Finite Group Databasebeta

← #10829244

Subgroups — #10829244

Order 768.

1184 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3C2×#5026C211yesmaximal, characteristic, complemented
4#82415C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71699C212yesmaximal, complemented
7#71695C212yesmaximal, complemented
8#71627C211yesmaximal, characteristic, complemented
9#71615C211yesmaximal, characteristic, complemented
10#71560C212yesmaximal, complemented
11#71555C212yesmaximal, complemented
12#24130—31nomaximal, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2212yes—
23C22×#215C2211yescharacteristic
24C2×#930C2212yes—
25C2×C4×D12C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2212yes—
28#5223—21no—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5087—24no—
33#5026—24no—
34#4992—21no—
35C2×#809C2211yescharacteristic
36C2×#809C2212yes—
37C2×#807C2211yescharacteristic
38C2×#807C2212yes—
39#4956—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554D311yescharacteristic, complemented
47C2×#474—31no—
48C2×#440—31no—
49#2535—31no—
50#2036—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1852—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4C2312yes—
62C22×#67—22no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yescomplemented
67C3×#125—28no—
68C3×C22⋊Q8—216no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yescomplemented
73C3×C4×D4—28no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—22no—
90C2×#215—22no—
91C2×#215—22no—
92C2×#215—22no—
93#930—24no—
94#930—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C4×D12—22no—
103C4×D12—22no—
104#835—22no—
105#835—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#830—24no—
110#830—42no—
111#809—22no—
112#809—24no—
113#807—22no—
114#807—24no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118#554—24no—
119#554—24no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—21no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8C2×D312yes—
124C2×#121—32no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4C2×D312yes—
130#474—64no—
131#440—64no—
132C2×#107—31no—
133C2×#107—32no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#397—61no—
143#392—61no—
144#386—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C23×D3—21no—
152C2×C3×Q8—24no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—82no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—28no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—82no—
173C22×C3×C4—82no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—82no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12#4112yes—
189C2×C4×D3—42no—
190C2×C4×D3—82no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C2×D412yes—
199C3×C4⋊C4C2×D414yes—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C42—24no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205#217—82no—
206#215—28no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—42no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8#4112yes—
222C22×Q8—21no—
223C22×Q8C22×D311yescharacteristic
224C22×D4—31no—
225C22×D4—62no—
226C22×D4#6711yescharacteristic
227C23×C4—22no—
228C23×C4—61no—
229C23×C4C22×D312yes—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C22×C8—31no—
234C22×C8—31no—
235C22×C8—32no—
236C4⋊Q8—22no—
237C4⋊Q8#6714yescomplemented
238#125—28no—
239C22⋊Q8—216no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242#121—64no—
243#121—122no—
244C4×Q8—22no—
245C4×Q8#6714yescomplemented
246C4×D4—28no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—62no—
250C4×D4—62no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4#6711yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4#6712yes—
267C2×C42—21no—
268C2×C42C22×D311yescharacteristic
269#107—62no—
270#107—64no—
271C4⋊C8—62no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—64no—
276C22⋊C8—64no—
277C22⋊C8—122no—
278#97—62no—
279#95—62no—
280#95—61no—
281#95—61no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3#13714yes—
289C23×C3C22×D411yescharacteristic
290C22×D3—22no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—42no—
294C22×D3—42no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×Q8—84no—
300C3×D4—28no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—88no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12012yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4H(5,𝔽2)11yescharacteristic
326C2×C3×C42−1+411yescharacteristic
327#67—88no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333D12—22no—
334D12—22no—
335D12—41no—
336D12—41no—
337C4×D3—84no—
338C3⋊C8—22no—
339C3⋊C8—41no—
340C3⋊C8—42no—
341C24—61no—
342C24C2×#6712yes—
343C2×Q8—24no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—82no—
350C2×Q8C2×#6712yes—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—32no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—124no—
360C2×D4—124no—
361C2×D4—242no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—62no—
373C22×C4—61no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—122no—
377C22×C4—242no—
378C22×C4C2×#6712yes—
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—68no—
383C2×C8—32no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—62no—
387C2×C8—61no—
388C2×C8—62no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—62no—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4C2×#6712yes—
404C4⋊C4C2×#6714yes—
405C22⋊C4—28no—
406C22⋊C4—24no—
407C22⋊C4—124no—
408C22⋊C4—122no—
409C22⋊C4—242no—
410C42—24no—
411C42C2×#6712yes—
412C42C2×#6712yes—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#54611yescharacteristic
422C22×C3#54711yescharacteristic
423C22×C3C2×#13712yes—
424C22×C3#57612yes—
425C2×D3—42no—
426C2×D3—44no—
427C2×D3—82no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—44no—
436C3×C4#52511yescharacteristic
437C3×C4#53711yescharacteristic
438C3×C4#54811yescharacteristic
439C3×C4#55011yescharacteristic
440Dic3—81no—
441Dic3—82no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—62no—
449C23—122no—
450C23—121no—
451C23—122no—
452C23—121no—
453C23#94214yes—
454C23C22×#6711yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—84no—
460D4—28no—
461D4—62no—
462D4—62no—
463D4—121no—
464D4—121no—
465D4—248no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—88no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—244no—
490C2×C4#95312yes—
491C2×C4#95312yes—
492C2×C4D3×#4111yescharacteristic
493C2×C4D3×#4111yescharacteristic
494C2×C4#100911yescharacteristic
495C2×C4#101011yescharacteristic
496C2×C4C22×#6711yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—62no—
499C8—121no—
500C8—122no—
501C2×C3—44no—
502C2×C3#240711yescharacteristic
503C2×C3#300411yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#331412yes—
506C2×C3#334612yes—
507D3—82no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—122no—
516C22—124no—
517C22—241no—
518C22—241no—
519C22—242no—
520C22C2×#94212yes—
521C22#582411yescharacteristic
522C22#582911yescharacteristic
523C22#583512yes—
524C22C2×#95311yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—44no—
531C4—241no—
532C4—242no—
533C4#582811yescharacteristic
534C4#583111yescharacteristic
535C4#583311yescharacteristic
536C4#583411yescharacteristic
537C3#2413011yescharacteristic, complemented, Sylow
538C2—44no—
539C2—242no—
540C2#8362512yes—
541C2#8363512yes—
542C2#8405911yescharacteristic
543C2#8409711yescharacteristic
544C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenterC2×#554D3
hypocenterC3#24130
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C3×#554C2semidirect
2. C2×#5087C2semidirect
3. C2×#5026C2semidirect
5. #71788C2semidirect
6. #71699C2semidirect
7. #71695C2semidirect
8. #71627C2semidirect
9. #71615C2semidirect
10. #71560C2semidirect
11. #71555C2semidirect
46. C2×#554D3semidirect
66. C3×C4⋊Q8D4semidirect
72. C3×C4×Q8D4semidirect
237. C4⋊Q8#67semidirect
245. C4×Q8#67semidirect
537. C3#24130semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×#50872
C2×#50262
#824152
#717882
#716992
#716952
#716272
#716152
#715602
#715552
#241303