Finite Group Databasebeta

← #10800212

Subgroups — #10800212

Order 768.

1256 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86219C211yesmaximal, characteristic
2C4×D3×M16C211yesmaximal, characteristic, complemented
3C4×#892C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic, complemented
5#71818C211yesmaximal, characteristic, complemented
6#70381C214yesmaximal, complemented
7Dic3×#104C214yesmaximal, complemented
8#69868C211yesmaximal, characteristic, complemented
9#69493C211yesmaximal, characteristic, complemented
10#21382—31nomaximal, Sylow
11C3×#532C2211yescharacteristic
12#5828C2212yes—
13Dic3×#41C2212yes—
14C2×D3×M16C2212yes—
15#5728C411yescharacteristic, complemented
16#5728C411yescharacteristic, complemented
17#5716C411yescharacteristic, complemented
18#5716C411yescharacteristic, complemented
19#5680C412yescomplemented
20#5680C412yescomplemented
21#5606—21no—
22#5606—21no—
23#5600—21no—
24#5600—21no—
25#5557—21no—
26#5557—21no—
27C4×#237—21no—
28C4×#237—21no—
29C2×#892—21no—
30C2×C42×D3C2211yescharacteristic
31#5470C2211yescharacteristic
32C2×C3×#104C2214yes—
33C3×C4×M16C2211yescharacteristic
34C3×C4×C4⋊C4C2211yescharacteristic
35C2×#837C2214yes—
36Dic3×M16C2212yes—
37Dic3×C4⋊C4—21no—
38C4×(C2×D3)⋊C4C2212yes—
39C4×C4⋊Dic3C2211yescharacteristic
40C42×Dic3C2212yes—
41C4×#208C2211yescharacteristic
42#4707—21no—
43D3×#97—21no—
44C4×ΓL(1,𝔽25)—21no—
45C4×D3×C8—21no—
46#4666—21no—
47#4648—21no—
48#4555C2214yes—
49#4523C2214yes—
50#3444—31no—
51C2×C4×M16—31no—
52C4×#117—31no—
53#1986—31no—
54#1986—31no—
55#1980—31no—
56#1980D311yescharacteristic, complemented
57#1871—34no—
58C4×#104—34no—
59C2×C3×#41C2312yes—
60C2×#237—21no—
61C2×#237—21no—
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3C2311yescharacteristic
65C2×C3×M16C2312yes—
66C3×C4⋊Q8C2×C411yescharacteristic
67C3×#127C2×C411yescharacteristic
68C3×#124C2×C412yes—
69C3×#121—21no—
70C3×#121—21no—
71C3×C4×D4—21no—
72C3×C4×D4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76#946C2×C412yes—
77#946C2×C412yes—
78Q8×Dic3C2×C412yes—
79Q8×Dic3C2×C412yes—
80#938—21no—
81#938—21no—
82#938C2×C412yes—
83#938C2×C412yes—
84#937—22no—
85#936—21no—
86#936—21no—
87#935—22no—
88D4×Dic3—22no—
89D4×Dic3—22no—
90D4×Dic3—21no—
91D4×Dic3—21no—
92D4×Dic3C2×C412yes—
93D4×Dic3C2×C412yes—
94C4×#67—22no—
95C4×#67—22no—
96C2×(C2×D3)⋊C4—22no—
97#926—22no—
98#926C2312yes—
99C2×C4⋊Dic3—21no—
100#924—21no—
101#924—21no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3C2312yes—
108C2×C4×Dic3C2311yescharacteristic
109C2×#208C2312yes—
110D3×M16—24no—
111D3×M16—24no—
112C2×ΓL(1,𝔽25)—22no—
113C2×D3×C8—22no—
114#897—21no—
115#897—21no—
116#892—28no—
117#892—21no—
118#892—21no—
119#890—21no—
120#890—21no—
121#887—21no—
122#887—21no—
123#883—22no—
124#883—22no—
125#67⋊C4—22no—
126#67⋊C4—22no—
127#879—22no—
128#879—22no—
129#877—22no—
130#877—22no—
131C42×D3C2×C411yescharacteristic
132C42×D3C2×C411yescharacteristic
133C42×D3C2×C411yescharacteristic
134C42×D3C2×C411yescharacteristic
135C42×D3D411yescharacteristic
136C42×D3D411yescharacteristic
137C42×D3D411yescharacteristic
138C42×D3D411yescharacteristic
139#870C2×C412yes—
140#869C2×C411yescharacteristic
141#869C2×C411yescharacteristic
142C4×Dic6—21no—
143C4×Dic6—21no—
144C3×#104—28no—
145C3×#97—21no—
146C3×C4×C8—21no—
147C3×#95—21no—
148#837—28no—
149C3⋊#95—21no—
150C3⋊#95—22no—
151#815—22no—
152C8×Dic3—22no—
153#803—21no—
154C4×C3⋊C8—21no—
155C22×M16—32no—
156#533—31no—
157#533—31no—
158#533—31no—
159#532—31no—
160#532—31no—
161#532C2×D311yescharacteristic
162#529—32no—
163#529—32no—
164#529—61no—
165#529—61no—
166#529—61no—
167#529—61no—
168#525—32no—
169#525—61no—
170#525—61no—
171C4×#41—32no—
172C4×#41—61no—
173C4×#41—61no—
174C2×#117—61no—
175C22×C42—31no—
176C2×#104—34no—
177C2×#104C2×D314yes—
178C4×M16—32no—
179C4×M16—31no—
180C4×M16—61no—
181C4×M16—61no—
182C4×M16C2×D311yescharacteristic
183C2×#97—61no—
184C2×C4×C8—61no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—61no—
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C22⋊C4—32no—
189C4×C22⋊C4—61no—
190#376—61no—
191C43—32no—
192#339—34no—
193#339—34no—
194C23×D3C2×D411yescharacteristic
195C3×#41—24no—
196C3×#41—24no—
197C2×C3×Q8C22×C412yes—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C22×C412yes—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C2×#67—22no—
208C2×#67—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215#237—44no—
216#237—44no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—24no—
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×Dic6—21no—
234C2×Dic6—21no—
235C3×M16—22no—
236C3×M16#4114yescomplemented
237C2×C3×C8—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C42—24no—
244C3×C42C22×C411yescharacteristic
245C3×C42C22×C411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C3×C42C2×D411yescharacteristic
248#217—22no—
249#217—22no—
250#217—22no—
251#217—22no—
252#217—22no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—22no—
256C4⋊Dic3—24no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—22no—
260C4⋊Dic3—22no—
261C4⋊Dic3C22×C412yes—
262C4⋊Dic3C22×C412yes—
263Dic3⋊C4—22no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267C4×Dic3—28no—
268C4×Dic3—22no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—24no—
272C4×Dic3—44no—
273C4×Dic3C22⋊C411yescharacteristic
274C4×Dic3C22⋊C411yescharacteristic
275C4×Dic3C22⋊C411yescharacteristic
276C4×Dic3C22⋊C411yescharacteristic
277C4×Dic3C22⋊C411yescharacteristic
278C4×Dic3C22⋊C411yescharacteristic
279C4×Dic3C22⋊C411yescharacteristic
280C4×Dic3C22⋊C411yescharacteristic
281C4×Dic3C22×C411yescharacteristic
282C4×Dic3C22×C411yescharacteristic
283C4×Dic3C2×D411yescharacteristic
284C4×Dic3C2×D411yescharacteristic
285#208—22no—
286#208#4114yescomplemented
287C2×C3⋊C8—22no—
288ΓL(1,𝔽25)—28no—
289D3×C8—28no—
290C2×#41—61no—
291C2×#41—61no—
292C2×#41C22×D312yes—
293C23×C4—31no—
294C23×C4—31no—
295C23×C4—31no—
296C2×M16—32no—
297C2×M16—62no—
298C2×M16—64no—
299C2×M16—64no—
300C2×M16C22×D312yes—
301C22×C8—62no—
302C4⋊Q8—31no—
303C4⋊Q8—31no—
304C4⋊Q8C4×D311yescharacteristic
305#127C4×D311yescharacteristic
306#126—62no—
307#126—62no—
308#125—32no—
309#125—61no—
310#125—61no—
311#124—61no—
312#124—61no—
313#124C4×D312yes—
314#123—62no—
315#123—62no—
316#123—62no—
317C22⋊Q8—32no—
318C22⋊Q8—32no—
319C22⋊Q8—61no—
320C22⋊Q8—61no—
321C22⋊Q8—61no—
322C22⋊Q8—61no—
323#121—21no—
324#121—21no—
325#121—32no—
326#121—32no—
327#121—61no—
328#121—61no—
329#120—62no—
330C4×Q8—32no—
331C4×Q8—32no—
332C4×Q8—61no—
333C4×Q8—61no—
334C4×Q8—61no—
335C4×Q8—61no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—32no—
339C4×D4—32no—
340C4×D4—61no—
341C4×D4—61no—
342C4×D4—62no—
343C4×D4—62no—
344C4×D4—62no—
345C4×D4—62no—
346C4×D4—62no—
347C4×D4—62no—
348#117—32no—
349#117—62no—
350#117—62no—
351#117—62no—
352#117—61no—
353#117—61no—
354#117—68no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—61no—
357C2×C22⋊C4—62no—
358C2×C42—31no—
359C2×C42—31no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—31no—
366C2×C42—31no—
367C2×C42—31no—
368C2×C42—31no—
369C2×C42—32no—
370C2×C42—62no—
371C2×C42—62no—
372C2×C42—62no—
373C2×C42C22×D312yes—
374C2×C42C22×D311yescharacteristic
375#104—28no—
376#104—68no—
377#97—21no—
378#97—62no—
379#97—61no—
380C4×C8—21no—
381C4×C8—62no—
382C4×C8—61no—
383#95—21no—
384#95—62no—
385#95—61no—
386C23×C3—22no—
387C23×C3C22×D411yescharacteristic
388C22×D3—21no—
389C22×D3—21no—
390C22×D3—22no—
391C22×D3—41no—
392C22×D3C2×C22⋊C411yescharacteristic
393C22×D3C2×C22⋊C411yescharacteristic
394C3×Q8—24no—
395C3×D4—24no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C2×C3×C4—22no—
401C2×C3×C4—24no—
402C2×C3×C4—22no—
403C2×C3×C4—41no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—44no—
407C2×C3×C4C2×C22⋊C411yescharacteristic
408C2×C3×C4C2×C22⋊C411yescharacteristic
409C2×C3×C4C2×C22⋊C411yescharacteristic
410C2×C3×C4C2×C22⋊C411yescharacteristic
411C2×C3×C4C23×C411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C2×#4112yes—
414C2×C3×C4C2×#4112yes—
415#67—44no—
416#67—44no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—42no—
425C2×Dic3—42no—
426C2×Dic3—44no—
427C2×Dic3—44no—
428C2×Dic3C2×C22⋊C411yescharacteristic
429C2×Dic3C2×C22⋊C411yescharacteristic
430C2×Dic3C2×C22⋊C411yescharacteristic
431C2×Dic3C2×C22⋊C411yescharacteristic
432C2×Dic3C2×C22⋊C411yescharacteristic
433C2×Dic3C2×C22⋊C411yescharacteristic
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—22no—
439C4×D3—22no—
440C4×D3—22no—
441C4×D3—22no—
442C4×D3—22no—
443C4×D3—41no—
444C4×D3—41no—
445C4×D3#10414yescomplemented
446Dic6—42no—
447Dic6—42no—
448C3×C8—24no—
449C3⋊C8—24no—
450C24—31no—
451#41—24no—
452#41—24no—
453#41—124no—
454#41—124no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8C2×C4×D312yes—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4C2×C4×D312yes—
464C22×C4—22no—
465C22×C4—22no—
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—31no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—64no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—122no—
488C22×C4D3×D411yescharacteristic
489C22×C4D3×D411yescharacteristic
490C22×C4C23×D311yescharacteristic
491M16—22no—
492M16—34no—
493M16—68no—
494M16—62no—
495M16#23714yescomplemented
496C2×C8—22no—
497C2×C8—68no—
498C2×C8—62no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—32no—
503C4⋊C4—32no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—64no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—62no—
512C4⋊C4—62no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—68no—
518C22⋊C4—62no—
519C22⋊C4—62no—
520C22⋊C4—62no—
521C22⋊C4—62no—
522C22⋊C4—62no—
523C42—24no—
524C42—31no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—31no—
530C42—31no—
531C42—31no—
532C42—31no—
533C42—31no—
534C42—31no—
535C42—31no—
536C42—62no—
537C42—62no—
538C42—68no—
539C42—62no—
540C42—64no—
541C42—124no—
542C42C2×C4×D311yescharacteristic
543C42C2×C4×D311yescharacteristic
544C42D3×D411yescharacteristic
545C42D3×D411yescharacteristic
546C22×C3—21no—
547C22×C3—21no—
548C22×C3—42no—
549C22×C3—42no—
550C22×C3—42no—
551C22×C3C22×C22⋊C411yescharacteristic
552C22×C3#53211yescharacteristic
553C22×C3#53211yescharacteristic
554C2×D3—21no—
555C2×D3—21no—
556C2×D3—21no—
557C2×D3—21no—
558C2×D3—21no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3C2×#10412yes—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—44no—
566C3×C4—42no—
567C3×C4C2×#10412yes—
568C3×C4C4×#4112yes—
569C3×C4#52512yes—
570Dic3—21no—
571Dic3—21no—
572Dic3—21no—
573Dic3—21no—
574Dic3—44no—
575Dic3—42no—
576Dic3—42no—
577Dic3C2×#10412yes—
578Dic3#42112yes—
579C23—22no—
580C23—31no—
581C23—31no—
582C23—62no—
583C23—61no—
584C23—61no—
585C23—121no—
586C23C2×D3×D411yescharacteristic
587Q8—24no—
588Q8—122no—
589Q8—122no—
590D4—24no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—124no—
596D4—124no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—34no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—31no—
606C2×C4—31no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—124no—
630C2×C4—124no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4D3×C22⋊C411yescharacteristic
634C2×C4D3×C22⋊C411yescharacteristic
635C2×C4D3×C22⋊C411yescharacteristic
636C2×C4D3×C22⋊C411yescharacteristic
637C2×C4C22×C4×D311yescharacteristic
638C2×C4C2×D3×D411yescharacteristic
639C2×C4C2×#23712yes—
640C2×C4D3×#4112yes—
641C8—24no—
642C8—64no—
643C2×C3—21no—
644C2×C3—21no—
645C2×C3—82no—
646C2×C3#229711yescharacteristic
647C2×C3C22×#10411yescharacteristic
648C2×C3C2×#42111yescharacteristic
649D3—22no—
650D3—41no—
651C22—21no—
652C22—21no—
653C22—32no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—61no—
658C22—61no—
659C22—61no—
660C22—61no—
661C22—61no—
662C22—122no—
663C22—122no—
664C22C2×D3×C22⋊C411yescharacteristic
665C22#566811yescharacteristic
666C22#571611yescharacteristic
667C4—21no—
668C4—21no—
669C4—21no—
670C4—32no—
671C4—32no—
672C4—44no—
673C4—42no—
674C4—61no—
675C4—61no—
676C4—61no—
677C4—61no—
678C4—124no—
679C4—122no—
680C4—122no—
681C4D3×#10412yes—
682C4C4×#23712yes—
683C4#560612yes—
684C3#2138211yescharacteristic, complemented, Sylow
685C2—21no—
686C2—21no—
687C2—62no—
688C2—82no—
689C2—121no—
690C2#7629011yescharacteristic
691C2C2×D3×#10411yescharacteristic
692C2D3×#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1980C2
socleC22×C3C22×C22⋊C4
prime radical#86219C2
cosocleC22×C3×C4C24
hypercenter#1980D3
hypocenterC3#21382
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. C4×D3×M16C2semidirect
3. C4×#892C2semidirect
4. C3×#1980C2semidirect
5. #71818C2semidirect
6. #70381C2semidirect
7. Dic3×#104C2semidirect
8. #69868C2semidirect
9. #69493C2semidirect
15. #5728C4semidirect
16. #5728C4semidirect
17. #5716C4semidirect
18. #5716C4semidirect
19. #5680C4semidirect
20. #5680C4semidirect
56. #1980D3semidirect
236. C3×M16#41semidirect
286. #208#41semidirect
445. C4×D3#104semidirect
495. M16#237semidirect
684. C3#21382semidirect

Maximal subgroups

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

subgroupindex
#862192
C4×D3×M162
C4×#8922
C3×#19802
#718182
#703812
Dic3×#1042
#698682
#694932
#213823