Finite Group Databasebeta

← #10701557

Subgroups — #10701557

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#410C211yesmaximal, characteristic
2C2×#5236C211yesmaximal, characteristic
3C2×#5233C211yesmaximal, characteristic
4C2×#4997C211yesmaximal, characteristic, complemented
5#71593C211yesmaximal, characteristic, complemented
6#71580C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#10944—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×#926C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16C3×#410—22no—
17C3×#410C414yescomplemented
18C2×C3×#99—22no—
19#5236—22no—
20#5236C414yescomplemented
21C2×#837—22no—
22#5233—21no—
23#5233—21no—
24#5233—22no—
25C2×#835C412yescomplemented
26C2×#832C412yescomplemented
27#4997—22no—
28#4997—22no—
29C2×#803—21no—
30C2×C4×C3⋊C8—21no—
31#4529—22no—
32#4527—22no—
33#4523—22no—
34C2×#421—31no—
35C2×#416—31no—
36C2×#410D311yescharacteristic
37C2×#405—31no—
38#1740—31no—
39#1735—31no—
40#1733—31no—
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×#41—21no—
50C2×C3×#41—21no—
51C2×C3×#41C2×C412yes—
52C2×C3×#41C2×C412yes—
53C22×C3×Q8C2×C411yescharacteristic
54C22×C3×D4—21no—
55C22×C3×D4C2×C411yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C3×#117—21no—
59C3×#117—21no—
60C3×#117—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117C2×C412yes—
64C3×#117C2×C412yes—
65C3×#117D412yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×#217—21no—
73C2×#217—21no—
74#926—21no—
75#926—21no—
76#926—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926C2×C412yes—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2×C412yes—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208C2×C412yes—
90C2×#208D412yes—
91C2×#208D412yes—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8—21no—
97C22×C3⋊C8C2×C412yes—
98C22×C3⋊C8D411yescharacteristic
99C22×C3⋊C8D411yescharacteristic
100C22×C3⋊C8D412yes—
101C3×#99—28no—
102#837—28no—
103#835—22no—
104#835—22no—
105#832—22no—
106#832—22no—
107#803—24no—
108C4×C3⋊C8—24no—
109C22×#41C2×D311yescharacteristic
110C22×M16—31no—
111C23×C8—31no—
112C2×#117—31no—
113C2×#117—31no—
114C2×#117C2×D311yescharacteristic
115C2×#117C2×D311yescharacteristic
116#421—34no—
117#421—62no—
118C2×#104—62no—
119#416—62no—
120#416—61no—
121#416—61no—
122C2×#103—32no—
123C2×#102—32no—
124#410—22no—
125#410Dic314yescomplemented
126C2×#99—22no—
127#405—62no—
128#405—62no—
129C2×#97—61no—
130C2×C4×C8—61no—
131#342—62no—
132#340—62no—
133#339—62no—
134C24×C3—21no—
135C24×C3C2×D411yescharacteristic
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—22no—
142C3×#41—22no—
143C3×#41—22no—
144C3×#41—41no—
145C3×#41—41no—
146C3×#41—41no—
147C3×#41—41no—
148C3×#41—41no—
149C3×#41—41no—
150C3×#41—41no—
151C3×#41—41no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C4212yes—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—41no—
166C2×C3×D4—41no—
167C2×C3×D4—41no—
168C2×C3×D4—41no—
169C2×C3×D4C4212yes—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—41no—
175C22×C3×C4—41no—
176C22×C3×C4—41no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C22×C3×C4C22×C411yescharacteristic
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4#4111yescharacteristic
188C22×C3×C4#4111yescharacteristic
189C22×C3×C4#4111yescharacteristic
190C22×C3×C4#4111yescharacteristic
191C22×C3×C4#4111yescharacteristic
192C22×C3×C4#4111yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—41no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—41no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—41no—
202C3×C22⋊C4C22⋊C414yes—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—41no—
207#217—22no—
208#217—41no—
209#217—41no—
210#217—41no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—41no—
215C4×Dic3—22no—
216C4×Dic3—41no—
217C4×Dic3—41no—
218C4×Dic3—41no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208C22⋊C414yes—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8C22⋊C412yes—
235C2×C3⋊C8C22⋊C412yes—
236C2×#41—21no—
237C2×#41—21no—
238C2×#41—21no—
239C2×#41—21no—
240C2×#41—21no—
241C2×#41—21no—
242C2×#41—21no—
243C2×#41—21no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41C4×D312yes—
247C2×#41C2×Dic312yes—
248C22×Q8C4×D311yescharacteristic
249C22×D4—21no—
250C22×D4C4×D311yescharacteristic
251C23×C4—21no—
252C23×C4C22×D311yescharacteristic
253C2×M16—32no—
254C2×M16—32no—
255C2×M16—32no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C22×C8—32no—
260C22×C8—32no—
261C22×C8—31no—
262C22×C8—31no—
263C22×C8—61no—
264C22×C8—61no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8—61no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—32no—
274#117—61no—
275#117—61no—
276#117—61no—
277#117—61no—
278#117—61no—
279#117—61no—
280#117—61no—
281#117C2×Dic312yes—
282#117C2×Dic312yes—
283#117#6712yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—61no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4#6712yes—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—61no—
295C2×C42—61no—
296#104—68no—
297#103—62no—
298#103—62no—
299#102—62no—
300#102—62no—
301#99—28no—
302#97—64no—
303C4×C8—64no—
304C23×C3—22no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—82no—
309C23×C3C2×C22⋊C411yescharacteristic
310C23×C3C4×D411yescharacteristic
311C23×C3C4×D411yescharacteristic
312C23×C3#12111yescharacteristic
313C23×C3#12111yescharacteristic
314C23×C3#12411yescharacteristic
315C23×C3#12711yescharacteristic
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—41no—
319C3×Q8—41no—
320C3×D4—22no—
321C3×D4—22no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—44no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—82no—
339C2×C3×C4C2×C4211yescharacteristic
340C2×C3×C4C2×C22⋊C411yescharacteristic
341C2×C3×C4C2×C22⋊C411yescharacteristic
342C2×C3×C4#11711yescharacteristic
343C2×C3×C4#11711yescharacteristic
344C2×C3×C4#11711yescharacteristic
345C2×C3×C4C4×D411yescharacteristic
346C2×C3×C4C4×D411yescharacteristic
347C2×C3×C4C4×D411yescharacteristic
348C2×C3×C4C4×D411yescharacteristic
349C2×C3×C4C4×D411yescharacteristic
350C2×C3×C4C4×D411yescharacteristic
351C2×C3×C4C4×D411yescharacteristic
352C2×C3×C4C4×D411yescharacteristic
353C2×C3×C4C4×D411yescharacteristic
354C2×C3×C4C4×D411yescharacteristic
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12111yescharacteristic
357C2×C3×C4#12111yescharacteristic
358C2×C3×C4#12111yescharacteristic
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4#12311yescharacteristic
361C2×C3×C4#12311yescharacteristic
362C2×C3×C4#12311yescharacteristic
363C2×C3×C4#12411yescharacteristic
364C2×C3×C4#12411yescharacteristic
365C2×C3×C4#12611yescharacteristic
366C2×C3×C4#12611yescharacteristic
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—82no—
370C2×Dic3—82no—
371C3⋊C8—22no—
372C3⋊C8—22no—
373C3⋊C8—24no—
374C24—21no—
375C24C2×#6711yescharacteristic
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—22no—
383#41—22no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8C4×Dic312yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4C4×Dic312yes—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—62no—
420C22×C4—121no—
421C22×C4C2×C4×D311yescharacteristic
422C22×C4C2×C4×D311yescharacteristic
423C22×C4#23511yescharacteristic
424C22×C4#23511yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4#23711yescharacteristic
430C22×C4#23711yescharacteristic
431C22×C4#23711yescharacteristic
432C22×C4#23711yescharacteristic
433C22×C4C22×Dic311yescharacteristic
434C22×C4C2×#6711yescharacteristic
435M16—34no—
436M16—62no—
437M16—62no—
438M16—62no—
439C2×C8—32no—
440C2×C8—32no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—61no—
449C2×C8—62no—
450C2×C8—61no—
451C2×C8—62no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—41no—
456C4⋊C4—62no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—121no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—41no—
463C22⋊C4—62no—
464C22⋊C4—121no—
465C22⋊C4—121no—
466C22⋊C4—121no—
467C22⋊C4#21714yes—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—41no—
472C42—62no—
473C42—121no—
474C42—121no—
475C42—121no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—42no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3C4×C22⋊C411yescharacteristic
488C22×C3#38611yescharacteristic
489C22×C3#38811yescharacteristic
490C22×C3#38811yescharacteristic
491C22×C3#39011yescharacteristic
492C22×C3#39011yescharacteristic
493C22×C3#39011yescharacteristic
494C3×C4—44no—
495C3×C4—42no—
496C3×C4—82no—
497C3×C4C4×C22⋊C411yescharacteristic
498C3×C4C4×C22⋊C411yescharacteristic
499C3×C4#38611yescharacteristic
500C3×C4#38611yescharacteristic
501C3×C4#38811yescharacteristic
502C3×C4#38811yescharacteristic
503C3×C4#38811yescharacteristic
504C3×C4#38811yescharacteristic
505Dic3—82no—
506Dic3—82no—
507C23—22no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—82no—
512C23C4×#6711yescharacteristic
513C23C4×#6711yescharacteristic
514C23#93611yescharacteristic
515C23#93911yescharacteristic
516C23#93911yescharacteristic
517C23#94011yescharacteristic
518C23C2×#21711yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523D4—22no—
524D4—22no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—41no—
530D4—41no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—82no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—242no—
545C2×C4—242no—
546C2×C4#87711yescharacteristic
547C2×C4#87711yescharacteristic
548C2×C4#87911yescharacteristic
549C2×C4#87911yescharacteristic
550C2×C4D3×C22⋊C411yescharacteristic
551C2×C4D3×C22⋊C411yescharacteristic
552C2×C4#67⋊C411yescharacteristic
553C2×C4#67⋊C411yescharacteristic
554C2×C4#67⋊C411yescharacteristic
555C2×C4#67⋊C411yescharacteristic
556C2×C4#88311yescharacteristic
557C2×C4#88311yescharacteristic
558C2×C4#88411yescharacteristic
559C2×C4#88411yescharacteristic
560C2×C4#88511yescharacteristic
561C2×C4#88511yescharacteristic
562C2×C4C2×C4×Dic311yescharacteristic
563C2×C4#92611yescharacteristic
564C2×C4C4×#6711yescharacteristic
565C2×C4C4×#6711yescharacteristic
566C2×C4D4×Dic311yescharacteristic
567C2×C4D4×Dic311yescharacteristic
568C2×C4D4×Dic311yescharacteristic
569C2×C4D4×Dic311yescharacteristic
570C2×C4#93511yescharacteristic
571C2×C4#93711yescharacteristic
572C2×C4#93811yescharacteristic
573C2×C4#93811yescharacteristic
574C8—62no—
575C8—62no—
576C8—64no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—82no—
584C2×C3#143711yescharacteristic
585C2×C3#197612yes—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—42no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#494511yescharacteristic
598C22#494511yescharacteristic
599C22#495711yescharacteristic
600C22#495711yescharacteristic
601C22C4×#21711yescharacteristic
602C22#522011yescharacteristic
603C22#522111yescharacteristic
604C4—44no—
605C4—42no—
606C4—82no—
607C4—242no—
608C4—242no—
609C4Dic3×C22⋊C411yescharacteristic
610C4Dic3×C22⋊C411yescharacteristic
611C4#494411yescharacteristic
612C4#494411yescharacteristic
613C4#495111yescharacteristic
614C4#495111yescharacteristic
615C4#495611yescharacteristic
616C4#495611yescharacteristic
617C3#1094411yescharacteristic, complemented, Sylow
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—82no—
625C2#7004212yes—
626C2#7154511yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#410C2
socleC22×C3#390
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC2×#410D3
hypocenterC3#10944
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
4. C2×#4997C2semidirect
5. #71593C2semidirect
6. #71580C2semidirect
7. #71578C2semidirect
17. C3×#410C4semidirect
20. #5236C4semidirect
25. C2×#835C4semidirect
26. C2×#832C4semidirect
125. #410Dic3semidirect
617. C3#10944semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4102
C2×#52362
C2×#52332
C2×#49972
#715932
#715802
#715782
#109443