Finite Group Databasebeta

← #10824576

Subgroups — #10824576

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2320C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic
3C2×C4×#216C211yesmaximal, characteristic, complemented
4#73387C211yesmaximal, characteristic
5#69689C211yesmaximal, characteristic
6#69681C211yesmaximal, characteristic, complemented
7#69680C211yesmaximal, characteristic
8#69670C211yesmaximal, characteristic, complemented
9#69635C211yesmaximal, characteristic, complemented
10#69630C211yesmaximal, characteristic
11#69628C211yesmaximal, characteristic
12#69607C211yesmaximal, characteristic, complemented
13#69475C211yesmaximal, characteristic, complemented
14C4×#809C211yesmaximal, characteristic
15#69436C211yesmaximal, characteristic, complemented
16#23695—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#387—21no—
29C3×#383—21no—
30C3×#382—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×#376C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C4×#216—24no—
36#5026—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4939C2211yescharacteristic
46#4935C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×#803C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4670—21no—
53#4670—21no—
54#4668—21no—
55#4668—21no—
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#440—31no—
63C2×C4×Q16—31no—
64#2326—31no—
65#2320D311yescharacteristic, complemented
66#2083—31no—
67#2072—31no—
68#2068—31no—
69#2067—31no—
70#2056—31no—
71#2055—31no—
72#2042—31no—
73#2015—31no—
74#1768—31no—
75#1763—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C3×C4×Q8—22no—
82C3×C4×Q8D414yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
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
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#216—22no—
95C2×#216—22no—
96C2×#216—22no—
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#875—22no—
116#875—22no—
117C4×D12—22no—
118C4×D12—22no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132#835—22no—
133#835—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#810—22no—
137#810—22no—
138#810—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#807—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154#440—64no—
155C4×Q16—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391C2×D311yescharacteristic
172#391C2×D311yescharacteristic
173#390—31no—
174#389C2×D311yescharacteristic
175#389C2×D311yescharacteristic
176#388—61no—
177#388—61no—
178#387—21no—
179#387—21no—
180#385—61no—
181#385—61no—
182#384—31no—
183#383—21no—
184#382—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186#376C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
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×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12#4112yes—
220C2×C4×D3—42no—
221C2×C4×D3—81no—
222C2×C4×D3—81no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4C2×D412yes—
238C3×C42—24no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—81no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#216—28no—
247#215—28no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—81no—
252(C2×D3)⋊C4—81no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—81no—
258C4⋊Dic3—81no—
259Dic3⋊C4—42no—
260C4×Dic3—42no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8#4112yes—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×D4—31no—
276C23×C4—61no—
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16—62no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—31no—
286C22×C8—31no—
287#124—62no—
288#124—62no—
289C4×Q8—22no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8D1214yes—
293C4×D4—62no—
294C4×D4—62no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107—62no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#102—62no—
327#102—62no—
328#102—62no—
329#97—62no—
330C4×C8—62no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—61no—
339#95—61no—
340#95#6711yescharacteristic
341#95#6711yescharacteristic
342#95#6711yescharacteristic
343#95#6711yescharacteristic
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
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—41no—
383C2×C3×C4—41no—
384C2×C3×C4#12112yes—
385C2×C3×C4#12112yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401D12—22no—
402D12—22no—
403D12—41no—
404D12—41no—
405C4×D3—84no—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410C3⋊C8—22no—
411C3⋊C8—22no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—61no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8C2×D1212yes—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—68no—
451SD16—68no—
452C2×C8—32no—
453C2×C8—32no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4—241no—
479C4⋊C4C2×D1212yes—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—24no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—81no—
491C42—122no—
492C42C2×D1212yes—
493C42#23512yes—
494C22×C3C2×#12111yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53211yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
499C22×C3#57811yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52511yescharacteristic
519C3×C4#52511yescharacteristic
520C3×C4#52911yescharacteristic
521C3×C4#52911yescharacteristic
522Dic3—42no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C22×#6711yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—62no—
538Q8—62no—
539Q8—121no—
540Q8—121no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
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—62no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—244no—
583C2×C4#93012yes—
584C2×C4#93012yes—
585C2×C4C22×D1211yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—62no—
592C8—62no—
593C8—121no—
594C8—121no—
595C2×C3#239211yescharacteristic
596C2×C3#304411yescharacteristic
597C2×C3#304511yescharacteristic
598C2×C3#312011yescharacteristic
599C2×C3#312411yescharacteristic
600C2×C3#317511yescharacteristic
601C2×C3#317811yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22C2×#93011yescharacteristic
611C22#582811yescharacteristic
612C22#583211yescharacteristic
613C22#583511yescharacteristic
614C22#583711yescharacteristic
615C22#583811yescharacteristic
616C22#583811yescharacteristic
617C4—22no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—122no—
628C4—241no—
629C4—241no—
630C4#552411yescharacteristic
631C4#552411yescharacteristic
632C4#552811yescharacteristic
633C4#552811yescharacteristic
634C3#2369511yescharacteristic, complemented, Sylow
635C2—242no—
636C2#8227311yescharacteristic
637C2#8255411yescharacteristic
638C2#8256511yescharacteristic
639C2#8344511yescharacteristic
640C2#8344611yescharacteristic
641C2#8345711yescharacteristic
642C2#8346111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2320C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2320D3
hypocenterC3#23695
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
1. C3×#2320C2semidirect
3. C2×C4×#216C2semidirect
6. #69681C2semidirect
8. #69670C2semidirect
9. #69635C2semidirect
12. #69607C2semidirect
13. #69475C2semidirect
15. #69436C2semidirect
65. #2320D3semidirect
634. C3#23695semidirect

Maximal subgroups

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

subgroupindex
C3×#23202
C2×#50262
C2×C4×#2162
#733872
#696892
#696812
#696802
#696702
#696352
#696302
#696282
#696072
#694752
C4×#8092
#694362
#236953