Finite Group Databasebeta

← #10845182

Subgroups — #10845182

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
1C2×C3×#459C211yesmaximal, characteristic
2#83173C211yesmaximal, characteristic, complemented
3C2×#5071C211yesmaximal, characteristic, complemented
4#82386C211yesmaximal, characteristic, complemented
5#72040C211yesmaximal, characteristic, complemented
6#72040C211yesmaximal, characteristic, complemented
7#72037C211yesmaximal, characteristic, complemented
8#72037C211yesmaximal, characteristic, complemented
9#71609C211yesmaximal, characteristic, complemented
10#70469C211yesmaximal, characteristic, complemented
11#70461C211yesmaximal, characteristic, complemented
12#70166C211yesmaximal, characteristic, complemented
13#70165C211yesmaximal, characteristic, complemented
14#70154C211yesmaximal, characteristic, complemented
15#70153C211yesmaximal, characteristic, complemented
16#25330—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×D4×Dic3—21no—
22C2×D4×Dic3C2211yescharacteristic
23C2×D4×Dic3C2211yescharacteristic
24C2×#870C2211yescharacteristic
25C3×#459—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29#5221—21no—
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34#5220—21no—
35#5220—21no—
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#832C2211yescharacteristic
42#5210—21no—
43#5210—21no—
44C2×Dic3⋊C8C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46#5071—24no—
47#4977C2211yescharacteristic
48#4975C2211yescharacteristic
49#4948—21no—
50#4948—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#459D311yescharacteristic
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#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C23×C3×C4—21no—
83C23×Dic3—21no—
84C23×Dic3—21no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—22no—
89C2×C3×D8—41no—
90C2×C3×D8—41no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#127—22no—
94C3×#127D414yes—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107C2×#217—21no—
108C2×#217—21no—
109C2×#217—21no—
110D4×Dic3—22no—
111D4×Dic3—22no—
112D4×Dic3—41no—
113D4×Dic3—41no—
114D4×Dic3—44no—
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3D411yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3C2311yescharacteristic
127C2×C4×Dic3C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130#870—22no—
131#870—22no—
132C3×C4⋊C8—22no—
133C3×#102—22no—
134C3×#102—22no—
135#832—22no—
136#832—22no—
137#832—22no—
138#832—22no—
139#832—22no—
140#832—22no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95D411yescharacteristic
145C3⋊#95D411yescharacteristic
146Dic3⋊C8—41no—
147Dic3⋊C8—41no—
148#803—22no—
149C22×D8C2×D311yescharacteristic
150C22×D8C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic
152C2×#125—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—61no—
156C2×C4×D4C2×D311yescharacteristic
157#490—64no—
158#459—24no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
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×#97—31no—
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⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3—21no—
196C24×C3—21no—
197C2×C3×D4—24no—
198C2×C3×D4—24no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—42no—
215C2×C3×D4—42no—
216C2×C3×D4—81no—
217C2×C3×D4—81no—
218C2×C3×D4C2×D412yes—
219C2×C3×D4C2×D412yes—
220C2×C3×D4#4112yes—
221C22×C3×C4—21no—
222C22×C3×C4—42no—
223C22×C3×C4—81no—
224C22×C3×C4—81no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—42no—
233C22×Dic3—42no—
234C22×Dic3—42no—
235C22×Dic3—42no—
236C22×Dic3—81no—
237C22×Dic3—81no—
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D411yescharacteristic
242C3×D8—44no—
243C3×D8—44no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C22⋊C4—42no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255#217—42no—
256#217—42no—
257#217—42no—
258#217—42no—
259#217—42no—
260#217—42no—
261#217—81no—
262#217—81no—
263#217—81no—
264#217—81no—
265C4⋊Dic3—21no—
266C4⋊Dic3—22no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—41no—
274C4⋊Dic3#4112yes—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—81no—
278Dic3⋊C4—81no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—42no—
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—21no—
291C22×D4—21no—
292C22×D4C22×D311yescharacteristic
293C22×D4C22×D311yescharacteristic
294C22×D4C22×D311yescharacteristic
295C23×C4—21no—
296C23×C4—61no—
297C23×C4—61no—
298C2×D8—22no—
299C2×D8—22no—
300C2×D8—22no—
301C2×D8—22no—
302C2×D8—41no—
303C2×D8—41no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8C22×D311yescharacteristic
307C22×C8C22×D311yescharacteristic
308#127—22no—
309#127#6714yes—
310#125—62no—
311#125—62no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—121no—
317C4×D4—121no—
318C4×D4—124no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
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—31no—
342C2×C42—31no—
343C2×C42—61no—
344C2×C42C22×D311yescharacteristic
345C4⋊C8—22no—
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—62no—
357#95—31no—
358#95—31no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3C22×D411yescharacteristic
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—48no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—84no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—84no—
406C2×C3×C4#12112yes—
407C2×C3×C4#12112yes—
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×C3×C42−1+411yescharacteristic
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—84no—
438C2×Dic3—84no—
439C2×Dic3AGL(1,ℤ/8)12yes—
440C3×C8—41no—
441C3×C8—41no—
442C3⋊C8—41no—
443C3⋊C8—41no—
444C24—21no—
445C24—21no—
446C24—21no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—81no—
467C2×D4—81no—
468C2×D4#23712yes—
469C2×D4C2×#6712yes—
470C2×D4C2×#6712yes—
471C22×C4—21no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—42no—
477C22×C4—61no—
478C22×C4—61no—
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—44no—
493D8—44no—
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—32no—
509C4⋊C4—62no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—121no—
518C4⋊C4—122no—
519C4⋊C4—122no—
520C4⋊C4—241no—
521C4⋊C4—241no—
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—61no—
534C42—61no—
535C42—61no—
536C42—61no—
537C42—122no—
538C42#23712yes—
539C42C2×#6712yes—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—42no—
548C22×C3—42no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3—81no—
552C22×C3—81no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3C2×#12111yescharacteristic
562C22×C3#54611yescharacteristic
563C22×C3#54811yescharacteristic
564C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
565C22×C3#57511yescharacteristic
566C22×C3#57611yescharacteristic
567C22×C3#57611yescharacteristic
568C3×C4—22no—
569C3×C4—22no—
570C3×C4—81no—
571C3×C4#52911yescharacteristic
572C3×C4#54211yescharacteristic
573C3×C4#54611yescharacteristic
574C3×C4#54711yescharacteristic
575Dic3—22no—
576Dic3—41no—
577Dic3—41no—
578Dic3—41no—
579Dic3—42no—
580Dic3—81no—
581Dic3—81no—
582Dic3—81no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23C2×D3×D411yescharacteristic
605D4—22no—
606D4—22no—
607D4—22no—
608D4—22no—
609D4—22no—
610D4—22no—
611D4—48no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—41no—
618D4—84no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—32no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—62no—
639C2×C4—84no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
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#93912yes—
654C2×C4#93912yes—
655C2×C4C2×#23711yescharacteristic
656C2×C4C2×#23711yescharacteristic
657C2×C4#100411yescharacteristic
658C2×C4D3×#4111yescharacteristic
659C2×C4#101011yescharacteristic
660C2×C4C22×#6711yescharacteristic
661C8—41no—
662C8—41no—
663C8—121no—
664C8—121no—
665C2×C3—82no—
666C2×C3—82no—
667C2×C3—82no—
668C2×C3#242711yescharacteristic
669C2×C3#303611yescharacteristic
670C2×C3#304111yescharacteristic
671C2×C3#330911yescharacteristic
672C2×C3#331811yescharacteristic
673C2×C3#336611yescharacteristic
674C2×C3#337011yescharacteristic
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22#559811yescharacteristic
697C22#561211yescharacteristic
698C22#575711yescharacteristic
699C22#575711yescharacteristic
700C22D3×AGL(1,ℤ/8)11yescharacteristic
701C22#577311yescharacteristic
702C22C2×#93911yescharacteristic
703C4—22no—
704C4—22no—
705C4—62no—
706C4—81no—
707C4—121no—
708C4—121no—
709C4—121no—
710C4—122no—
711C4—241no—
712C4—241no—
713C4—241no—
714C4#560011yescharacteristic
715C4#560311yescharacteristic
716C4#580611yescharacteristic
717C4#582411yescharacteristic
718C3#2533011yescharacteristic, complemented, Sylow
719C2—82no—
720C2—82no—
721C2—82no—
722C2#8032111yescharacteristic
723C2#8032211yescharacteristic
724C2#8035411yescharacteristic
725C2#8035511yescharacteristic
726C2#8264711yescharacteristic
727C2#8382611yescharacteristic
728C2#8393111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#25330
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. #83173C2semidirect
3. C2×#5071C2semidirect
4. #82386C2semidirect
5. #72040C2semidirect
6. #72040C2semidirect
7. #72037C2semidirect
8. #72037C2semidirect
9. #71609C2semidirect
10. #70469C2semidirect
11. #70461C2semidirect
12. #70166C2semidirect
13. #70165C2semidirect
14. #70154C2semidirect
15. #70153C2semidirect
718. C3#25330semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
#831732
C2×#50712
#823862
#720402
#720372
#716092
#704692
#704612
#701662
#701652
#701542
#701532
#253303