Finite Group Databasebeta

← #10801798

Subgroups — #10801798

Order 768.

1010 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
1C2×#5122C211yesmaximal, characteristic, complemented
2#82744C211yesmaximal, characteristic, complemented
3C2×#5048C211yesmaximal, characteristic
4C2×#4817C211yesmaximal, characteristic, complemented
5#73872C211yesmaximal, characteristic
6C3×#1894C211yesmaximal, characteristic, complemented
7#71990C211yesmaximal, characteristic
8#70260C211yesmaximal, characteristic, complemented
9#70254C211yesmaximal, characteristic, complemented
10Dic3×#103C211yesmaximal, characteristic, complemented
11#69512C211yesmaximal, characteristic, complemented
12#68861C211yesmaximal, characteristic, complemented
13#68853C211yesmaximal, characteristic, complemented
14#68786C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#21531—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×#893C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5212C2211yescharacteristic
31#5122—24no—
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#5048C418yescomplemented
39#4988C2211yescharacteristic
40#4984—21no—
41#4984C2211yescharacteristic
42#4980—21no—
43Dic3×C4⋊C4C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4957—21no—
51#4956—21no—
52#4951—21no—
53#4951—21no—
54#4817—24no—
55#4670—21no—
56#4668—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#465—31no—
64C2×#440—31no—
65C2×#405—31no—
66#2324—31no—
67#2323—31no—
68#1973—31no—
69#1944—31no—
70#1934—31no—
71#1933—31no—
72#1895—31no—
73#1894D311yescharacteristic, complemented
74#1808—31no—
75#1776—31no—
76C4×#103—31no—
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8C2×C414yescomplemented
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98C2×#217D411yescharacteristic
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101C2×#215—22no—
102C2×#215—22no—
103C2×#215C2×C414yes—
104#930—22no—
105#930C2×C414yes—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110#926—22no—
111#926—22no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8—22no—
126C22×C3⋊C8—22no—
127C22×C3⋊C8C2×C414yescomplemented
128C22×C3⋊C8C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130#893—22no—
131#893—22no—
132C3×#103—22no—
133C3×#103—22no—
134C3×#95—21no—
135#835C2×C414yescomplemented
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#821—22no—
144#821—22no—
145#815—22no—
146#815D414yescomplemented
147C8×Dic3—22no—
148C8×Dic3—22no—
149#809C2×C414yes—
150#808C2×C414yescomplemented
151#807—22no—
152#807—22no—
153C22×SD16—31no—
154C22×M16C2×D311yescharacteristic
155C23×C8—31no—
156C2×C22⋊Q8C2×D311yescharacteristic
157C2×#121—31no—
158C2×C4×Q8—31no—
159C2×C4×D4—31no—
160C2×#117—31no—
161#465—38no—
162#440—64no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103C2×D311yescharacteristic
168C2×#103C2×D311yescharacteristic
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177#391—31no—
178#390—31no—
179#390—61no—
180#389—31no—
181#389—31no—
182#389—61no—
183#388—61no—
184#388—61no—
185#388—61no—
186#387—61no—
187#386—61no—
188#385—61no—
189#380—31no—
190#380C2×D311yescharacteristic
191C4×C4⋊C4—31no—
192C4×C4⋊C4—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8C22×C412yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C22×C412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4112yes—
218C22×C3×C4C2411yescharacteristic
219C2×#67—24no—
220C2×#67—81no—
221C2×#67—81no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12C22×C412yes—
235C2×C4×D3—42no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C3×M16—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C22×C412yes—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—81no—
254#217—22no—
255#217—22no—
256#217—81no—
257#215—28no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—22no—
264C4⋊Dic3—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—42no—
270C4⋊Dic3—81no—
271C4⋊Dic3C22×C412yes—
272Dic3⋊C4—81no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277C4×Dic3—81no—
278C4×Dic3C2×D412yes—
279C4×Dic3#4112yes—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—28no—
283C2×C3⋊C8C22×C412yes—
284C2×C3⋊C8C22×C412yes—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8#4112yes—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C22×D4—61no—
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C22×C8—31no—
299C22×C8—34no—
300C22×C8—31no—
301C22×C8—62no—
302C22×C8—62no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C22⋊Q8—22no—
306C22⋊Q8C4×D314yescomplemented
307#121—34no—
308#121—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×D4—62no—
312C4×D4—62no—
313#117—62no—
314#117—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42—61no—
341#107—62no—
342#107—62no—
343#106—34no—
344#103—22no—
345#103—22no—
346#103—34no—
347#102—34no—
348#102—62no—
349#102—62no—
350#97—34no—
351#97—62no—
352C4×C8—62no—
353C4×C8—62no—
354#95—21no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3C4×D412yes—
369C23×C3C22×D411yescharacteristic
370C22×D3—22no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—41no—
380C3×Q8—41no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4C4×D412yes—
400C2×C3×C4C23×C411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406#67—84no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3#12112yes—
427C2×Dic3#12112yes—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433C3×C8—22no—
434C3×C8—22no—
435C3⋊C8—22no—
436C3⋊C8C4×D414yes—
437C24—61no—
438C24D3×D411yescharacteristic
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D312yes—
445C2×D4—32no—
446C2×D4—64no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—241no—
471C22×C4—241no—
472C22×C4C2×C4×D312yes—
473C22×C4D3×D411yescharacteristic
474C22×C4#23712yes—
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477M16—28no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—32no—
483C2×C8—32no—
484C2×C8—32no—
485C2×C8—32no—
486C2×C8—68no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8D3×D412yes—
490C2×C8#23712yes—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—32no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—122no—
506C4⋊C4—241no—
507C4⋊C4—241no—
508C4⋊C4C2×C4×D312yes—
509C22⋊C4—24no—
510C22⋊C4—62no—
511C22⋊C4—62no—
512C22⋊C4—64no—
513C22⋊C4—81no—
514C22⋊C4—122no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C42—32no—
519C42—32no—
520C42—61no—
521C42—61no—
522C42—122no—
523C42—241no—
524C42—241no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3C2×C4×D411yescharacteristic
533C22×C3C2×#12111yescharacteristic
534C22×C3#53211yescharacteristic
535C22×C3#57611yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57811yescharacteristic
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C3×C4—22no—
547C3×C4—42no—
548C3×C4—42no—
549C3×C4—81no—
550C3×C4—81no—
551C3×C4C2×C4×D411yescharacteristic
552C3×C4C4×#4111yescharacteristic
553C3×C4#52511yescharacteristic
554C3×C4#52911yescharacteristic
555Dic3—22no—
556Dic3—22no—
557Dic3—42no—
558Dic3—41no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—62no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23#67⋊C412yes—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581D4—62no—
582D4—62no—
583D4—121no—
584D4—121no—
585D4—244no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—244no—
626C2×C4#67⋊C412yes—
627C2×C4C22×C4×D311yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—22no—
634C8—22no—
635C8—34no—
636C8—62no—
637C2×C3—42no—
638C2×C3C4×#12111yescharacteristic
639C2×C3#294911yescharacteristic
640C2×C3#295111yescharacteristic
641C2×C3#306211yescharacteristic
642C2×C3#306311yescharacteristic
643C2×C3#317811yescharacteristic
644C2×C3#317911yescharacteristic
645D3—82no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#67⋊C411yescharacteristic
661C22#559611yescharacteristic
662C22D3×#12111yescharacteristic
663C22#577411yescharacteristic
664C22#577811yescharacteristic
665C22#577811yescharacteristic
666C22#577911yescharacteristic
667C4—22no—
668C4—42no—
669C4—42no—
670C4—62no—
671C4—62no—
672C4—81no—
673C4—81no—
674C4—121no—
675C4—122no—
676C4—121no—
677C4—241no—
678C4—241no—
679C4C4×#23711yescharacteristic
680C4C4×D3×D411yescharacteristic
681C4#558611yescharacteristic
682C4#564911yescharacteristic
683C3#2153111yescharacteristic, complemented, Sylow
684C2—42no—
685C2—242no—
686C2#7466011yescharacteristic
687C2#7886811yescharacteristic
688C2#7894711yescharacteristic
689C2#8078911yescharacteristic
690C2#8080911yescharacteristic
691C2#8089111yescharacteristic
692C2#8090711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1894C2
socleC23×C3C22×D4
prime radicalC2×#930C22
cosocleC22×C3×C4C24
hypercenter#1894D3
hypocenterC3#21531
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#5122C2semidirect
2. #82744C2semidirect
4. C2×#4817C2semidirect
6. C3×#1894C2semidirect
8. #70260C2semidirect
9. #70254C2semidirect
10. Dic3×#103C2semidirect
11. #69512C2semidirect
12. #68861C2semidirect
13. #68853C2semidirect
14. #68786C2semidirect
15. #68780C2semidirect
38. #5048C4semidirect
73. #1894D3semidirect
89. C3×C22⋊Q8C2×C4semidirect
127. C22×C3⋊C8C2×C4semidirect
135. #835C2×C4semidirect
146. #815D4semidirect
150. #808C2×C4semidirect
306. C22⋊Q8C4×D3semidirect
683. C3#21531semidirect

Maximal subgroups

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

subgroupindex
C2×#51222
#827442
C2×#50482
C2×#48172
#738722
C3×#18942
#719902
#702602
#702542
Dic3×#1032
#695122
#688612
#688532
#687862
#687802
#215313