Finite Group Databasebeta

← #10701663

Subgroups — #10701663

Order 768.

1194 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#823C211yesmaximal, characteristic
2C22×#823C211yesmaximal, characteristic
3C2×Dic3×M16C211yesmaximal, characteristic, complemented
4C2×#4988C211yesmaximal, characteristic
5C2×C3×#343C211yesmaximal, characteristic, complemented
6#71979C211yesmaximal, characteristic, complemented
7C2×#4530C211yesmaximal, characteristic, complemented
8#10962—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×(C2×D3)⋊C4C414yes—
14C22×C4×Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C2×#823—22no—
18C2×#823—22no—
19C2×#823—22no—
20C2×#823—22no—
21C2×#823C414yescomplemented
22C2×#823C414yescomplemented
23Dic3×M16—22no—
24Dic3×M16—22no—
25C2×#815—21no—
26C2×C8×Dic3—21no—
27#4988—22no—
28#4988—21no—
29#4988—21no—
30C3×#343—22no—
31#4550—22no—
32#4530—22no—
33C22×#100—31no—
34C22×#100—31no—
35C2×C4×M16—31no—
36C2×#390—31no—
37C2×#343—31no—
38C2×#343D311yescharacteristic, complemented
39#1728—31no—
40C24×D3C2×C412yes—
41C23×C3×C4C2×C412yes—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2×C412yes—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12—21no—
52C22×D12—21no—
53C22×D12—21no—
54C22×D12—21no—
55C22×D12—21no—
56C22×D12C2×C412yes—
57C22×D12C2×C412yes—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16C2×C412yes—
67C2×C3×M16C2×C412yes—
68C2×C3×M16D412yes—
69C2×C3×M16D412yes—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—48no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3D412yes—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208C2×C412yes—
97C2×#208C2×C412yes—
98C2×#208D412yes—
99C2×#208D412yes—
100C22×C3⋊C8—21no—
101C22×C3⋊C8—21no—
102#823—28no—
103#823—28no—
104#815—24no—
105C8×Dic3—24no—
106C23×D4—31no—
107C22×M16—31no—
108C22×M16—31no—
109C22×M16C2×D311yescharacteristic
110C22×M16C2×D311yescharacteristic
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C22⋊C4—34no—
113C22×C42—31no—
114C2×#100—34no—
115C2×#100—34no—
116C2×#100—62no—
117C2×#100—62no—
118C2×#100—62no—
119C2×#100—62no—
120C4×M16—62no—
121C4×M16—62no—
122C2×#97—61no—
123C2×C4×C8—61no—
124#390—62no—
125#390—61no—
126#390—61no—
127#343—22no—
128#343—62no—
129#336—62no—
130C24×C3C22×C411yescharacteristic
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3—44no—
138C23×D3—44no—
139C23×D3C4214yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—44no—
144C22×C3×C4C22×C411yescharacteristic
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C22×C3×C4#4111yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4#4111yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—24no—
161C22×Dic3—42no—
162C22×Dic3C22⋊C412yes—
163C22×Dic3C22⋊C412yes—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—41no—
177C2×D12—41no—
178C2×D12—41no—
179C2×D12—41no—
180C2×D12—41no—
181C2×D12—41no—
182C2×D12—41no—
183C2×D12—41no—
184C2×D12—41no—
185C2×D12—41no—
186C2×D12—41no—
187C2×D12—41no—
188C2×D12—41no—
189C2×D12—41no—
190C2×D12—41no—
191C2×D12—41no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16—22no—
196C3×M16—22no—
197C3×M16—41no—
198C3×M16—41no—
199C3×M16C22⋊C414yescomplemented
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—41no—
209C3×C4⋊C4—41no—
210(C2×D3)⋊C4—48no—
211(C2×D3)⋊C4—48no—
212C4×Dic3—21no—
213C4×Dic3—21no—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C4×Dic3—21no—
217C4×Dic3—21no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208—22no—
223#208—41no—
224#208—41no—
225#208C22⋊C414yescomplemented
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—41no—
231C2×C3⋊C8—41no—
232C25—32no—
233C22×D4—32no—
234C22×D4—32no—
235C22×D4—61no—
236C22×D4—61no—
237C22×D4—61no—
238C22×D4—61no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4—61no—
242C22×D4—61no—
243C22×D4—61no—
244C22×D4—61no—
245C22×D4—61no—
246C22×D4—61no—
247C23×C4—32no—
248C23×C4C4×D312yes—
249C23×C4C22×D311yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—32no—
259C2×M16—32no—
260C2×M16—32no—
261C2×M16—32no—
262C2×M16—61no—
263C2×M16—61no—
264C2×M16—61no—
265C2×M16—61no—
266C2×M16—61no—
267C2×M16—61no—
268C2×M16—61no—
269C2×M16—61no—
270C2×M16C4×D312yes—
271C2×M16C4×D312yes—
272C2×M16D1212yes—
273C2×M16#6712yes—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—61no—
277C22×C8—61no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—128no—
287C2×C42—32no—
288C2×C42—31no—
289C2×C42—31no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294#100—68no—
295#100—68no—
296#97—64no—
297C4×C8—64no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3C2×C4211yescharacteristic
301C23×C3C2×C22⋊C411yescharacteristic
302C23×C3#11711yescharacteristic
303C23×C3C4×D411yescharacteristic
304C23×C3C4×D411yescharacteristic
305C23×C3C4×D411yescharacteristic
306C23×C3C4×D411yescharacteristic
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—48no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—44no—
316C22×D3—44no—
317C22×D3—48no—
318C22×D3—84no—
319C22×D3—84no—
320C22×D3—82no—
321C22×D3—82no—
322C22×D3—82no—
323C22×D3—82no—
324C2×C3×C4—42no—
325C2×C3×C4—44no—
326C2×C3×C4—82no—
327C2×C3×C4—82no—
328C2×C3×C4C2×C22⋊C411yescharacteristic
329C2×C3×C4C2×C22⋊C411yescharacteristic
330C2×C3×C4#11711yescharacteristic
331C2×C3×C4#11711yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4C4×D411yescharacteristic
339C2×C3×C4C4×D411yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4#12111yescharacteristic
347C2×C3×C4#12311yescharacteristic
348C2×C3×C4#12311yescharacteristic
349C2×C3×C4#12311yescharacteristic
350C2×C3×C4#12411yescharacteristic
351C2×C3×C4#12411yescharacteristic
352C2×C3×C4#12411yescharacteristic
353C2×C3×C4#12611yescharacteristic
354C2×C3×C4#12611yescharacteristic
355C2×C3×C4#12711yescharacteristic
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—44no—
362C2×Dic3—44no—
363C2×Dic3#10014yescomplemented
364D12—42no—
365D12—42no—
366D12—42no—
367D12—42no—
368D12—42no—
369D12—42no—
370D12—42no—
371D12—42no—
372C3×C8—24no—
373C3×C8—42no—
374C3⋊C8—24no—
375C3⋊C8—42no—
376C24—34no—
377C24—62no—
378C24—62no—
379C24—62no—
380C24—62no—
381C24—62no—
382C24—62no—
383C24—124no—
384C24—124no—
385C24C2×C4×D311yescharacteristic
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—121no—
399C2×D4—121no—
400C2×D4—121no—
401C2×D4—121no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4—121no—
405C2×D4—121no—
406C2×D4—121no—
407C2×D4—121no—
408C2×D4—121no—
409C2×D4—121no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4—121no—
413C2×D4—121no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—44no—
420C22×C4—64no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—122no—
424C22×C4C2×C4×D311yescharacteristic
425C22×C4C2×C4×D311yescharacteristic
426C22×C4C2×D1211yescharacteristic
427C22×C4#23511yescharacteristic
428C22×C4#23511yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4#23711yescharacteristic
434C22×C4#23711yescharacteristic
435C22×C4#23911yescharacteristic
436C22×C4#23911yescharacteristic
437C22×C4C2×#6711yescharacteristic
438M16—22no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—34no—
444M16—41no—
445M16—41no—
446M16—62no—
447M16—62no—
448M16—62no—
449M16—62no—
450M16—62no—
451M16—121no—
452M16—121no—
453M16(C2×D3)⋊C414yescomplemented
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—62no—
464C2×C8—121no—
465C2×C8—121no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C22⋊C4—128no—
471C22⋊C4—128no—
472C42—61no—
473C42—61no—
474C42—62no—
475C42—61no—
476C42—61no—
477C42—62no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3C4×C22⋊C411yescharacteristic
487C22×C3C4×C22⋊C411yescharacteristic
488C22×C3C4×C22⋊C411yescharacteristic
489C22×C3#38611yescharacteristic
490C22×C3#38811yescharacteristic
491C22×C3#38811yescharacteristic
492C22×C3#39011yescharacteristic
493C22×C3C2×#10012yes—
494C22×C3#41312yes—
495C2×D3—44no—
496C2×D3—48no—
497C2×D3—84no—
498C2×D3—84no—
499C2×D3—84no—
500C2×D3—82no—
501C2×D3—82no—
502C2×D3—84no—
503C2×D3—84no—
504C3×C4—82no—
505C3×C4#38611yescharacteristic
506C3×C4#38611yescharacteristic
507C3×C4#38811yescharacteristic
508C3×C4#38811yescharacteristic
509C3×C4#38811yescharacteristic
510C3×C4#38811yescharacteristic
511C3×C4#39011yescharacteristic
512C3×C4#39011yescharacteristic
513Dic3—24no—
514Dic3—42no—
515C23—22no—
516C23—22no—
517C23—64no—
518C23—64no—
519C23—64no—
520C23—128no—
521C23—128no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—124no—
527C23—124no—
528C23—244no—
529C23—244no—
530C23—242no—
531C23—242no—
532C23—242no—
533C23—242no—
534C23C42×D311yescharacteristic
535C23C4×D1211yescharacteristic
536C23D3×C22⋊C411yescharacteristic
537C23#67⋊C411yescharacteristic
538C23#89211yescharacteristic
539C23#89311yescharacteristic
540C23C4×#6711yescharacteristic
541D4—122no—
542D4—122no—
543D4—122no—
544D4—122no—
545D4—122no—
546D4—122no—
547D4—122no—
548D4—122no—
549C2×C4—34no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—124no—
560C2×C4—124no—
561C2×C4#87211yescharacteristic
562C2×C4C4×D1211yescharacteristic
563C2×C4D3×C22⋊C411yescharacteristic
564C2×C4#67⋊C411yescharacteristic
565C2×C4#67⋊C411yescharacteristic
566C2×C4#67⋊C411yescharacteristic
567C2×C4#88211yescharacteristic
568C2×C4#88411yescharacteristic
569C2×C4#88411yescharacteristic
570C2×C4#88511yescharacteristic
571C2×C4#88511yescharacteristic
572C2×C4#88611yescharacteristic
573C2×C4#89211yescharacteristic
574C2×C4#89311yescharacteristic
575C2×C4#89311yescharacteristic
576C2×C4#89311yescharacteristic
577C2×C4#89411yescharacteristic
578C2×C4#89411yescharacteristic
579C2×C4#89511yescharacteristic
580C2×C4#89511yescharacteristic
581C2×C4#89811yescharacteristic
582C2×C4#89811yescharacteristic
583C2×C4C2×(C2×D3)⋊C411yescharacteristic
584C2×C4C4×#6711yescharacteristic
585C2×C4#93911yescharacteristic
586C2×C4#93911yescharacteristic
587C2×C4#94011yescharacteristic
588C2×C4#94711yescharacteristic
589C8—24no—
590C8—42no—
591C8—64no—
592C8—122no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#143711yescharacteristic
598C2×C3C4×#10012yes—
599C2×C3#190012yes—
600C2×C3#196511yescharacteristic
601C2×C3#196711yescharacteristic
602D3—84no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—124no—
612C22—128no—
613C22—244no—
614C22—244no—
615C22—244no—
616C22—244no—
617C22—244no—
618C22—242no—
619C22—242no—
620C22#466511yescharacteristic
621C22#466611yescharacteristic
622C22#466611yescharacteristic
623C22#467011yescharacteristic
624C22D3×#10012yes—
625C22#475112yes—
626C22C4×(C2×D3)⋊C411yescharacteristic
627C22#495711yescharacteristic
628C22#499011yescharacteristic
629C4—64no—
630C4—82no—
631C4—122no—
632C4#466511yescharacteristic
633C4#466911yescharacteristic
634C4#466911yescharacteristic
635C4#467011yescharacteristic
636C4#495711yescharacteristic
637C4#495811yescharacteristic
638C4#498811yescharacteristic
639C4#499011yescharacteristic
640C3#1096211yescharacteristic, complemented, Sylow
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—244no—
646C2#6897212yes—
647C2#6899212yes—
648C2#6940811yescharacteristic
649C2#7006411yescharacteristic
650C2#7009911yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC23×C3C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#343C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenterC2×#343D3
hypocenterC3#10962
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. C2×Dic3×M16C2semidirect
5. C2×C3×#343C2semidirect
6. #71979C2semidirect
7. C2×#4530C2semidirect
21. C2×#823C4semidirect
22. C2×#823C4semidirect
38. C2×#343D3semidirect
199. C3×M16C22⋊C4semidirect
225. #208C22⋊C4semidirect
363. C2×Dic3#100semidirect
453. M16(C2×D3)⋊C4semidirect
640. C3#10962semidirect

Maximal subgroups

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

subgroupindex
C22×#8232
C2×Dic3×M162
C2×#49882
C2×C3×#3432
#719792
C2×#45302
#109623