4—31no—
291C23×C4—31no—
292C23×C4—31no—
293C2×M16—32no—
294C2×M16—62no—
295C2×M16—64no—
296C2×M16—64no—
297C2×M16C22×D312yes—
298C22×C8—62no—
299C4⋊Q8—31no—
300C4⋊Q8C4×D311yescharacteristic
301C4⋊Q8C4×D311yescharacteristic
302#127—31no—
303#126—62no—
304#126—62no—
305#125—61no—
306#125—61no—
307#125C4×D312yes—
308#124—32no—
309#124—61no—
310#124—61no—
311#123—62no—
312#123—62no—
313#123—62no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C22⋊Q8—32no—
317C22⋊Q8—32no—
318C22⋊Q8—61no—
319C22⋊Q8—61no—
320#121—32no—
321#121—32no—
322#121—61no—
323#121—61no—
324#121—61no—
325#121—61no—
326#120—62no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—32no—
330C4×Q8—32no—
331C4×Q8—61no—
332C4×Q8—61no—
333C4×D4—32no—
334C4×D4—32no—
335C4×D4—62no—
336C4×D4—62no—
337C4×D4—61no—
338C4×D4—61no—
339C4×D4—62no—
340C4×D4—62no—
341C4×D4—62no—
342C4×D4—62no—
343C4×D4—61no—
344C4×D4—61no—
345#117—62no—
346#117—62no—
347#117—62no—
348#117—61no—
349#117—61no—
350#117C22×D312yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—61no—
353C2×C22⋊C4—62no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—32no—
366C2×C42—62no—
367C2×C42C22×D312yes—
368C2×C42C22×D311yescharacteristic
369#104—68no—
370#104—68no—
371#97—21no—
372#97—62no—
373#97—61no—
374C4×C8—21no—
375C4×C8—62no—
376C4×C8—61no—
377#95—21no—
378#95—62no—
379#95—62no—
380#95—62no—
381#95—61no—
382C23×C3C22×D411yescharacteristic
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—41no—
388C22×D3C2×C22⋊C411yescharacteristic
389C22×D3C2×C22⋊C411yescharacteristic
390C3×Q8—42no—
391C3×Q8—42no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4C2×C22⋊C411yescharacteristic
400C2×C3×C4C2×C22⋊C411yescharacteristic
401C2×C3×C4C2×C22⋊C411yescharacteristic
402C2×C3×C4C2×C22⋊C411yescharacteristic
403C2×C3×C4C23×C411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4112yes—
406C2×C3×C4C2×#4112yes—
407#67—42no—
408#67—42no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—42no—
415C2×Dic3—42no—
416C2×Dic3—42no—
417C2×Dic3C2×C22⋊C411yescharacteristic
418C2×Dic3C2×C22⋊C411yescharacteristic
419C2×Dic3C2×C22⋊C411yescharacteristic
420C2×Dic3C2×C22⋊C411yescharacteristic
421C2×Dic3C2×C22⋊C411yescharacteristic
422C2×Dic3C2×C22⋊C411yescharacteristic
423D12—24no—
424D12—42no—
425D12—44no—
426D12—44no—
427D12—42no—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—41no—
437C4×D3—41no—
438C4×D3—41no—
439C4×D3—41no—
440C4×D3—44no—
441C4×D3—44no—
442C4×D3—44no—
443Dic6—24no—
444C3×C8—24no—
445C3⋊C8—24no—
446C24—31no—
447#41—64no—
448#41—64no—
449#41—124no—
450#41—124no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—32no—
454C2×D4—32no—
455C2×D4—62no—
456C2×D4—62no—
457C2×D4—62no—
458C2×D4—61no—
459C2×D4—61no—
460C22×C4—22no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—122no—
483C22×C4—122no—
484C22×C4D3×D411yescharacteristic
485C22×C4D3×D411yescharacteristic
486C22×C4C23×D311yescharacteristic
487M16—22no—
488M16—34no—
489M16—68no—
490M16—62no—
491M16#23914yescomplemented
492C2×C8—22no—
493C2×C8—68no—
494C2×C8—62no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—122no—
507C4⋊C4C2×C4×D312yes—
508C4⋊C4C2×C4×D312yes—
509C22⋊C4—22no—
510C22⋊C4—62no—
511C22⋊C4—62no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—62no—
518C22⋊C4—124no—
519C42—22no—
520C42—24no—
521C42—31no—
522C42—31no—
523C42—31no—
524C42—31no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—31no—
530C42—31no—
531C42—31no—
532C42—31no—
533C42—62no—
534C42—62no—
535C42—64no—
536C42C2×C4×D311yescharacteristic
537C42C2×C4×D311yescharacteristic
538C42D3×D411yescharacteristic
539C42D3×D411yescharacteristic
540C22×C3—21no—
541C22×C3—21no—
542C22×C3C22×C22⋊C411yescharacteristic
543C22×C3#53211yescharacteristic
544C22×C3#53211yescharacteristic
545C2×D3—21no—
546C2×D3—21no—
547C2×D3—21no—
548C2×D3—21no—
549C2×D3—21no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—42no—
555C2×D3#42112yes—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—82no—
562C3×C4#42112yes—
563C3×C4C4×#4112yes—
564C3×C4#52512yes—
565Dic3—21no—
566Dic3—21no—
567Dic3—21no—
568Dic3—21no—
569Dic3—21no—
570Dic3—21no—
571Dic3—42no—
572Dic3—82no—
573C23—31no—
574C23—31no—
575C23—62no—
576C23—61no—
577C23—61no—
578C23—62no—
579C23—121no—
580C23C2×D3×D411yescharacteristic
581Q8—42no—
582Q8—42no—
583Q8—64no—
584D4—64no—
585D4—122no—
586D4—124no—
587D4—124no—
588D4—122no—
589D4—122no—
590D4—122no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—124no—
620C2×C4—124no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—124no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4D3×C22⋊C411yescharacteristic
628C2×C4D3×C22⋊C411yescharacteristic
629C2×C4D3×C22⋊C411yescharacteristic
630C2×C4D3×C22⋊C411yescharacteristic
631C2×C4C22×C4×D311yescharacteristic
632C2×C4C2×D3×D411yescharacteristic
633C2×C4C2×#23912yes—
634C2×C4D3×#4112yes—
635C8—24no—
636C8—64no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3#229711yescharacteristic
640C2×C3C2×#42111yescharacteristic
641C2×C3C2×#42111yescharacteristic
642D3—22no—
643D3—41no—
644D3—82no—
645C22—21no—
646C22—21no—
647C22—32no—
648C22—61no—
649C22—61no—
650C22—61no—
651C22—61no—
652C22—61no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22C2×D3×C22⋊C411yescharacteristic
659C22#566811yescharacteristic
660C22#572511yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—42no—
666C4—61no—
667C4—61no—
668C4—61no—
669C4—61no—
670C4—61no—
671C4—61no—
672C4—82no—
673C4—122no—
674C4—242no—
675C4#482112yes—
676C4C4×#23912yes—
677C4#562712yes—
678C3#2138311yescharacteristic, complemented, Sylow
679C2—21no—
680C2—21no—
681C2—62no—
682C2—121no—
683C2—242no—
684C2#7647811yescharacteristic
685C2C2×#482111yescharacteristic
686C2D3×#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1986C2
socleC22×C3C22×C22⋊C4
prime radical#86257C2
cosocleC22×C3×C4C24
hypercenter#1986D3
hypocenterC3#21383
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. C4×D3×M16C2semidirect
3. #73752C2semidirect
4. C3×#1986C2semidirect
5. #71841C2semidirect
6. #69867C2semidirect
7. #69492C2semidirect
8. #68755C2semidirect
9. #68729C2semidirect
13. #5728C4semidirect
14. #5728C4semidirect
15. #5725C4semidirect
16. #5725C4semidirect
17. #5691C4semidirect
18. #5691C4semidirect
54. #1986D3semidirect
235. C3×M16#41semidirect
283. #208#41semidirect
491. M16#239semidirect
678. C3#21383semidirect

Maximal subgroups

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

subgroupindex
#862572
C4×D3×M162
#737522
C3×#19862
#718412
#698672
#694922
#687552
#687292
#213833