Finite Group Databasebeta

← #10858345

Subgroups — #10858345

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2454C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3C2×C4×#213C211yesmaximal, characteristic
4C4⋊C4×D12C211yesmaximal, characteristic, complemented
5#71739C211yesmaximal, characteristic, complemented
6#71606C211yesmaximal, characteristic, complemented
7#69657C211yesmaximal, characteristic
8#69657C211yesmaximal, characteristic
9#69656C211yesmaximal, characteristic, complemented
10#69652C211yesmaximal, characteristic, complemented
11#69626C211yesmaximal, characteristic, complemented
12#69626C211yesmaximal, characteristic, complemented
13#69618C211yesmaximal, characteristic, complemented
14#69607C211yesmaximal, characteristic, complemented
15#69482C211yesmaximal, characteristic, complemented
16#26382—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×D3×C4⋊C4—21no—
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C3×#399—21no—
25C3×#396—21no—
26C3×#390C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#385—21no—
31C3×#380—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5066—24no—
37C4×#213—24no—
38#4987—21no—
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4935C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4668—21no—
53#4668—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×C4×D8—31no—
64#2454D311yescharacteristic, complemented
65D4×C4⋊C4—31no—
66#2094—31no—
67#2083—31no—
68#2065—31no—
69#2059—31no—
70#1941—31no—
71#1916—31no—
72#1879—31no—
73#1879—31no—
74#1864—31no—
75#1864—31no—
76#1862—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C3×#125—22no—
84C3×#125—41no—
85C3×C4×D4—22no—
86C3×C4×D4—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
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⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#213—22no—
105C2×#213—22no—
106C2×#213Q814yes—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121D3×C4⋊C4—44no—
122C4×D12—22no—
123C4×D12—22no—
124C4×D12—22no—
125C4×D12—22no—
126C4×D12—41no—
127C4×D12—41no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132#832—22no—
133#832—22no—
134#832—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#809—22no—
138#809—22no—
139#809D414yes—
140#808—22no—
141#808—22no—
142#808—41no—
143#807—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146C4×C3⋊C8—22no—
147C22×D8—31no—
148C2×#125C2×D311yescharacteristic
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C22×C4⋊C4—61no—
154#478—64no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169#399—21no—
170#396—21no—
171#390C2×D311yescharacteristic
172#390C2×D311yescharacteristic
173#389—31no—
174#389—31no—
175#389C2×D311yescharacteristic
176#389C2×D311yescharacteristic
177#385—21no—
178#385—61no—
179#385—61no—
180#384—31no—
181#380—21no—
182#380—61no—
183C4×C4⋊C4C2×D311yescharacteristic
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192C24×C3—21no—
193C23×D3—21no—
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×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×D412yes—
220C2×D12C2×Q812yes—
221C2×D12#4112yes—
222C2×C4×D3—24no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—81no—
230C2×C4×D3—81no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—81no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254#213—28no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264C4⋊Dic3—81no—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
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⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C2×D412yes—
279C2×C3⋊C8C2×Q812yes—
280C22×D4—31no—
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—61no—
284C23×C4—61no—
285C23×C4—61no—
286C2×D8—34no—
287C2×D8—62no—
288C2×D8—62no—
289C22×C8—31no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—31no—
293#125—22no—
294#125—41no—
295C4×D4—22no—
296C4×D4—41no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—121no—
302C4×D4—121no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—124no—
315C2×C4⋊C4#6711yescharacteristic
316C2×C4⋊C4#6711yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C42C22×D311yescharacteristic
329C2×C42C22×D311yescharacteristic
330C2×C42C22×D311yescharacteristic
331#107—62no—
332#107—62no—
333#106—62no—
334#106—62no—
335#106—121no—
336C4⋊C8—62no—
337#102—34no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—121no—
343C4×C8—62no—
344#95—21no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95#6711yescharacteristic
349#95#6711yescharacteristic
350C23×C3—22no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—41no—
368C3×D4—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—84no—
395C2×C3×C4#13512yes—
396C2×C3×C4AGL(1,ℤ/8)12yes—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×Q811yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403C2×Dic3—22no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412D12—22no—
413D12—22no—
414D12—22no—
415D12C22⋊Q814yes—
416C4×D3—48no—
417C4×D3—84no—
418C4×D3—84no—
419C3⋊C8—22no—
420C3⋊C8—22no—
421C3⋊C8—41no—
422C3⋊C8—41no—
423C24—21no—
424C24—61no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—32no—
431C2×D4—32no—
432C2×D4—32no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4D3×Q812yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—81no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—241no—
454C22×C4—241no—
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C23×D311yescharacteristic
462D8—68no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C4⋊C4—241no—
496C4⋊C4D3×D412yes—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—64no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3#52511yescharacteristic
519C22×C3D4211yescharacteristic
520C22×C3D4×Q811yescharacteristic
521C22×C3C2×#13511yescharacteristic
522C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
523C22×C3#57611yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—24no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—42no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4C2×C22⋊Q811yescharacteristic
542C3×C4#53911yescharacteristic
543C3×C4#54711yescharacteristic
544C3×C4#55411yescharacteristic
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C22×#6711yescharacteristic
563D4—22no—
564D4—22no—
565D4—34no—
566D4—41no—
567D4—41no—
568D4—62no—
569D4—62no—
570D4—62no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—62no—
597C2×C4—84no—
598C2×C4—128no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—244no—
608C2×C4—244no—
609C2×C4#94912yes—
610C2×C4#95012yes—
611C2×C4C2×D3×D411yescharacteristic
612C2×C4#100411yescharacteristic
613C2×C4C2×D3×Q811yescharacteristic
614C2×C4#100711yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C8—62no—
618C8—62no—
619C8—121no—
620C8—121no—
621C2×C3—82no—
622C2×C3#244911yescharacteristic
623C2×C3#310511yescharacteristic
624C2×C3#311911yescharacteristic
625C2×C3#326411yescharacteristic
626C2×C3#328411yescharacteristic
627C2×C3#338011yescharacteristic
628C2×C3#338411yescharacteristic
629D3—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—64no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22D4×#6711yescharacteristic
645C22Q8×#6711yescharacteristic
646C22C2×#94911yescharacteristic
647C22C2×#95011yescharacteristic
648C22#582811yescharacteristic
649C22#583511yescharacteristic
650C22#583611yescharacteristic
651C4—22no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—122no—
661C4—241no—
662C4—241no—
663C4D3×C22⋊Q811yescharacteristic
664C4#563111yescharacteristic
665C4#563611yescharacteristic
666C4#563811yescharacteristic
667C3#2638211yescharacteristic, complemented, Sylow
668C2—82no—
669C2—124no—
670C2#8281711yescharacteristic
671C2#8314111yescharacteristic
672C2#8343411yescharacteristic
673C2#8350211yescharacteristic
674C2#8357911yescharacteristic
675C2#8362811yescharacteristic
676C2#8363011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2454C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2454D3
hypocenterC3#26382
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C3×#2454C2semidirect
2. C2×#5066C2semidirect
4. C4⋊C4×D12C2semidirect
5. #71739C2semidirect
6. #71606C2semidirect
9. #69656C2semidirect
10. #69652C2semidirect
11. #69626C2semidirect
12. #69626C2semidirect
13. #69618C2semidirect
14. #69607C2semidirect
15. #69482C2semidirect
64. #2454D3semidirect
667. C3#26382semidirect

Maximal subgroups

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

subgroupindex
C3×#24542
C2×#50662
C2×C4×#2132
C4⋊C4×D122
#717392
#716062
#696572
#696562
#696522
#696262
#696182
#696072
#694822
#263823