Finite Group Databasebeta

← #10801801

Subgroups — #10801801

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×C3×#405C211yesmaximal, characteristic, complemented
2C2×#5114C211yesmaximal, characteristic
3#82236C211yesmaximal, characteristic
4#82235C211yesmaximal, characteristic, complemented
5C2×#4710C211yesmaximal, characteristic, complemented
6#71990C211yesmaximal, characteristic
7#71989C211yesmaximal, characteristic, complemented
8#71971C211yesmaximal, characteristic, complemented
9#69826C211yesmaximal, characteristic, complemented
10#69820C211yesmaximal, characteristic, complemented
11#69814C211yesmaximal, characteristic, complemented
12#69768C211yesmaximal, characteristic, complemented
13#69767C211yesmaximal, characteristic, complemented
14C4×#816C211yesmaximal, characteristic, complemented
15C4×#821C211yesmaximal, characteristic, complemented
16#21531—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C22×#204C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#405—24no—
26C2×C3×#97C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30#5212C2211yescharacteristic
31#5212C2211yescharacteristic
32#5114C418yescomplemented
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#4990—21no—
43#4989—21no—
44#4978—21no—
45#4976—21no—
46#4957—21no—
47#4956—21no—
48#4945—21no—
49#4944—21no—
50#4939C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4929C2211yescharacteristic
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56#4710—24no—
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62#4551C2211yescharacteristic
63C2×#465—31no—
64C2×#440—31no—
65C2×#405D311yescharacteristic, complemented
66#2324—31no—
67#2323—31no—
68#1973—31no—
69#1944—31no—
70#1934—31no—
71#1933—31no—
72#1895—31no—
73#1894—31no—
74#1808—31no—
75#1776—31no—
76C4×#103—31no—
77C4×#102—31no—
78C23×C3×C4C2311yescharacteristic
79C22×#67—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C22×C3×C8—22no—
87C22×C3×C8—22no—
88C22×C3×C8C2×C414yescomplemented
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#117—22no—
94C3×#117—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
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—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116#924—22no—
117#924C2×C414yescomplemented
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×#204—22no—
126C2×#204—22no—
127C2×#204C2×C414yes—
128C4×D12—22no—
129C4×D12—22no—
130C4×Dic6—22no—
131C4×Dic6—22no—
132C3×#97—22no—
133C3×#97D414yescomplemented
134C3×C4×C8—22no—
135C3×C4×C8—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821—22no—
146#821C2×C414yes—
147#818—22no—
148#818—22no—
149#817C2×C414yescomplemented
150#816—22no—
151#816—22no—
152#816C2×C414yescomplemented
153C22×SD16—31no—
154C22×M16C2×D311yescharacteristic
155C23×C8C2×D311yescharacteristic
156C2×C22⋊Q8—31no—
157C2×#121—31no—
158C2×C4×Q8—31no—
159C2×C4×D4—31no—
160C2×#117C2×D311yescharacteristic
161#465—38no—
162#440—64no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172#405—24no—
173C2×#97C2×D311yescharacteristic
174C2×#97C2×D311yescharacteristic
175C2×C4×C8C2×D311yescharacteristic
176C2×C4×C8C2×D311yescharacteristic
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#380—31no—
191C4×C4⋊C4—31no—
192C4×C4⋊C4—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C22×C412yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4#4112yes—
212C22×C3×C4C2411yescharacteristic
213C2×#67—24no—
214C2×#67—81no—
215C2×#67—81no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12C22×C412yes—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C22×C412yes—
238C3×M16—28no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—28no—
246C2×C3×C8C22×C412yes—
247C2×C3×C8C22×C412yes—
248C2×C3×C8C2×D412yes—
249C2×C3×C8C2×D412yes—
250C2×C3×C8#4112yes—
251C2×C3×C8#4112yes—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42C2×D412yes—
259C3×C42#4112yes—
260#217—24no—
261#217—81no—
262#217—81no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3C22×C412yes—
278C4⋊Dic3C22×C412yes—
279Dic3⋊C4—24no—
280Dic3⋊C4—42no—
281Dic3⋊C4—81no—
282Dic3⋊C4—81no—
283C4×Dic3—42no—
284C4×Dic3—81no—
285C4×Dic3—81no—
286#204—28no—
287C22×Q8—31no—
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—22no—
299C22×C8—22no—
300C22×C8C4×D314yescomplemented
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C22⋊Q8—34no—
306C22⋊Q8—62no—
307#121—34no—
308#121—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×D4—62no—
312C4×D4—62no—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—31no—
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⋊C4—61no—
325C2×C4⋊C4D1211yescharacteristic
326C2×C4⋊C4D1211yescharacteristic
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4D1211yescharacteristic
335C2×C22⋊C4D1211yescharacteristic
336C2×C42—61no—
337C2×C42—61no—
338C2×C42—61no—
339C2×C42C22×D311yescharacteristic
340C2×C42C22×D311yescharacteristic
341#107—62no—
342#107—62no—
343#106—34no—
344#103—34no—
345#103—62no—
346#103—62no—
347#102—34no—
348#102—62no—
349#102—62no—
350#97—22no—
351#97#6714yescomplemented
352C4×C8—22no—
353C4×C8—22no—
354#95—61no—
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—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
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—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4C4×D412yes—
394C2×C3×C4#12112yes—
395C2×C3×C4#12112yes—
396C2×C3×C4C23×C411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402#67—84no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428C4×D3—84no—
429Dic6—22no—
430Dic6—22no—
431Dic6—41no—
432Dic6—41no—
433C3×C8—22no—
434C3×C8—22no—
435C3×C8—22no—
436C3×C8C4×D414yes—
437C24—61no—
438C24C2×D1211yescharacteristic
439C2×Q8—32no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
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—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4—241no—
468C22×C4C2×C4×D312yes—
469C22×C4C2×D1211yescharacteristic
470C22×C4C2×D1211yescharacteristic
471C22×C4C2×D1211yescharacteristic
472C22×C4C2×D1211yescharacteristic
473C22×C4C2×D1211yescharacteristic
474C22×C4#23512yes—
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477M16—28no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—28no—
485C2×C8C2×C4×D312yes—
486C2×C8C2×C4×D312yes—
487C2×C8#23512yes—
488C2×C8#23512yes—
489C2×C8C2×#6712yes—
490C2×C8C2×#6712yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—32no—
494C4⋊C4—32no—
495C4⋊C4—64no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—122no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C42—21no—
519C42—21no—
520C42—122no—
521C42—241no—
522C42—241no—
523C42#23512yes—
524C42C2×#6712yes—
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—22no—
548C3×C4—22no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4C2×C4×D411yescharacteristic
552C3×C4C4×#4111yescharacteristic
553C3×C4#52511yescharacteristic
554C3×C4#52911yescharacteristic
555Dic3—42no—
556Dic3—42no—
557Dic3—42no—
558Dic3—81no—
559Dic3—81no—
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—
575C23C4×D1212yes—
576C23C22×D1211yescharacteristic
577Q8—62no—
578Q8—62no—
579Q8—121no—
580Q8—121no—
581D4—62no—
582D4—62no—
583D4—121no—
584D4—121no—
585D4—244no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4C4×D1212yes—
625C2×C4#93012yes—
626C2×C4#93012yes—
627C2×C4C22×C4×D311yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×#23511yescharacteristic
631C2×C4C2×#23511yescharacteristic
632C2×C4C22×#6711yescharacteristic
633C8—22no—
634C8—22no—
635C8—22no—
636C8C4×#6714yes—
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×C4×D1211yescharacteristic
661C22#547711yescharacteristic
662C22#575111yescharacteristic
663C22#575111yescharacteristic
664C22#575211yescharacteristic
665C22#575311yescharacteristic
666C22C2×#93011yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—41no—
672C4—122no—
673C4—122no—
674C4—122no—
675C4—241no—
676C4—241no—
677C4—241no—
678C4—241no—
679C4C4×#23511yescharacteristic
680C4#547911yescharacteristic
681C4C2×C4×#6711yescharacteristic
682C4#579111yescharacteristic
683C3#2153111yescharacteristic, complemented, Sylow
684C2—42no—
685C2—242no—
686C2#7856211yescharacteristic
687C2#7857011yescharacteristic
688C2#7857211yescharacteristic
689C2#7857811yescharacteristic
690C2C4×#93011yescharacteristic
691C2#8380211yescharacteristic
692C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#405C2
socleC23×C3C22×D4
prime radicalC2×#930C22
cosocleC22×C3×C4C24
hypercenterC2×#405D3
hypocenterC3#21531
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×C3×#405C2semidirect
4. #82235C2semidirect
5. C2×#4710C2semidirect
7. #71989C2semidirect
8. #71971C2semidirect
9. #69826C2semidirect
10. #69820C2semidirect
11. #69814C2semidirect
12. #69768C2semidirect
13. #69767C2semidirect
14. C4×#816C2semidirect
15. C4×#821C2semidirect
32. #5114C4semidirect
65. C2×#405D3semidirect
88. C22×C3×C8C2×C4semidirect
117. #924C2×C4semidirect
133. C3×#97D4semidirect
149. #817C2×C4semidirect
152. #816C2×C4semidirect
300. C22×C8C4×D3semidirect
351. #97#67semidirect
683. C3#21531semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4052
C2×#51142
#822362
#822352
C2×#47102
#719902
#719892
#719712
#698262
#698202
#698142
#697682
#697672
C4×#8162
C4×#8212
#215313