Finite Group Databasebeta

← #10844832

Subgroups — #10844832

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2#83179C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4C2×#4828C211yesmaximal, characteristic, complemented
5#72040C211yesmaximal, characteristic, complemented
6#72040C211yesmaximal, characteristic, complemented
7#72039C211yesmaximal, characteristic, complemented
8#72038C211yesmaximal, characteristic, complemented
9#71765C211yesmaximal, characteristic, complemented
10#71609C211yesmaximal, characteristic, complemented
11#70455C211yesmaximal, characteristic, complemented
12#70188C211yesmaximal, characteristic, complemented
13#70186C211yesmaximal, characteristic, complemented
14#70180C211yesmaximal, characteristic, complemented
15#70179C211yesmaximal, characteristic, complemented
16#25304—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#936—21no—
22C2×C4×Dic6C2211yescharacteristic
23C3×#459—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27#5222—21no—
28#5222—21no—
29#5221—21no—
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#832C2211yescharacteristic
42#5210—21no—
43#5210—21no—
44C2×#818C2211yescharacteristic
45C2×#817C2211yescharacteristic
46C2×#815C2211yescharacteristic
47C2×Dic3⋊C8C2211yescharacteristic
48C2×Dic3⋊C8C2211yescharacteristic
49C2×C8×Dic3C2211yescharacteristic
50#4972C2211yescharacteristic
51#4972C2211yescharacteristic
52#4949—21no—
53#4947—21no—
54#4936C2211yescharacteristic
55C4×C4⋊Dic3C2211yescharacteristic
56C2×#803C2211yescharacteristic
57#4828—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#459D311yescharacteristic
63C2×#446—31no—
64#2580—31no—
65#2548—31no—
66#2067—31no—
67#2065—31no—
68#2054—31no—
69#2054—31no—
70#1972—31no—
71#1970—31no—
72#1961—31no—
73#1949—31no—
74#1936—31no—
75#1934—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—
83C22×Dic6D411yescharacteristic
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—41no—
89C2×C3×D8—41no—
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C3×#127—22no—
93C3×#127—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
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—
110C2×#217—21no—
111#936—44no—
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4D411yescharacteristic
120C2×Dic3⋊C4D411yescharacteristic
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×C4⋊C8D414yes—
129C3×#102—22no—
130C3×#102—22no—
131#832—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135#832—41no—
136#832—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#818—41no—
143#817—41no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148#803—22no—
149C22×D8C2×D311yescharacteristic
150C22×D8C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
155#459—24no—
156#446—64no—
157C2×#107—31no—
158C2×#106—31no—
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—
171C2×#97—31no—
172C2×C4×C8—31no—
173#399—61no—
174#394—61no—
175#394—61no—
176#393—61no—
177#390—31no—
178#390—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#389—31no—
185#389—31no—
186#385—61no—
187#385—61no—
188#382—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C24×C3—21no—
196C24×C3—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
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—42no—
217C2×C3×D4—42no—
218C2×C3×D4—81no—
219C2×C3×D4—81no—
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×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—41no—
239C3×D8—44no—
240C3×D8—44no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×D412yes—
246C2×C3×C8C2×D412yes—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C22⋊C4—42no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252#217—42no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—42no—
258#217—81no—
259#217—81no—
260#217—81no—
261#217—81no—
262#217—81no—
263#217—81no—
264C4⋊Dic3—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—81no—
276Dic3⋊C4—81no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—42no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8#4112yes—
288C22×Q8—31no—
289C22×D4—21no—
290C22×D4—21no—
291C22×D4C22×D311yescharacteristic
292C22×D4C22×D311yescharacteristic
293C22×D4C22×D311yescharacteristic
294C23×C4—21no—
295C2×D8—22no—
296C2×D8—22no—
297C2×D8—22no—
298C2×D8—22no—
299C2×D8—41no—
300C2×D8—41no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305#127—22no—
306#127—22no—
307#124—124no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×D4—22no—
311C4×D4—22no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
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×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—121no—
340#106—121no—
341C4⋊C8—62no—
342C4⋊C8—62no—
343C4⋊C8#6714yes—
344#102—22no—
345#102—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—121no—
351#102—121no—
352#97—62no—
353#97—62no—
354C4×C8—62no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
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×C3C22×D411yescharacteristic
382C3×D4—22no—
383C3×D4—22no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—41no—
394C3×D4—84no—
395C3×D4—84no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—84no—
404C2×C3×C4#12112yes—
405C2×C3×C4#12112yes—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C42−1+411yescharacteristic
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—21no—
422C2×Dic3—21no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
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×Dic3AGL(1,ℤ/8)12yes—
436Dic6—44no—
437C3×C8—22no—
438C3×C8—22no—
439C3⋊C8—22no—
440C3⋊C8—41no—
441C24—21no—
442C24—21no—
443C24—21no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—121no—
447C2×D4—21no—
448C2×D4—21no—
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—42no—
467C2×D4—42no—
468C2×D4—81no—
469C2×D4—81no—
470C2×D4#23712yes—
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×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—44no—
487D8—44no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—32no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8C2×#6712yes—
499C2×C8C2×#6712yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—241no—
514C4⋊C4—241no—
515C22⋊C4—42no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—241no—
523C22⋊C4—241no—
524C22⋊C4—241no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—61no—
529C42—61no—
530C42—61no—
531C42—61no—
532C42—122no—
533C42#23712yes—
534C42C2×#6712yes—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—81no—
551C22×C3—81no—
552C22×C3—81no—
553C22×C3—81no—
554C22×C3—81no—
555C22×C3—81no—
556C22×C3C2×#12111yescharacteristic
557C22×C3#54811yescharacteristic
558C22×C3#55011yescharacteristic
559C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
560C22×C3#57511yescharacteristic
561C22×C3#57611yescharacteristic
562C22×C3#57611yescharacteristic
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—81no—
566C3×C4#52911yescharacteristic
567C3×C4#54211yescharacteristic
568C3×C4#54611yescharacteristic
569C3×C4#54811yescharacteristic
570Dic3—22no—
571Dic3—41no—
572Dic3—41no—
573Dic3—41no—
574Dic3—81no—
575Dic3—81no—
576Dic3—81no—
577Dic3—81no—
578C23—22no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
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—
599C23C2×D3×D411yescharacteristic
600Q8—124no—
601D4—22no—
602D4—22no—
603D4—41no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
609D4—41no—
610D4—41no—
611D4—41no—
612D4—41no—
613D4—84no—
614D4—84no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—32no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—84no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4#93812yes—
648C2×C4#93812yes—
649C2×C4C2×#23711yescharacteristic
650C2×C4C2×#23711yescharacteristic
651C2×C4#100411yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C2×C4#101011yescharacteristic
654C2×C4C22×#6711yescharacteristic
655C8—22no—
656C8—22no—
657C8—62no—
658C8—121no—
659C2×C3—82no—
660C2×C3—161no—
661C2×C3—161no—
662C2×C3#242611yescharacteristic
663C2×C3#305511yescharacteristic
664C2×C3#305811yescharacteristic
665C2×C3#336611yescharacteristic
666C2×C3#337011yescharacteristic
667C2×C3#339911yescharacteristic
668C2×C3#340311yescharacteristic
669C22—42no—
670C22—42no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#559811yescharacteristic
691C22#559911yescharacteristic
692C22#575711yescharacteristic
693C22#575711yescharacteristic
694C22D3×AGL(1,ℤ/8)11yescharacteristic
695C22#577311yescharacteristic
696C22C2×#93811yescharacteristic
697C4—22no—
698C4—22no—
699C4—62no—
700C4—81no—
701C4—121no—
702C4—121no—
703C4—121no—
704C4—241no—
705C4—241no—
706C4—241no—
707C4—241no—
708C4#560011yescharacteristic
709C4#560311yescharacteristic
710C4#580611yescharacteristic
711C4#583111yescharacteristic
712C3#2530411yescharacteristic, complemented, Sylow
713C2—82no—
714C2—161no—
715C2—161no—
716C2#8032111yescharacteristic
717C2#8032211yescharacteristic
718C2#8032811yescharacteristic
719C2#8032911yescharacteristic
720C2#8264611yescharacteristic
721C2#8383611yescharacteristic
722C2#8395011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#25304
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. #83179C2semidirect
3. #82388C2semidirect
4. C2×#4828C2semidirect
5. #72040C2semidirect
6. #72040C2semidirect
7. #72039C2semidirect
8. #72038C2semidirect
9. #71765C2semidirect
10. #71609C2semidirect
11. #70455C2semidirect
12. #70188C2semidirect
13. #70186C2semidirect
14. #70180C2semidirect
15. #70179C2semidirect
712. C3#25304semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
#831792
#823882
C2×#48282
#720402
#720392
#720382
#717652
#716092
#704552
#701882
#701862
#701802
#701792
#253043