Finite Group Databasebeta

← #10835910

Subgroups — #10835910

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#433C211yesmaximal, characteristic
2#82262C211yesmaximal, characteristic
3#82235C211yesmaximal, characteristic
4C2×#4849C211yesmaximal, characteristic, complemented
5#70486C211yesmaximal, characteristic, complemented
6#70456C211yesmaximal, characteristic, complemented
7#70455C211yesmaximal, characteristic, complemented
8#69964C211yesmaximal, characteristic
9#69964C211yesmaximal, characteristic
10#69937C211yesmaximal, characteristic
11#69936C211yesmaximal, characteristic
12#69824C211yesmaximal, characteristic, complemented
13#69817C211yesmaximal, characteristic, complemented
14#69775C211yesmaximal, characteristic, complemented
15#69767C211yesmaximal, characteristic, complemented
16#24664—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22C22×Dic12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#433—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32#5212C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#4981—21no—
44#4981—21no—
45#4978—21no—
46#4976—21no—
47#4953—21no—
48#4953—21no—
49#4945—21no—
50#4944—21no—
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4929C2211yescharacteristic
54#4929C2211yescharacteristic
55#4929C2211yescharacteristic
56#4928C2211yescharacteristic
57#4849—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#462—31no—
63C2×#433D311yescharacteristic
64#2621—31no—
65#2323—31no—
66#2036—31no—
67#2035—31no—
68#2001—31no—
69#2001—31no—
70#1981—31no—
71#1971—31no—
72#1969—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1805—31no—
77C23×C3×C4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104#924—22no—
105#924—22no—
106#924—41no—
107#924—41no—
108#924—41no—
109#924—41no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×Dic12—22no—
118C2×Dic12—22no—
119C2×Dic12—22no—
120C2×Dic12—22no—
121C2×Dic12—41no—
122C2×Dic12—41no—
123#869—22no—
124#869—22no—
125C4×Dic6—22no—
126C4×Dic6—22no—
127C3×C4⋊C8—22no—
128C3×C4⋊C8D414yes—
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
131C3×#97—22no—
132C3×C4×C8—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#818—41no—
145#817—41no—
146#816—22no—
147#816—22no—
148#816—22no—
149#816—22no—
150#816—41no—
151#816—41no—
152#816—41no—
153#816—41no—
154C22×Q16—31no—
155C22×Q16—31no—
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8—31no—
159C2×C22⋊Q8—31no—
160C2×C4×Q8—31no—
161C2×#117C2×D311yescharacteristic
162#462—64no—
163#433—24no—
164C2×#107—31no—
165C2×#106—31no—
166C2×C4⋊C8C2×D311yescharacteristic
167C2×C4⋊C8C2×D311yescharacteristic
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103—31no—
175C2×#103—31no—
176C2×C22⋊C8C2×D311yescharacteristic
177C2×C22⋊C8C2×D311yescharacteristic
178C2×#97C2×D311yescharacteristic
179C2×C4×C8C2×D311yescharacteristic
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391—31no—
186#391—31no—
187#389—31no—
188#389—61no—
189#388—61no—
190#388—61no—
191#387—61no—
192#382—31no—
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6#4112yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D412yes—
249C2×C3×C8C2×D412yes—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258#217—24no—
259#217—42no—
260#217—42no—
261#217—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—42no—
272C4⋊Dic3—42no—
273C4⋊Dic3#4112yes—
274Dic3⋊C4—24no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278Dic3⋊C4—81no—
279C4×Dic3—42no—
280C4×Dic3—81no—
281Dic12—44no—
282Dic12—44no—
283C22×Q8—31no—
284C22×Q8—31no—
285C22×Q8—31no—
286C23×C4C22×D311yescharacteristic
287C2×Q16—62no—
288C2×Q16—62no—
289C2×Q16—62no—
290C2×Q16—62no—
291C2×Q16—121no—
292C2×Q16—121no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—62no—
301C22⋊Q8—121no—
302C22⋊Q8—121no—
303C22⋊Q8—121no—
304C22⋊Q8—121no—
305C4×Q8—62no—
306C4×Q8—62no—
307#117—22no—
308#117—22no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4D1211yescharacteristic
322C2×C4⋊C4D1211yescharacteristic
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4D1211yescharacteristic
328C2×C22⋊C4D1211yescharacteristic
329C2×C42—61no—
330C2×C42—61no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—121no—
334#106—121no—
335C4⋊C8—22no—
336C4⋊C8#6714yes—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#103—121no—
342#103—121no—
343#103—121no—
344#103—121no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347#97—22no—
348C4×C8—22no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3C22×D411yescharacteristic
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
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×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—41no—
423Dic6—41no—
424Dic6—41no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—22no—
431C3×C8—41no—
432C3×C8—41no—
433C24C2×D1211yescharacteristic
434C2×Q8—32no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4C2×D1211yescharacteristic
467C22×C4C2×D1211yescharacteristic
468C22×C4C2×D1211yescharacteristic
469C22×C4C2×D1211yescharacteristic
470C22×C4C2×D1211yescharacteristic
471C22×C4C23×D311yescharacteristic
472Q16—124no—
473Q16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8C2×#6712yes—
485C2×C8C2×#6712yes—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—32no—
489C4⋊C4—64no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4—122no—
504C4⋊C4—241no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—64no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—241no—
511C42—21no—
512C42—21no—
513C42—122no—
514C42—241no—
515C42#23512yes—
516C42C2×#6712yes—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3#53511yescharacteristic
527C22×C3#57711yescharacteristic
528C22×C3#57811yescharacteristic
529C22×C3#57811yescharacteristic
530C22×C3#57811yescharacteristic
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—41no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4#54611yescharacteristic
537C3×C4#55011yescharacteristic
538C3×C4#55311yescharacteristic
539C3×C4#55511yescharacteristic
540Dic3—42no—
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23C22×D1211yescharacteristic
556Q8—62no—
557Q8—62no—
558Q8—121no—
559Q8—121no—
560Q8—121no—
561Q8—121no—
562Q8—121no—
563Q8—121no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—62no—
586C2×C4—62no—
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—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4#93012yes—
610C2×C4#93012yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4#100911yescharacteristic
615C2×C4#100911yescharacteristic
616C2×C4C22×#6711yescharacteristic
617C8—22no—
618C8—22no—
619C8—41no—
620C8—41no—
621C2×C3—81no—
622C2×C3#241411yescharacteristic
623C2×C3#305911yescharacteristic
624C2×C3#306311yescharacteristic
625C2×C3#312011yescharacteristic
626C2×C3#312711yescharacteristic
627C2×C3#322311yescharacteristic
628C2×C3#322511yescharacteristic
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22#552411yescharacteristic
637C22#552511yescharacteristic
638C22#575111yescharacteristic
639C22#575311yescharacteristic
640C22#575311yescharacteristic
641C22#575311yescharacteristic
642C22C2×#93011yescharacteristic
643C4—22no—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—122no—
649C4—122no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4#553011yescharacteristic
657C4#553211yescharacteristic
658C4#582911yescharacteristic
659C4#583411yescharacteristic
660C3#2466411yescharacteristic, complemented, Sylow
661C2—81no—
662C2#7938511yescharacteristic
663C2#7939311yescharacteristic
664C2#7939611yescharacteristic
665C2#7939811yescharacteristic
666C2#8227611yescharacteristic
667C2#8380711yescharacteristic
668C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#433C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#433D3
hypocenterC3#24664
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
4. C2×#4849C2semidirect
5. #70486C2semidirect
6. #70456C2semidirect
7. #70455C2semidirect
12. #69824C2semidirect
13. #69817C2semidirect
14. #69775C2semidirect
15. #69767C2semidirect
660. C3#24664semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4332
#822622
#822352
C2×#48492
#704862
#704562
#704552
#699642
#699372
#699362
#698242
#698172
#697752
#697672
#246643