Finite Group Databasebeta

← #10845210

Subgroups — #10845210

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83039C211yesmaximal, characteristic, complemented
2C2×#5067C211yesmaximal, characteristic
3C2×#4848C211yesmaximal, characteristic
4#73653C211yesmaximal, characteristic
5C3×#1989C211yesmaximal, characteristic, complemented
6#70469C211yesmaximal, characteristic, complemented
7#69475C211yesmaximal, characteristic, complemented
8#68903C211yesmaximal, characteristic, complemented
9#68902C211yesmaximal, characteristic, complemented
10#68896C211yesmaximal, characteristic, complemented
11#68896C211yesmaximal, characteristic, complemented
12#68893C211yesmaximal, characteristic, complemented
13#68892C211yesmaximal, characteristic, complemented
14#68863C211yesmaximal, characteristic, complemented
15#68863C211yesmaximal, characteristic, complemented
16#25332—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#5067—24no—
32#4988C2211yescharacteristic
33#4988C2211yescharacteristic
34#4982—21no—
35#4982—21no—
36#4978—21no—
37#4978—21no—
38#4977C2211yescharacteristic
39#4975C2211yescharacteristic
40#4973C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48#4939—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4848—24no—
52#4665—21no—
53#4665—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#460—31no—
64#2565—31no—
65#2366—31no—
66#2073—31no—
67#2071—31no—
68#2055—31no—
69#2055—31no—
70#1989D311yescharacteristic, complemented
71#1940—31no—
72#1877—31no—
73#1877—31no—
74#1804—31no—
75#1802—31no—
76#1763—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#125—22no—
87C3×#125—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×(C2×D3)⋊C4—21no—
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⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×#204—22no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—22no—
117C2×#204—41no—
118C2×#204—41no—
119#893—22no—
120#893—22no—
121#893—41no—
122#893—41no—
123#874—22no—
124#874D414yes—
125C4×D12—44no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×C4⋊C8—22no—
129C3×#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#821—22no—
139#821—22no—
140Dic3⋊C8—41no—
141Dic3⋊C8—41no—
142#810—22no—
143#810—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#809—22no—
148#803—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#125C2×D311yescharacteristic
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—61no—
157#490—64no—
158#460—64no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171#398—61no—
172#398—61no—
173#391—31no—
174#391—31no—
175#390—31no—
176#390—31no—
177#390—61no—
178#387—61no—
179#387—61no—
180#386—61no—
181#386—61no—
182#384—31no—
183#382—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—24no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—24no—
221C2×D12—42no—
222C2×D12—42no—
223C2×D12—81no—
224C2×D12—81no—
225C2×D12C2×D412yes—
226C2×D12C2×D412yes—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—81no—
232C2×C4×D3—81no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6#4112yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4#4112yes—
255C3×C42—42no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—41no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282#204—44no—
283#204—44no—
284C22×Q8—31no—
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C23×C4—61no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×SD16—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#127—34no—
302#127—62no—
303#125—22no—
304#125—22no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—124no—
310C4×D4—121no—
311C4×D4—121no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—21no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335C4⋊C8—22no—
336C4⋊C8—121no—
337C4⋊C8—121no—
338#103—62no—
339#103—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346#97—62no—
347#95—21no—
348#95—31no—
349#95—31no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
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—41no—
387C2×C3×C4#12112yes—
388C2×C3×C4#12112yes—
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—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3AGL(1,ℤ/8)12yes—
413D12—22no—
414D12—22no—
415D12—22no—
416D12—22no—
417D12—48no—
418D12—41no—
419D12—41no—
420D12—41no—
421D12—41no—
422D12—84no—
423C4×D3—84no—
424C4×D3—84no—
425Dic6—22no—
426Dic6—22no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—41no—
430C3×C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—61no—
434C24—61no—
435C2×Q8—32no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479SD16—124no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—62no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—121no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4#23912yes—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C42—42no—
516C42—61no—
517C42—61no—
518C42—61no—
519C42—61no—
520C42—122no—
521C42#23512yes—
522C42D3×D412yes—
523C22×C3C2×#12111yescharacteristic
524C22×C3#54611yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57711yescharacteristic
529C22×C3#57711yescharacteristic
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4#52911yescharacteristic
551C3×C4#54211yescharacteristic
552C3×C4#54611yescharacteristic
553C3×C4#54711yescharacteristic
554Dic3—22no—
555Dic3—42no—
556Dic3—41no—
557Dic3—41no—
558Dic3—41no—
559Dic3—81no—
560C23—62no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—62no—
576Q8—62no—
577Q8—121no—
578Q8—121no—
579D4—62no—
580D4—62no—
581D4—62no—
582D4—62no—
583D4—128no—
584D4—121no—
585D4—121no—
586D4—121no—
587D4—121no—
588D4—244no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—32no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4—244no—
625C2×C4#88412yes—
626C2×C4#88412yes—
627C2×C4C2×#23511yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×#23911yescharacteristic
630C2×C4#100711yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4#100911yescharacteristic
633C8—41no—
634C8—41no—
635C8—121no—
636C8—121no—
637C2×C3#242711yescharacteristic
638C2×C3#303611yescharacteristic
639C2×C3#304211yescharacteristic
640C2×C3#331011yescharacteristic
641C2×C3#331711yescharacteristic
642C2×C3#336811yescharacteristic
643C2×C3#337111yescharacteristic
644D3—82no—
645D3—82no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#88411yescharacteristic
661C22#563011yescharacteristic
662C22#563311yescharacteristic
663C22#577011yescharacteristic
664C22#577011yescharacteristic
665C22D3×AGL(1,ℤ/8)11yescharacteristic
666C22#577711yescharacteristic
667C4—22no—
668C4—22no—
669C4—42no—
670C4—62no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—121no—
675C4—122no—
676C4—121no—
677C4—121no—
678C4—241no—
679C4#558011yescharacteristic
680C4#558211yescharacteristic
681C4#567611yescharacteristic
682C4#572611yescharacteristic
683C3#2533211yescharacteristic, complemented, Sylow
684C2—242no—
685C2—242no—
686C2#7662411yescharacteristic
687C2#7886311yescharacteristic
688C2#7891011yescharacteristic
689C2#8115311yescharacteristic
690C2#8118911yescharacteristic
691C2#8121811yescharacteristic
692C2#8124911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1989C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1989D3
hypocenterC3#25332
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
1. #83039C2semidirect
5. C3×#1989C2semidirect
6. #70469C2semidirect
7. #69475C2semidirect
8. #68903C2semidirect
9. #68902C2semidirect
10. #68896C2semidirect
11. #68896C2semidirect
12. #68893C2semidirect
13. #68892C2semidirect
14. #68863C2semidirect
15. #68863C2semidirect
70. #1989D3semidirect
683. C3#25332semidirect

Maximal subgroups

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

subgroupindex
#830392
C2×#50672
C2×#48482
#736532
C3×#19892
#704692
#694752
#689032
#689022
#688962
#688932
#688922
#688632
#253323