Finite Group Databasebeta

← #10845272

Subgroups — #10845272

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#464C211yesmaximal, characteristic
2#82955C211yesmaximal, characteristic
3C2×#5055C211yesmaximal, characteristic
4#82478C211yesmaximal, characteristic, complemented
5#72068C211yesmaximal, characteristic, complemented
6#72065C211yesmaximal, characteristic, complemented
7#72051C211yesmaximal, characteristic
8#72047C211yesmaximal, characteristic
9#71631C211yesmaximal, characteristic, complemented
10#70469C211yesmaximal, characteristic, complemented
11#70460C211yesmaximal, characteristic, complemented
12#70233C211yesmaximal, characteristic, complemented
13#70221C211yesmaximal, characteristic, complemented
14#70165C211yesmaximal, characteristic
15#70153C211yesmaximal, characteristic
16#25336—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#464—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#5055—24no—
41#4979—21no—
42#4979—21no—
43#4975C2211yescharacteristic
44#4975C2211yescharacteristic
45#4951—21no—
46#4951—21no—
47#4950—21no—
48Dic3×C22⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4928—21no—
52#4928—21no—
53C2×#803C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489—31no—
63C2×#464D311yescharacteristic
64#2567—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2056—31no—
69#2055—31no—
70#1988—31no—
71#1940—31no—
72#1878—31no—
73#1877—31no—
74#1803—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×Dic3—21no—
81C22×Dic6—21no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C2×C3×Q16—41no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16—41no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#124—22no—
91C3×#124D414yes—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103Q8×Dic3—22no—
104Q8×Dic3—41no—
105D4×Dic3—22no—
106D4×Dic3—41no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#869—22no—
126#869—22no—
127C3×C4⋊C8—22no—
128C3×#103—22no—
129C3×#102—22no—
130#835—22no—
131#835—22no—
132#835—22no—
133#835—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95D411yescharacteristic
144C3⋊#95D411yescharacteristic
145Dic3⋊C8—41no—
146Dic3⋊C8—41no—
147#803—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155#489—64no—
156#464—24no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169#395—61no—
170#395—61no—
171#391—31no—
172#391—31no—
173#391—31no—
174#390—31no—
175#389—61no—
176#389—61no—
177#388—61no—
178#388—61no—
179#385—61no—
180#384—31no—
181#384—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
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—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8#4112yes—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—42no—
223C22×Dic3—42no—
224C22×Dic3—81no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×Dic6—42no—
230C2×Dic6—81no—
231C2×Dic6—81no—
232C3×Q16—44no—
233C3×SD16—44no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C42—42no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248#217—42no—
249#217—42no—
250#217—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—22no—
253C4⋊Dic3—22no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262C4⋊Dic3#4112yes—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8—61no—
280C22×Q8C22×D311yescharacteristic
281C22×Q8C22×D311yescharacteristic
282C22×D4C22×D311yescharacteristic
283C23×C4—61no—
284C2×Q16—22no—
285C2×Q16—22no—
286C2×Q16—41no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—41no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—62no—
296#124—22no—
297#124#6714yes—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—62no—
301C4×Q8—121no—
302C4×D4—62no—
303C4×D4—121no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—21no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329C4⋊C8—22no—
330C4⋊C8—121no—
331C4⋊C8—121no—
332#103—22no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—22no—
338#102—62no—
339#102—62no—
340#97—62no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—41no—
369C3×D4—41no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4#12112yes—
384C2×C3×C4#12112yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—84no—
414C2×Dic3#13712yes—
415C2×Dic3#13712yes—
416Dic6—84no—
417C3×C8—41no—
418C3×C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—122no—
433C2×Q8—241no—
434C2×Q8—241no—
435C2×Q8#23712yes—
436C2×Q8C2×#6712yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×#6712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—241no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—44no—
461SD16—44no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—32no—
477C4⋊C4—42no—
478C4⋊C4—62no—
479C4⋊C4—62no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—42no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42—122no—
504C42—241no—
505C42#23712yes—
506C42C2×#6712yes—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3C2×#12111yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3#54811yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3C2×#13711yescharacteristic
519C22×C3#57611yescharacteristic
520C22×C3#57711yescharacteristic
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—81no—
526C3×C4#52911yescharacteristic
527C3×C4#54211yescharacteristic
528C3×C4#54611yescharacteristic
529C3×C4#54711yescharacteristic
530Dic3—22no—
531Dic3—22no—
532Dic3—41no—
533Dic3—42no—
534Dic3—41no—
535Dic3—81no—
536Dic3—81no—
537Dic3—81no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D3×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—244no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—62no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—244no—
597C2×C4#93912yes—
598C2×C4#93912yes—
599C2×C4C2×#23711yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4#100411yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4#101011yescharacteristic
604C2×C4C22×#6711yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3—82no—
610C2×C3#242711yescharacteristic
611C2×C3#303911yescharacteristic
612C2×C3#304511yescharacteristic
613C2×C3#331411yescharacteristic
614C2×C3#331611yescharacteristic
615C2×C3#336911yescharacteristic
616C2×C3#337511yescharacteristic
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22#559811yescharacteristic
625C22#561211yescharacteristic
626C22#576911yescharacteristic
627C22#577011yescharacteristic
628C22D3×#13711yescharacteristic
629C22D3×#13711yescharacteristic
630C22C2×#93911yescharacteristic
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—62no—
636C4—62no—
637C4—81no—
638C4—121no—
639C4—122no—
640C4—121no—
641C4—241no—
642C4—241no—
643C4—241no—
644C4#560011yescharacteristic
645C4#560311yescharacteristic
646C4#580611yescharacteristic
647C4#582411yescharacteristic
648C3#2533611yescharacteristic, complemented, Sylow
649C2—82no—
650C2#8058211yescharacteristic
651C2#8058511yescharacteristic
652C2#8061911yescharacteristic
653C2#8062011yescharacteristic
654C2#8264711yescharacteristic
655C2#8390111yescharacteristic
656C2#8396711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#464C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#464D3
hypocenterC3#25336
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
4. #82478C2semidirect
5. #72068C2semidirect
6. #72065C2semidirect
9. #71631C2semidirect
10. #70469C2semidirect
11. #70460C2semidirect
12. #70233C2semidirect
13. #70221C2semidirect
648. C3#25336semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4642
#829552
C2×#50552
#824782
#720682
#720652
#720512
#720472
#716312
#704692
#704602
#702332
#702212
#701652
#701532
#253363