Finite Group Databasebeta

← #10702222

Subgroups — #10702222

Order 768.

1174 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×#5131C211yesmaximal, characteristic
2#82234C211yesmaximal, characteristic, complemented
3C2×#5001C211yesmaximal, characteristic
4C2×#4939C211yesmaximal, characteristic
5C2×C3×#339C211yesmaximal, characteristic, complemented
6C2×#4554C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#11063—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C22×(C2×D3)⋊C4—22no—
14C2×#926C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×#208C2211yescharacteristic
17#5131—22no—
18#5131C414yescomplemented
19C2×#821—22no—
20#5001—22no—
21#5001—22no—
22#4977—21no—
23#4975—21no—
24C2×#809—21no—
25C2×#809—21no—
26#4946—22no—
27#4939—22no—
28#4939—22no—
29#4928—22no—
30C3×#339—22no—
31#4554—22no—
32#4523—22no—
33C2×#418—31no—
34C2×#418—31no—
35#2317—31no—
36C2×#390—31no—
37C2×#343—31no—
38C2×#339—31no—
39C2×#339D311yescharacteristic, complemented
40C24×D3—21no—
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3D412yes—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12—21no—
52C22×D12—21no—
53C22×D12—21no—
54C22×D12—21no—
55C22×D12C2×C412yes—
56C22×D12D412yes—
57C22×D12Q812yes—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16C2×C412yes—
64C22×C3×C8—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×C3×C42D412yes—
79C2×#217—21no—
80C2×(C2×D3)⋊C4—28no—
81C2×(C2×D3)⋊C4—48no—
82#926—21no—
83#926—21no—
84#926C2×C412yes—
85#926D412yes—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3D412yes—
93C2×Dic3⋊C4—24no—
94C2×C4×Dic3—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—21no—
99C2×#208—21no—
100C2×#208D412yes—
101C22×C3⋊C8—21no—
102C3⋊#95—24no—
103#821—28no—
104#809—24no—
105#809—24no—
106C23×D4—31no—
107C22×M16—31no—
108C22×M16C2×D311yescharacteristic
109C2×#117—31no—
110C2×#117C2×D311yescharacteristic
111C22×C4⋊C4—31no—
112C22×C22⋊C4—62no—
113C22×C42C2×D311yescharacteristic
114#418—34no—
115#418—62no—
116#418—62no—
117#418—62no—
118C2×#102—62no—
119C2×#102—61no—
120C2×#102—61no—
121#390—62no—
122#390—62no—
123#389—62no—
124#385—62no—
125#384—61no—
126#382—61no—
127#343—62no—
128#339—22no—
129#339—62no—
130C24×C3C2×D411yescharacteristic
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—44no—
135C23×D3—44no—
136C23×D3—41no—
137C23×D3—41no—
138C23×D3—41no—
139C23×D3—41no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×Q811yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4#4111yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C22×C3×C4#4111yescharacteristic
160C22×C3×C4#4111yescharacteristic
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—44no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—41no—
182C2×D12—41no—
183C2×D12—41no—
184C2×D12—41no—
185C2×D12—41no—
186C2×D12—41no—
187C2×D12—41no—
188C2×D12—41no—
189C2×D12—41no—
190C2×D12—41no—
191C2×D12—41no—
192C2×D12—41no—
193C2×D12C4⋊C414yescomplemented
194C3×M16—22no—
195C3×M16—22no—
196C3×M16—41no—
197C3×M16—41no—
198C2×C3×C8—22no—
199C2×C3×C8—41no—
200C2×C3×C8—41no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—41no—
210C3×C42—41no—
211C3×C42—41no—
212C3×C42—41no—
213#217—22no—
214#217—22no—
215(C2×D3)⋊C4—416no—
216C4⋊Dic3—22no—
217C4⋊Dic3—41no—
218C4⋊Dic3—41no—
219C4⋊Dic3—41no—
220C4⋊Dic3—41no—
221C4⋊Dic3C22⋊C414yescomplemented
222Dic3⋊C4—84no—
223C4×Dic3—22no—
224C4×Dic3—22no—
225#208—22no—
226#208—22no—
227#208—41no—
228#208—41no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—41no—
231C2×C3⋊C8—41no—
232C25—61no—
233C22×D4—32no—
234C22×D4—32no—
235C22×D4—32no—
236C22×D4—61no—
237C22×D4—61no—
238C22×D4—61no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4—61no—
242C22×D4—61no—
243C22×D4—61no—
244C22×D4—61no—
245C22×D4—61no—
246C22×D4—61no—
247C23×C4—21no—
248C23×C4—32no—
249C23×C4C22×D311yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—32no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16C4×D312yes—
262C22×C8—21no—
263C22×C8—61no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—32no—
269#117—32no—
270#117—61no—
271#117—61no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4D1211yescharacteristic
281C2×C4⋊C4D1211yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—128no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—61no—
293C2×C42#6712yes—
294#102—64no—
295#102—64no—
296#102—68no—
297#95—64no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3C4×D411yescharacteristic
301C23×C3#12011yescharacteristic
302C23×C3#12011yescharacteristic
303C23×C3#12111yescharacteristic
304C23×C3#12111yescharacteristic
305C23×C3C22⋊Q811yescharacteristic
306C23×C3#12411yescharacteristic
307C22×D3—24no—
308C22×D3—44no—
309C22×D3—48no—
310C22×D3—44no—
311C22×D3—48no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—84no—
319C22×D3—82no—
320C22×D3—82no—
321C22×D3—84no—
322C22×D3—82no—
323C22×D3—82no—
324C2×C3×C4—24no—
325C2×C3×C4—44no—
326C2×C3×C4—42no—
327C2×C3×C4—44no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4C2×C22⋊C411yescharacteristic
334C2×C3×C4C2×C4⋊C411yescharacteristic
335C2×C3×C4#11711yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4C4×D411yescharacteristic
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12111yescharacteristic
345C2×C3×C4#12111yescharacteristic
346C2×C3×C4C22⋊Q811yescharacteristic
347C2×C3×C4C22⋊Q811yescharacteristic
348C2×C3×C4C22⋊Q811yescharacteristic
349C2×C3×C4C22⋊Q811yescharacteristic
350C2×C3×C4C22⋊Q811yescharacteristic
351C2×C3×C4#12311yescharacteristic
352C2×C3×C4#12311yescharacteristic
353C2×C3×C4#12311yescharacteristic
354C2×C3×C4#12311yescharacteristic
355C2×C3×C4#12311yescharacteristic
356C2×C3×C4#12411yescharacteristic
357C2×C3×C4#12411yescharacteristic
358C2×C3×C4#12611yescharacteristic
359C2×C3×C4#12611yescharacteristic
360C2×C3×C4#12711yescharacteristic
361C2×Dic3—24no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—44no—
365C2×Dic3—42no—
366C2×Dic3—44no—
367C2×Dic3—82no—
368D12—24no—
369D12—24no—
370D12—42no—
371D12—42no—
372D12—42no—
373D12—42no—
374D12—42no—
375D12—42no—
376C3×C8—42no—
377C3⋊C8—42no—
378C24—62no—
379C24—62no—
380C24—62no—
381C24—124no—
382C24—124no—
383C24—121no—
384C24—121no—
385C24—121no—
386C24—121no—
387C24D3×D411yescharacteristic
388C2×D4—34no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—121no—
405C2×D4—121no—
406C2×D4—121no—
407C2×D4—121no—
408C2×D4—121no—
409C2×D4—121no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4—121no—
413C2×D4—121no—
414C2×D4—121no—
415C2×D4—121no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—124no—
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×D1211yescharacteristic
430C22×C4#23511yescharacteristic
431C22×C4#23511yescharacteristic
432C22×C4#23511yescharacteristic
433C22×C4#23511yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D411yescharacteristic
438C22×C4#23711yescharacteristic
439C22×C4D3×Q811yescharacteristic
440C22×C4#23911yescharacteristic
441C22×C4C2×#6711yescharacteristic
442M16—22no—
443M16—22no—
444M16—41no—
445M16—41no—
446M16—62no—
447M16—62no—
448M16—121no—
449M16—121no—
450C2×C8—22no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—62no—
454C2×C8—121no—
455C2×C8—121no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—34no—
461C4⋊C4—62no—
462C4⋊C4—121no—
463C4⋊C4—121no—
464C4⋊C4—121no—
465C4⋊C4—121no—
466C4⋊C4—244no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—1216no—
472C42—22no—
473C42—22no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—62no—
479C42—62no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3#38511yescharacteristic
489C22×C3#38611yescharacteristic
490C22×C3#38811yescharacteristic
491C22×C3#39211yescharacteristic
492C22×C3#39311yescharacteristic
493C22×C3#39411yescharacteristic
494C22×C3#39611yescharacteristic
495C22×C3#45312yes—
496C22×C3#45412yes—
497C2×D3—48no—
498C2×D3—44no—
499C2×D3—84no—
500C2×D3—84no—
501C2×D3—84no—
502C2×D3—84no—
503C2×D3—82no—
504C2×D3—82no—
505C2×D3—84no—
506C3×C4—44no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4#38011yescharacteristic
510C3×C4#38511yescharacteristic
511C3×C4#38811yescharacteristic
512C3×C4#39011yescharacteristic
513C3×C4#39411yescharacteristic
514C3×C4#39611yescharacteristic
515C3×C4#39711yescharacteristic
516C3×C4#39911yescharacteristic
517Dic3—44no—
518Dic3—82no—
519C23—22no—
520C23—22no—
521C23—64no—
522C23—124no—
523C23—128no—
524C23—128no—
525C23—124no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—244no—
533C23—244no—
534C23—242no—
535C23—242no—
536C23—242no—
537C23—242no—
538C23#88211yescharacteristic
539C23#88411yescharacteristic
540C23#88411yescharacteristic
541C23#88511yescharacteristic
542C23#89311yescharacteristic
543C23#89611yescharacteristic
544C23#93711yescharacteristic
545D4—64no—
546D4—64no—
547D4—122no—
548D4—122no—
549D4—122no—
550D4—122no—
551D4—122no—
552D4—122no—
553C2×C4—24no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—64no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—124no—
566C2×C4—122no—
567C2×C4—124no—
568C2×C4—242no—
569C2×C4#87211yescharacteristic
570C2×C4C4×D1211yescharacteristic
571C2×C4#87511yescharacteristic
572C2×C4#87611yescharacteristic
573C2×C4D3×C22⋊C411yescharacteristic
574C2×C4#67⋊C411yescharacteristic
575C2×C4#88211yescharacteristic
576C2×C4#88311yescharacteristic
577C2×C4#88411yescharacteristic
578C2×C4#88411yescharacteristic
579C2×C4#88411yescharacteristic
580C2×C4#88511yescharacteristic
581C2×C4D3×C4⋊C411yescharacteristic
582C2×C4#89411yescharacteristic
583C2×C4#89411yescharacteristic
584C2×C4#89411yescharacteristic
585C2×C4#89511yescharacteristic
586C2×C4#89611yescharacteristic
587C2×C4#89611yescharacteristic
588C2×C4#89611yescharacteristic
589C2×C4#89711yescharacteristic
590C2×C4#89811yescharacteristic
591C2×C4C4×#6711yescharacteristic
592C2×C4#92911yescharacteristic
593C2×C4#93011yescharacteristic
594C2×C4#93911yescharacteristic
595C2×C4#94011yescharacteristic
596C2×C4#94611yescharacteristic
597C8—42no—
598C8—122no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3#143911yescharacteristic
604C2×C3#188312yes—
605C2×C3#200211yescharacteristic
606C2×C3#200311yescharacteristic
607C2×C3#202712yes—
608D3—84no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—128no—
618C22—124no—
619C22—244no—
620C22—244no—
621C22—244no—
622C22—244no—
623C22—244no—
624C22—242no—
625C22—242no—
626C22#466511yescharacteristic
627C22#466911yescharacteristic
628C22#467311yescharacteristic
629C22#467611yescharacteristic
630C22#482212yes—
631C22#482712yes—
632C22#495911yescharacteristic
633C22#496011yescharacteristic
634C22#498911yescharacteristic
635C4—44no—
636C4—81no—
637C4—81no—
638C4—124no—
639C4—242no—
640C4#466711yescharacteristic
641C4#466811yescharacteristic
642C4#467211yescharacteristic
643C4#467511yescharacteristic
644C4#494011yescharacteristic
645C4#495711yescharacteristic
646C4#499311yescharacteristic
647C4#499411yescharacteristic
648C3#1106311yescharacteristic, complemented, Sylow
649C2—21no—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—244no—
654C2#6876312yes—
655C2#6877612yes—
656C2#6942811yescharacteristic
657C2#7038511yescharacteristic
658C2#7038811yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#339C2
socleC23×C3#120
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenterC2×#339D3
hypocenterC3#11063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. #82234C2semidirect
5. C2×C3×#339C2semidirect
6. C2×#4554C2semidirect
7. C2×#4523C2semidirect
18. #5131C4semidirect
39. C2×#339D3semidirect
193. C2×D12C4⋊C4semidirect
221. C4⋊Dic3C22⋊C4semidirect
648. C3#11063semidirect

Maximal subgroups

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

subgroupindex
C2×#51312
#822342
C2×#50012
C2×#49392
C2×C3×#3392
C2×#45542
C2×#45232
#110633