Finite Group Databasebeta

← #10732154

Subgroups — #10732154

Order 768.

2826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic, complemented
2C2×#5223C212yesmaximal, complemented
3C2×#4993C212yesmaximal, complemented
4C2×#4960C212yesmaximal, complemented
5C2×#4949C212yesmaximal, complemented
6C2×#4957C214yesmaximal, complemented
7C2×#4939C211yesmaximal, characteristic, complemented
8C4×C3⋊#95C211yesmaximal, characteristic, complemented
9#15172—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C23×#67C2214yes—
12C23×D12C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2214yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2214yes—
23C22×C4×Dic3C2212yes—
24C3×#390—24no—
25#5223—28no—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28#4993—28no—
29#4960—28no—
30#4957—216no—
31#4949—28no—
32#4939—24no—
33C2×#399—32no—
34C2×#399—32no—
35C2×#392—32no—
36C2×#392—32no—
37C2×#390—31no—
38C2×#390—34no—
39C2×#390D311yescharacteristic, complemented
40C4×#95—31no—
41C25×C3C2312yes—
42C24×D3C2312yes—
43C22×C3×D4—24no—
44C22×C3×D4—22no—
45C22×C3×D4—24no—
46C22×C3×D4—42no—
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×#67—28no—
52C22×#67—28no—
53C22×#67—28no—
54C22×#67—28no—
55C22×#67—28no—
56C22×#67—44no—
57C22×#67—44no—
58C23×Dic3C2312yes—
59C23×Dic3C2314yes—
60C22×D12—24no—
61C22×D12—22no—
62C22×D12—24no—
63C22×D12—42no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—28no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C42—42no—
72C2×C3×C42D414yescomplemented
73C2×#217—28no—
74C2×#217—28no—
75C2×#217—28no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—28no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—28no—
82C2×(C2×D3)⋊C4—28no—
83C2×(C2×D3)⋊C4—28no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—24no—
86C2×Dic3⋊C4—28no—
87C2×Dic3⋊C4—28no—
88C2×Dic3⋊C4—28no—
89C2×C4×Dic3—42no—
90C2×C4×Dic3—42no—
91C2×C4×Dic3D418yescomplemented
92C3⋊#95D418yescomplemented
93C3⋊#95D418yescomplemented
94C23×D4—34no—
95C23×D4—31no—
96C23×D4C2×D311yescharacteristic
97C22×C4⋊C4—34no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4C2×D311yescharacteristic
100C22×C22⋊C4—34no—
101C22×C22⋊C4—34no—
102C22×C22⋊C4—32no—
103C22×C22⋊C4—32no—
104C22×C22⋊C4C2×D312yes—
105C22×C22⋊C4C2×D312yes—
106C22×C42—32no—
107C22×C42C2×D311yescharacteristic
108#399—68no—
109#399—68no—
110#392—68no—
111#392—68no—
112#390—24no—
113#390—616no—
114#390—64no—
115C2×#95—32no—
116C2×#95—32no—
117C24×C3—22no—
118C24×C3—24no—
119C24×C3—24no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—24no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—24no—
126C24×C3—22no—
127C24×C3#4114yes—
128C24×C3C2411yescharacteristic
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—24no—
132C23×D3—24no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—24no—
136C23×D3—24no—
137C23×D3—22no—
138C23×D3—22no—
139C23×D3#4114yes—
140C2×C3×D4—28no—
141C2×C3×D4—416no—
142C2×C3×D4—44no—
143C2×C3×D4—44no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D414yes—
157C22×C3×C4C2×D414yes—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C2×#67—232no—
163C2×#67—416no—
164C2×#67—416no—
165C2×#67—416no—
166C2×#67—416no—
167C2×#67—416no—
168C2×#67—416no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—24no—
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—24no—
175C22×Dic3—24no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3C2×D418yes—
179C22×Dic3C2×D418yes—
180C22×Dic3C2×D418yes—
181C2×D12—28no—
182C2×D12—416no—
183C2×D12—44no—
184C2×D12—44no—
185C3×C4⋊C4—44no—
186C3×C22⋊C4—48no—
187C3×C22⋊C4—48no—
188C3×C42—44no—
189#217—416no—
190(C2×D3)⋊C4—48no—
191(C2×D3)⋊C4—48no—
192(C2×D3)⋊C4—416no—
193C4⋊Dic3—44no—
194Dic3⋊C4—416no—
195C4×Dic3—48no—
196C25—32no—
197C25C22×D312yes—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—42no—
202C22×D4—64no—
203C22×D4—68no—
204C22×D4—68no—
205C22×D4—68no—
206C22×D4—68no—
207C22×D4—68no—
208C22×D4—64no—
209C22×D4—62no—
210C22×D4—122no—
211C22×D4—124no—
212C22×D4—124no—
213C23×C4—34no—
214C23×C4—32no—
215C23×C4C22×D312yes—
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D311yescharacteristic
218C23×C4C22×D311yescharacteristic
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—68no—
222C2×C4⋊C4—68no—
223C2×C4⋊C4—68no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—62no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—68no—
232C2×C22⋊C4—68no—
233C2×C22⋊C4—68no—
234C2×C22⋊C4—68no—
235C2×C22⋊C4—68no—
236C2×C22⋊C4—68no—
237C2×C22⋊C4—68no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C42—38no—
243C2×C42—42no—
244C2×C42—122no—
245C2×C42—122no—
246C2×C42#6714yescomplemented
247#95—38no—
248#95—38no—
249C23×C3—28no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—28no—
253C23×C3—24no—
254C23×C3—44no—
255C23×C3—42no—
256C23×C3—44no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—44no—
260C23×C3—42no—
261C23×C3—44no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—44no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—42no—
269C23×C3—44no—
270C23×C3—44no—
271C23×C3—42no—
272C23×C3C22×D412yes—
273C23×C3C22×D412yes—
274C23×C3C22×D412yes—
275C23×C3C22×D411yescharacteristic
276C23×C3C2×#4112yes—
277C23×C3C2×#4112yes—
278C23×C3H(5,𝔽2)12yes—
279C23×C3H(5,𝔽2)11yescharacteristic
280C23×C32−1+411yescharacteristic
281C22×D3—24no—
282C22×D3—28no—
283C22×D3—28no—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—42no—
287C22×D3—44no—
288C22×D3—44no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—44no—
293C22×D3—42no—
294C22×D3—44no—
295C22×D3—42no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3—44no—
299C22×D3—44no—
300C22×D3—42no—
301C22×D3—44no—
302C22×D3—42no—
303C22×D3—44no—
304C3×D4—416no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—28no—
314C2×C3×C4—28no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4#12714yes—
321C2×C3×C4#12714yes—
322C2×C3×C4#12714yes—
323#67—464no—
324C2×Dic3—28no—
325C2×Dic3—28no—
326C2×Dic3—28no—
327C2×Dic3—28no—
328C2×Dic3—28no—
329C2×Dic3—28no—
330C2×Dic3—44no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—44no—
335C2×Dic3#127116yes—
336D12—416no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—34no—
348C24—64no—
349C24—62no—
350C24—64no—
351C24—62no—
352C24—64no—
353C24—64no—
354C24—62no—
355C24—62no—
356C24—62no—
357C24—62no—
358C24#23514yes—
359C24C23×D311yescharacteristic
360C2×D4—28no—
361C2×D4—416no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—68no—
365C2×D4—632no—
366C2×D4—1216no—
367C2×D4—124no—
368C2×D4—1216no—
369C2×D4—1216no—
370C2×D4—1216no—
371C2×D4—1216no—
372C2×D4—1216no—
373C2×D4—1216no—
374C2×D4—124no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—38no—
388C22×C4—38no—
389C22×C4—38no—
390C22×C4—64no—
391C22×C4—64no—
392C22×C4—64no—
393C22×C4—64no—
394C22×C4—62no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4D3×D414yes—
400C22×C4D3×D414yes—
401C22×C4D3×D414yes—
402C22×C4C2×#6712yes—
403C22×C4C2×#6712yes—
404C22×C4C2×#6712yes—
405C4⋊C4—44no—
406C4⋊C4—1216no—
407C4⋊C4—124no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—1216no—
411C22⋊C4—1216no—
412C22⋊C4—128no—
413C22⋊C4—128no—
414C42—44no—
415C42—128no—
416C22×C3—44no—
417C22×C3—48no—
418C22×C3—44no—
419C22×C3—48no—
420C22×C3—44no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—84no—
425C22×C3—82no—
426C22×C3—84no—
427C22×C3#52511yescharacteristic
428C22×C3#52514yes—
429C22×C3#52511yescharacteristic
430C22×C3C2×#12714yes—
431C22×C3C2×#12711yescharacteristic
432C22×C3C2×#12711yescharacteristic
433C22×C3C2×#12711yescharacteristic
434C22×C3#53411yescharacteristic
435C22×C3#53412yes—
436C22×C3#53711yescharacteristic
437C22×C3#53712yes—
438C22×C3#54612yes—
439C22×C3#54614yes—
440C22×C3#54612yes—
441C22×C3#54614yes—
442C22×C3#54812yes—
443C22×C3#54812yes—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—48no—
447C2×D3—44no—
448C2×D3—48no—
449C2×D3—82no—
450C2×D3—82no—
451C2×D3—84no—
452C2×D3—82no—
453C2×D3—84no—
454C2×D3—82no—
455C3×C4—24no—
456C3×C4—24no—
457C3×C4—24no—
458C3×C4—82no—
459Dic3—216no—
460Dic3—82no—
461C23—28no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—64no—
485C23—64no—
486C23—68no—
487C23—64no—
488C23—68no—
489C23—122no—
490C23—122no—
491C23—124no—
492C23—124no—
493C23—124no—
494C23—124no—
495C23—122no—
496C23—124no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23—124no—
502C23—124no—
503C23—122no—
504C23—124no—
505C23—124no—
506C23—124no—
507C23C2×#23512yes—
508C23C2×D3×D412yes—
509C23C2×D3×D412yes—
510C23C2×D3×D412yes—
511C23#100412yes—
512C23D3×#4112yes—
513C23#100911yescharacteristic
514C23#101011yescharacteristic
515C23C22×#6711yescharacteristic
516D4—416no—
517D4—1216no—
518D4—1264no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—316no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—68no—
536C2×C4—68no—
537C2×C4—68no—
538C2×C4—68no—
539C2×C4—68no—
540C2×C4—68no—
541C2×C4—124no—
542C2×C4—124no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4#94014yes—
547C2×C4#94014yes—
548C2×C4#94014yes—
549C2×C3—84no—
550C2×C3#239711yescharacteristic
551C2×C3#253911yescharacteristic
552C2×C3#253914yes—
553C2×C3#253911yescharacteristic
554C2×C3#254011yescharacteristic
555C2×C3#254014yes—
556C2×C3#254011yescharacteristic
557C2×C3#265611yescharacteristic
558C2×C3#286711yescharacteristic
559D3—84no—
560C22—48no—
561C22—48no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—84no—
569C22—82no—
570C22—82no—
571C22—124no—
572C22—124no—
573C22—128no—
574C22—128no—
575C22—124no—
576C22—244no—
577C22—244no—
578C22—242no—
579C22—242no—
580C22—242no—
581C22—242no—
582C22#549014yes—
583C22#554714yes—
584C22#555012yes—
585C22#555212yes—
586C22#559712yes—
587C22#560512yes—
588C22#561014yes—
589C22#567612yes—
590C22#567712yes—
591C22D3×#12714yes—
592C22#579111yescharacteristic
593C22C2×#94011yescharacteristic
594C22C2×#94011yescharacteristic
595C22C2×#94011yescharacteristic
596C22#580411yescharacteristic
597C22#582811yescharacteristic
598C22#583311yescharacteristic
599C4—24no—
600C4—24no—
601C4—24no—
602C4—616no—
603C4—82no—
604C4—242no—
605C3#1517211yescharacteristic, complemented, Sylow
606C2—84no—
607C2—244no—
608C2#7634714yes—
609C2#7645414yes—
610C2#8211511yescharacteristic
611C2#8234711yescharacteristic
612C2#8237711yescharacteristic
613C2#8268011yescharacteristic
614C2#8270811yescharacteristic
615C2#8299811yescharacteristic
616C2#8318511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
hypercenterC2×#390D3
hypocenterC3#15172
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×C3×#390C2semidirect
2. C2×#5223C2semidirect
3. C2×#4993C2semidirect
4. C2×#4960C2semidirect
5. C2×#4949C2semidirect
6. C2×#4957C2semidirect
7. C2×#4939C2semidirect
8. C4×C3⋊#95C2semidirect
39. C2×#390D3semidirect
72. C2×C3×C42D4semidirect
91. C2×C4×Dic3D4semidirect
92. C3⋊#95D4semidirect
93. C3⋊#95D4semidirect
246. C2×C42#67semidirect
605. C3#15172semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C2×#52232
C2×#49932
C2×#49602
C2×#49492
C2×#49572
C2×#49392
C4×C3⋊#952
#151723