Finite Group Databasebeta

← #10876164

Subgroups — #10876164

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76290C211yesmaximal, characteristic
2#73753C211yesmaximal, characteristic, complemented
3C3×#1926C211yesmaximal, characteristic, complemented
4#70590C211yesmaximal, characteristic, complemented
5#70580C211yesmaximal, characteristic, complemented
6#70199C211yesmaximal, characteristic
7#70199C211yesmaximal, characteristic
8#70183C211yesmaximal, characteristic, complemented
9#70165C211yesmaximal, characteristic, complemented
10#69637C211yesmaximal, characteristic, complemented
11#69465C211yesmaximal, characteristic, complemented
12#68914C211yesmaximal, characteristic, complemented
13#68877C211yesmaximal, characteristic, complemented
14#68664C211yesmaximal, characteristic, complemented
15#68628C211yesmaximal, characteristic, complemented
16#27587—31nomaximal, Sylow
17C22×#237C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×#872—21no—
20C2×C42×D3C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4975C2211yescharacteristic
37#4975C2211yescharacteristic
38#4973C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4956—21no—
44#4956—21no—
45Dic3×C22⋊C4—21no—
46Dic3×C22⋊C4—21no—
47C4×C4⋊Dic3C2211yescharacteristic
48C4×Dic3⋊C4—21no—
49#4929C2211yescharacteristic
50#4667—21no—
51#4650—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2297—31no—
61#2285—31no—
62#2096—31no—
63#2093—31no—
64#2072—31no—
65#2071—31no—
66#1928—31no—
67#1928—31no—
68#1926—31no—
69#1926D311yescharacteristic, complemented
70#1854—31no—
71#1854—31no—
72#1817—31no—
73#1788—31no—
74#1788—31no—
75C22×C3×D4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C2×#237—22no—
79C2×#237—22no—
80C2×#237—22no—
81C2×#237—22no—
82C2×#237—41no—
83C2×#237—41no—
84C22×C4×D3D411yescharacteristic
85C22×C4×D3D411yescharacteristic
86C22×C4×D3C2311yescharacteristic
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3D411yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#892—41no—
117#892—41no—
118#872—44no—
119C42×D3—22no—
120C42×D3—22no—
121C3×#106—41no—
122C3×#102—22no—
123C3×#102—41no—
124C3×#95—21no—
125#832—22no—
126#832—41no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134(C2×D3)⋊C8—22no—
135(C2×D3)⋊C8—22no—
136#818—41no—
137#816—22no—
138#816—41no—
139#810—22no—
140#810—41no—
141#808—41no—
142#807—41no—
143C22×#41—31no—
144C2×#117—31no—
145C2×#117—61no—
146C22×C42—31no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106C2×D311yescharacteristic
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102C2×D311yescharacteristic
158C2×#102C2×D311yescharacteristic
159C2×C22⋊C8—31no—
160C2×C22⋊C8—31no—
161#391—31no—
162#391—31no—
163#390—31no—
164#390C2×D311yescharacteristic
165#386—61no—
166#386—61no—
167#384—31no—
168#384—31no—
169#382—61no—
170#380—31no—
171#380—31no—
172#380—61no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—61no—
175C4×C4⋊C4C2×D311yescharacteristic
176C4×C22⋊C4—61no—
177C4×C22⋊C4—61no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C24×C3—21no—
187C23×D3C2×D411yescharacteristic
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4#4112yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—42no—
202C2×#67—42no—
203C22×Dic3—24no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213#237—44no—
214#237—44no—
215#237—44no—
216#237—44no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—22no—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—41no—
227C2×C4×D3#4112yes—
228C2×C4×D3#4112yes—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6#4112yes—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—42no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—42no—
252#217—42no—
253#217—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×#41—62no—
280C2×#41—62no—
281C2×#41—62no—
282C2×#41—62no—
283C2×#41—121no—
284C2×#41—121no—
285C22×Q8—31no—
286C22×D4—61no—
287C22×D4C22×D311yescharacteristic
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—31no—
291C23×C4—61no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296#117—124no—
297#117—121no—
298#117—121no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—21no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—62no—
321C2×C42—62no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—121no—
326#107—121no—
327#106—41no—
328#106—121no—
329#103—62no—
330#103—62no—
331#103—121no—
332#103—121no—
333#102—22no—
334#102—41no—
335#102—62no—
336#102—121no—
337C22⋊C8—62no—
338C22⋊C8—62no—
339#95—21no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3C22×D411yescharacteristic
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3#12312yes—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—41no—
365C3×D4—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
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#13712yes—
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387#67—48no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—48no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3#12312yes—
408C2×Dic3#13512yes—
409C2×Dic3AGL(1,ℤ/8)12yes—
410C4×D3—28no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—44no—
416Dic6—22no—
417Dic6—22no—
418Dic6—41no—
419Dic6—41no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C24—21no—
425C24—31no—
426#41—124no—
427#41—124no—
428#41—124no—
429#41—124no—
430C2×Q8—32no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—64no—
442C2×D4—122no—
443C2×D4—122no—
444C2×D4#23512yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—64no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—121no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4D3×D411yescharacteristic
471C22×C4D3×D411yescharacteristic
472C22×C4C23×D311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C42—21no—
509C42—21no—
510C42—42no—
511C42—62no—
512C42—62no—
513C42—61no—
514C42—61no—
515C42—122no—
516C42—122no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12311yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3#53511yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
529C22×C3C2×#13711yescharacteristic
530C22×C3#57511yescharacteristic
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—22no—
534C2×D3—41no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—41no—
538C3×C4—22no—
539C3×C4—41no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#55111yescharacteristic
544C3×C4#55111yescharacteristic
545C3×C4#55111yescharacteristic
546C3×C4#55111yescharacteristic
547Dic3—22no—
548Dic3—22no—
549Dic3—22no—
550Dic3—42no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—22no—
555C23—32no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—61no—
568C23C2×D3×D411yescharacteristic
569Q8—62no—
570Q8—62no—
571Q8—121no—
572Q8—121no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—128no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—32no—
584C2×C4—32no—
585C2×C4—32no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—68no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—124no—
608C2×C4—128no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4#91512yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4#100411yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4#100911yescharacteristic
627C8—41no—
628C8—41no—
629C8—121no—
630C8—121no—
631C2×C3—82no—
632C2×C3#248211yescharacteristic
633C2×C3#309111yescharacteristic
634C2×C3#309211yescharacteristic
635C2×C3#310711yescharacteristic
636C2×C3#311511yescharacteristic
637C2×C3#320711yescharacteristic
638C2×C3#321611yescharacteristic
639D3—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—62no—
644C22—62no—
645C22—62no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—121no—
651C22—121no—
652C22—121no—
653C22—121no—
654C22#554711yescharacteristic
655C22#560711yescharacteristic
656C22D3×#12311yescharacteristic
657C22C2×#91511yescharacteristic
658C22D3×#13511yescharacteristic
659C22D3×AGL(1,ℤ/8)11yescharacteristic
660C22#577311yescharacteristic
661C4—22no—
662C4—41no—
663C4—62no—
664C4—62no—
665C4—62no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4—122no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#552711yescharacteristic
674C4#555611yescharacteristic
675C4#567911yescharacteristic
676C4#570511yescharacteristic
677C3#2758711yescharacteristic, complemented, Sylow
678C2—82no—
679C2—122no—
680C2#7586611yescharacteristic
681C2#7965511yescharacteristic
682C2#7987211yescharacteristic
683C2#8038911yescharacteristic
684C2#8054011yescharacteristic
685C2#8132511yescharacteristic
686C2#8137111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1926C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenter#1926D3
hypocenterC3#27587
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
2. #73753C2semidirect
3. C3×#1926C2semidirect
4. #70590C2semidirect
5. #70580C2semidirect
8. #70183C2semidirect
9. #70165C2semidirect
10. #69637C2semidirect
11. #69465C2semidirect
12. #68914C2semidirect
13. #68877C2semidirect
14. #68664C2semidirect
15. #68628C2semidirect
69. #1926D3semidirect
677. C3#27587semidirect

Maximal subgroups

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

subgroupindex
#762902
#737532
C3×#19262
#705902
#705802
#701992
#701832
#701652
#696372
#694652
#689142
#688772
#686642
#686282
#275873