Finite Group Databasebeta

← #10702278

Subgroups — #10702278

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#421C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5236C211yesmaximal, characteristic
4#82232C211yesmaximal, characteristic, complemented
5C2×#4557C211yesmaximal, characteristic, complemented
6C2×#4554C211yesmaximal, characteristic, complemented
7C2×#4531C211yesmaximal, characteristic, complemented
8#11072—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×#926C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C3×#421—22no—
17C3×#421—22no—
18C2×C3×#104—22no—
19#5236—22no—
20#5236—22no—
21C2×#837—22no—
22#5229C412yescomplemented
23#5221C412yescomplemented
24#5220—22no—
25#5212—22no—
26#4972—22no—
27#4942—22no—
28#4928—22no—
29#4557—22no—
30#4554—22no—
31#4531—22no—
32C2×#421—31no—
33C2×#421D311yescharacteristic
34#2316—31no—
35#2311—31no—
36C2×#343—31no—
37C2×H(3,ℤ/4)—31no—
38C2×H(3,ℤ/4)—31no—
39C2×C3×#41—21no—
40C2×C3×#41—21no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—21no—
46C2×C3×#41—21no—
47C2×C3×#41—21no—
48C2×C3×#41—21no—
49C2×C3×#41D412yes—
50C2×C3×#41Q812yes—
51C22×C3×Q8C2×C411yescharacteristic
52C22×C3×D4—21no—
53C22×C3×D4C2×C411yescharacteristic
54C23×C3×C4—21no—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3D412yes—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D412yes—
63C22×C3×C8—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42D412yes—
71C2×#217—21no—
72C2×#217—21no—
73C2×#217—24no—
74#926—21no—
75#926—21no—
76#926—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2×C412yes—
89C2×C4⋊Dic3D412yes—
90C2×Dic3⋊C4—24no—
91C2×C4×Dic3C2×C412yes—
92C2×C4×Dic3D412yes—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208D412yes—
99C22×C3⋊C8—21no—
100C3×#104—28no—
101#837—28no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C22×#41C2×D311yescharacteristic
105C22×M16—31no—
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C2×#117—31no—
109C2×#117C2×D311yescharacteristic
110C22×C4⋊C4—31no—
111#421—22no—
112#421—22no—
113#421—62no—
114#421—62no—
115C2×#104—22no—
116C2×#104—62no—
117#391—32no—
118#390—32no—
119#389—62no—
120#389—62no—
121#386—62no—
122#380—62no—
123#380—62no—
124#343—62no—
125H(3,ℤ/4)—62no—
126H(3,ℤ/4)—62no—
127C24×C3—21no—
128C24×C3C2×D411yescharacteristic
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—22no—
133C3×#41—22no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—41no—
138C3×#41—41no—
139C3×#41—41no—
140C3×#41—41no—
141C3×#41—41no—
142C3×#41—41no—
143C3×#41—41no—
144C3×#41—41no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8C4⋊C412yescomplemented
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—41no—
159C2×C3×D4—41no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4C4⋊C412yescomplemented
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×Q811yescharacteristic
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C22×C3×C4#4111yescharacteristic
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—82no—
190C22×Dic3C22⋊C414yes—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—41no—
194C3×M16—41no—
195C2×C3×C8—22no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—41no—
202C3×C42—22no—
203C3×C42—22no—
204#217—41no—
205#217—41no—
206#217—41no—
207#217—41no—
208#217—84no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—41no—
214C4⋊Dic3—41no—
215C4⋊Dic3—41no—
216C4⋊Dic3—41no—
217Dic3⋊C4—84no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C4×Dic3—22no—
222#208—22no—
223#208—22no—
224#208—41no—
225#208—41no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—41no—
228C2×C3⋊C8—41no—
229C2×#41—21no—
230C2×#41—21no—
231C2×#41—21no—
232C2×#41—21no—
233C2×#41—21no—
234C2×#41—21no—
235C2×#41—21no—
236C2×#41—21no—
237C2×#41—21no—
238C2×#41—21no—
239C2×#41Dic612yes—
240C2×#41#6712yes—
241C22×Q8C4×D311yescharacteristic
242C22×D4—21no—
243C22×D4C4×D311yescharacteristic
244C23×C4—21no—
245C23×C4—32no—
246C23×C4C22×D311yescharacteristic
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—32no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16#6712yes—
259C22×C8—21no—
260C22×C8—61no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117—61no—
271#117—61no—
272#117—61no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—32no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—64no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—64no—
287C2×C42—32no—
288C2×C42—32no—
289C2×C42#6712yes—
290#104—28no—
291#104—68no—
292#95—64no—
293#95—64no—
294C23×C3—22no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3C2×C22⋊C411yescharacteristic
302C23×C3#12011yescharacteristic
303C23×C3#12111yescharacteristic
304C23×C3C22⋊Q811yescharacteristic
305C23×C3#12311yescharacteristic
306C23×C3#12411yescharacteristic
307C23×C3#12711yescharacteristic
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—41no—
311C3×Q8—41no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—41no—
319C3×D4—41no—
320C2×C3×C4—22no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4C2×C4⋊C411yescharacteristic
331C2×C3×C4#11711yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4#12011yescharacteristic
338C2×C3×C4#12111yescharacteristic
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4C22⋊Q811yescharacteristic
346C2×C3×C4C22⋊Q811yescharacteristic
347C2×C3×C4C22⋊Q811yescharacteristic
348C2×C3×C4C22⋊Q811yescharacteristic
349C2×C3×C4C22⋊Q811yescharacteristic
350C2×C3×C4#12311yescharacteristic
351C2×C3×C4#12311yescharacteristic
352C2×C3×C4#12311yescharacteristic
353C2×C3×C4#12311yescharacteristic
354C2×C3×C4#12411yescharacteristic
355C2×C3×C4#12411yescharacteristic
356C2×C3×C4#12611yescharacteristic
357C2×C3×C4#12611yescharacteristic
358C2×Dic3—24no—
359C2×Dic3—44no—
360C2×Dic3—44no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—82no—
368C3×C8—42no—
369C3⋊C8—42no—
370C24—21no—
371C24D3×D411yescharacteristic
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8Dic3⋊C412yescomplemented
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4Dic3⋊C412yescomplemented
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—34no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—242no—
420C22×C4C2×Dic611yescharacteristic
421C22×C4C2×C4×D311yescharacteristic
422C22×C4#23511yescharacteristic
423C22×C4#23511yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4#23711yescharacteristic
428C22×C4#23711yescharacteristic
429C22×C4#23711yescharacteristic
430C22×C4#23711yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C2×#6711yescharacteristic
434M16—22no—
435M16—22no—
436M16—41no—
437M16—41no—
438M16—62no—
439M16—62no—
440M16—121no—
441M16—121no—
442C2×C8—22no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—62no—
446C2×C8—121no—
447C2×C8—121no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—121no—
455C4⋊C4—121no—
456C4⋊C4—121no—
457C4⋊C4—121no—
458C4⋊C4—244no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C22⋊C4—121no—
464C22⋊C4—121no—
465C22⋊C4—244no—
466C42—22no—
467C42—22no—
468C42—62no—
469C42—62no—
470C42—62no—
471C42—62no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—44no—
483C22×C3—42no—
484C22×C3—82no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3#38011yescharacteristic
488C22×C3#38611yescharacteristic
489C22×C3#39011yescharacteristic
490C22×C3#39211yescharacteristic
491C22×C3#39311yescharacteristic
492C22×C3#39711yescharacteristic
493C22×C3#39911yescharacteristic
494C3×C4—42no—
495C3×C4—82no—
496C3×C4#38511yescharacteristic
497C3×C4#38511yescharacteristic
498C3×C4#38811yescharacteristic
499C3×C4#38811yescharacteristic
500C3×C4#39411yescharacteristic
501C3×C4#39411yescharacteristic
502C3×C4#39611yescharacteristic
503C3×C4#39611yescharacteristic
504Dic3—44no—
505Dic3—82no—
506Dic3—82no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23#87811yescharacteristic
515C23D3×C22⋊C411yescharacteristic
516C23#88311yescharacteristic
517C23#88511yescharacteristic
518C23#93711yescharacteristic
519C23#93911yescharacteristic
520C23#94011yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—22no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—41no—
532D4—41no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—64no—
544C2×C4—124no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—242no—
553C2×C4#87711yescharacteristic
554C2×C4#87811yescharacteristic
555C2×C4#87811yescharacteristic
556C2×C4#87811yescharacteristic
557C2×C4#87911yescharacteristic
558C2×C4#87911yescharacteristic
559C2×C4#67⋊C411yescharacteristic
560C2×C4#67⋊C411yescharacteristic
561C2×C4#88311yescharacteristic
562C2×C4#88411yescharacteristic
563C2×C4#88411yescharacteristic
564C2×C4#88511yescharacteristic
565C2×C4C2×Dic3⋊C411yescharacteristic
566C2×C4#92411yescharacteristic
567C2×C4#92411yescharacteristic
568C2×C4C4×#6711yescharacteristic
569C2×C4C4×#6711yescharacteristic
570C2×C4#92911yescharacteristic
571C2×C4#93511yescharacteristic
572C2×C4#93511yescharacteristic
573C2×C4#93611yescharacteristic
574C2×C4#93711yescharacteristic
575C2×C4#93811yescharacteristic
576C2×C4#93811yescharacteristic
577C2×C4#93911yescharacteristic
578C2×C4#93911yescharacteristic
579C2×C4#93911yescharacteristic
580C2×C4#95311yescharacteristic
581C8—42no—
582C8—122no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—82no—
590C2×C3#143911yescharacteristic
591C2×C3#191111yescharacteristic
592C2×C3#191211yescharacteristic
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—44no—
604C22—42no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22#494211yescharacteristic
609C22#494711yescharacteristic
610C22#494911yescharacteristic
611C22#495211yescharacteristic
612C22#495611yescharacteristic
613C22#495711yescharacteristic
614C22#522311yescharacteristic
615C4—42no—
616C4—82no—
617C4—124no—
618C4—242no—
619C4—242no—
620C4#494411yescharacteristic
621C4#494511yescharacteristic
622C4#494611yescharacteristic
623C4#494811yescharacteristic
624C4#494811yescharacteristic
625C4#495911yescharacteristic
626C4#521011yescharacteristic
627C4#522211yescharacteristic
628C3#1107211yescharacteristic, complemented, Sylow
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—82no—
636C2#7040811yescharacteristic
637C2#7041111yescharacteristic
638C2#7154711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#421C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC2×#421D3
hypocenterC3#11072
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
4. #82232C2semidirect
5. C2×#4557C2semidirect
6. C2×#4554C2semidirect
7. C2×#4531C2semidirect
22. #5229C4semidirect
23. #5221C4semidirect
150. C2×C3×Q8C4⋊C4semidirect
162. C2×C3×D4C4⋊C4semidirect
393. C2×Q8Dic3⋊C4semidirect
405. C2×D4Dic3⋊C4semidirect
628. C3#11072semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4212
#840872
C2×#52362
#822322
C2×#45572
C2×#45542
C2×#45312
#110723