Finite Group Databasebeta

← #10732574

Subgroups — #10732574

Order 768.

1838 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic, complemented
2C2×#4988C211yesmaximal, characteristic
3C2×#4988C211yesmaximal, characteristic
4C2×#4977C211yesmaximal, characteristic, complemented
5C2×#4939C211yesmaximal, characteristic
6C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#71570C214yesmaximal, complemented
8#71563C211yesmaximal, characteristic, complemented
9#69427C214yesmaximal, complemented
10#15222—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2214yes—
18C22×(C2×D3)⋊C4C2214yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C3×#389—24no—
25C2×C3×#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29#4988—24no—
30#4988—24no—
31#4977—24no—
32#4939—24no—
33C4×C4⋊Dic3—24no—
34C2×#390—31no—
35C2×#390—31no—
36C2×#390—31no—
37C2×#389D311yescharacteristic, complemented
38C2×#382—31no—
39C2×C4×C4⋊C4—31no—
40#1439—34no—
41#1439—34no—
42C4⋊#95—31no—
43C24×D3C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C22×D12—22no—
53C22×D12—22no—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12—22no—
57C22×D12—22no—
58C22×D12Q814yes—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—44no—
66C2×C3×C4⋊C4D414yes—
67C2×C3×C42—22no—
68C2×C3×C42—22no—
69C2×C3×C42D414yes—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—28no—
73C2×(C2×D3)⋊C4—28no—
74C2×(C2×D3)⋊C4—28no—
75C2×(C2×D3)⋊C4—28no—
76C2×(C2×D3)⋊C4—44no—
77C2×(C2×D3)⋊C4—44no—
78C2×(C2×D3)⋊C4—44no—
79C2×(C2×D3)⋊C4—44no—
80C2×C4⋊Dic3—24no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3D414yes—
92C3×#95—24no—
93C3⋊#95—28no—
94C3⋊#95—24no—
95C3⋊#95—44no—
96C23×D4—31no—
97C22×C4⋊C4—31no—
98C22×C4⋊C4—32no—
99C22×C4⋊C4C2×D311yescharacteristic
100C22×C4⋊C4C2×D311yescharacteristic
101C22×C4⋊C4C2×D312yes—
102C22×C22⋊C4—32no—
103C22×C22⋊C4—34no—
104C22×C22⋊C4—34no—
105C22×C22⋊C4—32no—
106C22×C42—31no—
107C22×C42—31no—
108C22×C42C2×D311yescharacteristic
109#390—64no—
110#390—64no—
111#390—64no—
112#389—24no—
113#382—64no—
114C4×C4⋊C4—64no—
115C2×#95—32no—
116C2×#95—34no—
117C2×#95—34no—
118C2×#95C2×D312yes—
119C24×C3C2411yescharacteristic
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—22no—
123C23×D3—24no—
124C23×D3—24no—
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—22no—
128C23×D3—24no—
129C23×D3—24no—
130C23×D3C2×Q814yes—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4C2×D414yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×Q812yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4114yes—
157C22×C3×C4#4112yes—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
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—22no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—24no—
175C22×Dic3C2×D414yes—
176C22×Dic3#4114yes—
177C2×D12—28no—
178C2×D12—28no—
179C2×D12—28no—
180C2×D12—44no—
181C2×D12—44no—
182C2×D12—44no—
183C2×D12—44no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—48no—
186C3×C4⋊C4—44no—
187C3×C42—28no—
188(C2×D3)⋊C4—416no—
189(C2×D3)⋊C4—416no—
190(C2×D3)⋊C4—48no—
191(C2×D3)⋊C4—48no—
192C4⋊Dic3—216no—
193C4⋊Dic3—44no—
194C4×Dic3—28no—
195C4×Dic3—44no—
196C25—32no—
197C22×D4—34no—
198C22×D4—62no—
199C22×D4—62no—
200C22×D4—62no—
201C22×D4—62no—
202C22×D4—62no—
203C22×D4—62no—
204C23×C4—32no—
205C23×C4—32no—
206C23×C4—32no—
207C23×C4C22×D312yes—
208C23×C4C22×D312yes—
209C23×C4C22×D311yescharacteristic
210C23×C4C22×D311yescharacteristic
211C23×C4C22×D311yescharacteristic
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4D1214yes—
226C2×C22⋊C4—68no—
227C2×C22⋊C4—68no—
228C2×C22⋊C4—68no—
229C2×C22⋊C4—68no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—124no—
233C2×C22⋊C4—124no—
234C2×C22⋊C4—124no—
235C2×C22⋊C4—124no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—34no—
239C2×C42—62no—
240C2×C42—62no—
241C2×C42—62no—
242C2×C42—62no—
243C2×C42—62no—
244C2×C42#6714yes—
245#95—24no—
246#95—64no—
247#95—68no—
248#95—124no—
249C23×C3C22×D411yescharacteristic
250C23×C3C22×D411yescharacteristic
251C23×C3C22×D411yescharacteristic
252C23×C3C22×Q811yescharacteristic
253C23×C3C2×#4111yescharacteristic
254C23×C3C2×#4111yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4112yes—
257C23×C3C2×#4112yes—
258C23×C3H(5,𝔽2)12yes—
259C23×C3H(5,𝔽2)12yes—
260C22×D3—24no—
261C22×D3—24no—
262C22×D3—24no—
263C22×D3—28no—
264C22×D3—28no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—44no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—44no—
272C22×D3—42no—
273C22×D3—42no—
274C22×D3—44no—
275C22×D3—42no—
276C22×D3—44no—
277C22×D3—44no—
278C22×D3—44no—
279C22×D3—44no—
280C22×D3—44no—
281C22×D3—44no—
282C22×D3—44no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—28no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—44no—
302C2×C3×C4#12114yes—
303C2×C3×C4#12114yes—
304C2×C3×C4#12114yes—
305C2×C3×C4C22⋊Q814yes—
306C2×C3×C4C22⋊Q814yes—
307C2×C3×C4C22⋊Q814yes—
308C2×C3×C4#12414yes—
309C2×C3×C4#12614yes—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—42no—
317C2×Dic3—42no—
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—44no—
329D12—416no—
330C24—34no—
331C24—62no—
332C24—62no—
333C24—62no—
334C24—62no—
335C24—64no—
336C24—64no—
337C24—62no—
338C24—64no—
339C24—64no—
340C24—62no—
341C24C23×D311yescharacteristic
342C2×D4—68no—
343C2×D4—68no—
344C2×D4—68no—
345C2×D4—124no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4—124no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—64no—
382C22×C4—64no—
383C22×C4—62no—
384C22×C4C2×D1214yes—
385C22×C4C2×D1212yes—
386C22×C4#23512yes—
387C22×C4D3×D412yes—
388C22×C4#23712yes—
389C22×C4D3×Q812yes—
390C22×C4#23914yes—
391C22×C4#23912yes—
392C22×C4C2×#6712yes—
393C22×C4C2×#6712yes—
394C22×C4C2×#6712yes—
395C4⋊C4—28no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—616no—
399C4⋊C4—124no—
400C22⋊C4—1216no—
401C22⋊C4—128no—
402C22⋊C4—1216no—
403C22⋊C4—128no—
404C42—28no—
405C42—68no—
406C42—124no—
407C22×C3C2×#12111yescharacteristic
408C22×C3C2×#12111yescharacteristic
409C22×C3C2×#12111yescharacteristic
410C22×C3C2×C22⋊Q811yescharacteristic
411C22×C3C2×C22⋊Q811yescharacteristic
412C22×C3C2×C22⋊Q811yescharacteristic
413C22×C3#52512yes—
414C22×C3C2×#12411yescharacteristic
415C22×C3C2×#12611yescharacteristic
416C22×C3#52912yes—
417C22×C3#53211yescharacteristic
418C22×C3#53511yescharacteristic
419C22×C3#53511yescharacteristic
420C22×C3#53512yes—
421C22×C3#54012yes—
422C22×C3#54612yes—
423C22×C3#55012yes—
424C22×C3#55312yes—
425C22×C3#55412yes—
426C22×C3#55412yes—
427C22×C3#55512yes—
428C22×C3#55912yes—
429C22×C3#55912yes—
430C2×D3—44no—
431C2×D3—44no—
432C2×D3—48no—
433C2×D3—44no—
434C2×D3—48no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—84no—
438C2×D3—84no—
439C2×D3—82no—
440C2×D3—82no—
441C3×C4—24no—
442C3×C4—42no—
443C3×C4—44no—
444C3×C4—82no—
445C3×C4#39618yes—
446Dic3—44no—
447Dic3—44no—
448Dic3—82no—
449C23—68no—
450C23—68no—
451C23—64no—
452C23—64no—
453C23—64no—
454C23—124no—
455C23—124no—
456C23—122no—
457C23—122no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—124no—
462C23—122no—
463C23—124no—
464C23—122no—
465C23—122no—
466C23—124no—
467C23—124no—
468C23—124no—
469C23—124no—
470C23—124no—
471C23—124no—
472C23C22×D1211yescharacteristic
473C23C2×#23511yescharacteristic
474C23C2×D3×D411yescharacteristic
475C23C2×#23711yescharacteristic
476C23#100412yes—
477C23C2×D3×Q811yescharacteristic
478C23C2×#23911yescharacteristic
479C23C2×#23912yes—
480C23D3×#4112yes—
481C23#100912yes—
482C23C22×#6711yescharacteristic
483D4—1216no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—122no—
510C2×C4—122no—
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—124no—
522C2×C4#89514yes—
523C2×C4#89614yes—
524C2×C4#89714yes—
525C2×C4#89814yes—
526C2×C4#93014yes—
527C2×C4#93914yes—
528C2×C4#94614yes—
529C2×C4#94714yes—
530C2×C3C2×#39611yescharacteristic
531C2×C3#241412yes—
532C2×C3#245412yes—
533C2×C3#251312yes—
534C2×C3#254411yescharacteristic
535C2×C3#254811yescharacteristic
536C2×C3#256311yescharacteristic
537C2×C3#256611yescharacteristic
538C2×C3#258012yes—
539C2×C3#261911yescharacteristic
540C2×C3#285011yescharacteristic
541D3—84no—
542C22—124no—
543C22—128no—
544C22—124no—
545C22—124no—
546C22—128no—
547C22—244no—
548C22—242no—
549C22—242no—
550C22—244no—
551C22—242no—
552C22—242no—
553C22C2×#89511yescharacteristic
554C22C2×#89611yescharacteristic
555C22C2×#89711yescharacteristic
556C22C2×#89811yescharacteristic
557C22#555412yes—
558C22#555912yes—
559C22#557712yes—
560C22#558512yes—
561C22#560712yes—
562C22#561912yes—
563C22#562412yes—
564C22#562712yes—
565C22#569012yes—
566C22#569512yes—
567C22#573012yes—
568C22#573412yes—
569C22C2×#93011yescharacteristic
570C22C2×#93911yescharacteristic
571C22#580511yescharacteristic
572C22C2×#94611yescharacteristic
573C22C2×#94711yescharacteristic
574C22#583011yescharacteristic
575C22#583211yescharacteristic
576C4—24no—
577C4—44no—
578C4—42no—
579C4—82no—
580C4—124no—
581C4—124no—
582C4—242no—
583C4#499418yes—
584C3#1522211yescharacteristic, complemented, Sylow
585C2—244no—
586C2#7601612yes—
587C2#7604812yes—
588C2#7607512yes—
589C2#7620912yes—
590C2C2×#499411yescharacteristic
591C2#8238111yescharacteristic
592C2#8247611yescharacteristic
593C2#8267611yescharacteristic
594C2#8280911yescharacteristic
595C2#8304011yescharacteristic
596C2#8326011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#389C2
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3#15222
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C2×C3×#389C2semidirect
4. C2×#4977C2semidirect
6. C2×C4×C4⋊Dic3C2semidirect
7. #71570C2semidirect
8. #71563C2semidirect
9. #69427C2semidirect
37. C2×#389D3semidirect
584. C3#15222semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49882
C2×#49772
C2×#49392
C2×C4×C4⋊Dic32
#715702
#715632
#694272
#152223