Finite Group Databasebeta

← #10703038

Subgroups — #10703038

Order 768.

962 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#104C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5232C211yesmaximal, characteristic
4C2×#4554C211yesmaximal, characteristic, complemented
5#71985C211yesmaximal, characteristic, complemented
6#71535C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#11195—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C2×#926C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C2×C3×#104—22no—
17C2×C3×#104—22no—
18C2×C3×#104—22no—
19#5232—22no—
20#5232C414yescomplemented
21#5229—21no—
22C2×#835—21no—
23#5221—21no—
24#5220—22no—
25C2×#832—21no—
26#5212—21no—
27#5212—21no—
28C2×C3⋊#95—22no—
29#4554—22no—
30#4551—21no—
31#4551—21no—
32#4523—22no—
33C22×#104D311yescharacteristic
34C2×#417—31no—
35#2311—31no—
36C2×#343—31no—
37#1736—31no—
38C2×#339—31no—
39C4⋊#95—31no—
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×#41—21no—
50C2×C3×#41C2×C412yes—
51C2×C3×#41D412yes—
52C22×C3×Q8D411yescharacteristic
53C22×C3×D4—21no—
54C22×C3×D4D411yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3D412yes—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16D412yes—
66C22×C3×C8—21no—
67C2×C3×C42—21no—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42D412yes—
73C2×#217—21no—
74C2×#217—22no—
75C2×#217—22no—
76#926—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—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⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2×C412yes—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4×Dic3—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—21no—
99C2×#208—21no—
100C2×#208C2×C412yes—
101C22×C3⋊C8—21no—
102C3×#104—28no—
103#835—24no—
104#832—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—44no—
108C22×#41C2×D311yescharacteristic
109C22×M16—31no—
110C22×M16C2×D311yescharacteristic
111C2×#117—31no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C42C2×D311yescharacteristic
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—22no—
118#417—34no—
119#417—62no—
120C2×#103—61no—
121C2×#102—61no—
122#391—61no—
123#390—61no—
124#386—62no—
125#380—61no—
126#380—61no—
127C2×#95—62no—
128#343—62no—
129#340—61no—
130#340—61no—
131#339—62no—
132C24×C3—21no—
133C24×C3C2×D411yescharacteristic
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—22no—
142C3×#41—41no—
143C3×#41—41no—
144C3×#41—41no—
145C3×#41—41no—
146C3×#41—41no—
147C3×#41—41no—
148C3×#41—41no—
149C3×#41—41no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8C22⋊C412yes—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—41no—
164C2×C3×D4—41no—
165C2×C3×D4—41no—
166C2×C3×D4—41no—
167C2×C3×D4C22⋊C412yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
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×C4—41no—
180C22×C3×C4—41no—
181C22×C3×C4—41no—
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4#4111yescharacteristic
189C22×C3×C4#4111yescharacteristic
190C22×C3×C4#4111yescharacteristic
191C22×C3×C4#4111yescharacteristic
192C22×C3×C4#4111yescharacteristic
193C22×C3×C4#4111yescharacteristic
194C22×C3×C4#4111yescharacteristic
195C22×C3×C4#4111yescharacteristic
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—41no—
207C22×Dic3—41no—
208C3×M16—22no—
209C3×M16—22no—
210C3×M16—41no—
211C3×M16—41no—
212C2×C3×C8—22no—
213C2×C3×C8—41no—
214C2×C3×C8—41no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—41no—
218C3×C42—41no—
219#217—41no—
220#217—41no—
221#217—82no—
222#217—82no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—41no—
230C4⋊Dic3—41no—
231C4⋊Dic3—41no—
232C4⋊Dic3—41no—
233C4⋊Dic3—41no—
234C4⋊Dic3—41no—
235C4×Dic3—41no—
236C4×Dic3—41no—
237#208—22no—
238#208—22no—
239#208—41no—
240#208—41no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—41no—
243C2×C3⋊C8—41no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41—21no—
247C2×#41—21no—
248C2×#41—21no—
249C2×#41—21no—
250C2×#41—21no—
251C2×#41—21no—
252C2×#41—21no—
253C2×#41—21no—
254C2×#41C4×D312yes—
255C2×#41#6712yes—
256C22×Q8D1211yescharacteristic
257C22×D4—21no—
258C22×D4D1211yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—32no—
262C23×C4—61no—
263C23×C4C22×D311yescharacteristic
264C2×M16—21no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—32no—
270C2×M16—61no—
271C2×M16—61no—
272C2×M16—61no—
273C2×M16—61no—
274C2×M16—61no—
275C2×M16#6712yes—
276C22×C8—21no—
277C22×C8—61no—
278#117—61no—
279#117—61no—
280#117—61no—
281#117—61no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—61no—
305C2×C42#6712yes—
306#104—28no—
307#103—64no—
308#102—64no—
309#95—62no—
310#95—62no—
311#95—124no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C4×D411yescharacteristic
324C23×C3#12011yescharacteristic
325C23×C3#12111yescharacteristic
326C23×C3#12111yescharacteristic
327C23×C3#12111yescharacteristic
328C23×C3#12311yescharacteristic
329C23×C3#12411yescharacteristic
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4C2×C22⋊C411yescharacteristic
357C2×C3×C4#11711yescharacteristic
358C2×C3×C4#11711yescharacteristic
359C2×C3×C4C4×D411yescharacteristic
360C2×C3×C4C4×D411yescharacteristic
361C2×C3×C4C4×D411yescharacteristic
362C2×C3×C4#12011yescharacteristic
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4#12111yescharacteristic
365C2×C3×C4#12111yescharacteristic
366C2×C3×C4#12111yescharacteristic
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4#12311yescharacteristic
369C2×C3×C4#12311yescharacteristic
370C2×C3×C4#12311yescharacteristic
371C2×C3×C4#12311yescharacteristic
372C2×C3×C4#12311yescharacteristic
373C2×C3×C4#12311yescharacteristic
374C2×C3×C4#12311yescharacteristic
375C2×C3×C4#12311yescharacteristic
376C2×C3×C4#12311yescharacteristic
377C2×C3×C4#12411yescharacteristic
378C2×C3×C4#12411yescharacteristic
379C2×C3×C4#12411yescharacteristic
380C2×C3×C4#12611yescharacteristic
381C2×C3×C4#12611yescharacteristic
382C2×C3×C4#12611yescharacteristic
383C2×C3×C4#12611yescharacteristic
384C2×Dic3—44no—
385C2×Dic3—44no—
386C2×Dic3—42no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—41no—
391C2×Dic3—42no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—42no—
395C2×Dic3—41no—
396C2×Dic3—82no—
397C2×Dic3—81no—
398C2×Dic3—81no—
399C2×Dic3—82no—
400C3×C8—42no—
401C3⋊C8—42no—
402C24—21no—
403C24D3×D411yescharacteristic
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411#41—22no—
412#41—41no—
413#41—41no—
414#41—41no—
415#41—41no—
416#41—41no—
417#41—41no—
418#41—41no—
419#41—41no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8(C2×D3)⋊C412yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4(C2×D3)⋊C412yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—121no—
463C22×C4—121no—
464C22×C4C2×C4×D311yescharacteristic
465C22×C4C2×D1211yescharacteristic
466C22×C4#23511yescharacteristic
467C22×C4#23511yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4#23711yescharacteristic
470C22×C4#23711yescharacteristic
471C22×C4#23711yescharacteristic
472C22×C4#23711yescharacteristic
473C22×C4#23711yescharacteristic
474C22×C4#23711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478M16—22no—
479M16—22no—
480M16—41no—
481M16—41no—
482M16—62no—
483M16—62no—
484M16—121no—
485M16—121no—
486C2×C8—22no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—62no—
490C2×C8—121no—
491C2×C8—121no—
492C4⋊C4—62no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—62no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C4⋊C4—121no—
503C4⋊C4—121no—
504C22⋊C4—121no—
505C22⋊C4—121no—
506C22⋊C4—242no—
507C22⋊C4—242no—
508C42—22no—
509C42—22no—
510C42—41no—
511C42—41no—
512C42—121no—
513C42—121no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3#38611yescharacteristic
534C22×C3#38811yescharacteristic
535C22×C3#38811yescharacteristic
536C22×C3#39411yescharacteristic
537C22×C3#39411yescharacteristic
538C22×C3#39411yescharacteristic
539C22×C3#39711yescharacteristic
540C22×C3#45311yescharacteristic
541C22×C3#45411yescharacteristic
542C22×C3#45511yescharacteristic
543C22×C3#45611yescharacteristic
544C3×C4—44no—
545C3×C4—42no—
546C3×C4#38111yescharacteristic
547C3×C4#38611yescharacteristic
548C3×C4#38811yescharacteristic
549C3×C4#38811yescharacteristic
550C3×C4#39411yescharacteristic
551C3×C4#39711yescharacteristic
552C3×C4#39711yescharacteristic
553C3×C4#39711yescharacteristic
554Dic3—82no—
555Dic3—82no—
556Dic3—81no—
557Dic3—81no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23#67⋊C411yescharacteristic
570C23#88211yescharacteristic
571C23#88311yescharacteristic
572C23#88511yescharacteristic
573C23#93811yescharacteristic
574C23#93911yescharacteristic
575C23#93911yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580D4—22no—
581D4—22no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—41no—
587D4—41no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—124no—
603C2×C4—124no—
604C2×C4—122no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—122no—
610C2×C4—121no—
611C2×C4—122no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—242no—
615C2×C4—241no—
616C2×C4—241no—
617C2×C4—242no—
618C2×C4#87711yescharacteristic
619C2×C4#87711yescharacteristic
620C2×C4#87911yescharacteristic
621C2×C4#87911yescharacteristic
622C2×C4#87911yescharacteristic
623C2×C4#87911yescharacteristic
624C2×C4#67⋊C411yescharacteristic
625C2×C4#88311yescharacteristic
626C2×C4#88511yescharacteristic
627C2×C4#88611yescharacteristic
628C2×C4#88611yescharacteristic
629C2×C4#88611yescharacteristic
630C2×C4C2×(C2×D3)⋊C411yescharacteristic
631C2×C4C4×#6711yescharacteristic
632C2×C4C4×#6711yescharacteristic
633C2×C4#92911yescharacteristic
634C2×C4#93011yescharacteristic
635C2×C4#93011yescharacteristic
636C2×C4#93511yescharacteristic
637C2×C4#93511yescharacteristic
638C2×C4#93511yescharacteristic
639C2×C4#93511yescharacteristic
640C2×C4#93611yescharacteristic
641C2×C4#93611yescharacteristic
642C2×C4#93811yescharacteristic
643C2×C4#93911yescharacteristic
644C2×C4#93911yescharacteristic
645C2×C4#95311yescharacteristic
646C8—42no—
647C8—122no—
648C2×C3—21no—
649C2×C3—21no—
650C2×C3—21no—
651C2×C3—21no—
652C2×C3—82no—
653C2×C3#144011yescharacteristic
654C2×C3#188211yescharacteristic
655C2×C3#188311yescharacteristic
656C2×C3#203711yescharacteristic
657C2×C3#203811yescharacteristic
658C2×C3#203911yescharacteristic
659C2×C3#204011yescharacteristic
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22#482211yescharacteristic
680C22#482511yescharacteristic
681C22#482611yescharacteristic
682C22#482711yescharacteristic
683C22#494411yescharacteristic
684C22#494511yescharacteristic
685C22#495211yescharacteristic
686C22#495811yescharacteristic
687C22#496111yescharacteristic
688C22#496111yescharacteristic
689C22#522211yescharacteristic
690C4—44no—
691C4—42no—
692C4—242no—
693C4—241no—
694C4—241no—
695C4—242no—
696C4#494311yescharacteristic
697C4#494411yescharacteristic
698C4#494511yescharacteristic
699C4#495211yescharacteristic
700C4#495311yescharacteristic
701C4#495311yescharacteristic
702C4#521311yescharacteristic
703C4#522211yescharacteristic
704C3#1119511yescharacteristic, complemented, Sylow
705C2—21no—
706C2—21no—
707C2—21no—
708C2—21no—
709C2—82no—
710C2#7035911yescharacteristic
711C2#7036011yescharacteristic
712C2#7036111yescharacteristic
713C2#7036211yescharacteristic
714C2#7036511yescharacteristic
715C2#7036611yescharacteristic
716C2#7154911yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC22×C3×#104C2
socleC23×C3#120
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC22×#104D3
hypocenterC3#11195
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
4. C2×#4554C2semidirect
5. #71985C2semidirect
6. #71535C2semidirect
7. C2×#4523C2semidirect
20. #5232C4semidirect
704. C3#11195semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1042
#840872
C2×#52322
C2×#45542
#719852
#715352
C2×#45232
#111953