Finite Group Databasebeta

← #10740370

Subgroups — #10740370

Order 768.

1586 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic
3C2×#4985C211yesmaximal, characteristic
4C2×#4983C211yesmaximal, characteristic
5#72134C211yesmaximal, characteristic
6#71536C211yesmaximal, characteristic
7#71536C211yesmaximal, characteristic
8#69424C212yesmaximal
9#69414C212yesmaximal
10#69409C212yesmaximal
11#69406C212yesmaximal
12#15783—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×#944—24no—
18C2×#944—24no—
19C2×#944—24no—
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C3×#391—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4985—24no—
41#4983—24no—
42C22×C4⋊Q8—31no—
43C2×#400—31no—
44C2×#398—31no—
45C2×#391D311yescharacteristic
46#1446—32no—
47#1446—31no—
48#1442—32no—
49#1441—32no—
50#1436—31no—
51#1436—31no—
52#1436—32no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—24no—
56C22×C3×Q8—42no—
57C22×C3×Q8D414yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—22no—
72C22×Dic6—24no—
73C22×Dic6—42no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—42no—
79#944—232no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4D414yes—
94C2×Dic3⋊C4D414yes—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C3×#95—24no—
99C3×#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C3⋊#95Q818yescomplemented
111C23×Q8—31no—
112C23×Q8C2×D311yescharacteristic
113C2×C4⋊Q8—64no—
114C2×C4⋊Q8—64no—
115C2×C4⋊Q8—64no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C42—31no—
123C22×C42C2×D311yescharacteristic
124#400—64no—
125#398—64no—
126#391—24no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95C2×D312yes—
139C2×#95C2×D312yes—
140C24×C3C2411yescharacteristic
141C2×C3×Q8—28no—
142C2×C3×Q8—28no—
143C2×C3×Q8—44no—
144C2×C3×Q8—48no—
145C2×C3×Q8—48no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q814yes—
174C22×C3×C4#4114yes—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3C2×D412yes—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×Q814yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3#4114yes—
209C2×Dic6—28no—
210C2×Dic6—48no—
211C2×Dic6—48no—
212C2×Dic6—44no—
213C2×Dic6—44no—
214C3×C4⋊C4—44no—
215C3×C42—44no—
216Dic3⋊C4—28no—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219Dic3⋊C4—28no—
220Dic3⋊C4—48no—
221C4×Dic3—28no—
222C22×Q8—24no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—42no—
226C22×Q8—62no—
227C22×Q8—62no—
228C22×Q8—64no—
229C22×Q8—62no—
230C22×Q8—122no—
231C22×Q8#6714yes—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4C22×D312yes—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D311yescharacteristic
243C4⋊Q8—632no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C42—22no—
263C2×C42—42no—
264C2×C42—62no—
265C2×C42—62no—
266C2×C42—62no—
267#95—24no—
268#95—24no—
269#95—38no—
270#95—64no—
271#95—64no—
272#95—64no—
273#95—64no—
274#95—64no—
275#95—62no—
276#95—62no—
277#95—122no—
278#95—122no—
279#95—122no—
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C22×Q812yes—
284C23×C3C2×#4112yes—
285C23×C3C2×#4112yes—
286C23×C3H(5,𝔽2)12yes—
287C23×C32−1+412yes—
288C23×C32−1+412yes—
289C3×Q8—416no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
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×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4#12014yes—
318C2×C3×C4#12314yes—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3#12414yes—
352C2×Dic3C4⋊Q814yes—
353Dic6—416no—
354C24C23×D311yescharacteristic
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—68no—
361C2×Q8—128no—
362C2×Q8—128no—
363C2×Q8—124no—
364C2×Q8—124no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—34no—
390C22×C4—34no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—34no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4#23514yes—
422C22×C4D3×D412yes—
423C22×C4D3×D412yes—
424C22×C4D3×Q814yes—
425C22×C4C2×#6712yes—
426C22×C4C2×#6712yes—
427C22×C4C2×#6712yes—
428C4⋊C4—44no—
429C4⋊C4—68no—
430C4⋊C4—68no—
431C4⋊C4—68no—
432C4⋊C4—68no—
433C4⋊C4—128no—
434C42—44no—
435C42—68no—
436C22×C3C2×#12011yescharacteristic
437C22×C3C2×#12311yescharacteristic
438C22×C3C2×#12411yescharacteristic
439C22×C3C2×C4⋊Q811yescharacteristic
440C22×C3#53312yes—
441C22×C3#53411yescharacteristic
442C22×C3#53611yescharacteristic
443C22×C3#53611yescharacteristic
444C22×C3#54112yes—
445C22×C3#54212yes—
446C22×C3#54412yes—
447C22×C3#54612yes—
448C22×C3#54612yes—
449C22×C3#54812yes—
450C22×C3#54812yes—
451C22×C3D4×Q812yes—
452C22×C3D4×Q812yes—
453C22×C3#55412yes—
454C22×C3#55612yes—
455C22×C3#55712yes—
456C22×C3#56312yes—
457C3×C4—24no—
458C3×C4—24no—
459C3×C4—42no—
460C3×C4—44no—
461C3×C4—82no—
462Dic3—24no—
463Dic3—24no—
464Dic3—42no—
465Dic3—42no—
466Dic3—44no—
467Dic3—82no—
468C23C2×#23512yes—
469C23C2×D3×D411yescharacteristic
470C23C2×D3×D411yescharacteristic
471C23#100412yes—
472C23C2×D3×Q812yes—
473C23#100712yes—
474C23D3×#4112yes—
475C23#101012yes—
476C23C22×#6711yescharacteristic
477Q8—416no—
478Q8—1216no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—34no—
494C2×C4—34no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—62no—
509C2×C4—64no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—64no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
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#92914yes—
541C2×C4#93714yes—
542C2×C3#240511yescharacteristic
543C2×C3#243411yescharacteristic
544C2×C3#243611yescharacteristic
545C2×C3#244612yes—
546C2×C3#249212yes—
547C2×C3#255012yes—
548C2×C3#271312yes—
549C2×C3#271612yes—
550C2×C3#272912yes—
551C22#549012yes—
552C22#549012yes—
553C22#556712yes—
554C22#556812yes—
555C22#557812yes—
556C22#558312yes—
557C22#562312yes—
558C22#563612yes—
559C22#564012yes—
560C22#564612yes—
561C22D3×#12411yescharacteristic
562C22#567511yescharacteristic
563C22#567712yes—
564C22D3×C4⋊Q811yescharacteristic
565C22#572411yescharacteristic
566C22#572712yes—
567C22C2×#92911yescharacteristic
568C22C2×#93711yescharacteristic
569C22#580411yescharacteristic
570C22Q8×#6712yes—
571C22#583112yes—
572C4—24no—
573C4—24no—
574C4—44no—
575C4—42no—
576C4—64no—
577C4—64no—
578C4—82no—
579C4—122no—
580C4—122no—
581C4—124no—
582C4—242no—
583C3#1578311yescharacteristic, complemented, Sylow
584C2#7651712yes—
585C2#7653712yes—
586C2#7656812yes—
587C2#7657012yes—
588C2#8212611yescharacteristic
589C2#8246812yes—
590C2#8280512yes—
591C2#8296111yescharacteristic
592C2#8323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#391C2
hypercenterC2×#391D3
hypocenterC3#15783
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
110. C3⋊#95Q8semidirect
583. C3#15783semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×C3×#3912
C2×#49852
C2×#49832
#721342
#715362
#694242
#694142
#694092
#694062
#157833