Finite Group Databasebeta

← #10824342

Subgroups — #10824342

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#944C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3C2×C4×#204C211yesmaximal, characteristic
4#73617C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70284C211yesmaximal, characteristic, complemented
7#70282C211yesmaximal, characteristic
8#70280C211yesmaximal, characteristic
9#70247C211yesmaximal, characteristic, complemented
10C4×Dic3⋊C8C211yesmaximal, characteristic, complemented
11C4×#810C211yesmaximal, characteristic, complemented
12#68912C211yesmaximal, characteristic, complemented
13#68902C211yesmaximal, characteristic
14#68863C211yesmaximal, characteristic
15#68835C211yesmaximal, characteristic, complemented
16#23677—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#872—21no—
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C4×#215—24no—
36#4988C2211yescharacteristic
37#4988C2211yescharacteristic
38#4976—21no—
39#4976—21no—
40#4975C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C4×(C2×D3)⋊C4—21no—
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4—21no—
51C42×Dic3C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53C4×#204—24no—
54#4669—21no—
55#4669—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×SD16—31no—
64C4×C4⋊Q8—31no—
65#2293—31no—
66#2084—31no—
67#2071—31no—
68#2066—31no—
69#2066—31no—
70#2055—31no—
71#2055—31no—
72#2043—31no—
73#2013—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#103D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92#944—22no—
93#944—22no—
94C2×#215—22no—
95C2×#215—22no—
96C2×#215—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×#204—22no—
121C2×#204—22no—
122C2×#204—22no—
123C4×D12—22no—
124C4×D12D414yes—
125#872—44no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#103—22no—
129C3×#103—22no—
130C3×C4×C8—22no—
131C3×#95—21no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#817—22no—
141#816—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#810—22no—
145#810—22no—
146#809—22no—
147#808—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×C4⋊Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155C2×#117—61no—
156C4×SD16—64no—
157C4×SD16—64no—
158C2×#106—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8—31no—
171C2×C4×C8C2×D311yescharacteristic
172#391—31no—
173#391—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389—61no—
179#389—61no—
180#388—61no—
181#388—61no—
182#384—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C22⋊C4—61no—
187C43—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12C2×D412yes—
221C2×C4×D3—24no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6#4112yes—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8#4112yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C42—42no—
244C3×C42—42no—
245C3×C42C2×D412yes—
246C3×C42#4112yes—
247#215—28no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3C2×D412yes—
257Dic3⋊C4—24no—
258Dic3⋊C4—24no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264C4×Dic3—22no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—22no—
270C4×Dic3—41no—
271C4×Dic3—42no—
272C4×Dic3SD1614yescomplemented
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8#4112yes—
279#204—28no—
280C22×Q8—31no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—31no—
283C23×C4—61no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—62no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—62no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—62no—
299C4×Q8—62no—
300C4×D4—34no—
301C4×D4—62no—
302#117—124no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#106—62no—
332#106—62no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#103—22no—
336#103—22no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#102—62no—
343C4×C8—22no—
344C4×C8—62no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3C22×D411yescharacteristic
353C22×D3—22no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—41no—
363C3×Q8—41no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#12112yes—
379C2×C3×C4#12112yes—
380C2×C3×C4C2×SD1612yes—
381C2×C3×C4#13712yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3C2×SD1612yes—
407C2×Dic3C2×SD1612yes—
408C2×Dic3#13512yes—
409D12—22no—
410D12—22no—
411D12—41no—
412D12—41no—
413C4×D3—48no—
414Dic6—22no—
415Dic6—22no—
416Dic6—41no—
417Dic6—41no—
418C3×C8—22no—
419C3×C8—41no—
420C3⋊C8—22no—
421C3⋊C8—41no—
422C24—61no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—32no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8#23512yes—
435C2×D4—32no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458SD16—68no—
459SD16—68no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—32no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8#23512yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—32no—
479C4⋊C4—42no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C22⋊C4—64no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C42—34no—
496C42—42no—
497C42—42no—
498C42—62no—
499C42—62no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42—61no—
504C42—121no—
505C42—122no—
506C42#23512yes—
507C42D3×D412yes—
508C22×C3C2×#12111yescharacteristic
509C22×C3#52511yescharacteristic
510C22×C3#53211yescharacteristic
511C22×C3C22×SD1611yescharacteristic
512C22×C3C2×#13511yescharacteristic
513C22×C3C2×#13711yescharacteristic
514C22×C3#57511yescharacteristic
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—42no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C2×D3—81no—
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4#46114yes—
528C3×C4#52511yescharacteristic
529C3×C4#52511yescharacteristic
530C3×C4#52911yescharacteristic
531C3×C4#52911yescharacteristic
532Dic3—22no—
533Dic3—22no—
534Dic3—22no—
535Dic3—42no—
536Dic3—41no—
537Dic3—42no—
538Dic3—81no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23C2×D3×D411yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—62no—
552Q8—62no—
553Q8—121no—
554Q8—121no—
555D4—62no—
556D4—62no—
557D4—121no—
558D4—121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—32no—
564C2×C4—32no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—128no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4#88412yes—
596C2×C4#88412yes—
597C2×C4D3×SD1612yes—
598C2×C4#91812yes—
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×#23511yescharacteristic
601C2×C4C2×#23511yescharacteristic
602C2×C4C2×D3×D411yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C8—22no—
606C8—41no—
607C8—62no—
608C8—121no—
609C2×C3#239211yescharacteristic
610C2×C3C2×#46111yescharacteristic
611C2×C3#303511yescharacteristic
612C2×C3#310311yescharacteristic
613C2×C3#310811yescharacteristic
614C2×C3#316711yescharacteristic
615C2×C3#317211yescharacteristic
616D3—82no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22C2×#88411yescharacteristic
625C22#554711yescharacteristic
626C22#554811yescharacteristic
627C22C2×D3×SD1611yescharacteristic
628C22C2×#91811yescharacteristic
629C22D3×#13511yescharacteristic
630C22#576811yescharacteristic
631C4—22no—
632C4—42no—
633C4—62no—
634C4—62no—
635C4—62no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—122no—
640C4—121no—
641C4—122no—
642C4—241no—
643C4#481814yes—
644C4#547911yescharacteristic
645C4#552811yescharacteristic
646C4#554711yescharacteristic
647C4#554711yescharacteristic
648C3#2367711yescharacteristic, complemented, Sylow
649C2—242no—
650C2#7358111yescharacteristic
651C2C2×#481811yescharacteristic
652C2#7886111yescharacteristic
653C2#7965311yescharacteristic
654C2#7966211yescharacteristic
655C2#7972311yescharacteristic
656C2#7972611yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC4×#103D3
hypocenterC3#23677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. C4×#944C2semidirect
5. C3×C4×#103C2semidirect
6. #70284C2semidirect
9. #70247C2semidirect
10. C4×Dic3⋊C8C2semidirect
11. C4×#810C2semidirect
12. #68912C2semidirect
15. #68835C2semidirect
76. C4×#103D3semidirect
272. C4×Dic3SD16semidirect
648. C3#23677semidirect

Maximal subgroups

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

subgroupindex
C4×#9442
C2×C4×#2152
C2×C4×#2042
#736172
C3×C4×#1032
#702842
#702822
#702802
#702472
C4×Dic3⋊C82
C4×#8102
#689122
#689022
#688632
#688352
#236773