Finite Group Databasebeta

← #10844604

Subgroups — #10844604

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#489C211yesmaximal, characteristic
2#83229C211yesmaximal, characteristic, complemented
3C2×#5091C211yesmaximal, characteristic, complemented
4#82971C211yesmaximal, characteristic
5#71721C211yesmaximal, characteristic
6#70262C211yesmaximal, characteristic, complemented
7#70246C211yesmaximal, characteristic, complemented
8#70242C211yesmaximal, characteristic, complemented
9#70229C211yesmaximal, characteristic, complemented
10#70194C211yesmaximal, characteristic
11#70178C211yesmaximal, characteristic
12#70173C211yesmaximal, characteristic
13#70162C211yesmaximal, characteristic
14#69836C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#25286—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×Q8×Dic3—21no—
20C2×#944C2211yescharacteristic
21C2×#936C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#489—24no—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29#5229—21no—
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#5091—24no—
41#4984C2211yescharacteristic
42#4984C2211yescharacteristic
43#4980—21no—
44#4980—21no—
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4948—21no—
52#4948—21no—
53#4944—21no—
54#4944—21no—
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#489D311yescharacteristic
63C2×#477—31no—
64#2545—31no—
65#2365—31no—
66#2068—31no—
67#2067—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1925—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8D414yes—
85C3×#124—22no—
86C3×#124—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97Q8×Dic3—44no—
98#944—22no—
99#944—41no—
100#936—22no—
101#936—41no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C4×Dic6—22no—
121C4×Dic6—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#97—22no—
127#835—22no—
128#835—41no—
129#832—22no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#816—22no—
141#816—22no—
142Dic3⋊C8—41no—
143Dic3⋊C8—41no—
144#810—22no—
145#810—22no—
146#807—22no—
147#807—22no—
148C4⋊(C3⋊C8)—22no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124—31no—
152C2×#124C2×D311yescharacteristic
153C2×C4×Q8—31no—
154C2×C4×Q8—61no—
155#489—24no—
156#477—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103C2×D311yescharacteristic
169C2×#103C2×D311yescharacteristic
170C2×#102—31no—
171C2×#102—31no—
172C2×#102C2×D311yescharacteristic
173C2×#102C2×D311yescharacteristic
174C2×#97C2×D311yescharacteristic
175#396—61no—
176#396—61no—
177#391—31no—
178#391—61no—
179#390—31no—
180#389—31no—
181#389—31no—
182#388—61no—
183#388—61no—
184#387—61no—
185#387—61no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C2×C3×Q8—24no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—42no—
203C2×C3×Q8—42no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6#4112yes—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C2×D412yes—
240C3×C4⋊C4C2×D412yes—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C42C2×D412yes—
244C3×C42#4112yes—
245#217—42no—
246#217—42no—
247#217—42no—
248#217—42no—
249C4⋊Dic3—21no—
250C4⋊Dic3—22no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—41no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3#4112yes—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C22×Q8—21no—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C22×D4C22×D311yescharacteristic
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—62no—
286C4⋊Q8—121no—
287C4⋊Q8#6714yes—
288#124—22no—
289#124—22no—
290#124—62no—
291#124—121no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—124no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#107—62no—
323C4⋊C8—62no—
324C4⋊C8—121no—
325C4⋊C8—121no—
326#103—22no—
327#103—22no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—22no—
335#102—22no—
336#102—62no—
337#102—121no—
338#97—22no—
339#95—31no—
340#95—31no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3C22×D411yescharacteristic
356C3×Q8—48no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#12112yes—
379C2×C3×C4#12112yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3AGL(1,ℤ/8)12yes—
410Dic6—22no—
411Dic6—22no—
412Dic6—41no—
413Dic6—41no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—21no—
419C2×Q8—24no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—32no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—32no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—121no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4C2×#6712yes—
483C4⋊C4C2×#6712yes—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—61no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42—122no—
495C42—122no—
496C42—122no—
497C42—122no—
498C42#23712yes—
499C42C2×#6712yes—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3C2×#12111yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3#54811yescharacteristic
510C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
511C22×C3#57511yescharacteristic
512C22×C3#57811yescharacteristic
513C22×C3#57811yescharacteristic
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—81no—
519C3×C4#52911yescharacteristic
520C3×C4#54211yescharacteristic
521C3×C4#54811yescharacteristic
522C3×C4#55011yescharacteristic
523Dic3—22no—
524Dic3—42no—
525Dic3—42no—
526Dic3—41no—
527Dic3—41no—
528Dic3—41no—
529Dic3—81no—
530Dic3—81no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×D3×D411yescharacteristic
539Q8—48no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—62no—
545Q8—62no—
546Q8—121no—
547Q8—121no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—32no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4#93912yes—
590C2×C4#93912yes—
591C2×C4C2×#23711yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#100911yescharacteristic
595C2×C4#101011yescharacteristic
596C2×C4C22×#6711yescharacteristic
597C8—41no—
598C8—41no—
599C8—121no—
600C8—121no—
601C2×C3—161no—
602C2×C3#242611yescharacteristic
603C2×C3#303611yescharacteristic
604C2×C3#304411yescharacteristic
605C2×C3#330811yescharacteristic
606C2×C3#331311yescharacteristic
607C2×C3#336711yescharacteristic
608C2×C3#337711yescharacteristic
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22#567611yescharacteristic
617C22#567711yescharacteristic
618C22D3×AGL(1,ℤ/8)11yescharacteristic
619C22#577311yescharacteristic
620C22#577911yescharacteristic
621C22#577911yescharacteristic
622C22C2×#93911yescharacteristic
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—62no—
628C4—81no—
629C4—122no—
630C4—121no—
631C4—121no—
632C4—121no—
633C4—122no—
634C4—241no—
635C4—241no—
636C4#568011yescharacteristic
637C4#568211yescharacteristic
638C4#583111yescharacteristic
639C4#583411yescharacteristic
640C3#2528611yescharacteristic, complemented, Sylow
641C2—161no—
642C2#8167011yescharacteristic
643C2#8167711yescharacteristic
644C2#8169711yescharacteristic
645C2#8169911yescharacteristic
646C2#8299511yescharacteristic
647C2#8393111yescharacteristic
648C2#8397011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#489D3
hypocenterC3#25286
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. #83229C2semidirect
3. C2×#5091C2semidirect
6. #70262C2semidirect
7. #70246C2semidirect
8. #70242C2semidirect
9. #70229C2semidirect
14. #69836C2semidirect
15. #69813C2semidirect
640. C3#25286semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
#832292
C2×#50912
#829712
#717212
#702622
#702462
#702422
#702292
#701942
#701782
#701732
#701622
#698362
#698132
#252863