Finite Group Databasebeta

← #10972864

Subgroups — #10972864

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×#5479C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic, complemented
3C3×#2283C211yesmaximal, characteristic, complemented
4#74308C214yesmaximal, complemented
5#74246C214yesmaximal, complemented
6#74116C214yesmaximal, complemented
7#73949C214yesmaximal, complemented
8#73719C212yesmaximal, complemented
9#73710C212yesmaximal, complemented
10#73519C211yesmaximal, characteristic
11#73517C211yesmaximal, characteristic, complemented
12#73491C211yesmaximal, characteristic
13#73490C211yesmaximal, characteristic, complemented
14#73388C212yesmaximal, complemented
15#73387C212yesmaximal, complemented
16#35246—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C2×#929C2212yes—
21C2×C4×#67C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2212yes—
26#5479—28no—
27C2×#876C2212yes—
28C2×#875C2212yes—
29C2×#875C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×#872C2212yes—
32C2×#870C2211yescharacteristic
33C2×#869C2212yes—
34C2×C4×Dic6—28no—
35C2×C4×Dic6C2212yes—
36C2×C4×Dic6C2212yes—
37C2×C4×Dic6C2212yes—
38C2×C4×Dic6C2212yes—
39C2×C4×Dic6C2211yescharacteristic
40C3×#383C2212yes—
41C3×#382C2212yes—
42C3×#381C2212yes—
43C3×#380C2212yes—
44C3×C4×C4⋊C4C2212yes—
45C3×C4×C22⋊C4C2211yescharacteristic
46C3×C4×C22⋊C4C2211yescharacteristic
47C3×#376C2212yes—
48#4994C2218yes—
49#4981C2218yes—
50#4958C2214yes—
51#4958C2214yes—
52#4950C2214yes—
53#4950C2214yes—
54#4939C2212yes—
55#4936C2212yes—
56#4935C2212yes—
57#4929C2212yes—
58#4929C2212yes—
59#4929C2212yes—
60#4674C2214yes—
61#4673C2214yes—
62#4670C2218yes—
63#4670C2218yes—
64#4668C2218yes—
65#4661C2214yes—
66#4656C2214yes—
67#4653C2218yes—
68#4653C2218yes—
69#4653C2218yes—
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#2502—31no—
80#2436—31no—
81#2434—31no—
82#2327—32no—
83#2326—32no—
84#2283D311yescharacteristic, complemented
85C23×C3×C4C2311yescharacteristic
86C23×C3×C4C2311yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C22×#67C2312yes—
89C23×Dic3C2312yes—
90C23×Dic3C2312yes—
91C22×D12C2311yescharacteristic
92C22×C4×D3C2312yes—
93C22×Dic6—24no—
94C22×Dic6—28no—
95C22×Dic6C2312yes—
96C22×Dic6C2311yescharacteristic
97C22×Dic6C2312yes—
98C2×C3×C4⋊C4C2314yes—
99C2×C3×C22⋊C4C2312yes—
100C2×C3×C22⋊C4C2312yes—
101C2×C3×C42D418yes—
102C2×C3×C42C2312yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×#217C2312yes—
109#930—24no—
110#929—28no—
111C4×#67—28no—
112C2×(C2×D3)⋊C4C2318yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2312yes—
117C2×(C2×D3)⋊C4C2318yes—
118C2×C4⋊Dic3—24no—
119C2×C4⋊Dic3C2318yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2318yes—
125#924—24no—
126C2×Dic3⋊C4—28no—
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2312yes—
133C2×C4×Dic3—28no—
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137#876—28no—
138#875—28no—
139#875—24no—
140C4×D12—24no—
141#872—28no—
142#870—24no—
143#869—28no—
144C4×Dic6—28no—
145C4×Dic6—28no—
146C4×Dic6—28no—
147C4×Dic6—28no—
148C4×Dic6—24no—
149C3×#95C2314yes—
150C3×#95C2314yes—
151C3×#95C2314yes—
152C3×#95C2312yes—
153C3×#95C2312yes—
154C3⋊#95C2318yes—
155C3⋊#95C2318yes—
156C3⋊#95C2318yes—
157C3⋊#95C2318yes—
158C23×Q8—31no—
159C2×C4⋊Q8—32no—
160#529—68no—
161C2×#126—32no—
162C2×#125—31no—
163C2×#124—32no—
164C2×#124—31no—
165C2×#123—32no—
166C2×C22⋊Q8—31no—
167C2×#121—31no—
168C2×C4×Q8—32no—
169C2×C4×Q8—32no—
170C2×C4×Q8—31no—
171C2×C4×Q8—32no—
172C2×C4×Q8—32no—
173C2×C4×Q8—68no—
174C2×C4×D4—32no—
175C2×C4×D4—31no—
176C2×#117—32no—
177C22×C4⋊C4—32no—
178C22×C4⋊C4—31no—
179C22×C42—32no—
180C22×C42C2×D311yescharacteristic
181#398—38no—
182#398—34no—
183#397—34no—
184#396—38no—
185#395—34no—
186#393—34no—
187#391—32no—
188#391—32no—
189#391—32no—
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—34no—
199#385—34no—
200#385—38no—
201#384—32no—
202#383C2×D312yes—
203#382—32no—
204#382C2×D312yes—
205#381C2×D312yes—
206#380C2×D312yes—
207C4×C4⋊C4C2×D312yes—
208C4×C22⋊C4C2×D311yescharacteristic
209C4×C22⋊C4C2×D311yescharacteristic
210#376C2×D312yes—
211C24×C3C2411yescharacteristic
212C23×D3C2412yes—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D414yes—
217C22×C3×C4C2×D414yes—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2414yes—
220C22×C3×C4C2414yes—
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C2×#67—24no—
228C2×#67—28no—
229C22×Dic3—24no—
230C22×Dic3—28no—
231C22×Dic3—24no—
232C22×Dic3—44no—
233C22×Dic3C2412yes—
234C22×Dic3C2414yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C22×Dic3C2414yes—
238C2×D12—22no—
239C2×D12—24no—
240C2×C4×D3—24no—
241C2×C4×D3—44no—
242C2×Dic6—232no—
243C2×Dic6—24no—
244C2×Dic6—24no—
245C2×Dic6—24no—
246C2×Dic6—28no—
247C2×Dic6—22no—
248C2×Dic6—24no—
249C2×Dic6#4118yes—
250C3×C4⋊C4—44no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C42—24no—
254C3×C42—24no—
255C3×C42C2×D414yes—
256C3×C42C2×D414yes—
257C3×C42C2×D414yes—
258C3×C42C2×D414yes—
259#217—24no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—216no—
265(C2×D3)⋊C4—48no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—22no—
269C4⋊Dic3—22no—
270C4⋊Dic3—216no—
271C4⋊Dic3—48no—
272Dic3⋊C4—28no—
273Dic3⋊C4—28no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278C4×Dic3—28no—
279C4×Dic3—28no—
280C4×Dic3—24no—
281C22×Q8—32no—
282C22×Q8—32no—
283C22×Q8—31no—
284C22×Q8—68no—
285C22×Q8—64no—
286C22×D4—32no—
287C22×D4—31no—
288C23×C4—32no—
289C23×C4—32no—
290C23×C4—32no—
291C23×C4C22×D311yescharacteristic
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C4⋊Q8—68no—
295#126—68no—
296#125—64no—
297#124—68no—
298#124—64no—
299#123—68no—
300C22⋊Q8—64no—
301#121—64no—
302C4×Q8—68no—
303C4×Q8—68no—
304C4×Q8—68no—
305C4×Q8—68no—
306C4×Q8—64no—
307C4×D4—68no—
308C4×D4—64no—
309#117—68no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—38no—
321C2×C4⋊C4—38no—
322C2×C4⋊C4—68no—
323C2×C4⋊C4—64no—
324C2×C4⋊C4C22×D314yes—
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—32no—
335C2×C42—34no—
336C2×C42—34no—
337C2×C42—68no—
338C2×C42D1218yes—
339C2×C42C22×D312yes—
340C2×C42C22×D312yes—
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D311yescharacteristic
345#95—38no—
346#95—38no—
347#95—38no—
348#95—38no—
349#95C22×D314yes—
350#95C22×D314yes—
351#95C22×D312yes—
352#95C22×D312yes—
353#95C22×D314yes—
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—
363C2×C3×C4—28no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—28no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—44no—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D412yes—
382C2×C3×C4C22×D412yes—
383C2×C3×C4C2×#4114yes—
384#67—48no—
385C2×Dic3—216no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—28no—
396C2×Dic3—48no—
397C2×Dic3C2×#4118yes—
398D12—44no—
399C4×D3—48no—
400Dic6—216no—
401Dic6—216no—
402Dic6—44no—
403C24—32no—
404C24C23×D311yescharacteristic
405C2×Q8—38no—
406C2×Q8—68no—
407C2×Q8—632no—
408C2×Q8—64no—
409C2×Q8—64no—
410C2×Q8—62no—
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—32no—
421C22×C4—34no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—42no—
425C22×C4—68no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—124no—
430C22×C4—124no—
431C22×C4C2×D1214yes—
432C22×C4C2×D1214yes—
433C22×C4C2×D1214yes—
434C22×C4C23×D314yes—
435C22×C4C23×D314yes—
436C22×C4C23×D311yescharacteristic
437C22×C4C23×D311yescharacteristic
438C22×C4C23×D311yescharacteristic
439C22×C4C23×D311yescharacteristic
440C22×C4C23×D311yescharacteristic
441C22×C4C23×D311yescharacteristic
442C4⋊C4—44no—
443C4⋊C4—616no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—62no—
451C4⋊C4—64no—
452C4⋊C4—62no—
453C4⋊C4—64no—
454C4⋊C4—128no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—616no—
458C22⋊C4—64no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—128no—
464C42—24no—
465C42—24no—
466C42—68no—
467C42—68no—
468C42—64no—
469C42C2×D1214yes—
470C42C2×D1214yes—
471C42C2×D1214yes—
472C42C2×D1214yes—
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—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—44no—
491C3×C4—44no—
492C3×C4#52514yes—
493Dic3—28no—
494Dic3—42no—
495Dic3—42no—
496Dic3—42no—
497Dic3—44no—
498C23—22no—
499C23—22no—
500C23—62no—
501C23—64no—
502C23—64no—
503C23—64no—
504C23C22×D1212yes—
505C23#101014yes—
506C23C24×D311yescharacteristic
507Q8—616no—
508Q8—616no—
509Q8—124no—
510D4—128no—
511D4—124no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—38no—
526C2×C4—44no—
527C2×C4—616no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—68no—
537C2×C4—64no—
538C2×C4—128no—
539C2×C4—128no—
540C2×C4C22×D1212yes—
541C2×C4C22×D1212yes—
542C2×C4C22×D1212yes—
543C2×C4C22×D1212yes—
544C2×C4C22×D1212yes—
545C2×C4C22×D1212yes—
546C2×C4D3×#4114yes—
547C2×C3—24no—
548C2×C3C2×#52511yescharacteristic
549C2×C3C2×#53611yescharacteristic
550C2×C3#344911yescharacteristic
551C2×C3#351614yes—
552D3—82no—
553C22—28no—
554C22—22no—
555C22—22no—
556C22—128no—
557C22—122no—
558C22—122no—
559C22—122no—
560C22#552614yes—
561C22C23×D1211yescharacteristic
562C22C2×D3×#4112yes—
563C22C2×#101012yes—
564C22#596412yes—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—44no—
571C4—44no—
572C4—68no—
573C4—122no—
574C4—122no—
575C4—124no—
576C4—122no—
577C4#552414yes—
578C3#3524611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—242no—
581C2C2×#552611yescharacteristic
582C2C2×#552411yescharacteristic
583C2#8573511yescharacteristic
584C2#8612414yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2283C2
prime radicalC2×#930C22
hypercenter#2283D3
hypocenterC3#35246
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic6C2semidirect
3. C3×#2283C2semidirect
4. #74308C2semidirect
5. #74246C2semidirect
6. #74116C2semidirect
7. #73949C2semidirect
8. #73719C2semidirect
9. #73710C2semidirect
11. #73517C2semidirect
13. #73490C2semidirect
14. #73388C2semidirect
15. #73387C2semidirect
84. #2283D3semidirect
578. C3#35246semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
C22×C4×Dic62
C3×#22832
#743082
#742462
#741162
#739492
#737192
#737102
#735192
#735172
#734912
#734902
#733882
#733872
#352463