Finite Group Databasebeta

← #10702269

Subgroups — #10702269

Order 768.

1202 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82095C211yesmaximal, characteristic
2C2×#4943C211yesmaximal, characteristic
3C2×#4943C211yesmaximal, characteristic
4C2×#4943C211yesmaximal, characteristic
5C2×C3×#342C211yesmaximal, characteristic
6C2×#4527C211yesmaximal, characteristic
7C2×#4527C211yesmaximal, characteristic
8#11070—31nomaximal, Sylow
9C24×Dic3C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×#217C2211yescharacteristic
14C22×#217C2211yescharacteristic
15C22×#217C2211yescharacteristic
16#5210—22no—
17C2×C3⋊#95—22no—
18C2×C3⋊#95—22no—
19C2×C3⋊#95—22no—
20#4946—22no—
21#4943—22no—
22#4943—22no—
23#4943—22no—
24#4943—22no—
25#4943—22no—
26#4943C414yescomplemented
27#4942—22no—
28C3×#342—22no—
29#4527—22no—
30#4527—22no—
31#2315—31no—
32C2×#381—31no—
33C2×#381—31no—
34C2×#381—31no—
35C2×#342—31no—
36C2×#342—31no—
37C2×#342D311yescharacteristic
38C25×C3C2311yescharacteristic
39C23×C3×C4—21no—
40C23×C3×C4—21no—
41C23×C3×C4D412yes—
42C23×Dic3—21no—
43C23×Dic3—21no—
44C23×Dic3—21no—
45C23×Dic3—21no—
46C23×Dic3—21no—
47C23×Dic3—21no—
48C23×Dic3—21no—
49C23×Dic3—21no—
50C23×Dic3—21no—
51C23×Dic3—21no—
52C23×Dic3—21no—
53C23×Dic3—21no—
54C23×Dic3—21no—
55C23×Dic3C2×C412yes—
56C23×Dic3D412yes—
57C23×Dic3D412yes—
58C23×Dic3D412yes—
59C23×Dic3Q812yes—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4C2×C412yes—
77C2×C3×C22⋊C4D412yes—
78C2×#217—21no—
79C2×#217—21no—
80C2×#217—21no—
81C2×#217—21no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217C2×C412yes—
94C2×#217D412yes—
95C2×#217D412yes—
96C2×Dic3⋊C4—24no—
97C3⋊#95—24no—
98C3⋊#95—28no—
99C3⋊#95—28no—
100C3⋊#95—28no—
101C24×C4—31no—
102C22×C22⋊C4—31no—
103C22×C22⋊C4—31no—
104C22×C22⋊C4—31no—
105C22×C22⋊C4C2×D311yescharacteristic
106C22×C22⋊C4C2×D311yescharacteristic
107C22×C22⋊C4C2×D311yescharacteristic
108#385—62no—
109#385—62no—
110#381—34no—
111#381—62no—
112#381—62no—
113#381—62no—
114#381—62no—
115#381—62no—
116#380—62no—
117C2×#95—62no—
118C2×#95—62no—
119C2×#95—62no—
120#342—22no—
121#342—62no—
122#342—62no—
123C24×C3—44no—
124C24×C3C22×C411yescharacteristic
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D411yescharacteristic
127C24×C3C2×D411yescharacteristic
128C24×C3C2×D411yescharacteristic
129C24×C3C2×D411yescharacteristic
130C24×C3C2×D411yescharacteristic
131C24×C3C2×D411yescharacteristic
132C24×C3C2×Q811yescharacteristic
133C24×C3#4111yescharacteristic
134C24×C3#4111yescharacteristic
135C24×C3#4111yescharacteristic
136C24×C3#4111yescharacteristic
137C24×C3#4111yescharacteristic
138C24×C3#4111yescharacteristic
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—44no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—41no—
176C22×Dic3—41no—
177C22×Dic3—41no—
178C22×Dic3—41no—
179C22×Dic3—41no—
180C22×Dic3—41no—
181C22×Dic3—41no—
182C22×Dic3—41no—
183C22×Dic3—41no—
184C22×Dic3—41no—
185C22×Dic3—41no—
186C22×Dic3—41no—
187C22×Dic3—41no—
188C22×Dic3—41no—
189C22×Dic3—41no—
190C22×Dic3—41no—
191C22×Dic3—44no—
192C22×Dic3—44no—
193C22×Dic3—44no—
194C22×Dic3—41no—
195C22×Dic3—41no—
196C22×Dic3C22⋊C414yes—
197C22×Dic3C4⋊C414yescomplemented
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—41no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210#217—22no—
211#217—22no—
212#217—22no—
213#217—22no—
214#217—22no—
215#217—22no—
216#217—41no—
217#217—41no—
218#217—41no—
219#217—41no—
220#217—41no—
221#217—41no—
222Dic3⋊C4—84no—
223C25C22×D311yescharacteristic
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4—32no—
230C23×C4—32no—
231C23×C4—61no—
232C23×C4—61no—
233C23×C4—61no—
234C23×C4—61no—
235C23×C4—61no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4—61no—
240C23×C4—61no—
241C23×C4—61no—
242C23×C4—61no—
243C23×C4—61no—
244C23×C4D1212yes—
245C2×C4⋊C4—64no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—32no—
264C2×C22⋊C4—32no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—61no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4C4×D312yes—
281C2×C22⋊C4#6712yes—
282#95—68no—
283#95—68no—
284#95—68no—
285#95—64no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3C2×C22⋊C411yescharacteristic
302C23×C3C2×C4⋊C411yescharacteristic
303C23×C3#11711yescharacteristic
304C23×C3C4×D411yescharacteristic
305C23×C3C4×D411yescharacteristic
306C23×C3C4×D411yescharacteristic
307C23×C3C4×D411yescharacteristic
308C23×C3#12011yescharacteristic
309C23×C3#12011yescharacteristic
310C23×C3#12011yescharacteristic
311C23×C3#12111yescharacteristic
312C23×C3#12111yescharacteristic
313C23×C3#12111yescharacteristic
314C23×C3#12111yescharacteristic
315C23×C3#12111yescharacteristic
316C23×C3#12111yescharacteristic
317C23×C3#12111yescharacteristic
318C23×C3#12111yescharacteristic
319C23×C3C22⋊Q811yescharacteristic
320C23×C3C22⋊Q811yescharacteristic
321C23×C3C22⋊Q811yescharacteristic
322C23×C3C22⋊Q811yescharacteristic
323C23×C3C22⋊Q811yescharacteristic
324C23×C3C22⋊Q811yescharacteristic
325C23×C3#12311yescharacteristic
326C23×C3#12311yescharacteristic
327C23×C3#12311yescharacteristic
328C23×C3#12311yescharacteristic
329C23×C3#12311yescharacteristic
330C23×C3#12411yescharacteristic
331C23×C3#12411yescharacteristic
332C23×C3#12411yescharacteristic
333C23×C3#12611yescharacteristic
334C23×C3#12611yescharacteristic
335C23×C3#12711yescharacteristic
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—44no—
343C2×C3×C4—82no—
344C2×C3×C4—82no—
345C2×C3×C4—82no—
346C2×C3×C4—82no—
347C2×C3×C4—82no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—44no—
365C2×Dic3—44no—
366C2×Dic3—44no—
367C2×Dic3—44no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—44no—
371C2×Dic3—82no—
372C2×Dic3—82no—
373C2×Dic3—82no—
374C2×Dic3—82no—
375C24—44no—
376C24C2×C4×D311yescharacteristic
377C24C2×D1211yescharacteristic
378C24#23511yescharacteristic
379C24#23511yescharacteristic
380C24#23511yescharacteristic
381C24#23511yescharacteristic
382C24D3×D411yescharacteristic
383C24D3×D411yescharacteristic
384C24D3×D411yescharacteristic
385C24D3×D411yescharacteristic
386C24D3×D411yescharacteristic
387C24#23711yescharacteristic
388C24D3×Q811yescharacteristic
389C24#23911yescharacteristic
390C24C2×#6711yescharacteristic
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—121no—
430C22×C4—121no—
431C22×C4—121no—
432C22×C4—121no—
433C22×C4—121no—
434C22×C4—121no—
435C22×C4—121no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4—121no—
439C22×C4—121no—
440C22×C4—121no—
441C22×C4—121no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—121no—
445C22×C4—124no—
446C22×C4—124no—
447C22×C4—124no—
448C22×C4—121no—
449C22×C4—121no—
450C4⋊C4—244no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C22⋊C4—121no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3#38011yescharacteristic
510C22×C3#38511yescharacteristic
511C22×C3#38511yescharacteristic
512C22×C3#38611yescharacteristic
513C22×C3#38811yescharacteristic
514C22×C3#38811yescharacteristic
515C22×C3#39011yescharacteristic
516C22×C3#39211yescharacteristic
517C22×C3#39311yescharacteristic
518C22×C3#39411yescharacteristic
519C22×C3#39411yescharacteristic
520C22×C3#39611yescharacteristic
521C22×C3#39611yescharacteristic
522C22×C3#39711yescharacteristic
523C22×C3#39911yescharacteristic
524C22×C3#45814yes—
525C3×C4—82no—
526C3×C4—82no—
527C3×C4—82no—
528Dic3—44no—
529Dic3—44no—
530Dic3—82no—
531Dic3—82no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23#87211yescharacteristic
548C23C4×D1211yescharacteristic
549C23#87511yescharacteristic
550C23#87611yescharacteristic
551C23D3×C22⋊C411yescharacteristic
552C23#67⋊C411yescharacteristic
553C23#88211yescharacteristic
554C23#88211yescharacteristic
555C23#88311yescharacteristic
556C23#88411yescharacteristic
557C23#88411yescharacteristic
558C23#88411yescharacteristic
559C23#88411yescharacteristic
560C23#88411yescharacteristic
561C23#88511yescharacteristic
562C23#88511yescharacteristic
563C23D3×C4⋊C411yescharacteristic
564C23#89311yescharacteristic
565C23#89411yescharacteristic
566C23#89411yescharacteristic
567C23#89411yescharacteristic
568C23#89511yescharacteristic
569C23#89611yescharacteristic
570C23#89611yescharacteristic
571C23#89611yescharacteristic
572C23#89611yescharacteristic
573C23#89711yescharacteristic
574C23#89811yescharacteristic
575C23C4×#6711yescharacteristic
576C23#92911yescharacteristic
577C23#93011yescharacteristic
578C23#93711yescharacteristic
579C23#93911yescharacteristic
580C23#94011yescharacteristic
581C23#94611yescharacteristic
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—124no—
611C2×C4—124no—
612C2×C4—124no—
613C2×C4—124no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—124no—
617C2×C4—242no—
618C2×C4—242no—
619C2×C4—242no—
620C2×C4—242no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3#143911yescharacteristic
634C2×C3#189012yes—
635C2×C3#202212yes—
636C2×C3#203012yes—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22#466511yescharacteristic
672C22#466711yescharacteristic
673C22#466811yescharacteristic
674C22#466911yescharacteristic
675C22#467211yescharacteristic
676C22#467311yescharacteristic
677C22#467511yescharacteristic
678C22#467611yescharacteristic
679C22#474214yes—
680C22#494011yescharacteristic
681C22#495711yescharacteristic
682C22#495911yescharacteristic
683C22#496011yescharacteristic
684C22#498911yescharacteristic
685C22#499311yescharacteristic
686C22#499411yescharacteristic
687C4—82no—
688C4—82no—
689C4—82no—
690C4—124no—
691C4—124no—
692C4—242no—
693C4—242no—
694C3#1107011yescharacteristic, complemented, Sylow
695C2—21no—
696C2—21no—
697C2—21no—
698C2—21no—
699C2—21no—
700C2—21no—
701C2—21no—
702C2—21no—
703C2—21no—
704C2—21no—
705C2—21no—
706C2—21no—
707C2#6895612yes—
708C2#6896312yes—
709C2#6942811yescharacteristic
710C2#7003112yes—
subgroupG/H
centerC23#882
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC2×C3×#342C2
socleC23×C3#120
prime radical, cosocleC25×C3C23
hypercenterC2×#342D3
hypocenterC3#11070
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
26. #4943C4semidirect
197. C22×Dic3C4⋊C4semidirect
694. C3#11070semidirect

Maximal subgroups

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

subgroupindex
#820952
C2×#49432
C2×C3×#3422
C2×#45272
#110703