Finite Group Databasebeta

← #10823417

Subgroups — #10823417

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5026C212yesmaximal, complemented
3C4×#930C211yesmaximal, characteristic
4#71792C211yesmaximal, characteristic, complemented
5#71791C211yesmaximal, characteristic, complemented
6#71782C211yesmaximal, characteristic, complemented
7#71781C211yesmaximal, characteristic, complemented
8#71699C212yesmaximal
9#71695C212yesmaximal, complemented
10#71631C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#23574—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#215C2212yes—
24C2×#930C2211yescharacteristic
25C2×C4×#67—21no—
26C2×C4×D12C2211yescharacteristic
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#835C2212yes—
30#5213—21no—
31#5212C2211yescharacteristic
32C2×#830C2212yes—
33#5026—28no—
34C2×#809C2212yes—
35C2×#807C2212yes—
36#4940—21no—
37#4939C2211yescharacteristic
38C4×(C2×D3)⋊C4—21no—
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×#803C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533D311yescharacteristic, complemented
47C2×#440—32no—
48C4×#121—31no—
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2023—31no—
53#2013—31no—
54#2000—32no—
55#1763—32no—
56#1749—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—24no—
67C3×#125—28no—
68C3×C22⋊Q8—28no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—24no—
71C3×C4×Q8—28no—
72C3×C4×Q8—24no—
73C3×C4×Q8D418yescomplemented
74C3×#117—28no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—22no—
83C2×C3×C42—24no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#215—24no—
90C2×#215—24no—
91C2×#215—24no—
92#930—22no—
93#930—22no—
94C4×#67—44no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106C4×D12—24no—
107#835—24no—
108#835—24no—
109#835—24no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#830—24no—
114#809—24no—
115#807—24no—
116#803—24no—
117C22×SD16—32no—
118#533—28no—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2×D312yes—
124C2×#121—31no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2×D312yes—
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C2×#117—21no—
130C22×C42C2×D311yescharacteristic
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#102—32no—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#390—31no—
140#389—31no—
141#388—61no—
142#386—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C22⋊C4—61no—
146#340—32no—
147#340—32no—
148#336—31no—
149#336—31no—
150C24×C3C2×D411yescharacteristic
151C23×D3—21no—
152C2×C3×Q8—24no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8C2×D414yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—41no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—42no—
181C2×#67—42no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—21no—
188C2×D12#4112yes—
189C2×C4×D3—42no—
190C2×C4×D3—42no—
191C2×C4×D3—42no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C2×D414yes—
199C3×C22⋊C4—28no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—28no—
203C3×C42—22no—
204C3×C42C2×D414yes—
205#217—42no—
206#215—216no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—42no—
210(C2×D3)⋊C4—42no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3#4112yes—
215Dic3⋊C4—42no—
216C4×Dic3—42no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8#4114yes—
223C22×Q8—21no—
224C22×Q8C22×D312yes—
225C22×D4—31no—
226C22×D4—61no—
227C23×C4—61no—
228C23×C4#6711yescharacteristic
229C23×C4#6711yescharacteristic
230C23×C4C22×D311yescharacteristic
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—64no—
234C22×C8—32no—
235C22×C8—32no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238#125—28no—
239C22⋊Q8—28no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242#121—62no—
243#121—62no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8#6718yescomplemented
247C4×D4—64no—
248C4×D4—124no—
249#117—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C22×D312yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—61no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—61no—
269C2×C42#6711yescharacteristic
270C2×C42#6711yescharacteristic
271C2×C42C22×D311yescharacteristic
272C2×C42C22×D311yescharacteristic
273#107—64no—
274#103—64no—
275#103—64no—
276#103—64no—
277#102—64no—
278C22⋊C8—64no—
279#97—64no—
280#95—61no—
281#95—61no—
282#95—61no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3#13712yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—48no—
301C3×Q8—42no—
302C3×Q8—42no—
303C2×C3×C4—28no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12012yes—
326C2×C3×C4AGL(1,ℤ/8)12yes—
327C2×C3×C4#13712yes—
328C2×C3×C4C22×D412yes—
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4112yes—
331C2×C3×C4C2×#4111yescharacteristic
332#67—48no—
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342D12—24no—
343D12—42no—
344C4×D3—84no—
345C3⋊C8—24no—
346C3⋊C8—42no—
347C24—61no—
348C24C2×#6711yescharacteristic
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8C2×#6714yes—
358C2×D4—32no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—61no—
362C2×D4—62no—
363C2×D4—122no—
364C2×D4—122no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—41no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C23×D311yescharacteristic
389SD16—616no—
390C2×C8—34no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—61no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C4⋊C4C2×#6714yes—
408C22⋊C4—28no—
409C22⋊C4—64no—
410C22⋊C4—122no—
411C22⋊C4—122no—
412C22⋊C4—122no—
413C22⋊C4—122no—
414C42—22no—
415C42—22no—
416C42—28no—
417C42—22no—
418C42—122no—
419C42C2×#6714yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3C2×#12011yescharacteristic
428C22×C3#52511yescharacteristic
429C22×C3#53211yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3C2×#13711yescharacteristic
432C22×C3C2×#13711yescharacteristic
433C22×C3#57511yescharacteristic
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C2×D3—81no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—41no—
446C3×C4—44no—
447C3×C4—44no—
448C3×C4#52512yes—
449C3×C4#53212yes—
450Dic3—42no—
451Dic3—81no—
452Dic3—81no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—62no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—121no—
465C23—121no—
466C23#94212yes—
467C23C22×#6711yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473D4—64no—
474D4—128no—
475D4—122no—
476C2×C4—28no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—62no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—244no—
507C2×C4#94212yes—
508C2×C4#94912yes—
509C2×C4#95312yes—
510C2×C4#95312yes—
511C2×C4D3×#4111yescharacteristic
512C2×C4D3×#4112yes—
513C2×C4D3×#4111yescharacteristic
514C2×C4C22×#6712yes—
515C8—64no—
516C8—122no—
517C2×C3—42no—
518C2×C3#238811yescharacteristic
519C2×C3#300411yescharacteristic
520C2×C3#300611yescharacteristic
521C2×C3#311011yescharacteristic
522C2×C3#311611yescharacteristic
523C2×C3#318111yescharacteristic
524C2×C3#318211yescharacteristic
525D3—82no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—241no—
537C22—241no—
538C22—241no—
539C22—241no—
540C22C2×#94211yescharacteristic
541C22C2×#94211yescharacteristic
542C22C2×#94911yescharacteristic
543C22#582211yescharacteristic
544C22#582811yescharacteristic
545C22#583211yescharacteristic
546C22C2×#95311yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—41no—
553C4—44no—
554C4—122no—
555C4—241no—
556C4—241no—
557C4#582812yes—
558C4#583212yes—
559C3#2357411yescharacteristic, complemented, Sylow
560C2—42no—
561C2—242no—
562C2#8334611yescharacteristic
563C2#8334711yescharacteristic
564C2#8335311yescharacteristic
565C2#8335411yescharacteristic
566C2#8405911yescharacteristic
567C2#8408211yescharacteristic
568C2#8409511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#533D3
hypocenterC3#23574
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. C2×#5026C2semidirect
4. #71792C2semidirect
5. #71791C2semidirect
6. #71782C2semidirect
7. #71781C2semidirect
9. #71695C2semidirect
10. #71631C2semidirect
11. #71529C2semidirect
46. C2×#533D3semidirect
73. C3×C4×Q8D4semidirect
246. C4×Q8#67semidirect
559. C3#23574semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50262
C4×#9302
#717922
#717912
#717822
#717812
#716992
#716952
#716312
#715292
#235743