Finite Group Databasebeta

← #10845351

Subgroups — #10845351

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×D4C211yesmaximal, characteristic
2#82383C211yesmaximal, characteristic
3#82375C211yesmaximal, characteristic
4C2×#4829C211yesmaximal, characteristic, complemented
5#71972C211yesmaximal, characteristic, complemented
6#71972C211yesmaximal, characteristic, complemented
7#71971C211yesmaximal, characteristic, complemented
8#71970C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69940C211yesmaximal, characteristic, complemented
11#69938C211yesmaximal, characteristic, complemented
12#69937C211yesmaximal, characteristic, complemented
13#69936C211yesmaximal, characteristic, complemented
14#69778C211yesmaximal, characteristic, complemented
15C4×#816C211yesmaximal, characteristic, complemented
16#25343—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×C8×D4—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5222—21no—
30#5222—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45#4985—21no—
46#4983—21no—
47#4945—21no—
48#4945—21no—
49#4944—21no—
50#4944—21no—
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4829—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#479—31no—
63C2×C8×D4D311yescharacteristic
64#2562—31no—
65#2536—31no—
66#2077—31no—
67#2075—31no—
68#2036—31no—
69#2035—31no—
70#1991—31no—
71#1949—31no—
72#1892—31no—
73#1892—31no—
74#1808—31no—
75#1807—31no—
76C4×#103—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8—22no—
82C22×C3×C8—22no—
83C22×C3×C8—22no—
84C22×C3×C8—22no—
85C22×C3×C8—41no—
86C22×C3×C8—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112#924—22no—
113#924—22no—
114#924—41no—
115#924—41no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123#870—22no—
124#870—22no—
125C4×Dic6—22no—
126C4×Dic6—22no—
127C3×C4⋊C8D414yes—
128C3×C22⋊C8—22no—
129C3×C22⋊C8—22no—
130C3×C4×C8—22no—
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—
140#818—22no—
141#818—41no—
142#817—22no—
143#817—41no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148C23×C8C2×D311yescharacteristic
149C23×C8C2×D311yescharacteristic
150C2×#125—31no—
151C2×C22⋊Q8—31no—
152C2×C22⋊Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
155#479—64no—
156C8×D4—24no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8C2×D311yescharacteristic
167C2×C22⋊C8C2×D311yescharacteristic
168C2×C4×C8C2×D311yescharacteristic
169#400—61no—
170#398—61no—
171#394—61no—
172#394—61no—
173#388—61no—
174#388—61no—
175#388—61no—
176#388—61no—
177#382—31no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C24×C3C2×D411yescharacteristic
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
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×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—44no—
239C2×C3×C8—44no—
240C2×C3×C8C2×D412yes—
241C2×C3×C8C2×D412yes—
242C2×C3×C8#4112yes—
243C3×C4⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248#217—42no—
249#217—42no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—42no—
254#217—81no—
255#217—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277C22×Q8—31no—
278C22×D4D1211yescharacteristic
279C23×C4C22×D311yescharacteristic
280C23×C4C22×D311yescharacteristic
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—41no—
286C22×C8—41no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#125—62no—
292#125—62no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—121no—
296C22⋊Q8—121no—
297C4×Q8—62no—
298C4×Q8—62no—
299C4×D4—22no—
300C4×D4—22no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4D1211yescharacteristic
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4D1211yescharacteristic
323C2×C22⋊C4D1211yescharacteristic
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—121no—
329#106—62no—
330#106—121no—
331C4⋊C8#6714yes—
332#103—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336C22⋊C8—22no—
337C22⋊C8—22no—
338C4×C8—22no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3#13512yes—
361C23×C3#13512yes—
362C23×C3C22×D411yescharacteristic
363C3×D4—28no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
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—41no—
375C2×C3×C4—41no—
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#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
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—
414Dic6—22no—
415Dic6—22no—
416Dic6—41no—
417Dic6—41no—
418C3×C8—22no—
419C3×C8—22no—
420C3×C8—22no—
421C3×C8—41no—
422C24C2×D1211yescharacteristic
423C24C2×D1211yescharacteristic
424C2×Q8—32no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4C2×D1211yescharacteristic
454C22×C4C2×D1211yescharacteristic
455C22×C4C2×D1211yescharacteristic
456C22×C4C2×D1211yescharacteristic
457C22×C4C23×D311yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8#23712yes—
470C2×C8C2×#6712yes—
471C2×C8C2×#6712yes—
472C4⋊C4—22no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
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—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C42—122no—
503C42—241no—
504C42#23512yes—
505C42C2×#6712yes—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3C2×#12111yescharacteristic
521C22×C3#54611yescharacteristic
522C22×C3#54711yescharacteristic
523C22×C3C2×#13511yescharacteristic
524C22×C3C2×#13511yescharacteristic
525C22×C3#57711yescharacteristic
526C22×C3#57811yescharacteristic
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4#52911yescharacteristic
532C3×C4#54211yescharacteristic
533C3×C4#54611yescharacteristic
534C3×C4#54811yescharacteristic
535Dic3—42no—
536Dic3—81no—
537Dic3—81no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23#90412yes—
556C23#90412yes—
557C23C22×D1211yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—121no—
561Q8—121no—
562D4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—62no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
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—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#93012yes—
606C2×C4#93012yes—
607C2×C4C2×#23511yescharacteristic
608C2×C4C2×#23711yescharacteristic
609C2×C4#100411yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4#101011yescharacteristic
612C2×C4C22×#6711yescharacteristic
613C8—22no—
614C8—22no—
615C8—22no—
616C8—41no—
617C2×C3—42no—
618C2×C3—42no—
619C2×C3#242711yescharacteristic
620C2×C3#305011yescharacteristic
621C2×C3#306311yescharacteristic
622C2×C3#330011yescharacteristic
623C2×C3#330111yescharacteristic
624C2×C3#333611yescharacteristic
625C2×C3#333711yescharacteristic
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22#555411yescharacteristic
641C22#555511yescharacteristic
642C22C2×#90411yescharacteristic
643C22C2×#90411yescharacteristic
644C22#575111yescharacteristic
645C22#575311yescharacteristic
646C22C2×#93011yescharacteristic
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—122no—
652C4—241no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4—241no—
657C4—241no—
658C4—241no—
659C4#555711yescharacteristic
660C4#556111yescharacteristic
661C4#580611yescharacteristic
662C4#583111yescharacteristic
663C3#2534311yescharacteristic, complemented, Sylow
664C2—42no—
665C2—42no—
666C2#7941311yescharacteristic
667C2#7941411yescharacteristic
668C2#7944911yescharacteristic
669C2#7945011yescharacteristic
670C2#8240811yescharacteristic
671C2#8374211yescharacteristic
672C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×D4C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×C8×D4D3
hypocenterC3#25343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
4. C2×#4829C2semidirect
5. #71972C2semidirect
6. #71972C2semidirect
7. #71971C2semidirect
8. #71970C2semidirect
9. #70455C2semidirect
10. #69940C2semidirect
11. #69938C2semidirect
12. #69937C2semidirect
13. #69936C2semidirect
14. #69778C2semidirect
15. C4×#816C2semidirect
663. C3#25343semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×D42
#823832
#823752
C2×#48292
#719722
#719712
#719702
#704552
#699402
#699382
#699372
#699362
#697782
C4×#8162
#253433