Finite Group Databasebeta

← #10972871

Subgroups — #10972871

Order 768.

2294 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5624C211yesmaximal, characteristic
2C22×#887C211yesmaximal, characteristic, complemented
3C3×#2502C211yesmaximal, characteristic, complemented
4#82760C211yesmaximal, characteristic, complemented
5#82180C211yesmaximal, characteristic
6#77825C214yesmaximal, complemented
7#77810C214yesmaximal, complemented
8#76534C212yesmaximal, complemented
9#76485C212yesmaximal, complemented
10#74588C214yesmaximal, complemented
11#74486C214yesmaximal, complemented
12#74338C212yesmaximal, complemented
13#74321C212yesmaximal, complemented
14#73491C211yesmaximal, characteristic
15#73437C211yesmaximal, characteristic, complemented
16#35246—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#944C2212yes—
23C2×#930C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27#5624—28no—
28C2×#898C2212yes—
29C2×#893C2211yescharacteristic
30C2×#892C2212yes—
31C2×#890C2211yescharacteristic
32C2×#887—28no—
33C2×#887C2212yes—
34C2×#887C2212yes—
35C2×#887C2212yes—
36C2×#887C2212yes—
37C2×#886C2212yes—
38C2×#67⋊C4C2212yes—
39C2×#875C2212yes—
40C3×#398C2214yes—
41C3×#391C2212yes—
42C3×#385C2214yes—
43C3×#384C2212yes—
44C3×C4×C22⋊C4C2211yescharacteristic
45#5213C2214yes—
46C4×#217C2211yescharacteristic
47#4992C2214yes—
48#4990C2218yes—
49#4989C2218yes—
50#4988C2212yes—
51#4983C2218yes—
52#4980C2218yes—
53#4979C2214yes—
54#4977C2212yes—
55#4973C2212yes—
56#4958C2214yes—
57#4946C2214yes—
58#4943C2212yes—
59#4942C2212yes—
60#4932C2212yes—
61C4×Dic3⋊C4C2212yes—
62#4929C2212yes—
63#4676C2218yes—
64#4673C2214yes—
65#4669C2218yes—
66#4655C2218yes—
67#4652C2218yes—
68#4651C2212yes—
69#4650C2212yes—
70C2×#529—31no—
71C22×C4×Q8—31no—
72#2853—34no—
73#2637—34no—
74#2599—34no—
75#2579—34no—
76#2565—32no—
77#2545—32no—
78#2504—31no—
79#2502D311yescharacteristic, complemented
80#2436—31no—
81#2434—31no—
82#2327—32no—
83#2326—32no—
84#2283—31no—
85C22×C3×Q8C2311yescharacteristic
86C23×C3×C4C2311yescharacteristic
87C23×C3×C4C2312yes—
88C22×#67C2312yes—
89C23×Dic3C2312yes—
90C23×Dic3C2311yescharacteristic
91C23×Dic3C2311yescharacteristic
92C22×D12C2311yescharacteristic
93C22×C4×D3C2312yes—
94C22×Dic6—24no—
95C22×Dic6—28no—
96C22×Dic6C2312yes—
97C22×Dic6C2312yes—
98C3×C22⋊Q8—24no—
99C2×C3×C4⋊C4—24no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2318yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C22⋊C4C2312yes—
107C2×C3×C42C2312yes—
108C2×#217C2312yes—
109#947—24no—
110Q8×Dic3—24no—
111#944—28no—
112#930—24no—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2318yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2312yes—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2318yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2312yes—
122C2×Dic3⋊C4—28no—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2318yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2314yes—
128C2×C4×Dic3—28no—
129C2×C4×Dic3D418yes—
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2314yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2311yescharacteristic
138#898—28no—
139#893—24no—
140#892—28no—
141#890—24no—
142#887—28no—
143#887—28no—
144#887—28no—
145#887—28no—
146#886—28no—
147#67⋊C4—28no—
148#875—28no—
149C3×#95C2312yes—
150C3×#95C2318yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2318yes—
156C3⋊#95C2318yes—
157C3⋊#95C2318yes—
158C23×Q8—31no—
159C2×C4⋊Q8—32no—
160#529—68no—
161C2×#126—32no—
162C2×#125—31no—
163C2×#124—31no—
164C2×#124—32no—
165C2×#123—32no—
166C2×C22⋊Q8C2×D311yescharacteristic
167C2×#121—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—32no—
170C2×C4×Q8—32no—
171C2×C4×Q8—32no—
172C2×C4×Q8—32no—
173C2×C4×Q8—68no—
174C2×C4×D4—31no—
175C2×C4×D4—32no—
176C2×#117—32no—
177C22×C4⋊C4—32no—
178C22×C4⋊C4C2×D311yescharacteristic
179C22×C42—31no—
180C22×C42—32no—
181#398—38no—
182#398C2×D314yes—
183#397—34no—
184#396—38no—
185#395—34no—
186#393—34no—
187#391—32no—
188#391—32no—
189#391C2×D312yes—
190#390—32no—
191#388—38no—
192#388—38no—
193#387—38no—
194#387—38no—
195#387—38no—
196#386—34no—
197#386—34no—
198#385—38no—
199#385—34no—
200#385C2×D314yes—
201#384C2×D312yes—
202#383—32no—
203#382—32no—
204#382—32no—
205#381—32no—
206#380—32no—
207C4×C4⋊C4—32no—
208C4×C22⋊C4—31no—
209C4×C22⋊C4C2×D311yescharacteristic
210#376—32no—
211C24×C3C2411yescharacteristic
212C23×D3C2412yes—
213C2×C3×Q8—22no—
214C2×C3×Q8—24no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—44no—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2414yes—
220C22×C3×C4C2412yes—
221C22×C3×C4C2414yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C2×#67—24no—
226C2×#67—28no—
227C22×Dic3—24no—
228C22×Dic3—24no—
229C22×Dic3—28no—
230C22×Dic3—42no—
231C22×Dic3C2×D414yes—
232C22×Dic3C2×D414yes—
233C22×Dic3C2414yes—
234C22×Dic3C2412yes—
235C22×Dic3C2411yescharacteristic
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2411yescharacteristic
239C22×Dic3C2414yes—
240C2×D12—22no—
241C2×D12—24no—
242C2×C4×D3—24no—
243C2×C4×D3—44no—
244C2×Dic6—232no—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C2×Dic6—28no—
249C2×Dic6#4118yes—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—48no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C42—24no—
258#217—24no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—216no—
264(C2×D3)⋊C4—48no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4—28no—
270Dic3⋊C4—28no—
271Dic3⋊C4—216no—
272Dic3⋊C4—44no—
273C4×Dic3—24no—
274C4×Dic3—24no—
275C4×Dic3—28no—
276C4×Dic3—28no—
277C4×Dic3C2×D414yes—
278C4×Dic3C2×D414yes—
279C4×Dic3C2×D414yes—
280C4×Dic3C2×D414yes—
281C22×Q8—32no—
282C22×Q8—32no—
283C22×Q8—68no—
284C22×Q8—64no—
285C22×Q8C22×D311yescharacteristic
286C22×D4—32no—
287C22×D4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—32no—
291C23×C4—32no—
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D312yes—
294C4⋊Q8—68no—
295#126—68no—
296#125—64no—
297#124—68no—
298#124—64no—
299#123—68no—
300C22⋊Q8—24no—
301#121—64no—
302C4×Q8—68no—
303C4×Q8—68no—
304C4×Q8—68no—
305C4×Q8—68no—
306C4×Q8—64no—
307C4×D4—64no—
308C4×D4—68no—
309#117—68no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—38no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—68no—
320C2×C4⋊C4C22×D312yes—
321C2×C4⋊C4C22×D312yes—
322C2×C4⋊C4C22×D312yes—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D318yes—
325C2×C22⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—32no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—38no—
331C2×C22⋊C4—38no—
332C2×C22⋊C4C22×D312yes—
333C2×C22⋊C4C22×D312yes—
334C2×C42—38no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—32no—
340C2×C42—34no—
341C2×C42—34no—
342C2×C42—32no—
343C2×C42—68no—
344C2×C42C22×D312yes—
345#95—34no—
346#95—34no—
347#95—32no—
348#95—34no—
349#95—38no—
350#95—38no—
351#95—38no—
352#95C22×D312yes—
353#95C22×D318yes—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3C22×D412yes—
357C23×C32−1+414yes—
358C23×C3C2511yescharacteristic
359C22×D3—22no—
360C22×D3—24no—
361C22×D3—24no—
362C22×D3—24no—
363C3×Q8—44no—
364C2×C3×C4—22no—
365C2×C3×C4—28no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—28no—
374C2×C3×C4—24no—
375C2×C3×C4—48no—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C2×#4114yes—
379#67—48no—
380C2×Dic3—216no—
381C2×Dic3—28no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—44no—
394C2×Dic3C22×D412yes—
395C2×Dic3C22×D412yes—
396C2×Dic3C22×D412yes—
397C2×Dic3C22×D412yes—
398C2×Dic3C2×#4118yes—
399D12—44no—
400C4×D3—48no—
401Dic6—216no—
402Dic6—216no—
403C24—32no—
404C24C23×D311yescharacteristic
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—38no—
408C2×Q8—68no—
409C2×Q8—632no—
410C2×Q8—64no—
411C2×Q8—64no—
412C2×Q8—64no—
413C2×D4—64no—
414C2×D4—68no—
415C2×D4—62no—
416C2×D4—64no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—34no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—34no—
424C22×C4—31no—
425C22×C4—34no—
426C22×C4—34no—
427C22×C4—32no—
428C22×C4—44no—
429C22×C4—64no—
430C22×C4—68no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—124no—
434C22×C4—122no—
435C22×C4D3×D414yes—
436C22×C4C23×D314yes—
437C22×C4C23×D312yes—
438C22×C4C23×D314yes—
439C22×C4C23×D312yes—
440C22×C4C23×D311yescharacteristic
441C22×C4C23×D311yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—48no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—64no—
450C4⋊C4—64no—
451C4⋊C4—64no—
452C4⋊C4—616no—
453C4⋊C4—62no—
454C4⋊C4—124no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—64no—
458C22⋊C4—64no—
459C22⋊C4—64no—
460C22⋊C4—616no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—128no—
464C42—24no—
465C42—34no—
466C42—34no—
467C42—34no—
468C42—34no—
469C42—68no—
470C42—68no—
471C42—64no—
472C42—64no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—28no—
476C22×C3#53614yes—
477C22×C3C23×D411yescharacteristic
478C22×C3C22×#4112yes—
479C22×C3C2×2−1+412yes—
480C22×C3#58512yes—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—48no—
485C3×C4—22no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—44no—
489C3×C4—44no—
490C3×C4#52514yes—
491Dic3—28no—
492Dic3—22no—
493Dic3—22no—
494Dic3—22no—
495Dic3—22no—
496Dic3—42no—
497Dic3—44no—
498C23—22no—
499C23—22no—
500C23—62no—
501C23—64no—
502C23—64no—
503C23—64no—
504C23C2×D3×D412yes—
505C23#100714yes—
506C23C24×D311yescharacteristic
507Q8—44no—
508Q8—616no—
509Q8—616no—
510D4—128no—
511D4—124no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—38no—
528C2×C4—48no—
529C2×C4—616no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—68no—
537C2×C4—62no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—128no—
543C2×C4—124no—
544C2×C4C2×D3×D412yes—
545C2×C4C2×D3×D412yes—
546C2×C4D3×#4114yes—
547C2×C3—24no—
548C2×C3C2×#52511yescharacteristic
549C2×C3C2×#53611yescharacteristic
550C2×C3#344911yescharacteristic
551C2×C3#351614yes—
552D3—82no—
553C22—22no—
554C22—28no—
555C22—22no—
556C22—122no—
557C22—128no—
558C22—122no—
559C22—122no—
560C22#572414yes—
561C22C22×D3×D411yescharacteristic
562C22C2×#100712yes—
563C22C2×D3×#4112yes—
564C22#596612yes—
565C4—22no—
566C4—44no—
567C4—42no—
568C4—44no—
569C4—42no—
570C4—62no—
571C4—62no—
572C4—62no—
573C4—62no—
574C4—68no—
575C4—124no—
576C4—122no—
577C4#565614yes—
578C3#3524611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—242no—
581C2C2×#565611yescharacteristic
582C2C2×#572411yescharacteristic
583C2#8631311yescharacteristic
584C2#8693214yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2502C2
prime radicalC2×#930C22
hypercenter#2502D3
hypocenterC3#35246
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2502

Complements

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

NH ≅ G/Ntype
2. C22×#887C2semidirect
3. C3×#2502C2semidirect
4. #82760C2semidirect
6. #77825C2semidirect
7. #77810C2semidirect
8. #76534C2semidirect
9. #76485C2semidirect
10. #74588C2semidirect
11. #74486C2semidirect
12. #74338C2semidirect
13. #74321C2semidirect
15. #73437C2semidirect
79. #2502D3semidirect
578. C3#35246semidirect

Maximal subgroups

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

subgroupindex
C2×#56242
C22×#8872
C3×#25022
#827602
#821802
#778252
#778102
#765342
#764852
#745882
#744862
#743382
#743212
#734912
#734372
#352463