Finite Group Databasebeta

← #10858111

Subgroups — #10858111

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C211yesmaximal, characteristic, complemented
2C2×#4766C211yesmaximal, characteristic, complemented
3#76433C211yesmaximal, characteristic
4#74327C211yesmaximal, characteristic, complemented
5C3×#2094C211yesmaximal, characteristic, complemented
6#70579C211yesmaximal, characteristic, complemented
7#70568C211yesmaximal, characteristic, complemented
8#70566C211yesmaximal, characteristic, complemented
9#70203C211yesmaximal, characteristic
10#70201C211yesmaximal, characteristic
11#70169C211yesmaximal, characteristic, complemented
12#69653C211yesmaximal, characteristic, complemented
13#69568C211yesmaximal, characteristic, complemented
14#68667C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#26366—31nomaximal, Sylow
17C2×#938C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×#67⋊C4—21no—
24C2×C3×#106C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4994—21no—
36#4993—21no—
37#4987C2211yescharacteristic
38#4980—21no—
39#4976—21no—
40#4972C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4956—21no—
46#4946—21no—
47#4944—21no—
48C4×(C2×D3)⋊C4C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4861—24no—
53#4766—24no—
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#428—31no—
64#2441—31no—
65#2361—31no—
66#2094D311yescharacteristic, complemented
67#2088—31no—
68#2067—31no—
69#2059—31no—
70#1948—31no—
71#1918—31no—
72#1895—31no—
73#1875—31no—
74#1864—31no—
75#1853—31no—
76#1851—31no—
77C22×C3×D4C2311yescharacteristic
78C22×#67—21no—
79C23×Dic3—21no—
80C22×C4×D3—21no—
81C22×C4×D3C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94#938—22no—
95#938Q814yes—
96D4×Dic3—22no—
97D4×Dic3—41no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4D411yescharacteristic
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C2×ΓL(1,𝔽25)—22no—
118C2×ΓL(1,𝔽25)—22no—
119C2×ΓL(1,𝔽25)—22no—
120#892—22no—
121#892—41no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4—41no—
124#890—22no—
125#890—41no—
126#67⋊C4—44no—
127C3×#106—22no—
128C3×#106—41no—
129C3×#102—22no—
130C3×#102—41no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140(C2×D3)⋊C8—22no—
141#818D414yescomplemented
142Dic3⋊C8—22no—
143#808—22no—
144#808—22no—
145#807—22no—
146C4⋊(C3⋊C8)—22no—
147C22×M16—31no—
148C2×#125—31no—
149C2×#121—31no—
150C2×C4×D4—31no—
151C2×C4×D4—61no—
152C2×#117—31no—
153C22×C4⋊C4—31no—
154#478—64no—
155#428—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106C2×D311yescharacteristic
161C2×#106C2×D311yescharacteristic
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C22⋊C8—31no—
169#399—61no—
170#396—61no—
171#390C2×D311yescharacteristic
172#389—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#385—61no—
180#384C2×D311yescharacteristic
181#380—31no—
182C4×C4⋊C4—31no—
183C4×C22⋊C4—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192C24×C3—21no—
193C23×D3C2×D411yescharacteristic
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4C2×Q812yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C2×#67—24no—
207C2×#67—42no—
208C2×#67—42no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—42no—
214C22×Dic3—42no—
215C22×Dic3—81no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—81no—
226C2×C4×D3C2×Q812yes—
227C2×C4×D3#4112yes—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8C2×D412yes—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—81no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—81no—
244C3×C42—21no—
245C3×C42—21no—
246#217—24no—
247#217—42no—
248(C2×D3)⋊C4—22no—
249(C2×D3)⋊C4—41no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—81no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3C2×D412yes—
259C4⋊Dic3C2×D412yes—
260C4⋊Dic3C2×Q812yes—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8#4112yes—
276ΓL(1,𝔽25)—28no—
277C22×D4—61no—
278C22×D4C22×D311yescharacteristic
279C23×C4—31no—
280C23×C4—61no—
281C23×C4—61no—
282C2×M16—62no—
283C2×M16—62no—
284C2×M16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#125—62no—
290#125—121no—
291#121—34no—
292#121—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296#117—62no—
297#117—121no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—121no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—34no—
328#107—62no—
329#106—22no—
330#106—41no—
331#106—62no—
332#106—62no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#102—22no—
336#102—41no—
337#102—62no—
338#102—62no—
339C22⋊C8—62no—
340#95—31no—
341#95—31no—
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×D3C22⋊Q812yes—
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×C4AGL(1,ℤ/8)12yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×Q811yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387#67—48no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
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—41no—
408C2×Dic3—84no—
409C2×Dic3C22⋊Q812yes—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—84no—
415C3×C8—22no—
416C3×C8—41no—
417C3⋊C8—22no—
418C3⋊C8—41no—
419C24—21no—
420C24—31no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—64no—
427C2×D4—122no—
428C2×D4—122no—
429C2×D4D3×Q812yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—31no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—241no—
451C22×C4—241no—
452C22×C4D3×D411yescharacteristic
453C22×C4D3×D411yescharacteristic
454C22×C4C23×D311yescharacteristic
455M16—68no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—32no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8D3×D412yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—32no—
475C4⋊C4—32no—
476C4⋊C4—32no—
477C4⋊C4—42no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—81no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C22⋊C4—42no—
491C22⋊C4—64no—
492C22⋊C4—62no—
493C22⋊C4—81no—
494C22⋊C4—121no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—61no—
501C42—61no—
502C42—122no—
503C42—122no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3C2×C22⋊Q811yescharacteristic
512C22×C3D4211yescharacteristic
513C22×C3#54711yescharacteristic
514C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57711yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C3×C4—22no—
526C3×C4—41no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4#52511yescharacteristic
531C3×C4#53911yescharacteristic
532C3×C4D4×Q811yescharacteristic
533C3×C4#55411yescharacteristic
534Dic3—22no—
535Dic3—42no—
536Dic3—41no—
537Dic3—42no—
538Dic3—41no—
539Dic3—42no—
540Dic3—81no—
541C23—22no—
542C23—32no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23C2×D3×D411yescharacteristic
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4—128no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—32no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—62no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4—244no—
602C2×C4#91412yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4#100411yescharacteristic
605C2×C4C2×D3×Q811yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4#101011yescharacteristic
609C8—22no—
610C8—41no—
611C8—62no—
612C8—121no—
613C2×C3—82no—
614C2×C3#244911yescharacteristic
615C2×C3#307711yescharacteristic
616C2×C3#308411yescharacteristic
617C2×C3#327111yescharacteristic
618C2×C3#327211yescharacteristic
619C2×C3#334511yescharacteristic
620C2×C3#335111yescharacteristic
621D3—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22D3×C22⋊Q811yescharacteristic
637C22#565511yescharacteristic
638C22#571711yescharacteristic
639C22C2×#91411yescharacteristic
640C22#576211yescharacteristic
641C22#577311yescharacteristic
642C22#577511yescharacteristic
643C4—22no—
644C4—41no—
645C4—62no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—121no—
650C4—122no—
651C4—122no—
652C4—122no—
653C4—121no—
654C4—241no—
655C4#563211yescharacteristic
656C4#563611yescharacteristic
657C4#565611yescharacteristic
658C4#566011yescharacteristic
659C3#2636611yescharacteristic, complemented, Sylow
660C2—82no—
661C2—122no—
662C2#7810411yescharacteristic
663C2#8106011yescharacteristic
664C2#8106311yescharacteristic
665C2#8132111yescharacteristic
666C2#8133711yescharacteristic
667C2#8177011yescharacteristic
668C2#8185911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2094C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2094D3
hypocenterC3#26366
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
1. C2×#4861C2semidirect
2. C2×#4766C2semidirect
4. #74327C2semidirect
5. C3×#2094C2semidirect
6. #70579C2semidirect
7. #70568C2semidirect
8. #70566C2semidirect
11. #70169C2semidirect
12. #69653C2semidirect
13. #69568C2semidirect
14. #68667C2semidirect
15. #68621C2semidirect
66. #2094D3semidirect
141. #818D4semidirect
659. C3#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
C2×#47662
#764332
#743272
C3×#20942
#705792
#705682
#705662
#702032
#702012
#701692
#696532
#695682
#686672
#686212
#263663