Finite Group Databasebeta

← #10732161

Subgroups — #10732161

Order 768.

1530 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4982C212yesmaximal
2C2×#4984C211yesmaximal, characteristic
3C2×#4656C212yesmaximal
4C2×#4646C211yesmaximal, characteristic
5C3×#1444C211yesmaximal, characteristic
6#71568C211yesmaximal, characteristic
7#71563C212yesmaximal
8#69420C211yesmaximal, characteristic
9#69419C211yesmaximal, characteristic
10Dic3⋊#95C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15173—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C4⋊Dic3C2212yes—
17C22×C4⋊Dic3C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4984—24no—
37#4982—28no—
38#4656—28no—
39#4646—24no—
40C2×#398—32no—
41C2×#395—32no—
42C2×#391—31no—
43C2×#389—31no—
44#1444—31no—
45#1444—31no—
46#1444—31no—
47#1444D311yescharacteristic
48C4⋊#95—32no—
49C4⋊#95—32no—
50C4×#95—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—24no—
66C22×Dic6—42no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—42no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—42no—
79C2×C4⋊Dic3—42no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—42no—
87C2×Dic3⋊C4—42no—
88C2×C4×Dic3—42no—
89C2×C4×Dic3—42no—
90C2×C4×Dic3—42no—
91C2×C4×Dic3D414yes—
92C2×C4×Dic3Q818yescomplemented
93C3×#95—22no—
94C3×#95—24no—
95C3×#95Q814yescomplemented
96C3⋊#95—22no—
97C3⋊#95—24no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95D418yes—
104C3⋊#95Q814yescomplemented
105C23×Q8—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—32no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C4⋊C4C2×D312yes—
114C22×C42—32no—
115C22×C42—31no—
116#398—68no—
117#395—68no—
118#391—64no—
119#389—64no—
120C2×#95—32no—
121C2×#95—31no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95C2×D312yes—
130C2×#95C2×D311yescharacteristic
131C2×#95C2×D311yescharacteristic
132C24×C3C2411yescharacteristic
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
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—21no—
154C22×C3×C4C2×D414yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×Q814yes—
157C22×C3×C4C2×Q812yes—
158C22×C3×C4C2×Q812yes—
159C22×C3×C4C2×Q814yes—
160C22×C3×C4#4114yes—
161C22×C3×C4#4112yes—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3C2×D414yes—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×Q814yes—
193C22×Dic3C2×Q814yes—
194C22×Dic3C2×Q814yes—
195C22×Dic3#4112yes—
196C2×Dic6—28no—
197C2×Dic6—48no—
198C2×Dic6—48no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—48no—
203C4⋊Dic3—48no—
204C4⋊Dic3—44no—
205C4⋊Dic3—48no—
206Dic3⋊C4—48no—
207Dic3⋊C4—48no—
208Dic3⋊C4—44no—
209C4×Dic3—44no—
210C4×Dic3—48no—
211C22×Q8—62no—
212C22×Q8—62no—
213C22×Q8—64no—
214C22×Q8—62no—
215C22×Q8—122no—
216C23×C4—31no—
217C23×C4—32no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—32no—
221C23×C4—31no—
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D312yes—
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D312yes—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4—122no—
246C2×C4⋊C4—122no—
247C2×C4⋊C4—122no—
248C2×C42—34no—
249C2×C42—38no—
250C2×C42—122no—
251C2×C42—122no—
252C2×C42—122no—
253#95—22no—
254#95—24no—
255#95—38no—
256#95—34no—
257#95—64no—
258#95—62no—
259#95—62no—
260#95—64no—
261#95—64no—
262#95—62no—
263#95—64no—
264#95Dic614yescomplemented
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D412yes—
267C23×C3C22×Q811yescharacteristic
268C23×C3C22×Q812yes—
269C23×C3C22×Q811yescharacteristic
270C23×C3C2×#4112yes—
271C23×C3C2×#4111yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3H(5,𝔽2)11yescharacteristic
274C23×C3H(5,𝔽2)12yes—
275C23×C32−1+411yescharacteristic
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4C4⋊Q814yes—
304C2×C3×C4C4⋊Q818yes—
305C2×Dic3—22no—
306C2×Dic3—24no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3#12714yes—
337C2×Dic3C4⋊Q814yes—
338C2×Dic3C4⋊Q818yes—
339Dic6—416no—
340C24C23×D311yescharacteristic
341C2×Q8—68no—
342C2×Q8—128no—
343C2×Q8—124no—
344C2×Q8—128no—
345C2×Q8—124no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—34no—
372C22×C4—34no—
373C22×C4—34no—
374C22×C4—32no—
375C22×C4—62no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—61no—
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—61no—
401C22×C4C2×Dic614yes—
402C22×C4C2×Dic612yes—
403C22×C4D3×D414yes—
404C22×C4D3×D412yes—
405C22×C4#23714yes—
406C22×C4D3×Q812yes—
407C22×C4D3×Q814yes—
408C22×C4#23912yes—
409C4⋊C4—28no—
410C4⋊C4—48no—
411C4⋊C4—128no—
412C4⋊C4—128no—
413C4⋊C4—128no—
414C4⋊C4—124no—
415C4⋊C4—124no—
416C4⋊C4—128no—
417C42—128no—
418C42—124no—
419C22×C3#52511yescharacteristic
420C22×C3#52511yescharacteristic
421C22×C3C2×#12711yescharacteristic
422C22×C3C2×C4⋊Q811yescharacteristic
423C22×C3C2×C4⋊Q812yes—
424C22×C3C2×C4⋊Q811yescharacteristic
425C22×C3C2×C4⋊Q812yes—
426C22×C3#53312yes—
427C22×C3#53312yes—
428C22×C3#53411yescharacteristic
429C22×C3#53711yescharacteristic
430C22×C3#54312yes—
431C22×C3#54412yes—
432C22×C3#54612yes—
433C22×C3#54612yes—
434C22×C3#54612yes—
435C22×C3#54812yes—
436C22×C3#55412yes—
437C22×C3#55411yescharacteristic
438C22×C3#55411yescharacteristic
439C22×C3#55412yes—
440C22×C3#55711yescharacteristic
441C22×C3#55711yescharacteristic
442C3×C4—24no—
443C3×C4—28no—
444C3×C4—42no—
445C3×C4—44no—
446C3×C4—81no—
447Dic3—24no—
448Dic3—24no—
449Dic3—28no—
450Dic3—42no—
451Dic3—82no—
452Dic3—81no—
453C23C22×Dic611yescharacteristic
454C23C2×D3×D411yescharacteristic
455C23C2×D3×D412yes—
456C23C2×#23712yes—
457C23#100411yescharacteristic
458C23C2×D3×Q812yes—
459C23C2×D3×Q811yescharacteristic
460C23C2×#23911yescharacteristic
461C23D3×#4111yescharacteristic
462C23#100912yes—
463C23#101011yescharacteristic
464Q8—1216no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—34no—
481C2×C4—34no—
482C2×C4—38no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—62no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4#88818yes—
527C2×C4#88814yes—
528C2×C3#239911yescharacteristic
529C2×C3#253911yescharacteristic
530C2×C3#254011yescharacteristic
531C2×C3#254912yes—
532C2×C3#254912yes—
533C2×C3#255012yes—
534C2×C3#255012yes—
535C2×C3#258311yescharacteristic
536C2×C3#258411yescharacteristic
537C2×C3#265911yescharacteristic
538C2×C3#286811yescharacteristic
539C22C2×#88812yes—
540C22C2×#88811yescharacteristic
541C22#553911yescharacteristic
542C22#554312yes—
543C22#557011yescharacteristic
544C22#558611yescharacteristic
545C22#559112yes—
546C22#559812yes—
547C22#560811yescharacteristic
548C22#562212yes—
549C22#562711yescharacteristic
550C22#563012yes—
551C22#563412yes—
552C22#563512yes—
553C22#565311yescharacteristic
554C22#565412yes—
555C22#568511yescharacteristic
556C22#568912yes—
557C22D3×#12711yescharacteristic
558C22D3×C4⋊Q811yescharacteristic
559C22D3×C4⋊Q812yes—
560C22#572812yes—
561C22#573011yescharacteristic
562C4—28no—
563C4—24no—
564C4—42no—
565C4—44no—
566C4—64no—
567C4—64no—
568C4—68no—
569C4—81no—
570C4—122no—
571C4—242no—
572C4—241no—
573C3#1517311yescharacteristic, complemented, Sylow
574C2#7435611yescharacteristic
575C2#7447212yes—
576C2#7448811yescharacteristic
577C2#7606112yes—
578C2#7607911yescharacteristic
579C2#7668711yescharacteristic
580C2#7717111yescharacteristic
581C2#7725312yes—
582C2#7778612yes—
583C2#7796611yescharacteristic
584C2#7810711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1444C2
hypercenter#1444D3
hypocenterC3#15173
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
92. C2×C4×Dic3Q8semidirect
95. C3×#95Q8semidirect
104. C3⋊#95Q8semidirect
264. #95Dic6semidirect
573. C3#15173semidirect

Maximal subgroups

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

subgroupindex
C2×#49822
C2×#49842
C2×#46562
C2×#46462
C3×#14442
#715682
#715632
#694202
#694192
Dic3⋊#952
Dic3×#952
#151733