Finite Group Databasebeta

← #10833996

Subgroups — #10833996

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#433C211yesmaximal, characteristic
2#82263C211yesmaximal, characteristic
3#82230C211yesmaximal, characteristic
4C2×#4739C211yesmaximal, characteristic
5#70489C211yesmaximal, characteristic, complemented
6#70488C211yesmaximal, characteristic, complemented
7#70487C211yesmaximal, characteristic, complemented
8#70487C211yesmaximal, characteristic, complemented
9#69963C211yesmaximal, characteristic
10#69931C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic
12#69824C211yesmaximal, characteristic, complemented
13#69817C211yesmaximal, characteristic, complemented
14#69775C211yesmaximal, characteristic, complemented
15#69767C211yesmaximal, characteristic, complemented
16#24514—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×Dic12C2211yescharacteristic
22C22×Dic12C2211yescharacteristic
23C3×#433—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4985—21no—
42#4983—21no—
43#4973—21no—
44#4973—21no—
45#4952—21no—
46#4952—21no—
47#4943—21no—
48#4943—21no—
49#4929C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52#4929C2211yescharacteristic
53#4929C2211yescharacteristic
54#4929C2211yescharacteristic
55#4928C2211yescharacteristic
56#4928C2211yescharacteristic
57#4739—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#451—31no—
63C2×#433D311yescharacteristic
64#2625—31no—
65#2310—31no—
66#2068—31no—
67#2066—31no—
68#2056—31no—
69#2056—31no—
70#1971—31no—
71#1969—31no—
72#1935—31no—
73#1933—31no—
74#1873—31no—
75#1854—31no—
76#1787—31no—
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×Dic6—21no—
80C22×Dic6—22no—
81C22×Dic6—22no—
82C22×Dic6—22no—
83C22×Dic6—22no—
84C22×Dic6—41no—
85C22×Dic6—41no—
86C22×Dic6C2311yescharacteristic
87C22×Dic6C2311yescharacteristic
88C22×Dic6C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#117—22no—
94C3×#117—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104#926—41no—
105#926—41no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111#924—22no—
112#924—22no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×Dic12—22no—
118C2×Dic12—22no—
119C2×Dic12—22no—
120C2×Dic12—22no—
121C2×Dic12—41no—
122C2×Dic12—41no—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×C22⋊C8—22no—
126C3×C22⋊C8D414yes—
127C3×#97—22no—
128C3×C4×C8—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
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—
154C23×Q8—31no—
155C22×Q16—31no—
156C22×Q16—31no—
157C2×C22⋊Q8—31no—
158C2×#117—31no—
159C2×#117C2×D311yescharacteristic
160#451—64no—
161#433—24no—
162C2×#107—31no—
163C2×#106—31no—
164C2×C4⋊C8C2×D311yescharacteristic
165C2×C4⋊C8C2×D311yescharacteristic
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×C22⋊C8C2×D311yescharacteristic
175C2×C22⋊C8C2×D311yescharacteristic
176C2×#97C2×D311yescharacteristic
177C2×C4×C8C2×D311yescharacteristic
178#400—61no—
179#398—61no—
180#397—61no—
181#397—61no—
182#391—31no—
183#391—31no—
184#391—31no—
185#391—31no—
186#391—31no—
187#391—31no—
188#391—61no—
189#391—61no—
190#389—31no—
191#389—31no—
192#381—61no—
193#381—61no—
194#380—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×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D412yes—
210C22×C3×C4#4112yes—
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—24no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—42no—
222C22×Dic3—42no—
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—24no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—44no—
241C2×Dic6—44no—
242C2×Dic6—44no—
243C2×Dic6—44no—
244C2×Dic6—42no—
245C2×Dic6—42no—
246C2×Dic6#4112yes—
247C2×Dic6#4112yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8C2×D412yes—
259C2×C3×C8C2×D412yes—
260C3×C4⋊C4—22no—
261C3×C4⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—21no—
268#217—42no—
269#217—42no—
270#217—81no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281Dic3⋊C4—42no—
282Dic3⋊C4—81no—
283Dic3⋊C4—81no—
284C4×Dic3—42no—
285Dic12—44no—
286Dic12—44no—
287C22×Q8—31no—
288C22×Q8—31no—
289C22×Q8—31no—
290C22×Q8—62no—
291C22×Q8—62no—
292C22×Q8—61no—
293C22×Q8—62no—
294C22×Q8—62no—
295C22×Q8—121no—
296C22×Q8—121no—
297C23×C4—61no—
298C23×C4C22×D311yescharacteristic
299C2×Q16—62no—
300C2×Q16—62no—
301C2×Q16—62no—
302C2×Q16—62no—
303C2×Q16—121no—
304C2×Q16—121no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309C22⋊Q8—62no—
310C22⋊Q8—62no—
311#117—22no—
312#117—22no—
313#117—121no—
314#117—121no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4D1211yescharacteristic
324C2×C4⋊C4D1211yescharacteristic
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4D1211yescharacteristic
329C2×C22⋊C4D1211yescharacteristic
330C2×C42—61no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—121no—
334#106—121no—
335C4⋊C8—22no—
336C4⋊C8—22no—
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#6714yes—
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—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3#12112yes—
371C23×C3C22×D411yescharacteristic
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4#12112yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—48no—
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—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422Dic6—22no—
423Dic6—22no—
424Dic6—22no—
425Dic6—22no—
426Dic6—48no—
427Dic6—41no—
428Dic6—41no—
429Dic6—41no—
430Dic6—41no—
431Dic6—41no—
432Dic6—41no—
433Dic6—41no—
434Dic6—41no—
435C3×C8—22no—
436C3×C8—22no—
437C3×C8—41no—
438C3×C8—41no—
439C24C2×D1211yescharacteristic
440C2×Q8—32no—
441C2×Q8—32no—
442C2×Q8—64no—
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—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—124no—
460C2×Q8—124no—
461C2×Q8—124no—
462C2×Q8—124no—
463C2×Q8—122no—
464C2×Q8—122no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—64no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—122no—
479C22×C4—122no—
480C22×C4C2×D1211yescharacteristic
481C22×C4C2×D1211yescharacteristic
482C22×C4C2×D1211yescharacteristic
483C22×C4C2×D1211yescharacteristic
484C22×C4C2×D1211yescharacteristic
485C22×C4#23512yes—
486C22×C4C2×#6712yes—
487C22×C4C23×D311yescharacteristic
488Q16—124no—
489Q16—124no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8C2×#6712yes—
501C2×C8C2×#6712yes—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—122no—
515C4⋊C4—241no—
516C4⋊C4—241no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—241no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—122no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3C2×#12111yescharacteristic
535C22×C3#52511yescharacteristic
536C22×C3#53511yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57811yescharacteristic
539C22×C3#57811yescharacteristic
540C22×C3#57811yescharacteristic
541C3×C4—22no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4#54611yescharacteristic
547C3×C4#54611yescharacteristic
548C3×C4#55111yescharacteristic
549C3×C4#55111yescharacteristic
550Dic3—42no—
551Dic3—42no—
552Dic3—81no—
553Dic3—81no—
554Dic3—81no—
555Dic3—81no—
556Dic3—81no—
557Dic3—81no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#93012yes—
565C23C22×D1211yescharacteristic
566Q8—62no—
567Q8—62no—
568Q8—62no—
569Q8—62no—
570Q8—128no—
571Q8—121no—
572Q8—121no—
573Q8—121no—
574Q8—121no—
575Q8—121no—
576Q8—121no—
577Q8—121no—
578Q8—121no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—128no—
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—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4#93012yes—
623C2×C4C2×#23511yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4#100911yescharacteristic
627C2×C4#100911yescharacteristic
628C2×C4C22×#6711yescharacteristic
629C8—22no—
630C8—22no—
631C8—41no—
632C8—41no—
633C2×C3—42no—
634C2×C3#241111yescharacteristic
635C2×C3#305911yescharacteristic
636C2×C3#306311yescharacteristic
637C2×C3#312111yescharacteristic
638C2×C3#312711yescharacteristic
639C2×C3#322311yescharacteristic
640C2×C3#322511yescharacteristic
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22#552411yescharacteristic
649C22#552511yescharacteristic
650C22#575111yescharacteristic
651C22#575311yescharacteristic
652C22#575311yescharacteristic
653C22#575311yescharacteristic
654C22C2×#93011yescharacteristic
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—122no—
661C4—122no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4—241no—
666C4—241no—
667C4—241no—
668C4#552711yescharacteristic
669C4#552711yescharacteristic
670C4#582911yescharacteristic
671C4#582911yescharacteristic
672C3#2451411yescharacteristic, complemented, Sylow
673C2—42no—
674C2#7939311yescharacteristic
675C2#7939411yescharacteristic
676C2#7939611yescharacteristic
677C2#7939811yescharacteristic
678C2#8227511yescharacteristic
679C2#8380711yescharacteristic
680C2#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#24514
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
5. #70489C2semidirect
6. #70488C2semidirect
7. #70487C2semidirect
8. #70487C2semidirect
12. #69824C2semidirect
13. #69817C2semidirect
14. #69775C2semidirect
15. #69767C2semidirect
672. C3#24514semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4332
#822632
#822302
C2×#47392
#704892
#704882
#704872
#699632
#699312
#699292
#698242
#698172
#697752
#697672
#245143