Finite Group Databasebeta

← #10858104

Subgroups — #10858104

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
1C3×#2441C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3C2×#4995C211yesmaximal, characteristic, complemented
4#73467C211yesmaximal, characteristic
5#71739C211yesmaximal, characteristic, complemented
6#71586C211yesmaximal, characteristic
7#69630C211yesmaximal, characteristic, complemented
8#69626C211yesmaximal, characteristic, complemented
9#69624C211yesmaximal, characteristic, complemented
10#69600C211yesmaximal, characteristic, complemented
11#69587C211yesmaximal, characteristic, complemented
12#69568C211yesmaximal, characteristic, complemented
13#69517C211yesmaximal, characteristic
14#69505C211yesmaximal, characteristic, complemented
15#69450C211yesmaximal, characteristic, complemented
16#26366—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×#67⋊C4—21no—
23C2×C4×D12C2211yescharacteristic
24C3×#399—21no—
25C3×#396—21no—
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#387—21no—
30C3×#384C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C3×C4×C22⋊C4C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#5066—24no—
35#4995—24no—
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4958—21no—
47#4939C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4670—21no—
53#4668—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×#428—31no—
64#2441D311yescharacteristic, complemented
65#2361—31no—
66#2094—31no—
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—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C23×Dic3—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×#125—22no—
83C3×#125—41no—
84C3×#117—22no—
85C3×#117—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103#930—22no—
104#930Q814yes—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3—22no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—41no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×#208—22no—
118C2×#208—22no—
119C2×#208—22no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#67⋊C4—44no—
125C4×D12—22no—
126C4×D12—41no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132C3⋊#95—21no—
133C3⋊#95—21no—
134#830—22no—
135#809—22no—
136#809—22no—
137#809—22no—
138#809—41no—
139#808—22no—
140#808—22no—
141#808—22no—
142#808—41no—
143#807—22no—
144#807D414yescomplemented
145C4⋊(C3⋊C8)—22no—
146C4⋊(C3⋊C8)—22no—
147C22×M16—31no—
148C2×#125C2×D311yescharacteristic
149C2×#121—31no—
150C2×C4×D4—31no—
151C2×C4×D4—61no—
152C2×#117C2×D311yescharacteristic
153C22×C4⋊C4—31no—
154#478—64no—
155#428—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C22⋊C8—31no—
169#399—21no—
170#396—21no—
171#390—31no—
172#389—21no—
173#389—31no—
174#389C2×D311yescharacteristic
175#389C2×D311yescharacteristic
176#388—61no—
177#387—21no—
178#386—61no—
179#385—61no—
180#384C2×D311yescharacteristic
181#380C2×D311yescharacteristic
182C4×C4⋊C4—31no—
183C4×C22⋊C4C2×D311yescharacteristic
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192C24×C3C2×D411yescharacteristic
193C23×D3—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×Q812yes—
208C22×C3×C4#4112yes—
209C22×C3×C4C2411yescharacteristic
210C2×#67—24no—
211C2×#67—42no—
212C2×#67—42no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—81no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×Q812yes—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—81no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4C2×D412yes—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—41no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—81no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—42no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
260C4⋊Dic3C2×D412yes—
261C4⋊Dic3C2×Q812yes—
262Dic3⋊C4—42no—
263C4×Dic3—42no—
264#208—28no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8#4112yes—
277C22×D4—31no—
278C22×D4—61no—
279C23×C4—61no—
280C23×C4—61no—
281C23×C4C22×D311yescharacteristic
282C2×M16—62no—
283C2×M16—62no—
284C2×M16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289#125—22no—
290#125—41no—
291#121—34no—
292#121—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296#117—22no—
297#117—41no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4#6711yescharacteristic
322C2×C22⋊C4#6711yescharacteristic
323C2×C42—21no—
324C2×C42—61no—
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327#107—34no—
328#107—62no—
329#106—62no—
330#106—62no—
331#106—62no—
332#106—121no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—121no—
339C22⋊C8—62no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—61no—
344#95—61no—
345#95#6711yescharacteristic
346#95#6711yescharacteristic
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3C22⋊Q812yes—
354C23×C3C22×D411yescharacteristic
355C22×D3—22no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4C22⋊Q812yes—
392C2×C3×C4AGL(1,ℤ/8)12yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×Q811yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399#67—48no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—84no—
410D12—22no—
411D12—22no—
412D12—41no—
413D12—41no—
414C4×D3—84no—
415C3⋊C8—22no—
416C3⋊C8—22no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C24—61no—
420C24C2×#6711yescharacteristic
421C2×D4—32no—
422C2×D4—64no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—122no—
429C2×D4—122no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×Q812yes—
448C22×C4#23912yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C23×D311yescharacteristic
455M16—68no—
456C2×C8—32no—
457C2×C8—32no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—32no—
476C4⋊C4—32no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4D3×D412yes—
490C22⋊C4—22no—
491C22⋊C4—41no—
492C22⋊C4—42no—
493C22⋊C4—64no—
494C22⋊C4—81no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—122no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
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—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C3×C4—22no—
526C3×C4—22no—
527C3×C4—41no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—42no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4#52511yescharacteristic
535C3×C4#53911yescharacteristic
536C3×C4D4×Q811yescharacteristic
537C3×C4#55411yescharacteristic
538Dic3—42no—
539Dic3—42no—
540Dic3—81no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—62no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23#94612yes—
555C23C22×#6711yescharacteristic
556D4—62no—
557D4—62no—
558D4—128no—
559D4—121no—
560D4—121no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—244no—
600C2×C4—244no—
601C2×C4#94612yes—
602C2×C4#94912yes—
603C2×C4C2×D3×D411yescharacteristic
604C2×C4#100411yescharacteristic
605C2×C4C2×D3×Q811yescharacteristic
606C2×C4C2×#23911yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4#101011yescharacteristic
609C8—62no—
610C8—62no—
611C8—121no—
612C8—121no—
613C2×C3—42no—
614C2×C3#244911yescharacteristic
615C2×C3#307711yescharacteristic
616C2×C3#308411yescharacteristic
617C2×C3#327111yescharacteristic
618C2×C3#327211yescharacteristic
619C2×C3#334511yescharacteristic
620C2×C3#335111yescharacteristic
621D3—82no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—122no—
630C22—122no—
631C22—122no—
632C22—241no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22D4×#6711yescharacteristic
637C22C2×#94611yescharacteristic
638C22C2×#94911yescharacteristic
639C22#582211yescharacteristic
640C22#582411yescharacteristic
641C22#583611yescharacteristic
642C22#583711yescharacteristic
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—122no—
653C4—122no—
654C4—241no—
655C4#562711yescharacteristic
656C4#563211yescharacteristic
657C4#563611yescharacteristic
658C4#564511yescharacteristic
659C3#2636611yescharacteristic, complemented, Sylow
660C2—42no—
661C2—242no—
662C2#8276411yescharacteristic
663C2#8335011yescharacteristic
664C2#8347611yescharacteristic
665C2#8352311yescharacteristic
666C2#8358311yescharacteristic
667C2#8359611yescharacteristic
668C2#8360811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2441C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2441D3
hypocenterC3#26366
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C3×#2441C2semidirect
2. C2×#5066C2semidirect
3. C2×#4995C2semidirect
5. #71739C2semidirect
7. #69630C2semidirect
8. #69626C2semidirect
9. #69624C2semidirect
10. #69600C2semidirect
11. #69587C2semidirect
12. #69568C2semidirect
14. #69505C2semidirect
15. #69450C2semidirect
64. #2441D3semidirect
144. #807D4semidirect
659. C3#26366semidirect

Maximal subgroups

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

subgroupindex
C3×#24412
C2×#50662
C2×#49952
#734672
#717392
#715862
#696302
#696262
#696242
#696002
#695872
#695682
#695172
#695052
#694502
#263663