Finite Group Databasebeta

← #10703019

Subgroups — #10703019

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5110C211yesmaximal, characteristic
2C2×#4966C211yesmaximal, characteristic
3C2×#4966C211yesmaximal, characteristic
4C2×C3×#340C211yesmaximal, characteristic, complemented
5C2×#4552C211yesmaximal, characteristic, complemented
6C2×#4530C211yesmaximal, characteristic, complemented
7C2×#4530C211yesmaximal, characteristic, complemented
8#11192—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C23×C3×C8C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×#208C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5110—22no—
17#5110—22no—
18C2×(C2×D3)⋊C8—22no—
19C2×#810—21no—
20C2×#810—21no—
21#4966—22no—
22#4966—22no—
23#4966—22no—
24#4966C414yescomplemented
25C2×#809—21no—
26C2×#809—21no—
27C3×#340—22no—
28#4552—22no—
29#4530—22no—
30#4530—22no—
31C2×#417—31no—
32C2×#417—31no—
33C2×#408—31no—
34C2×#343—31no—
35C2×#343—31no—
36C2×#341—31no—
37C2×#340D311yescharacteristic, complemented
38C23×C3×C4—21no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235C2×C412yes—
53C2×#235D412yes—
54C22×D12D411yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6D411yescharacteristic
57C22×C3×C8—21no—
58C22×C3×C8—21no—
59C22×C3×C8—21no—
60C22×C3×C8—21no—
61C22×C3×C8—21no—
62C22×C3×C8—21no—
63C22×C3×C8D412yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2×C412yes—
75C2×C3×C4⋊C4D412yes—
76C2×#208—21no—
77C2×#208—21no—
78C2×#208—21no—
79C2×#208—21no—
80C2×#208—21no—
81C2×#208—21no—
82C2×#208—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208C2×C412yes—
92C2×#208D412yes—
93C2×#208D412yes—
94C22×C3⋊C8—21no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8—21no—
97(C2×D3)⋊C8—28no—
98#810—24no—
99#810—24no—
100#809—24no—
101#809—24no—
102C22×#41—31no—
103C22×M16—31no—
104C22×M16—31no—
105C22×M16—31no—
106C23×C8C2×D311yescharacteristic
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C4⋊C4C2×D311yescharacteristic
109#417—34no—
110#417—62no—
111#417—62no—
112#417—62no—
113C2×#103—61no—
114C2×#103—61no—
115C2×#102—61no—
116C2×#102—61no—
117#408—62no—
118#408—62no—
119C2×C22⋊C8—62no—
120#343—62no—
121#343—62no—
122#341—62no—
123#340—22no—
124C24×C3#4111yescharacteristic
125C23×D3—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—81no—
133C22×C3×C4—81no—
134C22×C3×C4—81no—
135C22×C3×C4—81no—
136C22×C3×C4C22×C411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C2×#67—22no—
151C2×#67—22no—
152C2×#67—41no—
153C2×#67—41no—
154C2×#67—41no—
155C2×#67—41no—
156C22×Dic3—21no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164#235—22no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171#235—41no—
172#235—41no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C22⋊C412yes—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C22⋊C412yes—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—41no—
195C2×C3×C8—41no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206#208—22no—
207#208—22no—
208#208—22no—
209#208—22no—
210#208—22no—
211#208—22no—
212#208—41no—
213#208—41no—
214#208—41no—
215#208—41no—
216#208—41no—
217#208—41no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—41no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C2×#41—32no—
228C2×#41—32no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—61no—
239C22×Q8—31no—
240C22×D4—31no—
241C22×D4—61no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×M16—32no—
247C2×M16—32no—
248C2×M16—32no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16—61no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—61no—
271C22×C8—61no—
272C22×C8—61no—
273C22×C8#6712yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C4×D312yes—
285C2×C4⋊C4D1212yes—
286#103—64no—
287#103—64no—
288#102—64no—
289#102—64no—
290C22⋊C8—68no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3C4×D411yescharacteristic
296C23×C3#12111yescharacteristic
297C23×C3#12311yescharacteristic
298C23×C3#12311yescharacteristic
299C23×C3#12411yescharacteristic
300C23×C3#12611yescharacteristic
301C23×C3#12611yescharacteristic
302C22×D3—22no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—82no—
312C2×C3×C4—82no—
313C2×C3×C4—81no—
314C2×C3×C4—81no—
315C2×C3×C4—81no—
316C2×C3×C4—81no—
317C2×C3×C4—82no—
318C2×C3×C4—82no—
319C2×C3×C4C2×C22⋊C411yescharacteristic
320C2×C3×C4#11711yescharacteristic
321C2×C3×C4#11711yescharacteristic
322C2×C3×C4C4×D411yescharacteristic
323C2×C3×C4C4×D411yescharacteristic
324C2×C3×C4C4×D411yescharacteristic
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4#12011yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4#12111yescharacteristic
329C2×C3×C4#12111yescharacteristic
330C2×C3×C4#12111yescharacteristic
331C2×C3×C4#12111yescharacteristic
332C2×C3×C4#12111yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12311yescharacteristic
335C2×C3×C4#12311yescharacteristic
336C2×C3×C4#12311yescharacteristic
337C2×C3×C4#12311yescharacteristic
338C2×C3×C4#12311yescharacteristic
339C2×C3×C4#12311yescharacteristic
340C2×C3×C4#12311yescharacteristic
341C2×C3×C4#12311yescharacteristic
342C2×C3×C4#12411yescharacteristic
343C2×C3×C4#12411yescharacteristic
344C2×C3×C4#12411yescharacteristic
345C2×C3×C4#12611yescharacteristic
346C2×C3×C4#12611yescharacteristic
347#67—42no—
348#67—42no—
349#67—42no—
350#67—42no—
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354D12—22no—
355D12—22no—
356D12—41no—
357D12—41no—
358C4×D3—42no—
359C4×D3—42no—
360C4×D3—42no—
361C4×D3—42no—
362Dic6—22no—
363Dic6—22no—
364Dic6—41no—
365Dic6—41no—
366C3×C8—42no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C3⋊C8—42no—
370C24—61no—
371C24#23711yescharacteristic
372#41—62no—
373#41—62no—
374#41—62no—
375#41—62no—
376#41—62no—
377#41—62no—
378#41—62no—
379#41—62no—
380#41—121no—
381#41—121no—
382#41—121no—
383#41—121no—
384#41—121no—
385#41—121no—
386#41—121no—
387#41—121no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×D4—32no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—121no—
403C2×D4—121no—
404C2×D4—121no—
405C2×D4—121no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4—121no—
422C22×C4—121no—
423C22×C4C2×C4×D311yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4#23511yescharacteristic
426C22×C4#23511yescharacteristic
427C22×C4#23511yescharacteristic
428C22×C4#23511yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4#23711yescharacteristic
434C22×C4#23911yescharacteristic
435C22×C4#23911yescharacteristic
436C22×C4C2×#6711yescharacteristic
437M16—62no—
438M16—62no—
439M16—62no—
440M16—62no—
441M16—62no—
442M16—62no—
443M16—121no—
444M16—121no—
445M16—121no—
446M16—121no—
447M16—121no—
448M16—121no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—62no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—121no—
460C2×C8—121no—
461C2×C8—121no—
462C2×C8—121no—
463C2×C8—121no—
464C2×C8—121no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3#38611yescharacteristic
486C22×C3#38811yescharacteristic
487C22×C3#38811yescharacteristic
488C22×C3#39411yescharacteristic
489C22×C3#39711yescharacteristic
490C22×C3#39711yescharacteristic
491C22×C3#39711yescharacteristic
492C22×C3#47111yescharacteristic
493C22×C3#47111yescharacteristic
494C22×C3#47211yescharacteristic
495C22×C3#47311yescharacteristic
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C3×C4—82no—
504C3×C4—82no—
505C3×C4#38111yescharacteristic
506C3×C4#38611yescharacteristic
507C3×C4#38811yescharacteristic
508C3×C4#38811yescharacteristic
509C3×C4#39411yescharacteristic
510C3×C4#39411yescharacteristic
511C3×C4#39411yescharacteristic
512C3×C4#39711yescharacteristic
513Dic3—42no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—62no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23#67⋊C411yescharacteristic
526C23#88311yescharacteristic
527C23#88511yescharacteristic
528C23#88611yescharacteristic
529C23#89811yescharacteristic
530C23#89811yescharacteristic
531C23#93811yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—121no—
535Q8—121no—
536D4—62no—
537D4—62no—
538D4—122no—
539D4—122no—
540D4—122no—
541D4—122no—
542D4—121no—
543D4—121no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—62no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4#87211yescharacteristic
562C2×C4C4×D1211yescharacteristic
563C2×C4#87511yescharacteristic
564C2×C4#87611yescharacteristic
565C2×C4D3×C22⋊C411yescharacteristic
566C2×C4#88211yescharacteristic
567C2×C4#88311yescharacteristic
568C2×C4#88311yescharacteristic
569C2×C4#88411yescharacteristic
570C2×C4#88411yescharacteristic
571C2×C4#88411yescharacteristic
572C2×C4#88511yescharacteristic
573C2×C4#89211yescharacteristic
574C2×C4#89311yescharacteristic
575C2×C4#89411yescharacteristic
576C2×C4#89411yescharacteristic
577C2×C4#89411yescharacteristic
578C2×C4#89411yescharacteristic
579C2×C4#89411yescharacteristic
580C2×C4#89511yescharacteristic
581C2×C4#89511yescharacteristic
582C2×C4#89811yescharacteristic
583C2×C4C4×#6711yescharacteristic
584C2×C4#92911yescharacteristic
585C2×C4#93011yescharacteristic
586C2×C4#93711yescharacteristic
587C2×C4#93911yescharacteristic
588C2×C4#94711yescharacteristic
589C8—42no—
590C8—122no—
591C8—122no—
592C8—122no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#144011yescharacteristic
598C2×C3#190711yescharacteristic
599C2×C3#190811yescharacteristic
600C2×C3#205111yescharacteristic
601C2×C3#205211yescharacteristic
602C2×C3#207811yescharacteristic
603C2×C3#207911yescharacteristic
604D3—82no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22#466911yescharacteristic
625C22#467011yescharacteristic
626C22#467411yescharacteristic
627C22#467511yescharacteristic
628C22#495611yescharacteristic
629C22#496111yescharacteristic
630C22#499211yescharacteristic
631C22#516011yescharacteristic
632C22#517711yescharacteristic
633C22#517711yescharacteristic
634C22#519211yescharacteristic
635C4—82no—
636C4—82no—
637C4—122no—
638C4#466411yescharacteristic
639C4#466511yescharacteristic
640C4#467211yescharacteristic
641C4#467511yescharacteristic
642C4#494011yescharacteristic
643C4#495911yescharacteristic
644C4#499011yescharacteristic
645C4#499111yescharacteristic
646C3#1119211yescharacteristic, complemented, Sylow
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—242no—
652C2#6942911yescharacteristic
653C2#7019611yescharacteristic
654C2#7021811yescharacteristic
655C2#7026411yescharacteristic
656C2#7028611yescharacteristic
657C2#7059311yescharacteristic
658C2#7064711yescharacteristic
subgroupG/H
centerC23#938
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#340C2
socleC23×C3#121
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#340D3
hypocenterC3#11192
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. C2×C3×#340C2semidirect
5. C2×#4552C2semidirect
6. C2×#4530C2semidirect
7. C2×#4530C2semidirect
24. #4966C4semidirect
37. C2×#340D3semidirect
646. C3#11192semidirect

Maximal subgroups

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

subgroupindex
C2×#51102
C2×#49662
C2×C3×#3402
C2×#45522
C2×#45302
#111923