Finite Group Databasebeta

← #10800219

Subgroups — #10800219

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#86219C211yesmaximal, characteristic
2C4×D3×M16C211yesmaximal, characteristic, complemented
3#73752C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic, complemented
5#71818C211yesmaximal, characteristic, complemented
6#70381C214yesmaximal, complemented
7#70354C214yesmaximal, complemented
8#69868C211yesmaximal, characteristic, complemented
9#69493C211yesmaximal, characteristic, complemented
10#21383—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×C42×D3C2211yescharacteristic
30#5470C2211yescharacteristic
31C2×C3×#104C2214yes—
32C3×C4×M16C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34C2×#837C2214yes—
35Dic3×M16C2212yes—
36#4987—21no—
37#4977—21no—
38C4×(C2×D3)⋊C4C2212yes—
39C4×C4⋊Dic3C2211yescharacteristic
40#4932C2212yes—
41C4×#208C2211yescharacteristic
42#4707—21no—
43D3×#97—21no—
44C4×ΓL(1,𝔽25)—21no—
45C4×D3×C8—21no—
46#4664—21no—
47#4651—21no—
48#4555C2214yes—
49#4523C2214yes—
50#3444—31no—
51C2×C4×M16—31no—
52#2283—31no—
53#1986—31no—
54#1986—31no—
55#1980—31no—
56#1980D311yescharacteristic, complemented
57#1871—34no—
58#1759—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×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3C2312yes—
106C2×C4×Dic3C2311yescharacteristic
107C2×#208C2312yes—
108D3×M16—24no—
109D3×M16—24no—
110C2×ΓL(1,𝔽25)—22no—
111C2×D3×C8—22no—
112#897—21no—
113#897—21no—
114#892—21no—
115#892—21no—
116#890—21no—
117#890—21no—
118#887—21no—
119#887—21no—
120#883—22no—
121#883—22no—
122#67⋊C4—22no—
123#67⋊C4—22no—
124#879—22no—
125#879—22no—
126#877—22no—
127#877—22no—
128C42×D3C2×C411yescharacteristic
129C42×D3C2×C411yescharacteristic
130C42×D3C2×C411yescharacteristic
131C42×D3C2×C411yescharacteristic
132C42×D3D411yescharacteristic
133C42×D3D411yescharacteristic
134C42×D3D411yescharacteristic
135C42×D3D411yescharacteristic
136#870C2×C412yes—
137#869C2×C411yescharacteristic
138#869C2×C411yescharacteristic
139C4×Dic6—21no—
140C4×Dic6—21no—
141C3×#104—28no—
142C3×#97—21no—
143C3×C4×C8—21no—
144C3×#95—21no—
145#837—28no—
146C3⋊#95—21no—
147C3⋊#95—22no—
148C3⋊#95—22no—
149C3⋊#95—22no—
150#815—22no—
151C8×Dic3—22no—
152#803—21no—
153C4×C3⋊C8—21no—
154C22×M16—32no—
155#533—31no—
156#533—31no—
157#533—31no—
158#532—31no—
159#532—31no—
160#532C2×D311yescharacteristic
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×#104C2×D314yes—
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#339—34no—
193C23×D3C2×D411yescharacteristic
194C3×#41—24no—
195C3×#41—24no—
196C2×C3×Q8C22×C412yes—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C22×C412yes—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C2×#67—22no—
207C2×#67—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214#237—44no—
215#237—44no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3C22⋊C411yescharacteristic
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×D3C2×D411yescharacteristic
231C2×C4×D3C2×D411yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—21no—
234C3×M16—22no—
235C3×M16#4114yescomplemented
236C2×C3×C8—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C42—24no—
243C3×C42C22×C411yescharacteristic
244C3×C42C22×C411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247#217—22no—
248#217—22no—
249#217—22no—
250#217—22no—
251#217—22no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—44no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—22no—
258C4⋊Dic3—22no—
259C4⋊Dic3—42no—
260C4⋊Dic3C22×C412yes—
261C4⋊Dic3C22×C412yes—
262Dic3⋊C4—22no—
263Dic3⋊C4—22no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266C4×Dic3—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—61no—
288C2×#41—61no—
289C2×#41C22×D312yes—
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⋊Q8—31no—
301C4⋊Q8C4×D311yescharacteristic
302#127C4×D311yescharacteristic
303#126—62no—
304#126—62no—
305#125—32no—
306#125—61no—
307#125—61no—
308#124—61no—
309#124—61no—
310#124C4×D312yes—
311#123—62no—
312#123—62no—
313#123—62no—
314C22⋊Q8—32no—
315C22⋊Q8—32no—
316C22⋊Q8—61no—
317C22⋊Q8—61no—
318C22⋊Q8—61no—
319C22⋊Q8—61no—
320#121—21no—
321#121—21no—
322#121—32no—
323#121—32no—
324#121—61no—
325#121—61no—
326#120—62no—
327C4×Q8—32no—
328C4×Q8—32no—
329C4×Q8—61no—
330C4×Q8—61no—
331C4×Q8—61no—
332C4×Q8—61no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—32no—
336C4×D4—32no—
337C4×D4—61no—
338C4×D4—61no—
339C4×D4—62no—
340C4×D4—62no—
341C4×D4—62no—
342C4×D4—62no—
343C4×D4—62no—
344C4×D4—62no—
345#117—32no—
346#117—62no—
347#117—62no—
348#117—62no—
349#117—61no—
350#117—61no—
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—28no—
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×C3—22no—
383C23×C3C22×D411yescharacteristic
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—22no—
387C22×D3—41no—
388C22×D3C2×C22⋊C411yescharacteristic
389C22×D3C2×C22⋊C411yescharacteristic
390C3×Q8—24no—
391C3×D4—24no—
392C3×D4—42no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—42no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—41no—
399C2×C3×C4—44no—
400C2×C3×C4—42no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4C2×C22⋊C411yescharacteristic
404C2×C3×C4C2×C22⋊C411yescharacteristic
405C2×C3×C4C2×C22⋊C411yescharacteristic
406C2×C3×C4C2×C22⋊C411yescharacteristic
407C2×C3×C4C23×C411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4112yes—
410C2×C3×C4C2×#4112yes—
411#67—44no—
412#67—44no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—44no—
420C2×Dic3—44no—
421C2×Dic3—42no—
422C2×Dic3—42no—
423C2×Dic3—42no—
424C2×Dic3C2×C22⋊C411yescharacteristic
425C2×Dic3C2×C22⋊C411yescharacteristic
426C2×Dic3C2×C22⋊C411yescharacteristic
427C2×Dic3C2×C22⋊C411yescharacteristic
428C2×Dic3C2×C22⋊C411yescharacteristic
429C2×Dic3C2×C22⋊C411yescharacteristic
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—41no—
439C4×D3—41no—
440C4×D3—41no—
441C4×D3—41no—
442Dic6—42no—
443Dic6—42no—
444C3×C8—24no—
445C3⋊C8—24no—
446C24—31no—
447#41—24no—
448#41—24no—
449#41—124no—
450#41—124no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8C2×C4×D312yes—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—62no—
458C2×D4—62no—
459C2×D4C2×C4×D312yes—
460C22×C4—22no—
461C22×C4—22no—
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—31no—
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#23714yescomplemented
492C2×C8—22no—
493C2×C8—68no—
494C2×C8—62no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—32no—
498C4⋊C4—32no—
499C4⋊C4—42no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—122no—
509C22⋊C4—22no—
510C22⋊C4—22no—
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—24no—
520C42—31no—
521C42—31no—
522C42—31no—
523C42—31no—
524C42—31no—
525C42—31no—
526C42—31no—
527C42—31no—
528C42—31no—
529C42—31no—
530C42—31no—
531C42—31no—
532C42—62no—
533C42—62no—
534C42—62no—
535C42—64no—
536C42C2×C4×D311yescharacteristic
537C42C2×C4×D311yescharacteristic
538C42D3×D411yescharacteristic
539C42D3×D411yescharacteristic
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3C22×C22⋊C411yescharacteristic
546C22×C3#53211yescharacteristic
547C22×C3#53211yescharacteristic
548C2×D3—21no—
549C2×D3—21no—
550C2×D3—21no—
551C2×D3—21no—
552C2×D3—21no—
553C2×D3—42no—
554C2×D3—42no—
555C2×D3#42112yes—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—42no—
560C3×C4—82no—
561C3×C4#42112yes—
562C3×C4C4×#4112yes—
563C3×C4#52512yes—
564Dic3—21no—
565Dic3—21no—
566Dic3—21no—
567Dic3—21no—
568Dic3—21no—
569Dic3—21no—
570Dic3—42no—
571Dic3—42no—
572Dic3—82no—
573C23—22no—
574C23—31no—
575C23—31no—
576C23—62no—
577C23—61no—
578C23—61no—
579C23—121no—
580C23C2×D3×D411yescharacteristic
581Q8—24no—
582Q8—122no—
583Q8—122no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—124no—
590D4—124no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—31no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—41no—
600C2×C4—44no—
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—62no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—124no—
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×#23712yes—
634C2×C4D3×#4112yes—
635C8—24no—
636C8—64no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—82no—
640C2×C3#229711yescharacteristic
641C2×C3C2×#42111yescharacteristic
642C2×C3C2×#42111yescharacteristic
643D3—22no—
644D3—41no—
645C22—21no—
646C22—21no—
647C22—32no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—61no—
652C22—61no—
653C22—61no—
654C22—61no—
655C22—61no—
656C22—122no—
657C22—122no—
658C22C2×D3×C22⋊C411yescharacteristic
659C22#566811yescharacteristic
660C22#571611yescharacteristic
661C4—21no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—61no—
666C4—61no—
667C4—61no—
668C4—61no—
669C4—61no—
670C4—61no—
671C4—82no—
672C4—122no—
673C4—122no—
674C4—242no—
675C4#482112yes—
676C4C4×#23712yes—
677C4#560612yes—
678C3#2138311yescharacteristic, complemented, Sylow
679C2—21no—
680C2—21no—
681C2—62no—
682C2—82no—
683C2—121no—
684C2#7629011yescharacteristic
685C2C2×#482111yescharacteristic
686C2D3×#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#21383
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. #73752C2semidirect
4. C3×#1980C2semidirect
5. #71818C2semidirect
6. #70381C2semidirect
7. #70354C2semidirect
8. #69868C2semidirect
9. #69493C2semidirect
15. #5728C4semidirect
16. #5728C4semidirect
17. #5716C4semidirect
18. #5716C4semidirect
19. #5680C4semidirect
20. #5680C4semidirect
56. #1980D3semidirect
235. C3×M16#41semidirect
283. #208#41semidirect
491. M16#237semidirect
678. C3#21383semidirect

Maximal subgroups

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

subgroupindex
#862192
C4×D3×M162
#737522
C3×#19802
#718182
#703812
#703542
#698682
#694932
#213833