Finite Group Databasebeta

← #10732220

Subgroups — #10732220

Order 768.

1750 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×Dic3×C4⋊C4C211yesmaximal, characteristic
3C2×#4646C212yesmaximal
4C2×#4646C212yesmaximal
5C3×#1441C211yesmaximal, characteristic
6#71567C211yesmaximal, characteristic
7#71565C211yesmaximal, characteristic
8#69414C212yesmaximal
9#69412C211yesmaximal, characteristic
10#69411C212yesmaximal
11Dic3⋊#95C211yesmaximal, characteristic
12#15180—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C2×#888—28no—
24C2×#888—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38Dic3×C4⋊C4—24no—
39#4646—28no—
40#4646—28no—
41C22×C4⋊Q8—31no—
42C2×#391—32no—
43C2×#391—32no—
44C2×C4×C4⋊C4—31no—
45#1442—31no—
46#1442—32no—
47#1442—31no—
48#1441—32no—
49#1441—31no—
50#1441D311yescharacteristic
51C4⋊#95—31no—
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
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—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4Q814yescomplemented
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3Q814yescomplemented
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3D414yes—
92C2×C4×Dic3D414yes—
93#888—232no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—22no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C23×Q8—32no—
108C2×C4⋊Q8—64no—
109C2×C4⋊Q8—68no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C42—31no—
117C22×C42—31no—
118C22×C42—31no—
119#391—68no—
120#391—68no—
121C4×C4⋊C4—64no—
122C2×#95—32no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95C2×D311yescharacteristic
133C2×#95C2×D312yes—
134C2×#95C2×D312yes—
135C24×C3C2411yescharacteristic
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
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—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×Q812yes—
163C22×C3×C4C2×Q812yes—
164C22×C3×C4C2×Q812yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4#4112yes—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—21no—
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—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D414yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×Q812yes—
197C22×Dic3C2×Q812yes—
198C22×Dic3#4112yes—
199C22×Dic3#4112yes—
200C22×Dic3#4114yes—
201C2×Dic6—216no—
202C2×Dic6—216no—
203C2×Dic6—216no—
204C2×Dic6—48no—
205C2×Dic6—48no—
206C2×Dic6—48no—
207C2×Dic6—48no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—28no—
210C4⋊Dic3—28no—
211C4⋊Dic3—28no—
212Dic3⋊C4—216no—
213Dic3⋊C4—48no—
214C4×Dic3—28no—
215C4×Dic3—28no—
216C4×Dic3—44no—
217C22×Q8—38no—
218C22×Q8—64no—
219C22×Q8—64no—
220C22×Q8—64no—
221C22×Q8—64no—
222C22×Q8—64no—
223C22×Q8—64no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—32no—
230C23×C4C22×D312yes—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D312yes—
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C4⋊Q8—632no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—34no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4Dic614yescomplemented
252C2×C42—34no—
253C2×C42—34no—
254C2×C42—64no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42—62no—
258C2×C42—64no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—64no—
263#95—62no—
264#95—64no—
265#95—64no—
266#95—64no—
267#95—64no—
268#95—62no—
269#95—64no—
270#95—64no—
271#95—62no—
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D412yes—
274C23×C3C22×D411yescharacteristic
275C23×C3C22×Q811yescharacteristic
276C23×C3C22×Q811yescharacteristic
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4112yes—
280C23×C3C2×#4111yescharacteristic
281C23×C32−1+412yes—
282C23×C32−1+412yes—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4C22⋊Q814yes—
313C2×C3×C4C4⋊Q814yescomplemented
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3#12014yes—
343C2×Dic3#12114yes—
344C2×Dic3#12118yes—
345C2×Dic3C22⋊Q814yes—
346C2×Dic3#12318yes—
347C2×Dic3#12414yes—
348Dic6—432no—
349C24C23×D311yescharacteristic
350C2×Q8—616no—
351C2×Q8—616no—
352C2×Q8—616no—
353C2×Q8—128no—
354C2×Q8—128no—
355C2×Q8—128no—
356C2×Q8—128no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—34no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—34no—
390C22×C4—32no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4C2×Dic612yes—
415C22×C4C2×Dic612yes—
416C22×C4C2×Dic612yes—
417C22×C4D3×D414yes—
418C22×C4D3×D412yes—
419C22×C4D3×D412yes—
420C22×C4#23712yes—
421C22×C4D3×Q812yes—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—68no—
425C4⋊C4—616no—
426C4⋊C4—68no—
427C4⋊C4—128no—
428C42—68no—
429C42—68no—
430C42—124no—
431C22×C3C2×#12011yescharacteristic
432C22×C3C2×#12111yescharacteristic
433C22×C3C2×#12112yes—
434C22×C3C2×C22⋊Q811yescharacteristic
435C22×C3C2×C22⋊Q811yescharacteristic
436C22×C3C2×#12312yes—
437C22×C3C2×#12411yescharacteristic
438C22×C3C2×C4⋊Q811yescharacteristic
439C22×C3#53211yescharacteristic
440C22×C3#53311yescharacteristic
441C22×C3#53311yescharacteristic
442C22×C3#53611yescharacteristic
443C22×C3#53611yescharacteristic
444C22×C3#53612yes—
445C22×C3#54112yes—
446C22×C3#54812yes—
447C22×C3#54812yes—
448C22×C3D4×Q812yes—
449C22×C3D4×Q812yes—
450C22×C3#55612yes—
451C22×C3#55612yes—
452C22×C3#55712yes—
453C22×C3#55712yes—
454C3×C4—24no—
455C3×C4—24no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4—82no—
459Dic3—24no—
460Dic3—42no—
461Dic3—44no—
462Dic3—42no—
463Dic3—82no—
464Dic3#39418yes—
465C23C22×Dic611yescharacteristic
466C23C2×D3×D412yes—
467C23C2×D3×D411yescharacteristic
468C23C2×D3×D411yescharacteristic
469C23C2×#23711yescharacteristic
470C23C2×D3×Q811yescharacteristic
471C23#100712yes—
472C23D3×#4111yescharacteristic
473C23D3×#4112yes—
474C23D3×#4111yescharacteristic
475C23#101012yes—
476Q8—1232no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—38no—
493C2×C4—34no—
494C2×C4—34no—
495C2×C4—34no—
496C2×C4—38no—
497C2×C4—34no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—64no—
513C2×C4—62no—
514C2×C4—64no—
515C2×C4—62no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—64no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4#87814yes—
541C2×C4#88814yescomplemented
542C2×C3C2×#39411yescharacteristic
543C2×C3#240911yescharacteristic
544C2×C3#240911yescharacteristic
545C2×C3#244612yes—
546C2×C3#244612yes—
547C2×C3#254512yes—
548C2×C3#254612yes—
549C2×C3#256811yescharacteristic
550C2×C3#256811yescharacteristic
551C2×C3#262111yescharacteristic
552C2×C3#285711yescharacteristic
553C22C2×#87811yescharacteristic
554C22C2×#88811yescharacteristic
555C22#551711yescharacteristic
556C22D4×Dic612yes—
557C22#556912yes—
558C22D3×#12011yescharacteristic
559C22#559611yescharacteristic
560C22#559812yes—
561C22D3×#12112yes—
562C22D3×#12111yescharacteristic
563C22#562211yescharacteristic
564C22D3×C22⋊Q811yescharacteristic
565C22#562812yes—
566C22#563312yes—
567C22#564112yes—
568C22#564811yescharacteristic
569C22D3×#12312yes—
570C22D3×#12411yescharacteristic
571C22#568512yes—
572C22#569612yes—
573C22#570212yes—
574C22#572411yescharacteristic
575C22#572712yes—
576C4—24no—
577C4—24no—
578C4—38no—
579C4—44no—
580C4—42no—
581C4—64no—
582C4—82no—
583C4—124no—
584C4—122no—
585C4—122no—
586C4—242no—
587C3#1518011yescharacteristic, complemented, Sylow
588C2#7400111yescharacteristic
589C2#7468911yescharacteristic
590C2#7498612yes—
591C2#7606012yes—
592C2#7654411yescharacteristic
593C2D3×#39411yescharacteristic
594C2#7717211yescharacteristic
595C2#7731512yes—
596C2#7774712yes—
597C2#7781811yescharacteristic
598C2#7818111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1441C2
hypercenter#1441D3
hypocenterC3#15180
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
74. C2×C3×C4⋊C4Q8semidirect
79. C2×C4⋊Dic3Q8semidirect
251. C2×C4⋊C4Dic6semidirect
313. C2×C3×C4C4⋊Q8semidirect
541. C2×C4#888semidirect
587. C3#15180semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×Dic3×C4⋊C42
C2×#46462
C3×#14412
#715672
#715652
#694142
#694122
#694112
Dic3⋊#952
#151803