Finite Group Databasebeta

← #10829974

Subgroups — #10829974

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5671C211yesmaximal, characteristic
2C2×#4795C211yesmaximal, characteristic, complemented
3C2×#4758C211yesmaximal, characteristic
4#76077C211yesmaximal, characteristic, complemented
5C3×#2042C211yesmaximal, characteristic
6#71729C211yesmaximal, characteristic
7#70239C211yesmaximal, characteristic, complemented
8#70237C211yesmaximal, characteristic, complemented
9#70225C211yesmaximal, characteristic, complemented
10#70216C211yesmaximal, characteristic
11#70215C211yesmaximal, characteristic
12#70153C211yesmaximal, characteristic
13#69632C211yesmaximal, characteristic, complemented
14#68685C211yesmaximal, characteristic, complemented
15#68619C211yesmaximal, characteristic, complemented
16#24189—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#936C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C22×Dic12C2211yescharacteristic
25#5671—24no—
26#5671—24no—
27C2×#887C2211yescharacteristic
28C2×#879—21no—
29C2×#879—21no—
30C2×#878—21no—
31C2×#877—21no—
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×#102C2211yescharacteristic
37C2×C3×#102C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#4983—21no—
46#4976—21no—
47#4975C2211yescharacteristic
48#4973C2211yescharacteristic
49#4972C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4929C2211yescharacteristic
54#4795—24no—
55#4758—24no—
56#4657—21no—
57#4646—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#462—31no—
66C2×C4×SD16—31no—
67#2582—31no—
68#2042—31no—
69#2042D311yescharacteristic
70#2015—31no—
71#2015—31no—
72#1984—31no—
73#1956—31no—
74#1878—31no—
75#1878—31no—
76#1865—31no—
77#1865—31no—
78#1802—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#124—22no—
88C3×#124—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100#944—22no—
101#944D414yes—
102C2×#216—22no—
103C2×#216—22no—
104C2×#216—41no—
105#936—22no—
106#936—22no—
107D4×Dic3—22no—
108D4×Dic3D414yes—
109C2×#214—22no—
110C2×#214—22no—
111C2×#214—22no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4D411yescharacteristic
124C2×Dic3⋊C4D411yescharacteristic
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C2×Dic12—22no—
133C2×Dic12—22no—
134C2×Dic12—41no—
135#887—22no—
136#887—22no—
137#879—28no—
138#879—28no—
139#878—28no—
140#877—28no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C4×Dic6—22no—
144C4×Dic6—22no—
145C3×#103—22no—
146C3×#103—41no—
147C3×#102—22no—
148C3×#102—41no—
149C3×#95—21no—
150#835—22no—
151#832—41no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156C3⋊#95—21no—
157#816—22no—
158#816—41no—
159Dic3⋊C8—22no—
160C8×Dic3—22no—
161#810—22no—
162#810—22no—
163#808—22no—
164C22×Q16—31no—
165C22×Q16—31no—
166C22×SD16—31no—
167#556—64no—
168#556—64no—
169C2×C4⋊Q8—31no—
170C2×#126—61no—
171C2×#126—61no—
172C2×#124—31no—
173C2×#124C2×D311yescharacteristic
174C2×C22⋊Q8—61no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×Q8—31no—
179C2×C4×D4—31no—
180C2×#117—61no—
181#462—64no—
182C4×SD16—64no—
183C2×#106—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103C2×D311yescharacteristic
191C2×#103C2×D311yescharacteristic
192C2×#102—31no—
193C2×#102C2×D311yescharacteristic
194C2×#102C2×D311yescharacteristic
195C2×C4×C8—31no—
196#398—61no—
197#398—61no—
198#391—31no—
199#391—31no—
200#391—61no—
201#389—31no—
202#389—61no—
203#389C2×D311yescharacteristic
204#384—31no—
205#340—31no—
206#340—31no—
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C24×C3—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8C2×D412yes—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4C2×D412yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—24no—
234C22×Dic3—21no—
235C22×Dic3—42no—
236C22×Dic3—42no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—22no—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6C2×D412yes—
255C2×Dic6#4112yes—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—81no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—24no—
269C3×C42—21no—
270C3×C42—21no—
271#217—24no—
272#217—24no—
273#217—24no—
274#216—44no—
275#214—28no—
276C4⋊Dic3—24no—
277C4⋊Dic3—21no—
278C4⋊Dic3—22no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3C2×D412yes—
282Dic3⋊C4—24no—
283Dic3⋊C4—22no—
284Dic3⋊C4—22no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4—42no—
288Dic3⋊C4—42no—
289Dic3⋊C4—81no—
290Dic3⋊C4—81no—
291C4×Dic3—24no—
292C4×Dic3—24no—
293C4×Dic3—21no—
294C4×Dic3—21no—
295C4×Dic3—42no—
296C4×Dic3C2×D412yes—
297C4×Dic3C2×D412yes—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8#4112yes—
304Dic12—44no—
305C22×Q8—31no—
306C22×Q8—31no—
307C22×Q8—31no—
308C22×Q8C22×D311yescharacteristic
309C22×D4C22×D311yescharacteristic
310C23×C4—61no—
311C2×Q16—62no—
312C2×Q16—62no—
313C2×Q16—62no—
314C2×Q16—62no—
315C2×Q16—121no—
316C2×Q16—121no—
317C2×SD16—62no—
318C2×SD16—62no—
319C2×SD16—62no—
320C22×C8—31no—
321C22×C8—31no—
322C22×C8C22×D311yescharacteristic
323C22×C8C22×D311yescharacteristic
324C4⋊Q8—34no—
325C4⋊Q8—62no—
326#126—68no—
327#126—68no—
328#124—22no—
329#124—22no—
330#124—62no—
331#124—62no—
332C22⋊Q8—68no—
333C4×Q8—62no—
334C4×Q8—62no—
335C4×Q8—62no—
336C4×Q8—62no—
337C4×Q8—62no—
338C4×Q8—62no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×D4—34no—
342C4×D4—62no—
343#117—68no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—61no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C42—31no—
366C2×C42—31no—
367C2×C42—61no—
368C2×C42—61no—
369C2×C42—61no—
370C2×C42C22×D311yescharacteristic
371#106—62no—
372C4⋊C8—62no—
373#103—22no—
374#103—41no—
375#103—62no—
376#103—62no—
377#103—62no—
378#103—62no—
379#103—121no—
380#102—22no—
381#102—41no—
382#102—121no—
383C4×C8—62no—
384#95—21no—
385#95—61no—
386#95—61no—
387#95—61no—
388#95—61no—
389#95—61no—
390C23×C3—22no—
391C23×C3—41no—
392C23×C3—41no—
393C23×C3—41no—
394C23×C3—41no—
395C23×C3—41no—
396C23×C3—41no—
397C23×C3C22×D411yescharacteristic
398C3×Q8—22no—
399C3×Q8—22no—
400C3×Q8—41no—
401C3×Q8—41no—
402C3×D4—22no—
403C3×D4—22no—
404C3×D4—41no—
405C3×D4—41no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4H(5,𝔽2)11yescharacteristic
426C2×C3×C42−1+411yescharacteristic
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—21no—
433C2×Dic3—22no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—48no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3#12012yes—
449C2×Dic3#12012yes—
450C2×Dic3#13512yes—
451C2×Dic3#13712yes—
452Dic6—22no—
453Dic6—22no—
454Dic6—22no—
455Dic6—22no—
456Dic6—41no—
457Dic6—41no—
458Dic6—41no—
459Dic6—41no—
460Dic6—44no—
461C3×C8—41no—
462C3×C8—41no—
463C3⋊C8—22no—
464C3⋊C8—41no—
465C24—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—32no—
472C2×Q8—32no—
473C2×Q8—62no—
474C2×Q8—62no—
475C2×Q8—62no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8D3×D412yes—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4D3×D412yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—64no—
501C22×C4—61no—
502C22×C4—61no—
503C22×C4—61no—
504C22×C4—61no—
505C22×C4—122no—
506C22×C4—122no—
507C22×C4D3×D411yescharacteristic
508C22×C4D3×D411yescharacteristic
509C22×C4C23×D311yescharacteristic
510Q16—124no—
511Q16—124no—
512SD16—68no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—32no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C2×C8—61no—
524C2×C8—61no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—32no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—64no—
536C4⋊C4—64no—
537C4⋊C4—61no—
538C4⋊C4—61no—
539C4⋊C4—61no—
540C4⋊C4—81no—
541C4⋊C4—122no—
542C4⋊C4—122no—
543C4⋊C4—241no—
544C4⋊C4—241no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—64no—
548C22⋊C4—64no—
549C22⋊C4—64no—
550C42—21no—
551C42—21no—
552C42—32no—
553C42—32no—
554C42—64no—
555C42—64no—
556C42—61no—
557C42—61no—
558C42—122no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—81no—
563C22×C3—81no—
564C22×C3—81no—
565C22×C3—81no—
566C22×C3C2×#12011yescharacteristic
567C22×C3#54811yescharacteristic
568C22×C3#55011yescharacteristic
569C22×C3C2×#13511yescharacteristic
570C22×C3C2×#13711yescharacteristic
571C22×C3#57711yescharacteristic
572C22×C3#57811yescharacteristic
573C3×C4—42no—
574C3×C4—41no—
575C3×C4—41no—
576C3×C4—81no—
577C3×C4—81no—
578C3×C4#52511yescharacteristic
579C3×C4#53711yescharacteristic
580C3×C4#54611yescharacteristic
581C3×C4#54711yescharacteristic
582Dic3—22no—
583Dic3—42no—
584Dic3—42no—
585Dic3—41no—
586Dic3—42no—
587Dic3—41no—
588Dic3—81no—
589Dic3#45214yes—
590C23—22no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23C2×D3×D411yescharacteristic
598Q8—22no—
599Q8—22no—
600Q8—41no—
601Q8—41no—
602Q8—62no—
603Q8—62no—
604Q8—62no—
605Q8—62no—
606Q8—124no—
607Q8—121no—
608Q8—121no—
609Q8—121no—
610Q8—121no—
611D4—22no—
612D4—22no—
613D4—41no—
614D4—41no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—32no—
623C2×C4—32no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—62no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—62no—
643C2×C4—61no—
644C2×C4—62no—
645C2×C4—128no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4C2×D3×D411yescharacteristic
656C2×C4C2×D3×D411yescharacteristic
657C2×C4#100411yescharacteristic
658C2×C4D3×#4111yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4#101011yescharacteristic
661C8—41no—
662C8—41no—
663C8—62no—
664C8—121no—
665C2×C3—82no—
666C2×C3#240711yescharacteristic
667C2×C3C2×#45211yescharacteristic
668C2×C3#301311yescharacteristic
669C2×C3#336411yescharacteristic
670C2×C3#336911yescharacteristic
671C2×C3#339411yescharacteristic
672C2×C3#339811yescharacteristic
673C22—42no—
674C22—42no—
675C22—42no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22D3×#12011yescharacteristic
681C22#559911yescharacteristic
682C22#567711yescharacteristic
683C22D3×#13511yescharacteristic
684C22#577111yescharacteristic
685C22#577511yescharacteristic
686C22D3×#13711yescharacteristic
687C4—34no—
688C4—42no—
689C4—41no—
690C4—41no—
691C4—62no—
692C4—81no—
693C4—81no—
694C4—122no—
695C4—122no—
696C4—121no—
697C4—122no—
698C4—121no—
699C4—241no—
700C4#559711yescharacteristic
701C4#561011yescharacteristic
702C4#565511yescharacteristic
703C4#565611yescharacteristic
704C3#2418911yescharacteristic, complemented, Sylow
705C2—82no—
706C2#7689411yescharacteristic
707C2#8008811yescharacteristic
708C2D3×#45211yescharacteristic
709C2#8052411yescharacteristic
710C2#8059011yescharacteristic
711C2#8163411yescharacteristic
712C2#8167111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2042C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2042D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. C2×#4795C2semidirect
4. #76077C2semidirect
7. #70239C2semidirect
8. #70237C2semidirect
9. #70225C2semidirect
13. #69632C2semidirect
14. #68685C2semidirect
15. #68619C2semidirect
704. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#56712
C2×#47952
C2×#47582
#760772
C3×#20422
#717292
#702392
#702372
#702252
#702162
#702152
#701532
#696322
#686852
#686192
#241893