Finite Group Databasebeta

← #10732293

Subgroups — #10732293

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×#4646C212yesmaximal
3C2×#4646C212yesmaximal
4C2×#4647C211yesmaximal, characteristic
5C3×#1442C211yesmaximal, characteristic
6#71567C211yesmaximal, characteristic
7#69415C211yesmaximal, characteristic
8#69412C211yesmaximal, characteristic
9#69411C212yesmaximal
10#69409C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15188—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×#888—28no—
24C2×#888—24no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38#4647—24no—
39#4646—28no—
40#4646—28no—
41C22×C4⋊Q8—31no—
42C2×#391—32no—
43C2×#391—32no—
44C2×#389—31no—
45#1442—31no—
46#1442—31no—
47#1442—31no—
48#1442D311yescharacteristic
49#1441—32no—
50#1441—32no—
51C4×#95—31no—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×Dic6—24no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6—24no—
69C22×Dic6D418yes—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4D414yes—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3D414yes—
91#888—232no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—22no—
95C3×#95Q814yescomplemented
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95Q814yescomplemented
105C23×Q8—32no—
106C2×C4⋊Q8—64no—
107C2×C4⋊Q8—68no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C42—31no—
116C22×C42—32no—
117#391—68no—
118#391—68no—
119#389—64no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95C2×D312yes—
130C2×#95C2×D312yes—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D311yescharacteristic
133C24×C3C2411yescharacteristic
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
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—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×Q814yes—
161C22×C3×C4C2×Q812yes—
162C22×C3×C4C2×Q812yes—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
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—22no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3C2×D412yes—
188C22×Dic3C2×D414yes—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D414yes—
191C22×Dic3C2×D412yes—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×Q814yes—
194C22×Dic3#4114yes—
195C22×Dic3#4112yes—
196C22×Dic3#4114yes—
197C2×Dic6—216no—
198C2×Dic6—216no—
199C2×Dic6—216no—
200C2×Dic6—48no—
201C2×Dic6—48no—
202C2×Dic6—48no—
203C2×Dic6—48no—
204C3×C4⋊C4—28no—
205C4⋊Dic3—28no—
206C4⋊Dic3—44no—
207Dic3⋊C4—216no—
208Dic3⋊C4—28no—
209Dic3⋊C4—44no—
210Dic3⋊C4—44no—
211C4×Dic3—28no—
212C4×Dic3—48no—
213C22×Q8—38no—
214C22×Q8—64no—
215C22×Q8—64no—
216C22×Q8—64no—
217C22×Q8—64no—
218C22×Q8—64no—
219C22×Q8—64no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—31no—
224C23×C4—32no—
225C23×C4—31no—
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D312yes—
228C23×C4C22×D311yescharacteristic
229C23×C4C22×D312yes—
230C23×C4C22×D311yescharacteristic
231C4⋊Q8—632no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C42—34no—
249C2×C42—64no—
250C2×C42—64no—
251C2×C42—64no—
252C2×C42—64no—
253#95—22no—
254#95—24no—
255#95—24no—
256#95—34no—
257#95—64no—
258#95—64no—
259#95—64no—
260#95—64no—
261#95—64no—
262#95—62no—
263#95—64no—
264#95—64no—
265#95Dic614yescomplemented
266C23×C3C22×D412yes—
267C23×C3C22×D411yescharacteristic
268C23×C3C22×D411yescharacteristic
269C23×C3C22×Q811yescharacteristic
270C23×C3C22×Q811yescharacteristic
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4112yes—
273C23×C3C2×#4112yes—
274C23×C32−1+411yescharacteristic
275C23×C32−1+412yes—
276C23×C32−1+411yescharacteristic
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4C4⋊Q814yes—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3#12014yes—
335C2×Dic3#12118yes—
336C2×Dic3#12114yes—
337C2×Dic3#12318yes—
338C2×Dic3#12414yes—
339C2×Dic3#12414yes—
340Dic6—432no—
341C24C23×D311yescharacteristic
342C2×Q8—616no—
343C2×Q8—616no—
344C2×Q8—616no—
345C2×Q8—128no—
346C2×Q8—128no—
347C2×Q8—128no—
348C2×Q8—128no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—34no—
375C22×C4—34no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—32no—
380C22×C4—34no—
381C22×C4—34no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4C2×Dic614yes—
407C22×C4C2×Dic612yes—
408C22×C4D3×D412yes—
409C22×C4D3×D414yes—
410C22×C4D3×D412yes—
411C22×C4D3×Q812yes—
412C4⋊C4—28no—
413C4⋊C4—68no—
414C4⋊C4—616no—
415C4⋊C4—68no—
416C4⋊C4—124no—
417C4⋊C4—124no—
418C4⋊C4—124no—
419C42—68no—
420C42—128no—
421C22×C3C2×#12011yescharacteristic
422C22×C3C2×#12112yes—
423C22×C3C2×#12111yescharacteristic
424C22×C3C2×#12312yes—
425C22×C3C2×#12411yescharacteristic
426C22×C3C2×#12411yescharacteristic
427C22×C3C2×C4⋊Q811yescharacteristic
428C22×C3#53312yes—
429C22×C3#53312yes—
430C22×C3#53611yescharacteristic
431C22×C3#53611yescharacteristic
432C22×C3#54112yes—
433C22×C3#54112yes—
434C22×C3#54712yes—
435C22×C3#54812yes—
436C22×C3#54812yes—
437C22×C3D4×Q812yes—
438C22×C3D4×Q811yescharacteristic
439C22×C3D4×Q811yescharacteristic
440C22×C3D4×Q812yes—
441C22×C3#55612yes—
442C22×C3#55711yescharacteristic
443C22×C3#55711yescharacteristic
444C3×C4—24no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4—42no—
448C3×C4—81no—
449Dic3—24no—
450Dic3—44no—
451Dic3—44no—
452Dic3—42no—
453Dic3—81no—
454Dic3#39418yes—
455C23C22×Dic611yescharacteristic
456C23C2×D3×D412yes—
457C23C2×D3×D411yescharacteristic
458C23C2×D3×D411yescharacteristic
459C23C2×D3×Q811yescharacteristic
460C23#100711yescharacteristic
461C23D3×#4112yes—
462C23D3×#4111yescharacteristic
463C23D3×#4112yes—
464C23#101012yes—
465C23#101011yescharacteristic
466Q8—1232no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—38no—
480C2×C4—34no—
481C2×C4—34no—
482C2×C4—38no—
483C2×C4—34no—
484C2×C4—34no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—62no—
512C2×C4—64no—
513C2×C4—122no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—121no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—122no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4#88814yes—
530C2×C3C2×#39411yescharacteristic
531C2×C3#242912yes—
532C2×C3#242912yes—
533C2×C3#243411yescharacteristic
534C2×C3#243611yescharacteristic
535C2×C3#244612yes—
536C2×C3#244612yes—
537C2×C3#254611yescharacteristic
538C2×C3#254711yescharacteristic
539C2×C3#264211yescharacteristic
540C2×C3#286211yescharacteristic
541C22C2×#88811yescharacteristic
542C22#551712yes—
543C22D4×Dic611yescharacteristic
544C22D4×Dic612yes—
545C22#556911yescharacteristic
546C22D3×#12011yescharacteristic
547C22D3×#12111yescharacteristic
548C22D3×#12112yes—
549C22#562811yescharacteristic
550C22#563212yes—
551C22#563312yes—
552C22#564811yescharacteristic
553C22D3×#12312yes—
554C22#565512yes—
555C22D3×#12411yescharacteristic
556C22D3×#12411yescharacteristic
557C22#567712yes—
558C22#568511yescharacteristic
559C22#568812yes—
560C22#569612yes—
561C22#570112yes—
562C22#570212yes—
563C22#572711yescharacteristic
564C4—24no—
565C4—38no—
566C4—44no—
567C4—42no—
568C4—44no—
569C4—64no—
570C4—81no—
571C4—124no—
572C4—124no—
573C4—122no—
574C4—241no—
575C3#1518811yescharacteristic, complemented, Sylow
576C2#7422111yescharacteristic
577C2#7441211yescharacteristic
578C2#7498612yes—
579C2#7550012yes—
580C2#7606011yescharacteristic
581C2#7678111yescharacteristic
582C2D3×#39411yescharacteristic
583C2#7751411yescharacteristic
584C2#7774412yes—
585C2#7813612yes—
586C2#7821411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1442C2
hypercenter#1442D3
hypocenterC3#15188
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
95. C3×#95Q8semidirect
104. C3⋊#95Q8semidirect
265. #95Dic6semidirect
575. C3#15188semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#46462
C2×#46472
C3×#14422
#715672
#694152
#694122
#694112
#694092
Dic3×#952
#151883