Finite Group Databasebeta

← #10823413

Subgroups — #10823413

Order 768.

1508 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic, complemented
2C4×#930C211yesmaximal, characteristic, complemented
3C2×#4711C212yesmaximal, complemented
4C3×#1749C211yesmaximal, characteristic, complemented
5#69961C212yesmaximal, complemented
6#69935C212yesmaximal, complemented
7#69812C212yesmaximal, complemented
8#68678C211yesmaximal, characteristic, complemented
9#68674C211yesmaximal, characteristic, complemented
10#68633C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#23573—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2212yes—
17C2×#930C2211yescharacteristic
18C2×C4×#67—21no—
19C22×D24C2212yes—
20#5477—28no—
21C2×#875—21no—
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×#869C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×#97C2212yes—
29#5213—21no—
30#5212C2211yescharacteristic
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#817C2212yes—
36#4940—21no—
37#4939C2211yescharacteristic
38C4×(C2×D3)⋊C4—21no—
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41#4711—28no—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#4551C2212yes—
45#4551C2212yes—
46C2×#532—31no—
47C2×#442—32no—
48C4×#121—31no—
49#2058—31no—
50#2041—31no—
51#2034—32no—
52#2023—31no—
53#2011—31no—
54#1999—32no—
55#1762—32no—
56#1749D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67—21no—
62C2×#235—28no—
63C2×#235—44no—
64C2×#235—44no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2312yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—22no—
70C22×C4×D3—21no—
71C22×Dic6—21no—
72C22×C3×C8C2312yes—
73C22×C3×C8C2312yes—
74C2×C3×C42—24no—
75C2×C3×C42D411yescharacteristic
76C2×C3×C42D411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×#217—21no—
80#930—28no—
81#930—24no—
82#930—22no—
83#930—24no—
84#930—22no—
85C4×#67—44no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C2×D24—24no—
100C2×D24—24no—
101C2×D24—24no—
102#875—28no—
103#874—24no—
104C4×D12—28no—
105C4×D12—24no—
106C4×D12—24no—
107C4×D12D418yescomplemented
108#869—24no—
109C3×C22⋊C8—24no—
110C3×#97—24no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#821—24no—
115#821—24no—
116#821—24no—
117#821—24no—
118#817—24no—
119C22×#41—61no—
120C22×D8—32no—
121#532—68no—
122C2×C4⋊Q8—31no—
123C2×#127—31no—
124C2×#124—61no—
125C2×#121—32no—
126C2×#121—31no—
127C2×#121—61no—
128C2×C4×D4—32no—
129C2×C4×D4—31no—
130C2×C4×D4—61no—
131C2×C4×D4—61no—
132C22×C42C2×D311yescharacteristic
133#442—68no—
134C2×#106—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×C22⋊C8C2×D312yes—
140C2×#97C2×D312yes—
141#390—31no—
142#389—31no—
143#388—61no—
144#386—61no—
145#380—31no—
146C4×C4⋊C4—31no—
147C4×C22⋊C4—61no—
148#340—32no—
149#340—32no—
150#336C2×D311yescharacteristic
151#336C2×D311yescharacteristic
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C23×D3—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—21no—
163C22×C3×C4—21no—
164C22×C3×C4—41no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C2×#67—28no—
172C2×#67—24no—
173C2×#67—42no—
174C2×#67—44no—
175C2×#67—44no—
176C2×#67—42no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181#235—432no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—24no—
189C2×D12—24no—
190C2×D12—22no—
191C2×D12—21no—
192C2×D12—42no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12—42no—
196C2×D12C2×D414yes—
197C2×D12#4112yescomplemented
198C2×C4×D3—28no—
199C2×C4×D3—44no—
200C2×C4×D3—44no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×Dic6—24no—
205C2×Dic6—42no—
206C2×Dic6—42no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8#4114yescomplemented
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42C2×D414yes—
217#217—42no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—42no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—42no—
224C4⋊Dic3—24no—
225C4⋊Dic3—22no—
226C4⋊Dic3—21no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3C2×D414yes—
230C4⋊Dic3#4112yescomplemented
231Dic3⋊C4—42no—
232C4×Dic3—42no—
233D24—216no—
234C2×#41—68no—
235C2×#41—124no—
236C2×#41—124no—
237C22×Q8—61no—
238C22×D4—32no—
239C22×D4—31no—
240C22×D4—61no—
241C22×D4—62no—
242C22×D4—61no—
243C22×D4—61no—
244C23×C4—62no—
245C23×C4—61no—
246C23×C4D1211yescharacteristic
247C23×C4D1211yescharacteristic
248C23×C4C22×D311yescharacteristic
249C2×D8—64no—
250C2×D8—64no—
251C2×D8—64no—
252C22×C8C22×D312yes—
253C22×C8C22×D312yes—
254C4⋊Q8—64no—
255#127—64no—
256#124—68no—
257#121—64no—
258#121—62no—
259#121—64no—
260#121—62no—
261#121—68no—
262C4×D4—38no—
263C4×D4—68no—
264C4×D4—64no—
265C4×D4—64no—
266C4×D4—124no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—24no—
281C2×C42—61no—
282C2×C42D1211yescharacteristic
283C2×C42D1211yescharacteristic
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D311yescharacteristic
286#106—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290#102—64no—
291C22⋊C8—24no—
292#97—24no—
293#95—61no—
294#95—61no—
295#95—61no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3AGL(1,ℤ/8)12yes—
303C23×C3C22×D411yescharacteristic
304C22×D3—24no—
305C22×D3—22no—
306C22×D3—41no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—42no—
318C2×C3×C4—28no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4#12012yes—
334C2×C3×C4#12012yes—
335C2×C3×C4AGL(1,ℤ/8)12yes—
336C2×C3×C4AGL(1,ℤ/8)12yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C2×#4112yes—
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341#67—416no—
342#67—48no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355D12—24no—
356D12—24no—
357D12—24no—
358D12—48no—
359D12—48no—
360D12—42no—
361D12—42no—
362D12—42no—
363C4×D3—416no—
364C4×D3—84no—
365Dic6—48no—
366C3×C8—24no—
367C3×C8—42no—
368C24—61no—
369C24—62no—
370C24C2×D1211yescharacteristic
371#41—1232no—
372C2×Q8—64no—
373C2×Q8—122no—
374C2×Q8—122no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—64no—
378C2×D4—64no—
379C2×D4—68no—
380C2×D4—64no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4—122no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—68no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—124no—
413C22×C4—124no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4C23×D311yescharacteristic
423D8—616no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8#23514yescomplemented
430C4⋊C4—34no—
431C4⋊C4—32no—
432C4⋊C4—64no—
433C4⋊C4—62no—
434C4⋊C4—61no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—122no—
438C22⋊C4—68no—
439C22⋊C4—64no—
440C22⋊C4—68no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—122no—
444C22⋊C4—122no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—122no—
449C42D3×D414yes—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3C2×#12011yescharacteristic
458C22×C3#52511yescharacteristic
459C22×C3#53211yescharacteristic
460C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
461C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
462C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
463C22×C3#57511yescharacteristic
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—44no—
467C2×D3—42no—
468C2×D3—44no—
469C2×D3—44no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—82no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—41no—
483C3×C4#52512yes—
484C3×C4#53212yes—
485Dic3—42no—
486Dic3—44no—
487Dic3—81no—
488Dic3—81no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—64no—
496C23—62no—
497C23—122no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—121no—
506C23—121no—
507C23—122no—
508C23—122no—
509C23#90812yes—
510C23C22×D1211yescharacteristic
511Q8—128no—
512D4—64no—
513D4—64no—
514D4—64no—
515D4—128no—
516D4—128no—
517D4—1216no—
518D4—128no—
519D4—122no—
520D4—122no—
521D4—122no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—64no—
538C2×C4—62no—
539C2×C4—1216no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—244no—
551C2×C4#88212yes—
552C2×C4#88212yes—
553C2×C4#90812yes—
554C2×C4#90812yes—
555C2×C4C2×#23512yes—
556C2×C4C2×D3×D412yes—
557C2×C4D3×#4111yescharacteristic
558C2×C4D3×#4111yescharacteristic
559C8—24no—
560C8—42no—
561C2×C3—42no—
562C2×C3#238811yescharacteristic
563C2×C3#300311yescharacteristic
564C2×C3#300611yescharacteristic
565C2×C3#310911yescharacteristic
566C2×C3#311511yescharacteristic
567C2×C3#318011yescharacteristic
568C2×C3#318411yescharacteristic
569D3—82no—
570D3—84no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—124no—
579C22—124no—
580C22—122no—
581C22—124no—
582C22—122no—
583C22—122no—
584C22—241no—
585C22—241no—
586C22—242no—
587C22—241no—
588C22—241no—
589C22—242no—
590C22—242no—
591C22—242no—
592C22#547711yescharacteristic
593C22C2×#88211yescharacteristic
594C22#552411yescharacteristic
595C22C2×#90811yescharacteristic
596C22C2×#90811yescharacteristic
597C22C2×#90811yescharacteristic
598C22#574811yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—41no—
604C4—124no—
605C4—122no—
606C4—241no—
607C4—241no—
608C4#554712yes—
609C4#554812yes—
610C3#2357311yescharacteristic, complemented, Sylow
611C2—42no—
612C2—244no—
613C2—242no—
614C2#7358011yescharacteristic
615C2#7850011yescharacteristic
616C2#7853511yescharacteristic
617C2#7863711yescharacteristic
618C2#7863911yescharacteristic
619C2#7934311yescharacteristic
620C2#7934711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
hypercenter#1749D3
hypocenterC3#23573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C4×#930C2semidirect
3. C2×#4711C2semidirect
4. C3×#1749C2semidirect
5. #69961C2semidirect
6. #69935C2semidirect
7. #69812C2semidirect
8. #68678C2semidirect
9. #68674C2semidirect
10. #68633C2semidirect
11. #68629C2semidirect
56. #1749D3semidirect
107. C4×D12D4semidirect
197. C2×D12#41semidirect
212. C2×C3×C8#41semidirect
230. C4⋊Dic3#41semidirect
429. C2×C8#235semidirect
610. C3#23573semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C4×#9302
C2×#47112
C3×#17492
#699612
#699352
#698122
#686782
#686742
#686332
#686292
#235733