Finite Group Databasebeta

← #10732629

Subgroups — #10732629

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C2×C3×#390C211yesmaximal, characteristic
3C2×#5221C211yesmaximal, characteristic
4C2×#5221C211yesmaximal, characteristic
5#71563C211yesmaximal, characteristic, complemented
6#71547C214yesmaximal, complemented
7#71546C214yesmaximal, complemented
8#71535C211yesmaximal, characteristic, complemented
9C4×C3⋊#95C211yesmaximal, characteristic, complemented
10#15229—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2214yes—
14C22×C3×C42C2211yescharacteristic
15C22×#217C2214yes—
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C2×#890—24no—
25C2×#890—24no—
26C2×#890—24no—
27C3×#390—24no—
28#5221—24no—
29#5221—24no—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2214yes—
34C22×#125—31no—
35C2×#390—31no—
36C2×#390—31no—
37C2×#390D311yescharacteristic
38#1439—34no—
39#1439—34no—
40C4⋊#95—31no—
41C4⋊#95—31no—
42C4×#95—31no—
43C25×C3C2312yes—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4Q814yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2312yes—
55C23×Dic3C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4D414yes—
62C2×C3×C22⋊C4—28no—
63C2×C3×C22⋊C4—28no—
64C2×C3×C22⋊C4—44no—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×C3×C42—22no—
68C2×#217—28no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—28no—
72C2×#217—44no—
73C2×#217—44no—
74C2×#217—44no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3D414yes—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3D414yes—
93#890—232no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—44no—
97C3⋊#95—44no—
98C23×D4C2×D311yescharacteristic
99C2×#125—64no—
100C2×#125—64no—
101C2×#125—64no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—32no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C22⋊C4—34no—
108C22×C22⋊C4—32no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4C2×D314yes—
111C22×C42—31no—
112C22×C42—31no—
113C22×C42C2×D311yescharacteristic
114#390—24no—
115#390—64no—
116#390—64no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—34no—
120C2×#95—34no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—24no—
125C24×C3—24no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—24no—
130C24×C3—24no—
131C24×C3C2×Q814yes—
132C24×C3C2411yescharacteristic
133C2×C3×D4—28no—
134C2×C3×D4—28no—
135C2×C3×D4—28no—
136C2×C3×D4—44no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D414yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×Q812yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4114yes—
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3C2×D412yes—
183C22×Dic3C2×D412yes—
184C22×Dic3C2×D414yes—
185C22×Dic3#4114yes—
186C3×C4⋊C4—28no—
187C3×C22⋊C4—416no—
188C3×C42—44no—
189#217—416no—
190#217—48no—
191#217—48no—
192C4⋊Dic3—216no—
193C4⋊Dic3—28no—
194C4⋊Dic3—44no—
195Dic3⋊C4—216no—
196C4×Dic3—28no—
197C4×Dic3—44no—
198C25C22×D312yes—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4Dic614yes—
206C23×C4—32no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—32no—
210C23×C4—32no—
211C23×C4C22×D312yes—
212C23×C4C22×D312yes—
213C23×C4C22×D311yescharacteristic
214#125—632no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—34no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4#6714yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—68no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—68no—
237C2×C22⋊C4—124no—
238C2×C22⋊C4—124no—
239C2×C22⋊C4—124no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—34no—
244C2×C42—62no—
245C2×C42—62no—
246C2×C42—62no—
247C2×C42—62no—
248C2×C42—62no—
249#95—64no—
250#95—64no—
251#95—124no—
252#95—124no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—28no—
257C23×C3—28no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—44no—
263C23×C3—44no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—44no—
269C23×C3—44no—
270C23×C3—44no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—44no—
275C23×C3—44no—
276C23×C3C22×D411yescharacteristic
277C23×C3C22×D411yescharacteristic
278C23×C3C22×D411yescharacteristic
279C23×C3C22×Q811yescharacteristic
280C23×C3C2×#4111yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4112yes—
284C23×C3C2×#4112yes—
285C23×C3H(5,𝔽2)12yes—
286C23×C3H(5,𝔽2)12yes—
287C3×D4—416no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—28no—
292C2×C3×C4—24no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—44no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4#12014yes—
301C2×C3×C4C22⋊Q814yes—
302C2×C3×C4C22⋊Q814yes—
303C2×C3×C4C22⋊Q814yes—
304C2×C3×C4#12414yes—
305C2×C3×C4#12414yes—
306C2×C3×C4#12414yes—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—24no—
316C2×Dic3—28no—
317C2×Dic3—24no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—44no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24C2×Dic614yes—
345C24C23×D311yescharacteristic
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4C2×Dic612yes—
391C22×C4#23512yes—
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C22×C4#23714yes—
395C22×C4#23712yes—
396C22×C4#23712yes—
397C22×C4C2×#6714yes—
398C22×C4C2×#6712yes—
399C4⋊C4—28no—
400C4⋊C4—616no—
401C4⋊C4—616no—
402C4⋊C4—68no—
403C4⋊C4—124no—
404C22⋊C4—416no—
405C22⋊C4—1216no—
406C22⋊C4—128no—
407C22⋊C4—128no—
408C42—44no—
409C42—68no—
410C42—124no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—48no—
415C22×C3—48no—
416C22×C3—82no—
417C22×C3—82no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—84no—
421C22×C3—84no—
422C22×C3C2×#12011yescharacteristic
423C22×C3C2×C22⋊Q811yescharacteristic
424C22×C3C2×C22⋊Q811yescharacteristic
425C22×C3C2×C22⋊Q811yescharacteristic
426C22×C3C2×#12411yescharacteristic
427C22×C3C2×#12411yescharacteristic
428C22×C3C2×#12411yescharacteristic
429C22×C3#52912yes—
430C22×C3#52912yes—
431C22×C3#53211yescharacteristic
432C22×C3#53211yescharacteristic
433C22×C3#53511yescharacteristic
434C22×C3#53511yescharacteristic
435C22×C3#54012yes—
436C22×C3#54012yes—
437C22×C3#54612yes—
438C22×C3#54612yes—
439C22×C3#55012yes—
440C22×C3#55012yes—
441C22×C3#55312yes—
442C22×C3#55312yes—
443C22×C3#55412yes—
444C22×C3#55412yes—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4#39318yes—
448Dic3—42no—
449Dic3—44no—
450Dic3—42no—
451Dic3—44no—
452Dic3—82no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23C22×Dic611yescharacteristic
477C23C2×#23511yescharacteristic
478C23C2×D3×D411yescharacteristic
479C23C2×D3×D411yescharacteristic
480C23C2×#23712yes—
481C23C2×#23711yescharacteristic
482C23C2×#23711yescharacteristic
483C23#100412yes—
484C23D3×#4112yes—
485C23#100912yes—
486C23C22×#6711yescharacteristic
487D4—416no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—68no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—124no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—124no—
526C2×C4—124no—
527C2×C4#87814yes—
528C2×C4#87814yes—
529C2×C4#88514yes—
530C2×C4#88514yes—
531C2×C4#92414yes—
532C2×C4#93614yes—
533C2×C4#93714yes—
534C2×C3—84no—
535C2×C3C2×#39311yescharacteristic
536C2×C3#243111yescharacteristic
537C2×C3#243111yescharacteristic
538C2×C3#245412yes—
539C2×C3#245412yes—
540C2×C3#249012yes—
541C2×C3#249012yes—
542C2×C3#256611yescharacteristic
543C2×C3#256611yescharacteristic
544C2×C3#264111yescharacteristic
545C2×C3#286111yescharacteristic
546C22—48no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—48no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—84no—
555C22—84no—
556C22—82no—
557C22C2×#87811yescharacteristic
558C22C2×#87811yescharacteristic
559C22C2×#88511yescharacteristic
560C22C2×#88511yescharacteristic
561C22#553912yes—
562C22#554312yes—
563C22#555712yes—
564C22#556212yes—
565C22#560012yes—
566C22#561412yes—
567C22#561812yes—
568C22#561912yes—
569C22#566811yescharacteristic
570C22#567411yescharacteristic
571C22#567612yes—
572C22#571511yescharacteristic
573C22#571611yescharacteristic
574C22#571812yes—
575C22C2×#92411yescharacteristic
576C22C2×#93611yescharacteristic
577C22C2×#93711yescharacteristic
578C22#580612yes—
579C22#583412yes—
580C4—44no—
581C4—44no—
582C4—122no—
583C4—124no—
584C4—124no—
585C4—122no—
586C4—242no—
587C4#494718yes—
588C3#1522911yescharacteristic, complemented, Sylow
589C2—84no—
590C2#7632512yes—
591C2#7636211yescharacteristic
592C2#7636311yescharacteristic
593C2#7637411yescharacteristic
594C2#7642611yescharacteristic
595C2#7644912yes—
596C2C2×#494711yescharacteristic
597C2#8236312yes—
598C2#8266112yes—
599C2#8295811yescharacteristic
600C2#8318711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#15229
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
5. #71563C2semidirect
6. #71547C2semidirect
7. #71546C2semidirect
8. #71535C2semidirect
9. C4×C3⋊#95C2semidirect
588. C3#15229semidirect

Maximal subgroups

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

subgroupindex
C22×#8902
C2×C3×#3902
C2×#52212
#715632
#715472
#715462
#715352
C4×C3⋊#952
#152293