Finite Group Databasebeta

← #10845178

Subgroups — #10845178

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82386C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic
3C2×#4713C211yesmaximal, characteristic, complemented
4#73653C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic, complemented
6#70212C211yesmaximal, characteristic, complemented
7#70212C211yesmaximal, characteristic, complemented
8#70208C211yesmaximal, characteristic, complemented
9#70208C211yesmaximal, characteristic, complemented
10#69836C211yesmaximal, characteristic, complemented
11#69491C211yesmaximal, characteristic, complemented
12#68903C211yesmaximal, characteristic, complemented
13#68902C211yesmaximal, characteristic, complemented
14#68893C211yesmaximal, characteristic, complemented
15#68892C211yesmaximal, characteristic, complemented
16#25330—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#213C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×#870C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31#5210—21no—
32#5210—21no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#5015—24no—
40#4988C2211yescharacteristic
41#4988C2211yescharacteristic
42#4977C2211yescharacteristic
43#4975C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46#4948—21no—
47#4948—21no—
48#4939—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4713—24no—
52#4665—21no—
53#4665—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#459—31no—
64#2566—31no—
65#2366—31no—
66#2073—31no—
67#2071—31no—
68#2054—31no—
69#2054—31no—
70#1989—31no—
71#1940—31no—
72#1879—31no—
73#1879—31no—
74#1804—31no—
75#1802—31no—
76#1762D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×D12—21no—
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×C4×D3—21no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#213—22no—
100C2×#213—22no—
101C2×#213—22no—
102C2×#213—22no—
103C2×#213—41no—
104C2×#213—41no—
105C2×(C2×D3)⋊C4—21no—
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×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#893—22no—
124#893—22no—
125#893—41no—
126#893—41no—
127#874—22no—
128#874D414yes—
129C4×D12—44no—
130#870—22no—
131#870—22no—
132C3×#102—22no—
133C3×#102—22no—
134C3×#97—22no—
135C3×#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144Dic3⋊C8—41no—
145Dic3⋊C8—41no—
146#809—22no—
147#809—22no—
148C4⋊(C3⋊C8)—22no—
149C22×D8—31no—
150C22×D8—31no—
151C2×#127—31no—
152C2×#125—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—61no—
156C2×C4×D4C2×D311yescharacteristic
157#490—64no—
158#459—64no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#97C2×D311yescharacteristic
171#396—61no—
172#396—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#390—31no—
177#390—61no—
178#386—61no—
179#386—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#382—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C23×D3—21no—
196C23×D3—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4#4112yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—81no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—24no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—24no—
230C2×D12—42no—
231C2×D12—42no—
232C2×D12—81no—
233C2×D12—81no—
234C2×D12C2×D412yes—
235C2×D12C2×D412yes—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—81no—
241C2×C4×D3—81no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4#4112yes—
254C3×C22⋊C4—42no—
255C3×C42—42no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258#217—42no—
259#217—42no—
260#217—81no—
261#217—81no—
262#213—44no—
263#213—44no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269(C2×D3)⋊C4—81no—
270C4⋊Dic3—22no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—41no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278Dic3⋊C4—81no—
279Dic3⋊C4—81no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C22×D4—31no—
291C22×D4—31no—
292C22×D4—61no—
293C22×D4—61no—
294C22×D4C22×D311yescharacteristic
295C23×C4—21no—
296C23×C4—61no—
297C23×C4—61no—
298C2×D8—62no—
299C2×D8—62no—
300C2×D8—62no—
301C2×D8—62no—
302C2×D8—121no—
303C2×D8—121no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8C22×D311yescharacteristic
307C22×C8C22×D311yescharacteristic
308#127—34no—
309#127—62no—
310#125—62no—
311#125—62no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—124no—
317C4×D4—121no—
318C4×D4—121no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—21no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42C22×D311yescharacteristic
345C4⋊C8—62no—
346C4⋊C8—121no—
347C4⋊C8—121no—
348#102—22no—
349#102—22no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#102—62no—
355#102—62no—
356#97—22no—
357#95—21no—
358#95—31no—
359#95—31no—
360#95—61no—
361#95—61no—
362C23×C3—22no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3C22×D411yescharacteristic
370C22×D3—22no—
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—41no—
387C3×D4—41no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—84no—
402C2×C3×C4#12112yes—
403C2×C3×C4#12112yes—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4H(5,𝔽2)11yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3AGL(1,ℤ/8)12yes—
428D12—22no—
429D12—22no—
430D12—22no—
431D12—22no—
432D12—48no—
433D12—41no—
434D12—41no—
435D12—41no—
436D12—41no—
437D12—84no—
438C4×D3—84no—
439C4×D3—84no—
440C3×C8—41no—
441C3×C8—41no—
442C3⋊C8—41no—
443C3⋊C8—41no—
444C24—21no—
445C24—61no—
446C24—61no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—32no—
453C2×D4—32no—
454C2×D4—64no—
455C2×D4—64no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—122no—
467C2×D4—122no—
468C2×D4—241no—
469C2×D4—241no—
470C2×D4#23512yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—42no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—241no—
488C22×C4—241no—
489C22×C4D3×D411yescharacteristic
490C22×C4D3×D411yescharacteristic
491C22×C4C23×D311yescharacteristic
492D8—124no—
493D8—124no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—62no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—121no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C4⋊C4—241no—
520C4⋊C4—241no—
521C4⋊C4#23512yes—
522C22⋊C4—42no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—241no—
530C22⋊C4—241no—
531C22⋊C4—241no—
532C22⋊C4—241no—
533C42—42no—
534C42—61no—
535C42—61no—
536C42—61no—
537C42—61no—
538C42#23512yes—
539C42D3×D412yes—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3C2×#12111yescharacteristic
548C22×C3#54611yescharacteristic
549C22×C3#54811yescharacteristic
550C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
551C22×C3#57511yescharacteristic
552C22×C3#57611yescharacteristic
553C22×C3#57611yescharacteristic
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C3×C4—22no—
569C3×C4—22no—
570C3×C4—42no—
571C3×C4—81no—
572C3×C4—81no—
573C3×C4#52911yescharacteristic
574C3×C4#54211yescharacteristic
575C3×C4#54611yescharacteristic
576C3×C4#54711yescharacteristic
577Dic3—22no—
578Dic3—41no—
579Dic3—41no—
580Dic3—41no—
581Dic3—81no—
582Dic3—81no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—62no—
591C23—62no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23C2×D3×D411yescharacteristic
605D4—22no—
606D4—22no—
607D4—41no—
608D4—41no—
609D4—62no—
610D4—62no—
611D4—62no—
612D4—62no—
613D4—128no—
614D4—121no—
615D4—121no—
616D4—121no—
617D4—121no—
618D4—244no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—32no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—84no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—244no—
652C2×C4—244no—
653C2×C4#88412yes—
654C2×C4#88412yes—
655C2×C4C2×#23511yescharacteristic
656C2×C4C2×#23511yescharacteristic
657C2×C4C2×#23511yescharacteristic
658C2×C4C2×D3×D411yescharacteristic
659C2×C4#100911yescharacteristic
660C2×C4#101011yescharacteristic
661C8—41no—
662C8—41no—
663C8—121no—
664C8—121no—
665C2×C3—82no—
666C2×C3#242711yescharacteristic
667C2×C3#303611yescharacteristic
668C2×C3#304111yescharacteristic
669C2×C3#330911yescharacteristic
670C2×C3#331811yescharacteristic
671C2×C3#336611yescharacteristic
672C2×C3#337011yescharacteristic
673D3—82no—
674D3—82no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—122no—
683C22—122no—
684C22—122no—
685C22—122no—
686C22—122no—
687C22—122no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22C2×#88411yescharacteristic
697C22#555011yescharacteristic
698C22#555211yescharacteristic
699C22D3×AGL(1,ℤ/8)11yescharacteristic
700C22#577411yescharacteristic
701C22#577411yescharacteristic
702C22#577711yescharacteristic
703C4—22no—
704C4—22no—
705C4—42no—
706C4—62no—
707C4—81no—
708C4—81no—
709C4—121no—
710C4—121no—
711C4—121no—
712C4—241no—
713C4—241no—
714C4#547911yescharacteristic
715C4#553411yescharacteristic
716C4#555011yescharacteristic
717C4#555111yescharacteristic
718C3#2533011yescharacteristic, complemented, Sylow
719C2—82no—
720C2—242no—
721C2—242no—
722C2#7348611yescharacteristic
723C2#7891011yescharacteristic
724C2#7894411yescharacteristic
725C2#7975311yescharacteristic
726C2#7979911yescharacteristic
727C2#7985111yescharacteristic
728C2#7988211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenter#1762D3
hypocenterC3#25330
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. #82386C2semidirect
3. C2×#4713C2semidirect
4. #73653C2semidirect
5. C3×#1762C2semidirect
6. #70212C2semidirect
7. #70212C2semidirect
8. #70208C2semidirect
9. #70208C2semidirect
10. #69836C2semidirect
11. #69491C2semidirect
12. #68903C2semidirect
13. #68902C2semidirect
14. #68893C2semidirect
15. #68892C2semidirect
76. #1762D3semidirect
718. C3#25330semidirect

Maximal subgroups

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

subgroupindex
#823862
C2×#50152
C2×#47132
#736532
C3×#17622
#702122
#702082
#698362
#694912
#689032
#689022
#688932
#688922
#253303