Finite Group Databasebeta

← #10831749

Subgroups — #10831749

Order 768.

1248 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic6C211yesmaximal, characteristic
2C2×#4875C211yesmaximal, characteristic, complemented
3C2×#4757C211yesmaximal, characteristic
4#76076C211yesmaximal, characteristic, complemented
5C3×#1862C211yesmaximal, characteristic
6#71612C211yesmaximal, characteristic
7#70617C211yesmaximal, characteristic, complemented
8#70171C211yesmaximal, characteristic
9#70167C211yesmaximal, characteristic
10#70154C211yesmaximal, characteristic
11#69604C211yesmaximal, characteristic, complemented
12#68910C211yesmaximal, characteristic, complemented
13#68806C211yesmaximal, characteristic, complemented
14#68689C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24342—31nomaximal, Sylow
17C23×Dic6—21no—
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#924—21no—
23D4×Dic6—24no—
24D4×Dic6—24no—
25C2×#888C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887C2211yescharacteristic
28C2×#878—21no—
29C2×#878—21no—
30C2×#869C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#107C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4980—21no—
42#4979—21no—
43#4977C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4929C2211yescharacteristic
50#4875—24no—
51#4757—24no—
52#4658—21no—
53#4652—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×D4×Q8—31no—
65C2×#477—31no—
66C2×#440—31no—
67#2581—31no—
68#2057—31no—
69#2057—31no—
70#2024—31no—
71#2024—31no—
72#1993—31no—
73#1985—31no—
74#1877—31no—
75#1877—31no—
76#1862—31no—
77#1862D311yescharacteristic
78#1804—31no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C22×Dic6—28no—
84C22×Dic6—21no—
85C22×Dic6—21no—
86C22×Dic6—21no—
87C22×Dic6—44no—
88C22×Dic6—44no—
89C22×Dic6D411yescharacteristic
90C22×Dic6C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103D4×Dic3—22no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3D414yes—
107C2×#214—22no—
108C2×#214—22no—
109C2×#214Q814yescomplemented
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115#924—28no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4D411yescharacteristic
121C2×Dic3⋊C4D411yescharacteristic
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#888—22no—
128#888—22no—
129#888—22no—
130#888D414yes—
131#887—22no—
132#887—22no—
133#878—28no—
134#878—28no—
135#869—22no—
136#869—22no—
137C4×Dic6—22no—
138C4×Dic6—22no—
139C3×#107—22no—
140C3×#102—22no—
141C3×#102—22no—
142C3×#95—21no—
143#832—22no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151#816—22no—
152#816—22no—
153#815—22no—
154Dic3⋊C8—22no—
155#810—22no—
156#807—22no—
157#807—22no—
158C23×Q8—61no—
159C22×SD16—31no—
160D4×Q8—64no—
161D4×Q8—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—31no—
164C2×C4⋊Q8—31no—
165C2×C22⋊Q8—61no—
166C2×C22⋊Q8—61no—
167C2×C22⋊Q8—61no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×D4—31no—
171C2×C4×D4—31no—
172C2×C4×D4C2×D311yescharacteristic
173#477—64no—
174#440—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#107C2×D311yescharacteristic
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#102—31no—
183C2×#102C2×D311yescharacteristic
184C2×#102C2×D311yescharacteristic
185C2×#97—31no—
186#400—61no—
187#395—61no—
188#391—31no—
189#391—31no—
190#389—31no—
191#389C2×D311yescharacteristic
192#387—61no—
193#387—61no—
194#382—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C24×C3—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4C2×D412yes—
210C2×C3×D4C2×Q812yes—
211C2×C3×D4#4112yes—
212C22×C3×C4—24no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—24no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—24no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—42no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—24no—
243C2×Dic6—24no—
244C2×Dic6—21no—
245C2×Dic6—24no—
246C2×Dic6—432no—
247C2×Dic6—42no—
248C2×Dic6—42no—
249C2×Dic6—42no—
250C2×Dic6—42no—
251C2×Dic6—42no—
252C2×Dic6—42no—
253C2×Dic6C2×Q812yescomplemented
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—42no—
266C3×C4⋊C4C2×D412yes—
267C3×C22⋊C4—24no—
268C3×C42—21no—
269C3×C42—21no—
270#217—24no—
271#217—24no—
272#214—28no—
273C4⋊Dic3—24no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—22no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3C2×D412yes—
280C4⋊Dic3C2×D412yes—
281Dic3⋊C4—24no—
282Dic3⋊C4—22no—
283Dic3⋊C4—22no—
284Dic3⋊C4—24no—
285Dic3⋊C4—42no—
286Dic3⋊C4—42no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—42no—
290C4×Dic3C2×D412yes—
291C4×Dic3C2×D412yes—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8C2×Q812yescomplemented
298C22×Q8—31no—
299C22×Q8—31no—
300C22×Q8—61no—
301C22×Q8—61no—
302C22×Q8—68no—
303C22×Q8—61no—
304C22×Q8—124no—
305C22×Q8—124no—
306C22×D4C22×D311yescharacteristic
307C23×C4—21no—
308C23×C4—61no—
309C23×C4—61no—
310C2×SD16—34no—
311C2×SD16—62no—
312C2×SD16—62no—
313C22×C8—31no—
314C22×C8—31no—
315C22×C8C22×D311yescharacteristic
316C22×C8C22×D311yescharacteristic
317C4⋊Q8—34no—
318C4⋊Q8—62no—
319C4⋊Q8—62no—
320C4⋊Q8—62no—
321C4⋊Q8—62no—
322C4⋊Q8—62no—
323C22⋊Q8—68no—
324C22⋊Q8—68no—
325C22⋊Q8—68no—
326C4×Q8—62no—
327C4×Q8—62no—
328C4×Q8—62no—
329C4×Q8—62no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—34no—
333C4×D4—62no—
334C4×D4—62no—
335C4×D4—62no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4C22×D311yescharacteristic
349C2×C4⋊C4C22×D311yescharacteristic
350C2×C4⋊C4C22×D311yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—61no—
357C2×C42C22×D311yescharacteristic
358#107—22no—
359#107—62no—
360#107—62no—
361C4⋊C8—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365#102—22no—
366#102—22no—
367#102—62no—
368#97—62no—
369#95—21no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3C22×D411yescharacteristic
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4C22⋊Q814yes—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—48no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4AGL(1,ℤ/8)12yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×Q811yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—48no—
424C2×Dic3—48no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3#12012yes—
435C2×Dic3#12012yes—
436C2×Dic3AGL(1,ℤ/8)12yes—
437Dic6—28no—
438Dic6—22no—
439Dic6—22no—
440Dic6—48no—
441Dic6—48no—
442Dic6—48no—
443Dic6—41no—
444Dic6—41no—
445C3×C8—41no—
446C3×C8—41no—
447C3⋊C8—22no—
448C3⋊C8—41no—
449C24—21no—
450C2×Q8—32no—
451C2×Q8—62no—
452C2×Q8—62no—
453C2×Q8—64no—
454C2×Q8—64no—
455C2×Q8—64no—
456C2×Q8—62no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—122no—
463C2×Q8—122no—
464C2×Q8—122no—
465C2×Q8—122no—
466C2×Q8—122no—
467C2×Q8—1232no—
468C2×Q8—122no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#23512yes—
473C2×D4D3×D412yes—
474C2×D4D3×Q812yes—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—64no—
486C22×C4—64no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—122no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4D3×D411yescharacteristic
496C22×C4D3×D411yescharacteristic
497C22×C4C23×D311yescharacteristic
498SD16—68no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—32no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—32no—
517C4⋊C4—32no—
518C4⋊C4—42no—
519C4⋊C4—62no—
520C4⋊C4—62no—
521C4⋊C4—62no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—122no—
530C4⋊C4—122no—
531C4⋊C4D3×D412yes—
532C22⋊C4—24no—
533C22⋊C4—64no—
534C22⋊C4—64no—
535C42—21no—
536C42—21no—
537C42—32no—
538C42—32no—
539C42—61no—
540C42—61no—
541C42—122no—
542C22×C3—24no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—42no—
548C22×C3—42no—
549C22×C3C2×#12011yescharacteristic
550C22×C3#54811yescharacteristic
551C22×C3D4×Q811yescharacteristic
552C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
553C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
554C22×C3#57811yescharacteristic
555C22×C3#57811yescharacteristic
556C3×C4—22no—
557C3×C4—42no—
558C3×C4—41no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4C2×C22⋊Q811yescharacteristic
562C3×C4#53611yescharacteristic
563C3×C4#54811yescharacteristic
564C3×C4D4×Q811yescharacteristic
565Dic3—22no—
566Dic3—22no—
567Dic3—22no—
568Dic3—42no—
569Dic3—42no—
570Dic3—42no—
571Dic3—41no—
572Dic3—81no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23C2×D3×D411yescharacteristic
581Q8—68no—
582Q8—62no—
583Q8—62no—
584Q8—128no—
585Q8—128no—
586Q8—128no—
587Q8—121no—
588Q8—121no—
589D4—22no—
590D4—22no—
591D4—22no—
592D4#89614yes—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—62no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—128no—
624C2×C4—128no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4#91112yes—
635C2×C4C2×#23511yescharacteristic
636C2×C4C2×D3×D411yescharacteristic
637C2×C4C2×D3×D411yescharacteristic
638C2×C4C2×D3×Q811yescharacteristic
639C2×C4#101011yescharacteristic
640C2×C4#101011yescharacteristic
641C8—41no—
642C8—41no—
643C8—62no—
644C8—121no—
645C2×C3—44no—
646C2×C3#240911yescharacteristic
647C2×C3#300311yescharacteristic
648C2×C3#301111yescharacteristic
649C2×C3#336711yescharacteristic
650C2×C3#336711yescharacteristic
651C2×C3#338611yescharacteristic
652C2×C3#338611yescharacteristic
653C22—24no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22#555211yescharacteristic
661C22D3×#12011yescharacteristic
662C22#563211yescharacteristic
663C22C2×#91111yescharacteristic
664C22#577111yescharacteristic
665C22D3×AGL(1,ℤ/8)11yescharacteristic
666C22#577911yescharacteristic
667C4—22no—
668C4—41no—
669C4—42no—
670C4—62no—
671C4—62no—
672C4—62no—
673C4—81no—
674C4—81no—
675C4—121no—
676C4—122no—
677C4—122no—
678C4—122no—
679C4—241no—
680C4C2×#89611yescharacteristic
681C4#555211yescharacteristic
682C4#563211yescharacteristic
683C4#564811yescharacteristic
684C3#2434211yescharacteristic, complemented, Sylow
685C2—44no—
686C2#7440511yescharacteristic
687C2#7957111yescharacteristic
688C2#7975111yescharacteristic
689C2#8000211yescharacteristic
690C2#8023011yescharacteristic
691C2#8096111yescharacteristic
692C2#8115911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1862C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1862D3
hypocenterC3#24342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. C2×#4875C2semidirect
4. #76076C2semidirect
7. #70617C2semidirect
11. #69604C2semidirect
12. #68910C2semidirect
13. #68806C2semidirect
14. #68689C2semidirect
15. #68620C2semidirect
109. C2×#214Q8semidirect
253. C2×Dic6C2×Q8semidirect
297. C2×C3⋊C8C2×Q8semidirect
684. C3#24342semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic62
C2×#48752
C2×#47572
#760762
C3×#18622
#716122
#706172
#701712
#701672
#701542
#696042
#689102
#688062
#686892
#686202
#243423