Finite Group Databasebeta

← #10858404

Subgroups — #10858404

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic
2#83029C211yesmaximal, characteristic, complemented
3#82334C211yesmaximal, characteristic
4C2×#4757C211yesmaximal, characteristic
5#71739C211yesmaximal, characteristic, complemented
6#71612C211yesmaximal, characteristic
7#70645C211yesmaximal, characteristic, complemented
8#70614C211yesmaximal, characteristic, complemented
9#70588C211yesmaximal, characteristic, complemented
10#70571C211yesmaximal, characteristic, complemented
11#70488C211yesmaximal, characteristic, complemented
12#70448C211yesmaximal, characteristic, complemented
13#70171C211yesmaximal, characteristic
14#70170C211yesmaximal, characteristic
15#70169C211yesmaximal, characteristic
16#26386—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C3×#478—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5210—21no—
31C2×C3⋊#95—21no—
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#815C2211yescharacteristic
36#4985—21no—
37#4984C2211yescharacteristic
38#4982—21no—
39#4978—21no—
40#4977C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972—21no—
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4946—21no—
49#4942—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928C2211yescharacteristic
53#4757—24no—
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478D311yescharacteristic
63C2×#440—31no—
64#2453—31no—
65#2349—31no—
66#2097—31no—
67#2085—31no—
68#2066—31no—
69#2059—31no—
70#1944—31no—
71#1917—31no—
72#1877—31no—
73#1877—31no—
74#1865—31no—
75#1864—31no—
76#1864—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#125—22no—
85C3×#125—41no—
86C3×C4×D4—22no—
87C3×C4×D4—41no—
88C2×C3×C4⋊C4—21no—
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×#214Q814yes—
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×#106—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#817—22no—
140#816—22no—
141#816—22no—
142#815—22no—
143#810—41no—
144#808—41no—
145#807—22no—
146C22×SD16—31no—
147C2×#125C2×D311yescharacteristic
148C2×C4×Q8—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4C2×D311yescharacteristic
152#478—24no—
153#440—64no—
154C2×#107—31no—
155C2×#107C2×D311yescharacteristic
156C2×#106—31no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102C2×D311yescharacteristic
165C2×#102C2×D311yescharacteristic
166C2×#97—31no—
167#400—61no—
168#398—61no—
169#391—31no—
170#391—31no—
171#389—31no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—61no—
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×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×D412yes—
196C2×C3×D4C2×Q812yes—
197C2×C3×D4#4112yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—24no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—42no—
213C22×Dic3—42no—
214C22×Dic3—81no—
215C22×Dic3—81no—
216C22×Dic3—81no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C2×Q812yes—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C2×D412yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4C2×D412yes—
241C3×C22⋊C4—42no—
242C3×C42—21no—
243C3×C42—21no—
244#217—24no—
245#217—42no—
246#214—28no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—42no—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—41no—
255Dic3⋊C4—41no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8C2×Q812yes—
271C22×Q8—31no—
272C22×D4C22×D311yescharacteristic
273C23×C4—21no—
274C23×C4—61no—
275C23×C4—61no—
276C2×SD16—34no—
277C2×SD16—62no—
278C2×SD16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#125—22no—
284#125—41no—
285C4×Q8—62no—
286C4×Q8—121no—
287C4×D4—22no—
288C4×D4—41no—
289C4×D4—62no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—22no—
316#107—62no—
317#106—22no—
318#106—62no—
319#106—121no—
320C4⋊C8—22no—
321#103—62no—
322#103—62no—
323#103—121no—
324#102—22no—
325#102—62no—
326#102#6714yes—
327#97—62no—
328#95—31no—
329#95—31no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3C22×D411yescharacteristic
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—22no—
351C3×D4C22⋊Q814yes—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
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—84no—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C42−1+411yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—41no—
389C2×Dic3—41no—
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—84no—
399C2×Dic3—84no—
400Dic6—22no—
401Dic6—22no—
402Dic6—41no—
403Dic6—41no—
404C3×C8—22no—
405C3×C8—41no—
406C3⋊C8—22no—
407C3⋊C8—41no—
408C24—21no—
409C2×Q8—32no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4D3×Q812yes—
419C2×D4#23912yes—
420C2×D4C2×#6712yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—64no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—81no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4—241no—
439C22×C4—241no—
440C22×C4—241no—
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4C23×D311yescharacteristic
444SD16—68no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—32no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8C2×#6712yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—32no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—121no—
471C4⋊C4—121no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—241no—
476C4⋊C4—241no—
477C4⋊C4C2×#6712yes—
478C22⋊C4—42no—
479C22⋊C4—64no—
480C22⋊C4—122no—
481C42—21no—
482C42—21no—
483C42—61no—
484C42—61no—
485C42—61no—
486C42—61no—
487C42—122no—
488C22×C3—24no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3#52511yescharacteristic
496C22×C3D4211yescharacteristic
497C22×C3D4×Q811yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57511yescharacteristic
500C22×C3#57711yescharacteristic
501C22×C3#57711yescharacteristic
502C3×C4—41no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4C2×C22⋊Q811yescharacteristic
508C3×C4#53911yescharacteristic
509C3×C4#54711yescharacteristic
510C3×C4#55411yescharacteristic
511Dic3—41no—
512Dic3—41no—
513Dic3—42no—
514Dic3—42no—
515Dic3—41no—
516Dic3—41no—
517Dic3—81no—
518Dic3—81no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D3×D411yescharacteristic
527Q8—62no—
528Q8—62no—
529Q8—121no—
530Q8—121no—
531D4—22no—
532D4—22no—
533D4—22no—
534D4#94614yes—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—84no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—244no—
576C2×C4—244no—
577C2×C4#100411yescharacteristic
578C2×C4C2×D3×Q811yescharacteristic
579C2×C4C2×#23911yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C2×C4#101011yescharacteristic
582C2×C4C22×#6711yescharacteristic
583C8—22no—
584C8—41no—
585C8—62no—
586C8—121no—
587C2×C3—44no—
588C2×C3#244911yescharacteristic
589C2×C3#311111yescharacteristic
590C2×C3#312311yescharacteristic
591C2×C3#326711yescharacteristic
592C2×C3#327511yescharacteristic
593C2×C3#338811yescharacteristic
594C2×C3#338811yescharacteristic
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22#562711yescharacteristic
603C22#563211yescharacteristic
604C22#576811yescharacteristic
605C22#577011yescharacteristic
606C22#577311yescharacteristic
607C22#577511yescharacteristic
608C22D4×#6711yescharacteristic
609C4—41no—
610C4—42no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—122no—
615C4—121no—
616C4—121no—
617C4—121no—
618C4—121no—
619C4—122no—
620C4—241no—
621C4—241no—
622C4#563611yescharacteristic
623C4#564511yescharacteristic
624C4C2×#94611yescharacteristic
625C4#582411yescharacteristic
626C3#2638611yescharacteristic, complemented, Sylow
627C2—44no—
628C2#8114211yescharacteristic
629C2#8114711yescharacteristic
630C2#8116011yescharacteristic
631C2#8116111yescharacteristic
632C2#8276411yescharacteristic
633C2#8392111yescharacteristic
634C2#8396211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#478C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#478D3
hypocenterC3#26386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. #83029C2semidirect
5. #71739C2semidirect
7. #70645C2semidirect
8. #70614C2semidirect
9. #70588C2semidirect
10. #70571C2semidirect
11. #70488C2semidirect
12. #70448C2semidirect
626. C3#26386semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4782
#830292
#823342
C2×#47572
#717392
#716122
#706452
#706142
#705882
#705712
#704882
#704482
#701712
#701702
#701692
#263863