Finite Group Databasebeta

← #10901360

Subgroups — #10901360

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5578C211yesmaximal, characteristic
2C2×#5020C211yesmaximal, characteristic, complemented
3C2×#4851C211yesmaximal, characteristic, complemented
4#74376C211yesmaximal, characteristic
5C3×#1803C211yesmaximal, characteristic, complemented
6#70475C211yesmaximal, characteristic
7#70246C212yesmaximal, complemented
8#70243C212yesmaximal
9#69588C211yesmaximal, characteristic, complemented
10#68827C212yesmaximal, complemented
11#68824C212yesmaximal
12#29688—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2211yescharacteristic
15C2×#944C2211yescharacteristic
16#5578—24no—
17#5578—24no—
18C2×#897—21no—
19C2×#897C2211yescharacteristic
20C2×#896—21no—
21C2×#889—21no—
22C2×#888C2211yescharacteristic
23C2×#872—21no—
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6—21no—
27C2×C4×Dic6C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C3×#384C2211yescharacteristic
31C2×#835C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#817C2212yes—
34C2×#816C2212yes—
35#5020—24no—
36#4987—21no—
37#4987C2212yes—
38#4984—21no—
39#4984C2212yes—
40#4973—21no—
41C2×#810C2212yes—
42C2×#808C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4851—24no—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#533—31no—
49C2×#475—31no—
50C2×#475—31no—
51#2857—31no—
52#2058—32no—
53#2043—32no—
54#2035—32no—
55#2035—32no—
56#1803—31no—
57#1803D311yescharacteristic, complemented
58#1797—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3D411yescharacteristic
62C22×C4×D3C2311yescharacteristic
63C22×Dic6—21no—
64C22×Dic6—21no—
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2312yes—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76#946—22no—
77#946—22no—
78#944—22no—
79#944—22no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98#897—22no—
99#897—22no—
100#897—44no—
101#896—28no—
102#889—28no—
103#888—22no—
104#888—22no—
105#872—28no—
106C42×D3—22no—
107C42×D3—22no—
108#869—22no—
109#869—22no—
110C4×Dic6—28no—
111C4×Dic6—22no—
112C4×Dic6D414yes—
113C3×C4⋊C8—22no—
114C3×#103—42no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118(C2×D3)⋊C8—42no—
119#817—24no—
120#816—24no—
121#810—42no—
122#808—24no—
123C4⋊(C3⋊C8)—22no—
124#533—64no—
125#533—64no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—31no—
128C2×C4⋊Q8—31no—
129C2×#125—61no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8—31no—
132C2×C22⋊Q8—61no—
133C2×C22⋊Q8—61no—
134C2×C4×Q8—31no—
135C2×C4×Q8—61no—
136C2×C4×Q8C2×D311yescharacteristic
137C2×#117—61no—
138C22×C42—31no—
139#475—64no—
140#475—64no—
141C2×#106—32no—
142C2×#106—32no—
143C2×C4⋊C8—31no—
144C2×C4⋊C8C2×D311yescharacteristic
145C2×#103—32no—
146C2×#103—32no—
147C2×#103—32no—
148C2×#103C2×D312yes—
149C2×C22⋊C8—32no—
150#391—61no—
151#389—32no—
152#389—61no—
153#384C2×D311yescharacteristic
154#380—32no—
155#380—61no—
156#340—32no—
157#340C2×D312yes—
158#336—32no—
159C23×D3C2×D411yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8C2×D412yes—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—22no—
188C2×C4×D3—41no—
189C2×C4×D3SD1614yes—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—24no—
195C2×Dic6—21no—
196C2×Dic6—42no—
197C2×Dic6—42no—
198C2×Dic6—42no—
199C2×Dic6—42no—
200C2×Dic6—42no—
201C2×Dic6C2×D412yes—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4C2×D412yes—
212C3×C42—24no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—44no—
218C4⋊Dic3—24no—
219C4⋊Dic3—21no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—42no—
223C4⋊Dic3—44no—
224C4⋊Dic3C2×D412yes—
225Dic3⋊C4—24no—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228Dic3⋊C4—24no—
229Dic3⋊C4—24no—
230C4×Dic3—22no—
231C4×Dic3—24no—
232C4×Dic3—22no—
233C4×Dic3—21no—
234C4×Dic3—21no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C22×Q8—31no—
239C22×Q8—61no—
240C22×Q8—61no—
241C22×Q8C22×D311yescharacteristic
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—31no—
245C22×C8—32no—
246C22×C8C22×D312yes—
247C4⋊Q8—62no—
248C4⋊Q8—62no—
249C4⋊Q8—62no—
250C4⋊Q8—62no—
251C4⋊Q8—62no—
252C4⋊Q8—62no—
253#125—68no—
254C22⋊Q8—62no—
255C22⋊Q8—68no—
256C22⋊Q8—62no—
257C22⋊Q8—62no—
258C22⋊Q8—62no—
259C22⋊Q8—124no—
260C4×Q8—22no—
261C4×Q8—34no—
262C4×Q8—68no—
263C4×Q8—62no—
264C4×Q8D1214yes—
265#117—68no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D312yes—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—62no—
284C2×C42—21no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—31no—
288C2×C42—61no—
289C2×C42—62no—
290C2×C42—62no—
291C2×C42C22×D311yescharacteristic
292#106—64no—
293#106—64no—
294C4⋊C8—22no—
295C4⋊C8—62no—
296#103—42no—
297#103—64no—
298#103—64no—
299#103—122no—
300C22⋊C8—122no—
301#95—62no—
302#95—62no—
303C23×C3C22×D411yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—21no—
310C22×D3C2×SD1612yes—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—41no—
314C3×Q8—41no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4#12012yes—
327C2×C3×C4#12012yes—
328C2×C3×C4C2×SD1612yes—
329C2×C3×C4#13712yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4H(5,𝔽2)11yescharacteristic
333C2×C3×C42−1+411yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335C2×C3×C42−1+411yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3C2×SD1612yes—
353C2×Dic3#13712yes—
354C4×D3—28no—
355C4×D3—44no—
356C4×D3—41no—
357C4×D3—41no—
358C4×D3—41no—
359C4×D3—41no—
360Dic6—22no—
361Dic6—22no—
362Dic6—48no—
363Dic6—41no—
364Dic6—41no—
365Dic6—84no—
366C3×C8—42no—
367C3⋊C8—42no—
368C24—31no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—32no—
375C2×Q8—64no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—122no—
382C2×Q8—122no—
383C2×Q8—122no—
384C2×Q8—122no—
385C2×Q8—122no—
386C2×Q8C2×D1212yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—34no—
393C22×C4—31no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—121no—
408C22×C4D3×D411yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4C23×D311yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—62no—
415C2×C8—62no—
416C2×C8—62no—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—32no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—64no—
425C4⋊C4—64no—
426C4⋊C4—64no—
427C4⋊C4—64no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—61no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—124no—
434C4⋊C4—122no—
435C4⋊C4C2×D1212yes—
436C22⋊C4—64no—
437C22⋊C4—64no—
438C22⋊C4—124no—
439C42—24no—
440C42—62no—
441C42—62no—
442C42—64no—
443C42—61no—
444C42—61no—
445C42C2×D1212yes—
446C42D3×D412yes—
447C22×C3C2×#12011yescharacteristic
448C22×C3#53611yescharacteristic
449C22×C3#53711yescharacteristic
450C22×C3C22×SD1611yescharacteristic
451C22×C3C2×#13711yescharacteristic
452C22×C3C2×#13711yescharacteristic
453C22×C3#57511yescharacteristic
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C3×C4—22no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—82no—
465C3×C4#44814yes—
466C3×C4#53611yescharacteristic
467C3×C4#53611yescharacteristic
468C3×C4#53711yescharacteristic
469C3×C4#53711yescharacteristic
470Dic3—22no—
471Dic3—22no—
472Dic3—42no—
473Dic3—41no—
474Dic3—42no—
475Dic3—82no—
476C23—32no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23C2×D3×D411yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—62no—
489Q8—62no—
490Q8—128no—
491Q8—121no—
492Q8—121no—
493Q8—244no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—68no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—124no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4#88212yes—
530C2×C4#88212yes—
531C2×C4D3×SD1612yes—
532C2×C4#91812yes—
533C2×C4C22×D1211yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4#100411yescharacteristic
536C2×C4#100711yescharacteristic
537C2×C4#101011yescharacteristic
538C2×C4#101011yescharacteristic
539C8—42no—
540C8—122no—
541C2×C3#261611yescharacteristic
542C2×C3C2×#44811yescharacteristic
543C2×C3#300711yescharacteristic
544C2×C3#322911yescharacteristic
545C2×C3#323711yescharacteristic
546C2×C3#324611yescharacteristic
547C2×C3#325511yescharacteristic
548D3—42no—
549C22—62no—
550C22—62no—
551C22—62no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22—121no—
556C22C2×#88211yescharacteristic
557C22#560811yescharacteristic
558C22#562811yescharacteristic
559C22C2×D3×SD1611yescharacteristic
560C22C2×#91811yescharacteristic
561C22#577311yescharacteristic
562C22D3×#13711yescharacteristic
563C4—22no—
564C4—42no—
565C4—42no—
566C4—62no—
567C4—62no—
568C4—82no—
569C4—122no—
570C4—121no—
571C4—122no—
572C4—242no—
573C4#480614yes—
574C4#551011yescharacteristic
575C4#552611yescharacteristic
576C4#560811yescharacteristic
577C4#562811yescharacteristic
578C3#2968811yescharacteristic, complemented, Sylow
579C2—122no—
580C2#7395011yescharacteristic
581C2C2×#480611yescharacteristic
582C2#7888511yescharacteristic
583C2#8036611yescharacteristic
584C2#8048711yescharacteristic
585C2#8101511yescharacteristic
586C2#8111411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1803D3
hypocenterC3#29688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
3. C2×#4851C2semidirect
5. C3×#1803C2semidirect
7. #70246C2semidirect
9. #69588C2semidirect
10. #68827C2semidirect
57. #1803D3semidirect
578. C3#29688semidirect

Maximal subgroups

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

subgroupindex
C2×#55782
C2×#50202
C2×#48512
#743762
C3×#18032
#704752
#702462
#702432
#695882
#688272
#688242
#296883