Finite Group Databasebeta

← #10845269

Subgroups — #10845269

Order 768.

840 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
1C2×#5185C211yesmaximal, characteristic
2C2×#4794C211yesmaximal, characteristic
3#76543C211yesmaximal, characteristic, complemented
4#76493C211yesmaximal, characteristic
5C3×#2056C211yesmaximal, characteristic, complemented
6#72065C211yesmaximal, characteristic, complemented
7#70479C211yesmaximal, characteristic
8#70452C211yesmaximal, characteristic, complemented
9#70280C211yesmaximal, characteristic
10#70221C211yesmaximal, characteristic, complemented
11#70220C211yesmaximal, characteristic, complemented
12#69687C211yesmaximal, characteristic
13#69670C211yesmaximal, characteristic, complemented
14#68902C211yesmaximal, characteristic
15#68800C211yesmaximal, characteristic, complemented
16#25336—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5185—24no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4988C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4935C2211yescharacteristic
44C4×Dic3⋊C4C2211yescharacteristic
45#4794—24no—
46#4670—21no—
47#4669—21no—
48#4668—21no—
49#4666—21no—
50#4656—21no—
51#4656—21no—
52#4647—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#464—31no—
64#2567—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2056D311yescharacteristic, complemented
69#2055—31no—
70#1988—31no—
71#1940—31no—
72#1878—31no—
73#1877—31no—
74#1803—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C2×C3×Q16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90#947—22no—
91#947D414yes—
92Q8×Dic3—22no—
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95Q8×Dic3—41no—
96C2×#215—22no—
97C2×#215—22no—
98C2×#215—41no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#888—22no—
124#888—22no—
125C4×D12—22no—
126C4×D12—41no—
127C3×C4⋊C8—41no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—22no—
135#835—22no—
136#835—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#821—22no—
143#821—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—41no—
147#809—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#489—64no—
156#464—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169#395—61no—
170#395—61no—
171#391—31no—
172#391C2×D311yescharacteristic
173#391C2×D311yescharacteristic
174#390—31no—
175#389—61no—
176#389—61no—
177#388—61no—
178#388—61no—
179#385—61no—
180#384—31no—
181#384—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
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×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C2×D412yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×Dic6—42no—
230C2×Dic6—81no—
231C2×Dic6—81no—
232C3×Q16—44no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C42—21no—
244C3×C42—21no—
245#215—44no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3—81no—
259C4⋊Dic3—81no—
260C4⋊Dic3#4112yes—
261Dic3⋊C4—22no—
262Dic3⋊C4—22no—
263Dic3⋊C4—42no—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—81no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8—61no—
280C22×Q8C22×D311yescharacteristic
281C22×Q8C22×D311yescharacteristic
282C22×D4—31no—
283C23×C4—61no—
284C2×Q16—22no—
285C2×Q16—22no—
286C2×Q16—41no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—62no—
296#124—34no—
297#124—62no—
298C4×Q8—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×Q8—121no—
302C4×D4—62no—
303C4×D4—121no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329C4⋊C8—41no—
330C4⋊C8—62no—
331C4⋊C8—121no—
332#103—22no—
333#103—22no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#97—62no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—31no—
346#95—31no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
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#13712yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3#12112yes—
409C2×Dic3#12112yes—
410C2×Dic3#13712yes—
411D12—22no—
412D12—22no—
413D12—41no—
414D12—41no—
415C4×D3—84no—
416Dic6—84no—
417C3×C8—41no—
418C3×C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C24—61no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—122no—
433C2×Q8—241no—
434C2×Q8—241no—
435C2×Q8D3×D412yes—
436C2×Q8#23712yes—
437C2×D4—32no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—241no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—44no—
461SD16—124no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—32no—
479C4⋊C4—62no—
480C4⋊C4—62no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—32no—
501C42—32no—
502C42—61no—
503C42—61no—
504C42—122no—
505C42—122no—
506C42—241no—
507C22×C3C2×#12111yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3#54811yescharacteristic
510C22×C3C2×#13711yescharacteristic
511C22×C3C2×#13711yescharacteristic
512C22×C3#57611yescharacteristic
513C22×C3#57711yescharacteristic
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C3×C4—22no—
522C3×C4—42no—
523C3×C4—42no—
524C3×C4—41no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4#52911yescharacteristic
528C3×C4#54211yescharacteristic
529C3×C4#54611yescharacteristic
530C3×C4#54711yescharacteristic
531Dic3—22no—
532Dic3—22no—
533Dic3—22no—
534Dic3—42no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—62no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23C2×D3×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—244no—
555D4—62no—
556D4—62no—
557D4—121no—
558D4—121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—32no—
567C2×C4—32no—
568C2×C4—32no—
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—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—244no—
598C2×C4#91812yes—
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4#100411yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4#101011yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3#242711yescharacteristic
610C2×C3#303911yescharacteristic
611C2×C3#304511yescharacteristic
612C2×C3#331411yescharacteristic
613C2×C3#331611yescharacteristic
614C2×C3#336911yescharacteristic
615C2×C3#337511yescharacteristic
616D3—82no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22#559811yescharacteristic
625C22D3×#12111yescharacteristic
626C22#561011yescharacteristic
627C22C2×#91811yescharacteristic
628C22#576911yescharacteristic
629C22D3×#13711yescharacteristic
630C22#577811yescharacteristic
631C4—22no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—62no—
636C4—62no—
637C4—62no—
638C4—81no—
639C4—81no—
640C4—121no—
641C4—122no—
642C4—241no—
643C4—241no—
644C4#559111yescharacteristic
645C4#564911yescharacteristic
646C4#565511yescharacteristic
647C4#566911yescharacteristic
648C3#2533611yescharacteristic, complemented, Sylow
649C2—242no—
650C2#7690211yescharacteristic
651C2#8046411yescharacteristic
652C2#8047711yescharacteristic
653C2#8058511yescharacteristic
654C2#8060211yescharacteristic
655C2#8077711yescharacteristic
656C2#8091711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2056D3
hypocenterC3#25336
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
3. #76543C2semidirect
5. C3×#2056C2semidirect
6. #72065C2semidirect
8. #70452C2semidirect
10. #70221C2semidirect
11. #70220C2semidirect
13. #69670C2semidirect
15. #68800C2semidirect
68. #2056D3semidirect
648. C3#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#51852
C2×#47942
#765432
#764932
C3×#20562
#720652
#704792
#704522
#702802
#702212
#702202
#696872
#696702
#689022
#688002
#253363