Finite Group Databasebeta

← #10844885

Subgroups — #10844885

Order 768.

788 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2582C211yesmaximal, characteristic, complemented
2#82484C211yesmaximal, characteristic
3C2×#5028C211yesmaximal, characteristic
4C2×#5024C211yesmaximal, characteristic, complemented
5#71631C211yesmaximal, characteristic, complemented
6#69681C211yesmaximal, characteristic, complemented
7#69680C211yesmaximal, characteristic
8#69679C211yesmaximal, characteristic
9#69677C211yesmaximal, characteristic, complemented
10#69635C211yesmaximal, characteristic, complemented
11#69630C211yesmaximal, characteristic
12#69616C211yesmaximal, characteristic, complemented
13#69615C211yesmaximal, characteristic
14#69581C211yesmaximal, characteristic
15#69567C211yesmaximal, characteristic, complemented
16#25307—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#875C2211yescharacteristic
22C3×#398—21no—
23C3×#398—21no—
24C3×#391—21no—
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389—21no—
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#384C2211yescharacteristic
31#5231—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5028—24no—
35#5024—24no—
36#4992—21no—
37#4990—21no—
38#4990—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×#446—31no—
64#2592—31no—
65#2582D311yescharacteristic, complemented
66#2068—31no—
67#2067—31no—
68#2056—31no—
69#2055—31no—
70#1973—31no—
71#1969—31no—
72#1959—31no—
73#1947—31no—
74#1935—31no—
75#1934—31no—
76#1763—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#216—22no—
101C2×#216—22no—
102C2×#216—41no—
103C2×#215—22no—
104C2×#215—22no—
105C2×#215—41no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#875—22no—
120#875—22no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127#835—22no—
128#835—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#810—22no—
135#810—22no—
136#810—41no—
137#809—22no—
138#809—22no—
139#809—41no—
140#807—41no—
141#807—41no—
142C4⋊(C3⋊C8)—22no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)D414yes—
145#803—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153#464—64no—
154#446—64no—
155C2×#107—31no—
156C2×#107—31no—
157C2×C4⋊C8—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×#97—31no—
170C2×C4×C8—31no—
171#398—21no—
172#398—21no—
173#397—61no—
174#393—61no—
175#391—21no—
176#391—31no—
177#391—31no—
178#391C2×D311yescharacteristic
179#391C2×D311yescharacteristic
180#390—31no—
181#390—31no—
182#389—21no—
183#389C2×D311yescharacteristic
184#389C2×D311yescharacteristic
185#388—61no—
186#388—61no—
187#384C2×D311yescharacteristic
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C23×D3—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—41no—
202C2×C3×Q8#4112yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
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⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250#216—44no—
251#215—44no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4×Dic3—81no—
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—21no—
272C2×C3⋊C8C2×D412yes—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8#4112yes—
275C22×Q8—31no—
276C22×Q8#6711yescharacteristic
277C22×Q8C22×D311yescharacteristic
278C22×D4—31no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—121no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289#124—62no—
290#124—62no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4#6711yescharacteristic
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—21no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#107—121no—
320#107—121no—
321C4⋊C8—34no—
322C4⋊C8—62no—
323C4⋊C8—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#102—62no—
330#102—62no—
331#102—121no—
332#97—62no—
333#97—62no—
334C4×C8—62no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—44no—
357C3×Q8—41no—
358C3×Q8—41no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
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—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
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—41no—
385C2×C3×C4—41no—
386C2×C3×C4#12112yes—
387C2×C3×C4#12112yes—
388C2×C3×C4#13712yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
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—
404D12—41no—
405D12—41no—
406D12—41no—
407D12—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—41no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8#23912yes—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—124no—
453SD16—124no—
454C2×C8—32no—
455C2×C8—32no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—241no—
489C22⋊C4—241no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—241no—
496C42D3×D412yes—
497C42#23712yes—
498C22×C3C2×#12111yescharacteristic
499C22×C3#54811yescharacteristic
500C22×C3#55011yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57611yescharacteristic
504C22×C3#57811yescharacteristic
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4#52911yescharacteristic
523C3×C4#54211yescharacteristic
524C3×C4#54611yescharacteristic
525C3×C4#54811yescharacteristic
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C22×#6711yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—44no—
540Q8—41no—
541Q8—41no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
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—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
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—41no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4#93912yes—
587C2×C4#93912yes—
588C2×C4#94212yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4C2×#23911yescharacteristic
592C2×C4#100711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#100911yescharacteristic
595C8—62no—
596C8—62no—
597C8—62no—
598C8—121no—
599C2×C3#242611yescharacteristic
600C2×C3#305611yescharacteristic
601C2×C3#306111yescharacteristic
602C2×C3#337511yescharacteristic
603C2×C3#337711yescharacteristic
604C2×C3#339811yescharacteristic
605C2×C3#340311yescharacteristic
606D3—161no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#93911yescharacteristic
615C22C2×#94211yescharacteristic
616C22#581711yescharacteristic
617C22#582211yescharacteristic
618C22#583411yescharacteristic
619C22#583511yescharacteristic
620C22#583811yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—241no—
632C4—241no—
633C4—241no—
634C4#562411yescharacteristic
635C4#563011yescharacteristic
636C4#563311yescharacteristic
637C4#564311yescharacteristic
638C3#2530711yescharacteristic, complemented, Sylow
639C2—481no—
640C2#8273211yescharacteristic
641C2#8275511yescharacteristic
642C2#8342111yescharacteristic
643C2#8368611yescharacteristic
644C2#8370011yescharacteristic
645C2#8370811yescharacteristic
646C2#8371611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2582C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2582D3
hypocenterC3#25307
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
1. C3×#2582C2semidirect
4. C2×#5024C2semidirect
5. #71631C2semidirect
6. #69681C2semidirect
9. #69677C2semidirect
10. #69635C2semidirect
12. #69616C2semidirect
15. #69567C2semidirect
65. #2582D3semidirect
638. C3#25307semidirect

Maximal subgroups

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

subgroupindex
C3×#25822
#824842
C2×#50282
C2×#50242
#716312
#696812
#696802
#696792
#696772
#696352
#696302
#696162
#696152
#695812
#695672
#253073