Finite Group Databasebeta

← #11233346

Subgroups — #11233346

Order 768.

1994 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5722C211yesmaximal, characteristic, complemented
2C2×#5732C211yesmaximal, characteristic
3C3×#2331C211yesmaximal, characteristic, complemented
4#76572C214yesmaximal
5#76503C214yesmaximal, complemented
6#76221C212yesmaximal, complemented
7#75997C212yesmaximal, complemented
8#75163C214yesmaximal, complemented
9#75005C214yesmaximal, complemented
10#74240C211yesmaximal, characteristic
11#74191C211yesmaximal, characteristic, complemented
12#73789C212yesmaximal, complemented
13#73647C212yesmaximal, complemented
14#73419C211yesmaximal, characteristic
15#73388C211yesmaximal, characteristic, complemented
16#45338—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#947C2212yes—
19C2×Q8×Dic3C2212yes—
20C2×#944C2214yes—
21#5732—28no—
22#5722—28no—
23C2×#898C2214yes—
24C2×#897C2212yes—
25C2×#897C2212yes—
26C2×#892C2212yes—
27C2×#888C2212yes—
28C2×#888C2212yes—
29C2×#887C2212yes—
30C2×#887C2212yes—
31C2×C4×D12C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C2×C4×Dic6C2211yescharacteristic
35C2×C4×Dic6C2211yescharacteristic
36C3×#391C2214yes—
37C3×#389C2212yes—
38C3×#387C2214yes—
39C3×#382C2211yescharacteristic
40C3×C4×C4⋊C4C2212yes—
41C3×#376C2211yescharacteristic
42#4994C2214yes—
43#4994C2214yes—
44#4987C2212yes—
45#4984C2212yes—
46#4984C2214yes—
47#4973C2212yes—
48#4973C2214yes—
49#4940C2214yes—
50#4940C2214yes—
51C4×(C2×D3)⋊C4C2212yes—
52C4×(C2×D3)⋊C4C2212yes—
53#4936C2211yescharacteristic
54C4×C4⋊Dic3C2214yes—
55#4929C2211yescharacteristic
56#4929C2214yes—
57#4929C2211yescharacteristic
58#4674C2212yes—
59#4673C2214yes—
60#4673C2214yes—
61#4673C2212yes—
62#4672C2212yes—
63#4672C2212yes—
64#4670C2214yes—
65#4670C2214yes—
66#4670C2214yes—
67#4661C2212yes—
68#4661C2214yes—
69#4659C2214yes—
70#4659C2212yes—
71#4659C2212yes—
72#4656C2212yes—
73#4653C2214yes—
74#4653C2214yes—
75#4653C2214yes—
76#4653C2214yes—
77#4653C2214yes—
78C2×Q82—31no—
79C2×#556—31no—
80#2731—34no—
81#2716—34no—
82#2581—31no—
83#2536—31no—
84#2492—32no—
85#2469—34no—
86#2446—34no—
87#2439—32no—
88#2429—32no—
89#2398—31no—
90#2362—32no—
91#2331D311yescharacteristic, complemented
92#2327—31no—
93C22×C3×Q8C2312yes—
94C22×D12C2311yescharacteristic
95C22×C4×D3C2312yes—
96C22×Dic6C2311yescharacteristic
97C22×Dic6C2311yescharacteristic
98C22×Dic6C2311yescharacteristic
99C22×Dic6C2314yes—
100C3×C4⋊Q8D418yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2314yes—
108C2×C3×C42C2311yescharacteristic
109#947—28no—
110Q8×Dic3—28no—
111#944—216no—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2314yes—
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2314yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2311yescharacteristic
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2314yes—
138C2×C4×Dic3C2312yes—
139#898—216no—
140#897—28no—
141#897—28no—
142#892—28no—
143#888—28no—
144#888—28no—
145#887—28no—
146#887—28no—
147C4×D12—24no—
148#869—24no—
149C4×Dic6—24no—
150C4×Dic6—24no—
151C4×Dic6—24no—
152C3×#95C2312yes—
153C3×#95C2312yes—
154C3×#95C2312yes—
155C3×#95C2314yes—
156C3×#95C2314yes—
157C3×#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C3⋊#95C2314yes—
163C3⋊#95C2312yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169Q82—68no—
170#556—68no—
171C2×C4⋊Q8—34no—
172C2×C4⋊Q8—32no—
173C2×C4⋊Q8—32no—
174C2×C4⋊Q8—31no—
175C2×C4⋊Q8C2×D311yescharacteristic
176C2×#126—34no—
177C2×#124—32no—
178C2×C22⋊Q8—32no—
179C2×C22⋊Q8—32no—
180C2×C4×Q8—32no—
181C2×C4×Q8—32no—
182C2×C4×Q8—32no—
183C2×C4×Q8—31no—
184C2×C4×Q8—31no—
185C2×C4×Q8—31no—
186C2×C4×D4—31no—
187C2×#117—32no—
188#400—32no—
189#400—32no—
190#400—34no—
191#398—34no—
192#398—32no—
193#397—32no—
194#396—34no—
195#396—34no—
196#395—32no—
197#394—32no—
198#394—32no—
199#393—32no—
200#393—34no—
201#393—34no—
202#391—32no—
203#391—34no—
204#391—31no—
205#391—31no—
206#391—34no—
207#391C2×D314yes—
208#389—32no—
209#389—34no—
210#389C2×D312yes—
211#388—34no—
212#388—34no—
213#388—34no—
214#388—34no—
215#388—34no—
216#387—34no—
217#387—34no—
218#387—34no—
219#387—34no—
220#387—34no—
221#387C2×D314yes—
222#382—31no—
223#382C2×D311yescharacteristic
224#380—32no—
225C4×C4⋊C4—34no—
226C4×C4⋊C4C2×D312yes—
227C4×C22⋊C4—32no—
228C4×C22⋊C4—32no—
229#376C2×D311yescharacteristic
230C23×D3C2412yes—
231C2×C3×Q8—44no—
232C2×C3×Q8C2×D418yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2414yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2414yes—
239C22×C3×C4C2411yescharacteristic
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242C22×Dic3C2412yes—
243C22×Dic3C2412yes—
244C22×Dic3C2412yes—
245C22×Dic3C2412yes—
246C22×Dic3C2412yes—
247C2×D12—22no—
248C2×D12—42no—
249C2×C4×D3—24no—
250C2×C4×D3—24no—
251C2×C4×D3—24no—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6—22no—
255C2×Dic6—24no—
256C2×Dic6—22no—
257C2×Dic6—22no—
258C2×Dic6—22no—
259C2×Dic6—28no—
260C2×Dic6—44no—
261C2×Dic6—44no—
262C2×Dic6#4114yes—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—44no—
265C3×C4⋊C4C2×D418yes—
266C3×C4⋊C4C2×D418yes—
267C3×C42—22no—
268C3×C42—22no—
269C3×C42—44no—
270C3×C42C2×D414yes—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—28no—
273(C2×D3)⋊C4—28no—
274(C2×D3)⋊C4—28no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—42no—
278(C2×D3)⋊C4—44no—
279(C2×D3)⋊C4—42no—
280C4⋊Dic3—22no—
281C4⋊Dic3—22no—
282C4⋊Dic3—22no—
283C4⋊Dic3—22no—
284C4⋊Dic3—28no—
285C4⋊Dic3—28no—
286C4⋊Dic3—28no—
287C4⋊Dic3—28no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—28no—
292Dic3⋊C4—28no—
293C4×Dic3—24no—
294C4×Dic3—24no—
295C4×Dic3—24no—
296C4×Dic3—28no—
297C4×Dic3—28no—
298C22×Q8—31no—
299C22×Q8—31no—
300C22×Q8—31no—
301C22×Q8—34no—
302C22×Q8C22×D312yes—
303C22×D4—31no—
304C23×C4—32no—
305C4⋊Q8—64no—
306C4⋊Q8—616no—
307C4⋊Q8—68no—
308C4⋊Q8—68no—
309C4⋊Q8D1218yes—
310#126—616no—
311#124—68no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314C4×Q8—68no—
315C4×Q8—68no—
316C4×Q8—68no—
317C4×Q8—64no—
318C4×Q8—64no—
319C4×Q8—64no—
320C4×D4—64no—
321#117—68no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D314yes—
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—32no—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—32no—
347C2×C22⋊C4—34no—
348C2×C42—34no—
349C2×C42—32no—
350C2×C42—32no—
351C2×C42—34no—
352C2×C42—32no—
353C2×C42C22×D311yescharacteristic
354C2×C42C22×D311yescharacteristic
355C2×C42C22×D314yes—
356C2×C42C22×D311yescharacteristic
357#95—32no—
358#95—32no—
359#95—32no—
360#95—32no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95—34no—
368#95C22×D312yes—
369#95C22×D312yes—
370#95C22×D314yes—
371#95C22×D312yes—
372#95C22×D312yes—
373#95C22×D314yes—
374C23×C3C2511yescharacteristic
375C22×D3—22no—
376C22×D3—22no—
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—22no—
382C3×Q8—48no—
383C2×C3×C4—22no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—24no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—22no—
396C2×C3×C4C22×D412yes—
397C2×C3×C4C22×D412yes—
398C2×C3×C4C22×D418yes—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C2×#4112yes—
401C2×C3×C42−1+412yes—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3C2×#4114yes—
420C2×Dic32−1+414yes—
421D12—44no—
422C4×D3—48no—
423Dic6—28no—
424Dic6—28no—
425Dic6—44no—
426Dic6—88no—
427C24—32no—
428C2×Q8—34no—
429C2×Q8—44no—
430C2×Q8—64no—
431C2×Q8—62no—
432C2×Q8—62no—
433C2×Q8—62no—
434C2×Q8—68no—
435C2×Q8—62no—
436C2×Q8—62no—
437C2×Q8—62no—
438C2×Q8—124no—
439C2×Q8—124no—
440C2×Q8C2×D1218yes—
441C2×D4—62no—
442C2×D4—122no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D311yescharacteristic
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D314yes—
459C22×C4C23×D311yescharacteristic
460C4⋊C4—24no—
461C4⋊C4—44no—
462C4⋊C4—68no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—68no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4C2×D1218yes—
476C4⋊C4C2×D1218yes—
477C22⋊C4—68no—
478C22⋊C4—64no—
479C22⋊C4—64no—
480C22⋊C4—68no—
481C22⋊C4—64no—
482C22⋊C4—68no—
483C22⋊C4—122no—
484C22⋊C4—124no—
485C22⋊C4—122no—
486C42—22no—
487C42—22no—
488C42—44no—
489C42—68no—
490C42—64no—
491C42—68no—
492C42—64no—
493C42—64no—
494C42C2×D1214yes—
495C22×C3C23×D411yescharacteristic
496C22×C3C22×#4111yescharacteristic
497C22×C3C2×2−1+411yescharacteristic
498C22×C3C2×2−1+412yes—
499C22×C3#58512yes—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C3×C4—28no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—44no—
512C3×C4—42no—
513C3×C4#54814yes—
514Dic3—24no—
515Dic3—24no—
516Dic3—42no—
517Dic3—42no—
518Dic3—42no—
519Dic3—42no—
520Dic3—42no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23C24×D311yescharacteristic
529Q8—48no—
530Q8—68no—
531Q8—68no—
532Q8—124no—
533Q8—248no—
534D4—124no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—34no—
549C2×C4—34no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—128no—
568C2×C4C22×D1212yes—
569C2×C4C22×D1212yes—
570C2×C4C22×D1218yes—
571C2×C4C22×D1212yes—
572C2×C4#100712yes—
573C2×C4D3×#4112yes—
574C2×C3C2×#54811yescharacteristic
575C2×C3#348511yescharacteristic
576C2×C3#348611yescharacteristic
577C2×C3#351812yes—
578C2×C3#355512yes—
579D3—82no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22C23×D1211yescharacteristic
588C22C2×#100711yescharacteristic
589C22C2×D3×#4111yescharacteristic
590C22#596612yes—
591C22D3×2−1+412yes—
592C4—22no—
593C4—28no—
594C4—22no—
595C4—42no—
596C4—42no—
597C4—44no—
598C4—64no—
599C4—64no—
600C4—122no—
601C4—122no—
602C4—122no—
603C4—122no—
604C4—122no—
605C4#557614yes—
606C3#4533811yescharacteristic, complemented, Sylow
607C2—242no—
608C2C2×#557611yescharacteristic
609C2#8585811yescharacteristic
610C2#8588111yescharacteristic
611C2#8652712yes—
612C2#8659012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2331C2
prime radicalC22×D12C23
hypercenter#2331D3
hypocenterC3#45338
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C2×#5722C2semidirect
3. C3×#2331C2semidirect
5. #76503C2semidirect
6. #76221C2semidirect
7. #75997C2semidirect
8. #75163C2semidirect
9. #75005C2semidirect
11. #74191C2semidirect
12. #73789C2semidirect
13. #73647C2semidirect
15. #73388C2semidirect
91. #2331D3semidirect
606. C3#45338semidirect

Maximal subgroups

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

subgroupindex
C2×#57222
C2×#57322
C3×#23312
#765722
#765032
#762212
#759972
#751632
#750052
#742402
#741912
#737892
#736472
#734192
#733882
#453383