Finite Group Databasebeta

← #10858194

Subgroups — #10858194

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic
2#83235C211yesmaximal, characteristic, complemented
3#82334C211yesmaximal, characteristic
4C2×#4758C211yesmaximal, characteristic
5#71783C211yesmaximal, characteristic, complemented
6#71612C211yesmaximal, characteristic
7#70645C211yesmaximal, characteristic, complemented
8#70642C211yesmaximal, characteristic, complemented
9#70620C211yesmaximal, characteristic, complemented
10#70611C211yesmaximal, characteristic, complemented
11#70489C211yesmaximal, characteristic, complemented
12#70447C211yesmaximal, characteristic, complemented
13#70171C211yesmaximal, characteristic
14#70169C211yesmaximal, characteristic
15#70168C211yesmaximal, characteristic
16#26372—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C3×#474—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29#5230—21no—
30#5229—21no—
31C2×#832C2211yescharacteristic
32#5210—21no—
33C2×C3⋊#95—21no—
34C2×#818C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4984C2211yescharacteristic
39#4982—21no—
40#4978—21no—
41#4977C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4946—21no—
49#4942—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928C2211yescharacteristic
53#4758—24no—
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#474D311yescharacteristic
63C2×C4×SD16—31no—
64#2451—31no—
65#2349—31no—
66#2097—31no—
67#2085—31no—
68#2068—31no—
69#2058—31no—
70#1942—31no—
71#1916—31no—
72#1877—31no—
73#1877—31no—
74#1864—31no—
75#1863—31no—
76#1862—31no—
77C22×C3×Q8—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—41no—
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—41no—
101C2×#214—22no—
102C2×#214—22no—
103C2×#214Q814yescomplemented
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×Dic3⋊C4D411yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#887—22no—
120#887—41no—
121C3×#107—22no—
122C3×#107—22no—
123C3×C4⋊C8—22no—
124C3×#102—22no—
125C3×#102D414yes—
126#832—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
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⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#818—22no—
140#816—22no—
141#816—22no—
142C8×Dic3—22no—
143#810—41no—
144#808—22no—
145#808—41no—
146C22×SD16—31no—
147C2×C4⋊Q8C2×D311yescharacteristic
148C2×C4×Q8—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4C2×D311yescharacteristic
152#474—24no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107C2×D311yescharacteristic
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102C2×D311yescharacteristic
165C2×#102C2×D311yescharacteristic
166C2×C4×C8—31no—
167#398—61no—
168#395—61no—
169#391—31no—
170#391—31no—
171#391—61no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—31no—
176#387—61no—
177#385—61no—
178#385—61no—
179#382—31no—
180#380—61no—
181C4×C4⋊C4—31no—
182C2×#95—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#336—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191C24×C3—21no—
192C2×C3×Q8—42no—
193C2×C3×Q8—42no—
194C2×C3×Q8—81no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4C2×D412yes—
199C2×C3×D4C2×Q812yes—
200C2×C3×D4#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—81no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—24no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C22×Dic3—81no—
219C22×Dic3—81no—
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×Dic6C2×Q812yes—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D412yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C42—21no—
245C3×C42—21no—
246#217—24no—
247#217—42no—
248#214—28no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—81no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258Dic3⋊C4—42no—
259Dic3⋊C4—81no—
260Dic3⋊C4—81no—
261Dic3⋊C4—81no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×Q812yes—
273C22×Q8—21no—
274C22×Q8—31no—
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C23×C4—61no—
278C23×C4—61no—
279C2×SD16—34no—
280C2×SD16—62no—
281C2×SD16—62no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C4⋊Q8—22no—
287C4⋊Q8—41no—
288C4×Q8—62no—
289C4×Q8—121no—
290C4×D4—22no—
291C4×D4—41no—
292C4×D4—62no—
293C4×D4—62no—
294C4×D4—62no—
295C4×D4—121no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—22no—
318#107—22no—
319#107—62no—
320#106—62no—
321#106—121no—
322C4⋊C8—22no—
323#103—62no—
324#103—62no—
325#103—121no—
326#102—22no—
327#102—62no—
328#102#6714yes—
329C4×C8—62no—
330#95—31no—
331#95—31no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3—22no—
343C23×C3—22no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—84no—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4C22⋊Q814yes—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—84no—
370C2×C3×C4#13512yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C42−1+411yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
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—84no—
401C2×Dic3—84no—
402C2×Dic3#13712yes—
403Dic6—22no—
404Dic6—22no—
405Dic6—41no—
406Dic6—41no—
407C3×C8—22no—
408C3×C8—41no—
409C3⋊C8—22no—
410C3⋊C8—41no—
411C24—21no—
412C2×Q8—32no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—81no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4D3×Q812yes—
425C2×D4#23912yes—
426C2×D4C2×#6712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—64no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—81no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—241no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—68no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8C2×#6712yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4C2×#6712yes—
483C22⋊C4—42no—
484C22⋊C4—64no—
485C22⋊C4—122no—
486C42—21no—
487C42—21no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—61no—
492C42—122no—
493C22×C3—24no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3#52511yescharacteristic
501C22×C3D4211yescharacteristic
502C22×C3D4×Q811yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×#13711yescharacteristic
505C22×C3#57611yescharacteristic
506C22×C3#57711yescharacteristic
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4C2×C22⋊Q811yescharacteristic
513C3×C4#53911yescharacteristic
514C3×C4#54711yescharacteristic
515C3×C4#55411yescharacteristic
516Dic3—22no—
517Dic3—41no—
518Dic3—41no—
519Dic3—42no—
520Dic3—42no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D3×D411yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—84no—
535Q8—121no—
536Q8—121no—
537D4—22no—
538D4—22no—
539D4—22no—
540D4#94614yes—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—32no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—84no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—244no—
581C2×C4—244no—
582C2×C4#91212yes—
583C2×C4#100411yescharacteristic
584C2×C4C2×D3×Q811yescharacteristic
585C2×C4C2×#23911yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4#101011yescharacteristic
588C2×C4C22×#6711yescharacteristic
589C8—22no—
590C8—41no—
591C8—62no—
592C8—121no—
593C2×C3—44no—
594C2×C3#244911yescharacteristic
595C2×C3#310711yescharacteristic
596C2×C3#311811yescharacteristic
597C2×C3#326611yescharacteristic
598C2×C3#328011yescharacteristic
599C2×C3#338111yescharacteristic
600C2×C3#338511yescharacteristic
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22#562711yescharacteristic
609C22#563211yescharacteristic
610C22C2×#91211yescharacteristic
611C22#575711yescharacteristic
612C22D3×#13711yescharacteristic
613C22#577811yescharacteristic
614C22D4×#6711yescharacteristic
615C4—22no—
616C4—41no—
617C4—42no—
618C4—62no—
619C4—81no—
620C4—81no—
621C4—121no—
622C4—121no—
623C4—122no—
624C4—122no—
625C4—121no—
626C4—241no—
627C4—241no—
628C4#563611yescharacteristic
629C4#564511yescharacteristic
630C4C2×#94611yescharacteristic
631C4#582411yescharacteristic
632C3#2637211yescharacteristic, complemented, Sylow
633C2—44no—
634C2#8094911yescharacteristic
635C2#8095211yescharacteristic
636C2#8095811yescharacteristic
637C2#8096011yescharacteristic
638C2#8276411yescharacteristic
639C2#8383911yescharacteristic
640C2#8398711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#474D3
hypocenterC3#26372
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. #83235C2semidirect
5. #71783C2semidirect
7. #70645C2semidirect
8. #70642C2semidirect
9. #70620C2semidirect
10. #70611C2semidirect
11. #70489C2semidirect
12. #70447C2semidirect
103. C2×#214Q8semidirect
632. C3#26372semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
#832352
#823342
C2×#47582
#717832
#716122
#706452
#706422
#706202
#706112
#704892
#704472
#701712
#701692
#701682
#263723