Finite Group Databasebeta

← #10732005

Subgroups — #10732005

Order 768.

1778 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2C2×#4979C211yesmaximal, characteristic
3C2×#4976C211yesmaximal, characteristic
4C2×#4656C211yesmaximal, characteristic
5C2×#4647C212yesmaximal
6C3×#1441C211yesmaximal, characteristic
7#71566C211yesmaximal, characteristic
8#69411C212yesmaximal
9#69409C212yesmaximal
10Dic3⋊#95C212yesmaximal
11Dic3⋊#95C211yesmaximal, characteristic
12#15158—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic3C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#887—28no—
26C2×#887—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4979—24no—
39#4976—24no—
40#4656—24no—
41#4647—28no—
42C22×C4×Q8—31no—
43C2×#395—31no—
44C2×#395—31no—
45C2×#389—32no—
46C2×#389—31no—
47#1441—32no—
48#1441—32no—
49#1441—31no—
50#1441D311yescharacteristic
51C4⋊#95—32no—
52C4⋊#95—31no—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6D414yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4Q818yescomplemented
93C2×Dic3⋊C4Q814yescomplemented
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3D418yes—
98C2×C4×Dic3D414yes—
99#887—232no—
100C3×#95—24no—
101C3×#95—22no—
102C3×#95—42no—
103C3⋊#95—24no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—42no—
110C3⋊#95—42no—
111C23×Q8—31no—
112C2×C4×Q8—64no—
113C2×C4×Q8—68no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C4⋊C4C2×D311yescharacteristic
123C22×C42—32no—
124C22×C42—31no—
125#395—64no—
126#395—64no—
127#389—68no—
128#389—64no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—31no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C2×#95C2×D312yes—
140C24×C3C2411yescharacteristic
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×Q812yes—
170C22×C3×C4C2×Q814yes—
171C22×C3×C4#4112yes—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3C2×D412yes—
198C22×Dic3C2×D414yes—
199C22×Dic3C2×D414yes—
200C22×Dic3C2×D414yes—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×Q814yes—
203C22×Dic3C2×Q812yes—
204C22×Dic3C2×Q814yes—
205C22×Dic3C2×Q812yes—
206C22×Dic3#4114yes—
207C22×Dic3#4112yes—
208C2×Dic6—28no—
209C2×Dic6—216no—
210C2×Dic6—28no—
211C2×Dic6—28no—
212C2×Dic6—216no—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—44no—
215C4⋊Dic3—48no—
216Dic3⋊C4—216no—
217Dic3⋊C4—216no—
218Dic3⋊C4—28no—
219Dic3⋊C4—44no—
220Dic3⋊C4—44no—
221Dic3⋊C4—48no—
222C4×Dic3—216no—
223C4×Dic3—28no—
224C22×Q8—34no—
225C22×Q8—64no—
226C22×Q8—64no—
227C22×Q8—62no—
228C22×Q8—62no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240C4×Q8—632no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—38no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C42—38no—
267C2×C42—34no—
268C2×C42—64no—
269C2×C42—64no—
270C2×C42—64no—
271#95—24no—
272#95—22no—
273#95—42no—
274#95—62no—
275#95—62no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—62no—
280#95—122no—
281#95—122no—
282C23×C3C22×D412yes—
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D411yescharacteristic
285C23×C3C22×Q812yes—
286C23×C3C22×Q811yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4112yes—
289C23×C3C2×#4111yescharacteristic
290C23×C3H(5,𝔽2)11yescharacteristic
291C23×C32−1+411yescharacteristic
292C23×C32−1+412yes—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4#12114yes—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3#12014yes—
355C2×Dic3#12118yes—
356C2×Dic3C22⋊Q818yes—
357C2×Dic3C22⋊Q814yes—
358C2×Dic3C4⋊Q818yes—
359C2×Dic3C4⋊Q814yes—
360Dic6—232no—
361C24C23×D311yescharacteristic
362C2×Q8—616no—
363C2×Q8—68no—
364C2×Q8—68no—
365C2×Q8—616no—
366C2×Q8—68no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—34no—
393C22×C4—32no—
394C22×C4—34no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—32no—
401C22×C4—34no—
402C22×C4—32no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4#23512yes—
429C22×C4D3×D414yes—
430C22×C4D3×D412yes—
431C22×C4D3×D412yes—
432C22×C4D3×Q814yes—
433C22×C4D3×Q812yes—
434C4⋊C4—28no—
435C4⋊C4—44no—
436C4⋊C4—616no—
437C4⋊C4—68no—
438C4⋊C4—616no—
439C4⋊C4—124no—
440C4⋊C4—128no—
441C4⋊C4—128no—
442C4⋊C4—124no—
443C42—68no—
444C42—616no—
445C22×C3C2×#12011yescharacteristic
446C22×C3C2×#12111yescharacteristic
447C22×C3C2×#12112yes—
448C22×C3C2×C22⋊Q812yes—
449C22×C3C2×C22⋊Q811yescharacteristic
450C22×C3#52511yescharacteristic
451C22×C3#52512yes—
452C22×C3C2×C4⋊Q812yes—
453C22×C3C2×C4⋊Q811yescharacteristic
454C22×C3#53612yes—
455C22×C3#53611yescharacteristic
456C22×C3#53712yes—
457C22×C3#53711yescharacteristic
458C22×C3D4×Q812yes—
459C22×C3D4×Q811yescharacteristic
460C22×C3#55011yescharacteristic
461C22×C3#55412yes—
462C22×C3#55411yescharacteristic
463C22×C3#55612yes—
464C22×C3#55611yescharacteristic
465C22×C3#55712yes—
466C22×C3#55712yes—
467C22×C3#55712yes—
468C3×C4—24no—
469C3×C4—44no—
470C3×C4—44no—
471C3×C4—42no—
472C3×C4—81no—
473Dic3—28no—
474Dic3—42no—
475Dic3—44no—
476Dic3—42no—
477Dic3—81no—
478Dic3#39518yescomplemented
479C23C2×#23511yescharacteristic
480C23C2×D3×D411yescharacteristic
481C23C2×D3×D412yes—
482C23C2×D3×D411yescharacteristic
483C23#100411yescharacteristic
484C23C2×D3×Q812yes—
485C23C2×D3×Q811yescharacteristic
486C23D3×#4112yes—
487C23D3×#4111yescharacteristic
488C23#101011yescharacteristic
489C23#101012yes—
490Q8—632no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—38no—
505C2×C4—34no—
506C2×C4—34no—
507C2×C4—38no—
508C2×C4—34no—
509C2×C4—38no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—62no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—121no—
550C2×C4—122no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4—122no—
557C2×C4#88414yes—
558C2×C3C2×#39511yescharacteristic
559C2×C3#253812yes—
560C2×C3#253811yescharacteristic
561C2×C3#254612yes—
562C2×C3#254611yescharacteristic
563C2×C3#258112yes—
564C2×C3#258111yescharacteristic
565C2×C3#258412yes—
566C2×C3#258411yescharacteristic
567C2×C3#261611yescharacteristic
568C2×C3#284711yescharacteristic
569C22C2×#88411yescharacteristic
570C22#551111yescharacteristic
571C22#553311yescharacteristic
572C22#554712yes—
573C22#556712yes—
574C22D3×#12011yescharacteristic
575C22#559712yes—
576C22#559911yescharacteristic
577C22D3×#12112yes—
578C22#560811yescharacteristic
579C22D3×C22⋊Q812yes—
580C22D3×C22⋊Q811yescharacteristic
581C22#563212yes—
582C22#563211yescharacteristic
583C22#563612yes—
584C22#564812yes—
585C22#565611yescharacteristic
586C22#567511yescharacteristic
587C22#568512yes—
588C22#568512yes—
589C22#570212yes—
590C22D3×C4⋊Q812yes—
591C22D3×C4⋊Q811yescharacteristic
592C4—24no—
593C4—38no—
594C4—44no—
595C4—44no—
596C4—42no—
597C4—68no—
598C4—81no—
599C4—124no—
600C4—122no—
601C4—122no—
602C4—241no—
603C3#1515811yescharacteristic, complemented, Sylow
604C2#7393611yescharacteristic
605C2#7493511yescharacteristic
606C2#7509012yes—
607C2#7608012yes—
608C2#7653511yescharacteristic
609C2#7697011yescharacteristic
610C2D3×#39511yescharacteristic
611C2#7731412yes—
612C2#7778512yes—
613C2#7779711yescharacteristic
614C2#7819111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1441C2
hypercenter#1441D3
hypocenterC3#15158
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
92. C2×Dic3⋊C4Q8semidirect
93. C2×Dic3⋊C4Q8semidirect
478. Dic3#395semidirect
603. C3#15158semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×#49792
C2×#49762
C2×#46562
C2×#46472
C3×#14412
#715662
#694112
#694092
Dic3⋊#952
#151583