Finite Group Databasebeta

← #10824369

Subgroups — #10824369

Order 768.

956 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#124C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3C2×C4×#213C211yesmaximal, characteristic
4#73299C211yesmaximal, characteristic, complemented
5#71728C211yesmaximal, characteristic
6#71722C211yesmaximal, characteristic, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic, complemented
9#69683C211yesmaximal, characteristic, complemented
10#69680C211yesmaximal, characteristic, complemented
11#69652C211yesmaximal, characteristic, complemented
12#69650C211yesmaximal, characteristic
13#69557C211yesmaximal, characteristic, complemented
14#69541C211yesmaximal, characteristic, complemented
15C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
16#23680—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#872—21no—
24C3×#391C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#388—21no—
27C3×#388—21no—
28C3×#382C2211yescharacteristic
29C3×C4×C22⋊C4—21no—
30C3×C4×C22⋊C4—21no—
31C3×C43C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C4×#215—24no—
39C4×#213—24no—
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4940—21no—
45#4940—21no—
46#4939C2211yescharacteristic
47#4939C2211yescharacteristic
48C4×(C2×D3)⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×D8—31no—
64C4×#124D311yescharacteristic, complemented
65#2293—31no—
66#2085—31no—
67#2073—31no—
68#2068—31no—
69#2065—31no—
70#2055—31no—
71#2054—31no—
72#2042—31no—
73#2012—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×#124—22no—
83C3×#124—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#215—22no—
105C2×#215—22no—
106C2×#215—22no—
107C2×#213—22no—
108C2×#213—22no—
109C2×#213—22no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C4×D12—22no—
126C4×D12D414yes—
127#872—44no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132#835—22no—
133#835—22no—
134#835—22no—
135#832—22no—
136#832—22no—
137#832—22no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#809—22no—
142#809—22no—
143#808—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146C4⋊(C3⋊C8)—22no—
147C4×C3⋊C8—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4C2×D311yescharacteristic
155C2×#117—61no—
156C4×SD16—64no—
157C4×D8—64no—
158C2×#107—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8—31no—
171C2×C4×C8—31no—
172#391C2×D311yescharacteristic
173#390—31no—
174#390—31no—
175#390C2×D311yescharacteristic
176#389—31no—
177#389—31no—
178#388—21no—
179#388—21no—
180#388—61no—
181#388—61no—
182#382C2×D311yescharacteristic
183C4×C4⋊C4—31no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—61no—
187C43C2×D311yescharacteristic
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4#4112yes—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12C2×D412yes—
230C2×C4×D3—24no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—22no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—41no—
251C3×C42—42no—
252C3×C42C2×D412yes—
253C3×C42#4112yes—
254#215—28no—
255#213—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3C2×D412yes—
267Dic3⋊C4—42no—
268Dic3⋊C4—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—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8#4112yes—
281C2×C3⋊C8#4112yes—
282C22×Q8C22×D311yescharacteristic
283C22×D4—31no—
284C22×D4C22×D311yescharacteristic
285C23×C4—21no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—62no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—31no—
297#124—22no—
298#124—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—34no—
304C4×D4—62no—
305#117—124no—
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⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—21no—
325C2×C42—61no—
326C2×C42#6711yescharacteristic
327C2×C42#6711yescharacteristic
328C2×C42#6711yescharacteristic
329C2×C42#6711yescharacteristic
330C2×C42C22×D311yescharacteristic
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—62no—
335C4⋊C8—62no—
336C4⋊C8—62no—
337#103—62no—
338#103—62no—
339#103—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345C4×C8—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—84no—
400C2×C3×C4#12112yes—
401C2×C3×C4#12112yes—
402C2×C3×C4#13512yes—
403C2×C3×C4#13512yes—
404C2×C3×C4AGL(1,ℤ/8)12yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×Dic3—22no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424C4×D3—48no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C3⋊C8—41no—
428C3⋊C8—41no—
429C24—21no—
430C24—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8#23512yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—32no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4#23512yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—61no—
456C22×C4—64no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—68no—
469D8—68no—
470C2×C8—32no—
471C2×C8—32no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—32no—
487C4⋊C4—42no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—64no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—42no—
514C42—122no—
515C42D3×D412yes—
516C42#23712yes—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3#53211yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3C2×#13511yescharacteristic
529C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
530C22×C3#57511yescharacteristic
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—42no—
543C3×C4—41no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4#46314yes—
547C3×C4#52511yescharacteristic
548C3×C4#52511yescharacteristic
549C3×C4#52911yescharacteristic
550C3×C4#52911yescharacteristic
551Dic3—42no—
552Dic3—81no—
553Dic3—81no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C22×#6711yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—62no—
578D4—62no—
579D4—121no—
580D4—121no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—62no—
604C2×C4—84no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—128no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4#93812yes—
615C2×C4#93812yes—
616C2×C4#94912yes—
617C2×C4#95012yes—
618C2×C4#95012yes—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×#23511yescharacteristic
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C8—62no—
626C8—62no—
627C8—121no—
628C8—121no—
629C2×C3—82no—
630C2×C3#239211yescharacteristic
631C2×C3C2×#46311yescharacteristic
632C2×C3#303511yescharacteristic
633C2×C3#310211yescharacteristic
634C2×C3#310611yescharacteristic
635C2×C3#316811yescharacteristic
636C2×C3#317111yescharacteristic
637D3—82no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22#579111yescharacteristic
653C22C2×#93811yescharacteristic
654C22C2×#94911yescharacteristic
655C22C2×#95011yescharacteristic
656C22C2×#95011yescharacteristic
657C22#582211yescharacteristic
658C22#583211yescharacteristic
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—42no—
665C4—81no—
666C4—81no—
667C4—122no—
668C4—241no—
669C4—241no—
670C4#505714yes—
671C4#553811yescharacteristic
672C4#553811yescharacteristic
673C4#554711yescharacteristic
674C4#554711yescharacteristic
675C3#2368011yescharacteristic, complemented, Sylow
676C2—82no—
677C2—242no—
678C2#8240111yescharacteristic
679C2#8252811yescharacteristic
680C2#8252911yescharacteristic
681C2C2×#505711yescharacteristic
682C2#8313411yescharacteristic
683C2#8313511yescharacteristic
684C2#8329611yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#124C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC4×#124D3
hypocenterC3#23680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C3×C4×#124C2semidirect
2. C2×C4×#215C2semidirect
4. #73299C2semidirect
6. #71722C2semidirect
7. C4×#835C2semidirect
8. C4×#832C2semidirect
9. #69683C2semidirect
10. #69680C2semidirect
11. #69652C2semidirect
13. #69557C2semidirect
14. #69541C2semidirect
15. C4×C4⋊(C3⋊C8)C2semidirect
64. C4×#124D3semidirect
675. C3#23680semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1242
C2×C4×#2152
C2×C4×#2132
#732992
#717282
#717222
C4×#8352
C4×#8322
#696832
#696802
#696522
#696502
#695572
#695412
C4×C4⋊(C3⋊C8)2
#236803