Finite Group Databasebeta

← #10702673

Subgroups — #10702673

Order 768.

1190 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#100C211yesmaximal, characteristic
2#84044C211yesmaximal, characteristic
3C2×#5221C211yesmaximal, characteristic
4C2×#5121C211yesmaximal, characteristic, complemented
5#71985C211yesmaximal, characteristic, complemented
6C2×#4555C211yesmaximal, characteristic, complemented
7C2×#4555C211yesmaximal, characteristic, complemented
8#11135—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×#217—22no—
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4×Dic3C2211yescharacteristic
17C2×C3×#100—22no—
18C2×C3×#100—22no—
19C2×C3×#100C414yescomplemented
20#5247—22no—
21#5221—22no—
22#5221—22no—
23#5221—21no—
24#5221—21no—
25#5221—22no—
26#5121—22no—
27#5121—22no—
28C2×#818—21no—
29C2×#817—21no—
30#4555—22no—
31#4555—22no—
32#4551—22no—
33C2×#428—31no—
34C22×#100D311yescharacteristic
35#2309—31no—
36C2×#390—31no—
37#1736—31no—
38C2×#339—31no—
39C2×#339—31no—
40C25×C3D412yes—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4C2×C412yes—
54C22×C3×D4Q812yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—21no—
57C23×Dic3—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16C2×C412yes—
67C2×C3×M16C2×C412yes—
68C2×C3×M16D412yes—
69C2×C3×M16D412yes—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C2×#217—28no—
73C2×#217—21no—
74C2×#217—21no—
75C2×#217—24no—
76C2×#217—24no—
77C2×#217—48no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D412yes—
92C2×C4⋊Dic3D412yes—
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3D411yescharacteristic
95C2×C4×Dic3—21no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3D412yes—
103C3×#100—28no—
104#818—24no—
105#817—24no—
106C23×D4C2×D311yescharacteristic
107C22×M16C2×D311yescharacteristic
108C22×M16C2×D311yescharacteristic
109C2×#117—31no—
110C2×#117—31no—
111C22×C4⋊C4—31no—
112C22×C22⋊C4—62no—
113C22×C42—31no—
114#428—62no—
115#428—62no—
116C2×#107—61no—
117C2×#106—61no—
118C2×#100—22no—
119C2×#100—22no—
120C2×#100Dic314yescomplemented
121#390—62no—
122#390—62no—
123#390—61no—
124#390—61no—
125#390—62no—
126#379—62no—
127#340—62no—
128#339—62no—
129#339—62no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—44no—
137C24×C3—44no—
138C24×C3C4⋊C414yes—
139C24×C3C2×D411yescharacteristic
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—41no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
155C2×C3×D4—41no—
156C2×C3×D4—41no—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4—41no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4—41no—
164C2×C3×D4—41no—
165C2×C3×D4—41no—
166C2×C3×D4—41no—
167C2×C3×D4—41no—
168C22×C3×C4C22×C411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×Q811yescharacteristic
176C22×C3×C4#4111yescharacteristic
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4#4111yescharacteristic
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—42no—
189C22×Dic3—41no—
190C22×Dic3—41no—
191C22×Dic3—44no—
192C22×Dic3—41no—
193C22×Dic3—41no—
194C3×M16—22no—
195C3×M16—22no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16—41no—
200C3×M16—41no—
201C3×M16C22⋊C414yescomplemented
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—41no—
207C2×C3×C8—41no—
208#217—416no—
209#217—41no—
210#217—41no—
211#217—41no—
212#217—41no—
213#217—84no—
214#217—84no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—41no—
224C4⋊Dic3—41no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—41no—
228C4×Dic3—41no—
229C4×Dic3—41no—
230C4×Dic3—41no—
231C4×Dic3—41no—
232C4×Dic3—41no—
233C25D1212yes—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4Dic612yes—
247C22×D4C2×Dic312yes—
248C23×C4—61no—
249C23×C4—61no—
250C23×C4C22×D311yescharacteristic
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16C2×Dic312yes—
260C2×M16C2×Dic312yes—
261C2×M16#6712yes—
262C2×M16#6712yes—
263C22×C8—21no—
264C22×C8—21no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117—61no—
271#117—61no—
272#117—61no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—128no—
288C2×C42—32no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42—61no—
295C2×C42—61no—
296#107—64no—
297#106—64no—
298#100—28no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—48no—
305C23×C3—44no—
306C23×C3—44no—
307C23×C3—48no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—84no—
315C23×C3—82no—
316C23×C3—82no—
317C23×C3—84no—
318C23×C3C2×C4⋊C411yescharacteristic
319C23×C3#12011yescharacteristic
320C23×C3#12011yescharacteristic
321C23×C3#12111yescharacteristic
322C23×C3#12111yescharacteristic
323C23×C3#12311yescharacteristic
324C23×C3#12311yescharacteristic
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C2×C3×C4C2×C22⋊C411yescharacteristic
334C2×C3×C4#11711yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
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×C4C22⋊Q811yescharacteristic
352C2×C3×C4#12311yescharacteristic
353C2×C3×C4#12311yescharacteristic
354C2×C3×C4#12311yescharacteristic
355C2×C3×C4#12411yescharacteristic
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—44no—
363C2×Dic3—42no—
364C2×Dic3—44no—
365C2×Dic3—42no—
366C2×Dic3—44no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—82no—
376C2×Dic3—82no—
377C3×C8—24no—
378C3×C8—42no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—44no—
386C24—44no—
387C24C4⋊Dic314yes—
388C24C2×D1211yescharacteristic
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C22×C4—64no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—62no—
423C22×C4—122no—
424C22×C4—121no—
425C22×C4—121no—
426C22×C4—124no—
427C22×C4—121no—
428C22×C4—121no—
429C22×C4C2×Dic611yescharacteristic
430C22×C4#23511yescharacteristic
431C22×C4#23511yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4#23711yescharacteristic
437C22×C4#23711yescharacteristic
438C22×C4#23711yescharacteristic
439C22×C4#23711yescharacteristic
440C22×C4C22×Dic311yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443M16—22no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—22no—
448M16—41no—
449M16—41no—
450M16#21714yescomplemented
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—41no—
456C2×C8—41no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—62no—
463C4⋊C4—61no—
464C4⋊C4—62no—
465C4⋊C4—121no—
466C4⋊C4—121no—
467C22⋊C4—1216no—
468C22⋊C4—121no—
469C22⋊C4—121no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C22⋊C4—244no—
473C22⋊C4—244no—
474C42—62no—
475C42—62no—
476C42—121no—
477C42—121no—
478C42—121no—
479C42—121no—
480C42—121no—
481C42—121no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—48no—
491C22×C3—48no—
492C22×C3—44no—
493C22×C3—84no—
494C22×C3—84no—
495C22×C3—84no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—84no—
499C22×C3#38011yescharacteristic
500C22×C3#38511yescharacteristic
501C22×C3#38511yescharacteristic
502C22×C3#39211yescharacteristic
503C22×C3#39411yescharacteristic
504C22×C3#39411yescharacteristic
505C22×C3#39711yescharacteristic
506C22×C3#45312yes—
507C22×C3#45512yes—
508C3×C4#38611yescharacteristic
509C3×C4#38811yescharacteristic
510C3×C4#38811yescharacteristic
511C3×C4#39011yescharacteristic
512C3×C4#39311yescharacteristic
513C3×C4#39611yescharacteristic
514C3×C4#39611yescharacteristic
515C3×C4#39911yescharacteristic
516Dic3—44no—
517Dic3—81no—
518Dic3—82no—
519Dic3—81no—
520Dic3—82no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—48no—
527C23—44no—
528C23—48no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—84no—
537C23—84no—
538C23—82no—
539C23—82no—
540C23#88211yescharacteristic
541C23#88211yescharacteristic
542C23#88611yescharacteristic
543C23#88611yescharacteristic
544C23C2×C4⋊Dic311yescharacteristic
545C23#93011yescharacteristic
546C23#93011yescharacteristic
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555C2×C4—64no—
556C2×C4—124no—
557C2×C4—124no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—124no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—242no—
570C2×C4—242no—
571C2×C4#87811yescharacteristic
572C2×C4#87811yescharacteristic
573C2×C4#87811yescharacteristic
574C2×C4#87811yescharacteristic
575C2×C4#87911yescharacteristic
576C2×C4#87911yescharacteristic
577C2×C4#88311yescharacteristic
578C2×C4#88311yescharacteristic
579C2×C4#88411yescharacteristic
580C2×C4#88411yescharacteristic
581C2×C4#88511yescharacteristic
582C2×C4#88511yescharacteristic
583C2×C4#92411yescharacteristic
584C2×C4#92411yescharacteristic
585C2×C4#92611yescharacteristic
586C2×C4D4×Dic311yescharacteristic
587C2×C4D4×Dic311yescharacteristic
588C2×C4D4×Dic311yescharacteristic
589C2×C4D4×Dic311yescharacteristic
590C2×C4#93511yescharacteristic
591C2×C4#93611yescharacteristic
592C2×C4#93711yescharacteristic
593C2×C4#93811yescharacteristic
594C2×C4#93811yescharacteristic
595C2×C4#93911yescharacteristic
596C2×C4#93911yescharacteristic
597C2×C4#94011yescharacteristic
598C2×C4C2×#21711yescharacteristic
599C8—24no—
600C8—42no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—84no—
606C2×C3#143911yescharacteristic
607C2×C3#185612yes—
608C2×C3#200211yescharacteristic
609C2×C3#200411yescharacteristic
610C2×C3#203712yes—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—48no—
620C22—44no—
621C22—48no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22#474612yes—
629C22#474712yes—
630C22#495011yescharacteristic
631C22#495011yescharacteristic
632C22#495311yescharacteristic
633C22#496011yescharacteristic
634C22#496111yescharacteristic
635C22#496111yescharacteristic
636C22#521211yescharacteristic
637C4—124no—
638C4—242no—
639C4—242no—
640C4—241no—
641C4—241no—
642C4#494711yescharacteristic
643C4#494811yescharacteristic
644C4#494811yescharacteristic
645C4#494911yescharacteristic
646C4#495111yescharacteristic
647C4#495111yescharacteristic
648C4#522011yescharacteristic
649C4#522111yescharacteristic
650C3#1113511yescharacteristic, complemented, Sylow
651C2—21no—
652C2—21no—
653C2—21no—
654C2—21no—
655C2—84no—
656C2#7006612yes—
657C2#7006812yes—
658C2#7008211yescharacteristic
659C2#7008411yescharacteristic
660C2#7154611yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC22×C3×#100C2
socleC23×C3#120
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenterC22×#100D3
hypocenterC3#11135
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#100

Complements

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

NH ≅ G/Ntype
4. C2×#5121C2semidirect
5. #71985C2semidirect
6. C2×#4555C2semidirect
7. C2×#4555C2semidirect
19. C2×C3×#100C4semidirect
120. C2×#100Dic3semidirect
201. C3×M16C22⋊C4semidirect
450. M16#217semidirect
650. C3#11135semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1002
#840442
C2×#52212
C2×#51212
#719852
C2×#45552
#111353