Finite Group Databasebeta

← #10731976

Subgroups — #10731976

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4975C211yesmaximal, characteristic
2C2×#4660C212yesmaximal
3C2×#4658C212yesmaximal
4C2×#4647C211yesmaximal, characteristic
5C3×#1445C211yesmaximal, characteristic
6#71569C211yesmaximal, characteristic
7#69423C211yesmaximal, characteristic
8#69422C211yesmaximal, characteristic
9Dic3⋊#95C212yesmaximal
10Dic3⋊#95C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15153—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36#4975—24no—
37#4660—28no—
38#4658—28no—
39#4647—24no—
40C2×#400—32no—
41C2×#400—32no—
42C2×#389—31no—
43C2×#384—31no—
44#1445—31no—
45#1445—31no—
46#1445—31no—
47#1445D311yescharacteristic
48C4⋊#95—32no—
49C4⋊#95—32no—
50C4×#95—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—24no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4Q814yes—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—24no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3D414yes—
88C3×#95—22no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—22no—
92C3⋊#95—22no—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95D418yes—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—32no—
103C22×C4⋊C4—31no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C42—31no—
110C22×C42—32no—
111#400—68no—
112#400—68no—
113#389—64no—
114#384—64no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95C2×D312yes—
125C2×#95C2×D312yes—
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D311yescharacteristic
128C24×C3C2411yescharacteristic
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4C2×D414yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×Q812yes—
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3C2×D414yes—
184C22×Dic3C2×D414yes—
185C22×Dic3C2×D412yes—
186C22×Dic3C2×D412yes—
187C22×Dic3C2×Q812yes—
188C22×Dic3C2×Q812yes—
189C22×Dic3#4112yes—
190C22×Dic3#4114yes—
191C22×Dic3#4114yes—
192C3×C4⋊C4—44no—
193C4⋊Dic3—44no—
194C4⋊Dic3—44no—
195Dic3⋊C4—28no—
196Dic3⋊C4—44no—
197Dic3⋊C4—48no—
198Dic3⋊C4—44no—
199Dic3⋊C4—48no—
200C4×Dic3—28no—
201C4×Dic3—48no—
202C23×C4—31no—
203C23×C4—31no—
204C23×C4—31no—
205C23×C4—32no—
206C23×C4—32no—
207C23×C4—31no—
208C23×C4C22×D312yes—
209C23×C4C22×D311yescharacteristic
210C23×C4C22×D312yes—
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D311yescharacteristic
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—34no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4—64no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C42—34no—
235C2×C42—64no—
236C2×C42—64no—
237C2×C42—64no—
238C2×C42—64no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—38no—
244#95—64no—
245#95—64no—
246#95—64no—
247#95—64no—
248#95—62no—
249#95—62no—
250#95—64no—
251#95—64no—
252C23×C3C22×D411yescharacteristic
253C23×C3C22×D412yes—
254C23×C3C22×Q811yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4111yescharacteristic
257C23×C3C2×#4111yescharacteristic
258C23×C3C2×#4112yes—
259C23×C3C2×#4112yes—
260C23×C32−1+411yescharacteristic
261C23×C32−1+411yescharacteristic
262C23×C32−1+412yes—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4#12414yes—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—22no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—42no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3C22⋊Q814yes—
321C2×Dic3#12314yes—
322C2×Dic3#12414yes—
323C2×Dic3#12518yes—
324C2×Dic3#12618yes—
325C2×Dic3#12714yes—
326C24C23×D311yescharacteristic
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—34no—
354C22×C4—32no—
355C22×C4—34no—
356C22×C4—34no—
357C22×C4—32no—
358C22×C4—34no—
359C22×C4—32no—
360C22×C4—61no—
361C22×C4—62no—
362C22×C4—61no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4#23512yes—
386C22×C4D3×D414yes—
387C22×C4D3×D412yes—
388C22×C4#23712yes—
389C22×C4D3×Q812yes—
390C4⋊C4—44no—
391C4⋊C4—68no—
392C4⋊C4—124no—
393C4⋊C4—128no—
394C4⋊C4—124no—
395C4⋊C4—128no—
396C4⋊C4—124no—
397C4⋊C4—124no—
398C42—68no—
399C42—128no—
400C22×C3C2×C22⋊Q811yescharacteristic
401C22×C3C2×#12311yescharacteristic
402C22×C3C2×#12411yescharacteristic
403C22×C3C2×#12411yescharacteristic
404C22×C3C2×#12512yes—
405C22×C3C2×#12612yes—
406C22×C3C2×#12711yescharacteristic
407C22×C3#53212yes—
408C22×C3#53212yes—
409C22×C3#53611yescharacteristic
410C22×C3#53611yescharacteristic
411C22×C3#54112yes—
412C22×C3#54112yes—
413C22×C3#54712yes—
414C22×C3#54712yes—
415C22×C3#54712yes—
416C22×C3D4×Q812yes—
417C22×C3#55212yes—
418C22×C3#55212yes—
419C22×C3#55611yescharacteristic
420C22×C3#55611yescharacteristic
421C22×C3#55711yescharacteristic
422C22×C3#55711yescharacteristic
423C3×C4—24no—
424C3×C4—44no—
425C3×C4—42no—
426C3×C4—82no—
427C3×C4—81no—
428Dic3—24no—
429Dic3—44no—
430Dic3—42no—
431Dic3—82no—
432Dic3—81no—
433Dic3#40018yescomplemented
434C23C2×#23511yescharacteristic
435C23C2×D3×D412yes—
436C23C2×D3×D411yescharacteristic
437C23C2×#23711yescharacteristic
438C23C2×D3×Q811yescharacteristic
439C23#100711yescharacteristic
440C23D3×#4112yes—
441C23D3×#4111yescharacteristic
442C23D3×#4112yes—
443C23#101012yes—
444C23#101011yescharacteristic
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—34no—
455C2×C4—34no—
456C2×C4—38no—
457C2×C4—38no—
458C2×C4—34no—
459C2×C4—34no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—62no—
480C2×C4—64no—
481C2×C4—62no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4#88514yes—
508C2×C3C2×#40011yescharacteristic
509C2×C3#249612yes—
510C2×C3#249612yes—
511C2×C3#250111yescharacteristic
512C2×C3#250311yescharacteristic
513C2×C3#250612yes—
514C2×C3#250612yes—
515C2×C3#254211yescharacteristic
516C2×C3#256811yescharacteristic
517C2×C3#264211yescharacteristic
518C2×C3#285811yescharacteristic
519C22C2×#88511yescharacteristic
520C22#553112yes—
521C22#554812yes—
522C22#556711yescharacteristic
523C22#556811yescharacteristic
524C22#559612yes—
525C22D3×C22⋊Q811yescharacteristic
526C22#562811yescharacteristic
527C22#563112yes—
528C22#563212yes—
529C22#564811yescharacteristic
530C22#565012yes—
531C22D3×#12311yescharacteristic
532C22#565512yes—
533C22#565512yes—
534C22#567111yescharacteristic
535C22D3×#12411yescharacteristic
536C22D3×#12512yes—
537C22#569612yes—
538C22#570112yes—
539C22D3×#12612yes—
540C22D3×#12711yescharacteristic
541C22#572111yescharacteristic
542C4—24no—
543C4—38no—
544C4—44no—
545C4—42no—
546C4—64no—
547C4—82no—
548C4—81no—
549C4—124no—
550C4—122no—
551C4—242no—
552C4—241no—
553C3#1515311yescharacteristic, complemented, Sylow
554C2#7407911yescharacteristic
555C2#7502312yes—
556C2#7502612yes—
557C2#7642411yescharacteristic
558C2#7654511yescharacteristic
559C2#7749012yes—
560C2#7776612yes—
561C2#7803011yescharacteristic
562C2#7803711yescharacteristic
563C2#7821411yescharacteristic
564C2D3×#40011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1445C2
hypercenter#1445D3
hypocenterC3#15153
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
433. Dic3#400semidirect
553. C3#15153semidirect

Maximal subgroups

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

subgroupindex
C2×#49752
C2×#46602
C2×#46582
C2×#46472
C3×#14452
#715692
#694232
#694222
Dic3⋊#952
Dic3×#952
#151533