Finite Group Databasebeta

← #10804133

Subgroups — #10804133

Order 768.

1554 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2D3×#381C211yesmaximal, characteristic, complemented
3#74050C211yesmaximal, characteristic, complemented
4C3×#1782C211yesmaximal, characteristic, complemented
5#72139C211yesmaximal, characteristic, complemented
6#70036C214yesmaximal, complemented
7#70029C214yesmaximal, complemented
8#69534C211yesmaximal, characteristic, complemented
9#69534C211yesmaximal, characteristic, complemented
10#21748—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13#5828—22no—
14#5828—22no—
15#5828—22no—
16#5828—22no—
17C2×#946C412yescomplemented
18C2×#938C412yescomplemented
19C2×#937C2212yes—
20C2×#935C2212yes—
21C2×C4×#67C414yescomplemented
22C2×#926C2211yescharacteristic
23C2×D3×C22⋊C4C2212yes—
24C2×#877C2212yes—
25C2×C3×#99C2214yes—
26C3×#381C2211yescharacteristic
27C3×#381C2211yescharacteristic
28C2×#834C2214yes—
29#5211C2211yescharacteristic
30#5211C2211yescharacteristic
31#4955C2212yes—
32#4943C2212yes—
33#4667—22no—
34#4666—22no—
35#4664—22no—
36D3×#95—22no—
37#4527C2214yes—
38#4527C2214yes—
39C2×#525—31no—
40#2280—31no—
41C2×#381—31no—
42#1899—34no—
43#1898—34no—
44#1820—31no—
45#1820—31no—
46#1782—31no—
47#1782D311yescharacteristic, complemented
48C24×D3C2311yescharacteristic
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—44no—
52C22×C3×Q8C2×C411yescharacteristic, complemented
53C22×C3×D4C2×C411yescharacteristic, complemented
54C22×C3×D4C2312yes—
55C23×C3×C4C2×C412yescomplemented
56C23×C3×C4C2311yescharacteristic
57C22×#67C2×C414yes—
58C23×Dic3C2311yescharacteristic
59C22×C4×D3—22no—
60C22×C4×D3—44no—
61C22×C4×D3C2×C412yes—
62C22×C4×D3C2×C412yes—
63C22×C4×D3D412yes—
64C22×C4×D3D412yes—
65C22×C4×D3D412yes—
66C22×C4×D3D412yes—
67C2×C3×C22⋊C4C2312yes—
68C2×C3×C22⋊C4C2312yes—
69C2×#217C2×C414yescomplemented
70C2×#217C2312yes—
71C2×#217C2312yes—
72C2×#217C2312yes—
73#946—22no—
74#946—22no—
75#946—22no—
76#946—22no—
77#938—22no—
78#938—22no—
79#938—22no—
80#938—22no—
81#937—22no—
82#937—24no—
83#937—22no—
84#935—24no—
85#935—22no—
86#935—22no—
87C4×#67—24no—
88C4×#67—24no—
89C4×#67—24no—
90C4×#67—24no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4C2×C414yes—
94#926—22no—
95#926—22no—
96C2×C4⋊Dic3C2×C412yescomplemented
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4C2×C414yescomplemented
99C2×C4×Dic3—22no—
100C2×C4×Dic3C2×C412yescomplemented
101D3×C22⋊C4—44no—
102#877—44no—
103C3×#99—44no—
104C3×#95—22no—
105C3×#95—22no—
106#834—44no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C22×#41C2×D311yescharacteristic
111C24×C4—31no—
112#525—62no—
113#525—62no—
114#525—62no—
115#525—62no—
116C2×#123—32no—
117C2×C22⋊Q8—32no—
118C2×#121—32no—
119C2×#120—32no—
120C2×C4×D4—34no—
121C2×#117—31no—
122C2×#117—32no—
123C22×C22⋊C4—32no—
124C2×#99—34no—
125C2×#99C2×D314yes—
126#381—31no—
127#381—31no—
128#381—32no—
129#381—62no—
130#381C2×D311yescharacteristic
131#381C2×D311yescharacteristic
132#380—62no—
133#379—32no—
134C4×C22⋊C4—62no—
135C2×#95—62no—
136#342—34no—
137#342—34no—
138C24×C3C22×C412yes—
139C24×C3C2411yescharacteristic
140C23×D3—24no—
141C23×D3—44no—
142C23×D3C22×C412yes—
143C23×D3C2×D411yescharacteristic
144C23×D3C2×D412yes—
145C23×D3C2×D411yescharacteristic
146C3×#41—48no—
147C3×#41—48no—
148C2×C3×Q8—22no—
149C2×C3×Q8—42no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—42no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4C22×C412yes—
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C22×C411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C2×#67—24no—
168C2×#67—24no—
169C2×#67—48no—
170C22×Dic3—24no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3C22×C412yes—
174C22×Dic3C22×C414yes—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×C4×D3—44no—
181C2×C4×D3—44no—
182C2×C4×D3—44no—
183C2×C4×D3—44no—
184C2×C4×D3—44no—
185C2×C4×D3—48no—
186C2×C4×D3C22⋊C412yes—
187C2×C4×D3C22⋊C412yes—
188C2×C4×D3C22⋊C412yes—
189C2×C4×D3C22⋊C412yes—
190C2×C4×D3C22⋊C412yes—
191C2×C4×D3C22⋊C412yes—
192C2×C4×D3C22⋊C412yes—
193C2×C4×D3C22⋊C412yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—24no—
196#217—22no—
197#217—22no—
198#217—24no—
199#217—24no—
200#217—24no—
201#217—24no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—48no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208Dic3⋊C4—24no—
209Dic3⋊C4—24no—
210Dic3⋊C4—44no—
211C4×Dic3—24no—
212C4×Dic3—44no—
213C25—31no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—44no—
217C22×Q8C4×D311yescharacteristic, complemented
218C22×D4—34no—
219C22×D4C4×D311yescharacteristic, complemented
220C22×D4C22×D312yes—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—32no—
224C23×C4—32no—
225C23×C4—32no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—62no—
229C23×C4—124no—
230C23×C4C4×D312yescomplemented
231C23×C4C22×D311yescharacteristic
232#123—64no—
233#123—62no—
234#123—62no—
235C22⋊Q8—62no—
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C22⋊Q8—62no—
239#121—62no—
240#121—62no—
241#121—62no—
242#121—62no—
243#120—62no—
244#120—64no—
245#120—62no—
246C4×D4—64no—
247C4×D4—64no—
248C4×D4—64no—
249C4×D4—64no—
250#117—62no—
251#117—62no—
252#117—124no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—34no—
255C2×C4⋊C4—62no—
256C2×C22⋊C4—34no—
257C2×C22⋊C4—32no—
258C2×C22⋊C4—34no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—32no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—124no—
264C2×C22⋊C4C22×D312yes—
265C2×C22⋊C4C22×D312yes—
266C2×C42—32no—
267C2×C42—62no—
268#99—44no—
269#99—124no—
270#95—22no—
271#95—22no—
272#95—64no—
273#95—62no—
274#95—62no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—42no—
279C23×C3—44no—
280C23×C3C23×C411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3H(5,𝔽2)14yes—
284C22×D3—22no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—22no—
288C22×D3—24no—
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—416no—
295C22×D3C2×C22⋊C412yes—
296C22×D3C2×C22⋊C412yes—
297C3×Q8—44no—
298C3×D4—44no—
299C3×D4—44no—
300C3×D4—44no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—48no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4C2×C22⋊C412yes—
314C2×C3×C4C2×C22⋊C412yes—
315#67—48no—
316#67—48no—
317C2×Dic3—24no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—44no—
323C2×Dic3—44no—
324C2×Dic3—44no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3C2×C22⋊C412yes—
328C2×Dic3C2×C22⋊C412yes—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—42no—
337C4×D3—88no—
338C24—31no—
339C24—32no—
340C24—32no—
341C24—31no—
342C24—64no—
343C24—124no—
344C24C2×C4×D312yes—
345C24C23×D311yescharacteristic
346#41—48no—
347#41—48no—
348C2×Q8—22no—
349C2×Q8—42no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—42no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—128no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—34no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4—124no—
384C22×C4—124no—
385C22×C4—124no—
386C22×C4—124no—
387C22×C4—124no—
388C22×C4—128no—
389C22×C4C2×C4×D312yes—
390C22×C4C2×C4×D311yescharacteristic
391C22×C4C2×C4×D311yescharacteristic
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—124no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—64no—
402C22⋊C4—64no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—64no—
408C22⋊C4—64no—
409C22⋊C4—124no—
410C22⋊C4—128no—
411C42—64no—
412C42—124no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—44no—
417C22×C3—42no—
418C22×C3—44no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3C22×C22⋊C411yescharacteristic
422C22×C3#51812yes—
423C22×C3#53412yes—
424C22×C3#53412yes—
425C2×D3—28no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—48no—
432C2×D3—48no—
433C2×D3—44no—
434C2×D3#41014yes—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—42no—
438C3×C4—82no—
439C3×C4—82no—
440Dic3—22no—
441Dic3—22no—
442Dic3—44no—
443Dic3—82no—
444Dic3—82no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—32no—
449C23—32no—
450C23—42no—
451C23—44no—
452C23—64no—
453C23—64no—
454C23—64no—
455C23—62no—
456C23—64no—
457C23—62no—
458C23—64no—
459C23—122no—
460C23—122no—
461C23—122no—
462C23—1216no—
463C23C22×C4×D311yescharacteristic
464C23C2×D3×D411yescharacteristic
465C23C2×D3×D411yescharacteristic
466C23#100414yes—
467Q8—44no—
468D4—44no—
469D4—44no—
470D4—44no—
471D4—128no—
472D4—128no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—64no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—124no—
501C2×C4—124no—
502C2×C4—124no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—248no—
506C2×C4D3×C22⋊C412yes—
507C2×C4D3×C22⋊C412yes—
508C2×C3—22no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—82no—
512C2×C3#230811yescharacteristic
513C2×C3C2×#41012yes—
514D3—24no—
515D3—42no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—34no—
520C22—44no—
521C22—44no—
522C22—42no—
523C22—68no—
524C22—62no—
525C22—62no—
526C22—62no—
527C22—62no—
528C22—62no—
529C22—82no—
530C22—82no—
531C22—128no—
532C22—128no—
533C22—124no—
534C22C2×D3×C22⋊C411yescharacteristic
535C22#548612yes—
536C22#558912yes—
537C22#558912yes—
538C4—22no—
539C4—22no—
540C4—42no—
541C4—62no—
542C4—62no—
543C4—82no—
544C4—82no—
545C4—124no—
546C4—242no—
547C4—242no—
548C3#2174811yescharacteristic, complemented, Sylow
549C2—22no—
550C2—22no—
551C2—22no—
552C2—64no—
553C2—82no—
554C2—122no—
555C2#7384911yescharacteristic
556C2D3×#41012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5828C2
cosocleC24×C3C24
hypercenter#1782D3
hypocenterC3#21748
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. D3×#381C2semidirect
3. #74050C2semidirect
4. C3×#1782C2semidirect
5. #72139C2semidirect
6. #70036C2semidirect
7. #70029C2semidirect
8. #69534C2semidirect
9. #69534C2semidirect
17. C2×#946C4semidirect
18. C2×#938C4semidirect
21. C2×C4×#67C4semidirect
47. #1782D3semidirect
52. C22×C3×Q8C2×C4semidirect
53. C22×C3×D4C2×C4semidirect
55. C23×C3×C4C2×C4semidirect
69. C2×#217C2×C4semidirect
96. C2×C4⋊Dic3C2×C4semidirect
98. C2×Dic3⋊C4C2×C4semidirect
100. C2×C4×Dic3C2×C4semidirect
217. C22×Q8C4×D3semidirect
219. C22×D4C4×D3semidirect
230. C23×C4C4×D3semidirect
548. C3#21748semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
D3×#3812
#740502
C3×#17822
#721392
#700362
#700292
#695342
#217483