Finite Group Databasebeta

← #10763545

Subgroups — #10763545

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2D3×#431C211yesmaximal, characteristic
3D3×#423C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic, complemented
5#71775C211yesmaximal, characteristic, complemented
6#70779C211yesmaximal, characteristic, complemented
7#70507C211yesmaximal, characteristic, complemented
8#68423C214yesmaximal, complemented
9#68418C214yesmaximal, complemented
10#17708—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#946—21no—
13C2×Q8×Dic3—21no—
14C2×D3×M16—21no—
15#5728—21no—
16#5728—21no—
17#5728—21no—
18#5728—21no—
19#5728C412yes—
20#5728C412yes—
21#5728C2212yes—
22#5728C2212yes—
23C2×#897C412yes—
24C2×#892—21no—
25C2×#890—21no—
26C2×#887C412yes—
27C2×C42×D3C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C3×#431C2211yescharacteristic
30C3×#423C2211yescharacteristic
31#5118—21no—
32#4925C2211yescharacteristic
33#4924C2211yescharacteristic
34#4837—21no—
35#4837C2212yes—
36#4833C2212yes—
37D3×C4⋊C8—21no—
38D3×C4⋊C8C2212yes—
39D3×C4⋊C8C2212yes—
40D3×C22⋊C8—21no—
41#4718—21no—
42#4685—21no—
43C4×D3×C8—21no—
44C3×#332C2214yescomplemented
45#4547C2214yescomplemented
46#4481C2214yescomplemented
47#4481C2214yescomplemented
48C2×#533—31no—
49C2×#431—31no—
50C2×#423—31no—
51#1550—34no—
52#1549—31no—
53#1549D311yescharacteristic, complemented
54#1548—31no—
55#1548—31no—
56#1542—34no—
57C22×C3×Q8—21no—
58C22×C4×D3C2×C411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×Dic6C2×C412yes—
62C2×C3×M16—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8C2×C412yes—
66C3×C4⋊Q8C2312yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2×C412yes—
69C2×C3×C42C2311yescharacteristic
70#946—22no—
71#946—22no—
72#946—22no—
73#946—22no—
74Q8×Dic3—22no—
75Q8×Dic3—22no—
76Q8×Dic3—22no—
77Q8×Dic3—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—22no—
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3C2×C411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3D411yescharacteristic
87C2×#208—21no—
88D3×M16—44no—
89C2×ΓL(1,𝔽25)—22no—
90C2×D3×C8—22no—
91C2×D3×C8—22no—
92#897—22no—
93#897—22no—
94#897C2×C414yes—
95#897C2×C414yes—
96#892—22no—
97#892—22no—
98#892—22no—
99#892—22no—
100#890—22no—
101#890—22no—
102#890—22no—
103#890—22no—
104#887—22no—
105#887—22no—
106#887C2×C414yes—
107#887C2×C414yes—
108C42×D3C2×C411yescharacteristic
109C42×D3C2×C411yescharacteristic
110C42×D3D411yescharacteristic
111C42×D3D411yescharacteristic
112C42×D3D411yescharacteristic
113C42×D3D411yescharacteristic
114C42×D3C2311yescharacteristic
115C42×D3C2311yescharacteristic
116#869—21no—
117#869—21no—
118#869C2×C412yes—
119#869C2312yes—
120C3×C4⋊C8—21no—
121C3×C4⋊C8C2312yes—
122C3×C4⋊C8C2312yes—
123C3×C22⋊C8—21no—
124C3×C4×C8—21no—
125#830—21no—
126(C2×D3)⋊C8—22no—
127Dic3⋊C8—22no—
128Dic3⋊C8—22no—
129Dic3⋊C8—22no—
130C8×Dic3—22no—
131C4⋊(C3⋊C8)—21no—
132C4⋊(C3⋊C8)C2312yes—
133C4⋊(C3⋊C8)C2312yes—
134C4×C3⋊C8—21no—
135C22×M16—61no—
136#533—32no—
137#533—32no—
138#533—32no—
139#533—32no—
140#533—61no—
141#533—61no—
142#533—61no—
143#533—61no—
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8C2×D311yescharacteristic
146C2×#125—61no—
147C2×C22⋊Q8—32no—
148C2×C22⋊Q8—61no—
149C2×C4×Q8—32no—
150C2×C4×Q8—61no—
151C2×#117—61no—
152C22×C42—31no—
153#431—31no—
154#431—32no—
155#431—61no—
156#431—61no—
157#431C2×D311yescharacteristic
158#423—31no—
159#423—32no—
160#423—61no—
161#423—61no—
162#423C2×D311yescharacteristic
163C2×C4⋊C8—32no—
164C2×C4⋊C8—32no—
165C2×C4⋊C8—61no—
166C2×C22⋊C8—61no—
167C2×C4×C8—61no—
168#333—34no—
169#333—34no—
170#332—34no—
171#332C2×D314yescomplemented
172C23×D3C22⋊C411yescharacteristic
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C22×C3×C4—22no—
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×Dic3—22no—
181C22×Dic3C22⋊C411yescharacteristic
182C22×Dic3C22⋊C411yescharacteristic
183C22×Dic3C22⋊C411yescharacteristic
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×C4×D3C22⋊C411yescharacteristic
187C2×C4×D3C22⋊C411yescharacteristic
188C2×C4×D3C22⋊C411yescharacteristic
189C2×C4×D3C22⋊C411yescharacteristic
190C2×C4×D3C22⋊C411yescharacteristic
191C2×C4×D3C22⋊C411yescharacteristic
192C2×C4×D3SD1614yes—
193C2×C4×D3Q1614yes—
194C2×C4×D3C22×C411yescharacteristic
195C2×C4×D3C22×C411yescharacteristic
196C2×C4×D3C2×D411yescharacteristic
197C2×C4×D3C2×D411yescharacteristic
198C2×C4×D3C2×D411yescharacteristic
199C2×C4×D3C2×D411yescharacteristic
200C2×Dic6—22no—
201C2×Dic6—44no—
202C2×Dic6C22×C414yes—
203C3×M16—42no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4C22×C414yes—
210C3×C42C22×C411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2411yescharacteristic
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—24no—
221C4⋊Dic3—24no—
222Dic3⋊C4—24no—
223Dic3⋊C4—24no—
224C4×Dic3—24no—
225C4×Dic3—24no—
226C4×Dic3C22⋊C411yescharacteristic
227C4×Dic3C22⋊C411yescharacteristic
228C4×Dic3C22⋊C411yescharacteristic
229C4×Dic3C22⋊C411yescharacteristic
230C4×Dic3C22⋊C411yescharacteristic
231C4×Dic3C22⋊C411yescharacteristic
232C4×Dic3C22×C411yescharacteristic
233C4×Dic3C22×C411yescharacteristic
234C4×Dic3C2×D411yescharacteristic
235C4×Dic3C2×D411yescharacteristic
236C4×Dic3C2×D411yescharacteristic
237C4×Dic3C2×D411yescharacteristic
238#208—42no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241ΓL(1,𝔽25)—44no—
242D3×C8—44no—
243D3×C8—44no—
244C22×Q8—21no—
245C22×Q8—32no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C2×M16—21no—
250C2×M16—62no—
251C2×M16—61no—
252C2×M16—124no—
253C22×C8—62no—
254C22×C8—62no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—32no—
258C4⋊Q8—32no—
259C4⋊Q8—61no—
260C4⋊Q8—61no—
261C4⋊Q8C4×D312yes—
262C4⋊Q8C22×D312yes—
263#125—62no—
264#125—62no—
265#125—62no—
266#125—62no—
267C22⋊Q8—34no—
268C22⋊Q8—34no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272C22⋊Q8—62no—
273C22⋊Q8—62no—
274C22⋊Q8—62no—
275C4×Q8—34no—
276C4×Q8—34no—
277C4×Q8—62no—
278C4×Q8—62no—
279C4×Q8—62no—
280C4×Q8—62no—
281C4×Q8—62no—
282C4×Q8—62no—
283#117—62no—
284#117—62no—
285#117—62no—
286#117—62no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4C4×D312yes—
293C2×C22⋊C4—62no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—62no—
306C2×C42C22×D311yescharacteristic
307C4⋊C8—21no—
308C4⋊C8—32no—
309C4⋊C8—32no—
310C4⋊C8—62no—
311C4⋊C8—62no—
312C4⋊C8—62no—
313C4⋊C8—61no—
314C4⋊C8C22×D312yes—
315C4⋊C8C22×D312yes—
316C22⋊C8—21no—
317C22⋊C8—62no—
318C22⋊C8—61no—
319C4×C8—21no—
320C4×C8—62no—
321C4×C8—61no—
322C23×C3C2×C22⋊C411yescharacteristic
323C22×D3—41no—
324C22×D3—41no—
325C22×D3#10314yes—
326C22×D3C2×C22⋊C411yescharacteristic
327C22×D3C2×C22⋊C411yescharacteristic
328C3×Q8—44no—
329C3×Q8—44no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—44no—
334C2×C3×C4C2×C22⋊C411yescharacteristic
335C2×C3×C4C2×C22⋊C411yescharacteristic
336C2×C3×C4C2×C22⋊C411yescharacteristic
337C2×C3×C4C2×SD1612yes—
338C2×C3×C4C2×Q1612yes—
339C2×C3×C4C23×C411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×Dic3—24no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—44no—
346C2×Dic3#10314yes—
347C2×Dic3C2×C22⋊C411yescharacteristic
348C2×Dic3C2×C22⋊C411yescharacteristic
349C2×Dic3C2×C22⋊C411yescharacteristic
350C2×Dic3C2×C22⋊C411yescharacteristic
351C2×Dic3C2×C22⋊C411yescharacteristic
352C2×Dic3C2×C22⋊C411yescharacteristic
353C4×D3—22no—
354C4×D3—22no—
355C4×D3—22no—
356C4×D3—22no—
357C4×D3—41no—
358C4×D3—41no—
359C4×D3—41no—
360C4×D3—41no—
361C4×D3#10314yes—
362C4×D3#10314yes—
363C4×D3C2×SD1612yes—
364C4×D3C2×SD1612yes—
365C4×D3C2×Q1612yes—
366C4×D3C2×Q1612yes—
367Dic6—44no—
368Dic6—44no—
369C3×C8—42no—
370C3×C8—42no—
371C3⋊C8—42no—
372C3⋊C8—42no—
373C24—31no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—34no—
377C2×Q8—44no—
378C2×Q8—62no—
379C2×Q8—124no—
380C22×C4—22no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—34no—
385C22×C4—34no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—62no—
399C22×C4—122no—
400C22×C4—122no—
401C22×C4C2×C4×D311yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4D3×D411yescharacteristic
404M16—42no—
405M16—124no—
406M16—122no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—124no—
412C2×C8—124no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—64no—
417C4⋊C4—64no—
418C4⋊C4—64no—
419C4⋊C4—64no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4C2×C4×D314yes—
425C22⋊C4—64no—
426C22⋊C4—64no—
427C42—31no—
428C42—31no—
429C42—31no—
430C42—31no—
431C42—31no—
432C42—31no—
433C42—31no—
434C42—31no—
435C42—31no—
436C42—31no—
437C42—31no—
438C42—31no—
439C42—64no—
440C42—64no—
441C42C2×C4×D311yescharacteristic
442C42D3×D411yescharacteristic
443C42D3×D411yescharacteristic
444C42C23×D311yescharacteristic
445C22×C3—21no—
446C22×C3—21no—
447C22×C3C2×#10312yes—
448C22×C3C22×C22⋊C411yescharacteristic
449C2×D3—21no—
450C2×D3—21no—
451C2×D3—21no—
452C2×D3—21no—
453C2×D3—44no—
454C2×D3C2×#10312yes—
455C2×D3C2×#10312yes—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—44no—
459C3×C4#41312yes—
460C3×C4C2×#10312yes—
461C3×C4C22×SD1611yescharacteristic
462C3×C4C22×Q1611yescharacteristic
463C3×C4#57512yes—
464Dic3—21no—
465Dic3—21no—
466Dic3—21no—
467Dic3—21no—
468Dic3—44no—
469Dic3C2×#10312yes—
470Dic3C2×#10312yes—
471C23—31no—
472C23—31no—
473C23—34no—
474C23—121no—
475C23—121no—
476C23D3×C22⋊C411yescharacteristic
477Q8—44no—
478Q8—44no—
479Q8—124no—
480Q8—124no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—34no—
487C2×C4—32no—
488C2×C4—32no—
489C2×C4—32no—
490C2×C4—32no—
491C2×C4—34no—
492C2×C4—34no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—31no—
497C2×C4—44no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—124no—
510C2×C4D3×C22⋊C411yescharacteristic
511C2×C4D3×C22⋊C411yescharacteristic
512C2×C4D3×C22⋊C411yescharacteristic
513C2×C4D3×SD1612yes—
514C2×C4D3×Q1612yes—
515C2×C4C22×C4×D311yescharacteristic
516C2×C4C2×D3×D411yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C8—42no—
519C8—42no—
520C8—122no—
521C8—122no—
522C2×C3—22no—
523C2×C3C2×#41311yescharacteristic
524C2×C3C22×#10311yescharacteristic
525C2×C3#289611yescharacteristic
526D3—22no—
527D3—22no—
528C22—21no—
529C22—21no—
530C22—32no—
531C22—32no—
532C22—61no—
533C22—61no—
534C22—61no—
535C22—61no—
536C22—124no—
537C22D3×#10312yes—
538C22C2×D3×C22⋊C411yescharacteristic
539C4—21no—
540C4—21no—
541C4—32no—
542C4—32no—
543C4—44no—
544C4—61no—
545C4—61no—
546C4—61no—
547C4—61no—
548C4—124no—
549C4#475112yes—
550C4D3×#10312yes—
551C4C2×D3×SD1611yescharacteristic
552C4C2×D3×Q1611yescharacteristic
553C4#577312yes—
554C3#1770811yescharacteristic, complemented, Sylow
555C2—22no—
556C2—62no—
557C2—62no—
558C2C2×#475111yescharacteristic
559C2C2×D3×#10311yescharacteristic
560C2#7892411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1549C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1549D3
hypocenterC3#17708
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. C3×#1549C2semidirect
5. #71775C2semidirect
6. #70779C2semidirect
7. #70507C2semidirect
8. #68423C2semidirect
9. #68418C2semidirect
44. C3×#332C22semidirect
45. #4547C22semidirect
46. #4481C22semidirect
47. #4481C22semidirect
53. #1549D3semidirect
171. #332C2×D3semidirect
554. C3#17708semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
D3×#4312
D3×#4232
C3×#15492
#717752
#707792
#705072
#684232
#684182
#177083