Finite Group Databasebeta

← #10828832

Subgroups — #10828832

Order 768.

1080 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5642C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3#82197C211yesmaximal, characteristic
4C22×#808C211yesmaximal, characteristic, complemented
5#71696C211yesmaximal, characteristic, complemented
6#70633C212yesmaximal, complemented
7#70618C212yesmaximal, complemented
8#70274C212yesmaximal, complemented
9#70237C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic
11#69924C211yesmaximal, characteristic, complemented
12#24095—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#946C2211yescharacteristic
16C2×Q8×Dic3C2211yescharacteristic
17C2×C4×#67C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20#5642—24no—
21#5642—24no—
22C2×#898—21no—
23C2×#892C2211yescharacteristic
24C2×#890—21no—
25C2×#890C2211yescharacteristic
26C2×#886—21no—
27C2×#877—21no—
28C3×#484—24no—
29C2×C3×#107C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×#835C2212yes—
33#5213—21no—
34#5212C2211yescharacteristic
35#5210—21no—
36C2×(C2×D3)⋊C8C2212yes—
37#4993—21no—
38#4987C2212yes—
39#4985—21no—
40#4972C2212yes—
41C2×#808—24no—
42C2×#808C2212yes—
43C2×#808C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#552—31no—
49C2×#484D311yescharacteristic, complemented
50C22×#106—31no—
51#2487—31no—
52#2059—32no—
53#2035—32no—
54#1892—31no—
55#1869—31no—
56#1865—32no—
57#1854—32no—
58#1823—31no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67D411yescharacteristic
63C23×Dic3—21no—
64C22×C4×D3C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8D414yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×#217—21no—
76C2×#217D411yescharacteristic
77#946—22no—
78#946—22no—
79Q8×Dic3—22no—
80Q8×Dic3—22no—
81C4×#67—22no—
82C4×#67—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4D411yescharacteristic
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D414yes—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4D411yescharacteristic
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8—22no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8C2312yes—
105#898—28no—
106#892—22no—
107#892—22no—
108#890—28no—
109#890—22no—
110#890—22no—
111#886—28no—
112#877—28no—
113C3×#107—24no—
114C3×#103—24no—
115C3×C22⋊C8—22no—
116#835—24no—
117C3⋊#95—21no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120(C2×D3)⋊C8—24no—
121#808—24no—
122#808—24no—
123C23×C8—31no—
124#552—64no—
125#552—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#125—61no—
129C2×#123—61no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C4×Q8—31no—
133C2×C4×D4—31no—
134C2×#117—31no—
135C2×#117—61no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#484—24no—
139C2×#107C2×D312yes—
140C2×#106—32no—
141C2×#106—32no—
142C2×#106—64no—
143C2×#103—32no—
144C2×#103C2×D312yes—
145C2×C22⋊C8—32no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#399—61no—
149#389—32no—
150#386—61no—
151#385—61no—
152#380—31no—
153#380—32no—
154#340—32no—
155#340—32no—
156#340—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C23×D3C2×D411yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—42no—
170C22×C3×C4—82no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—22no—
178C22×Dic3—24no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—42no—
183C22×Dic3—42no—
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—24no—
201#217—22no—
202#217—42no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—22no—
206(C2×D3)⋊C4—82no—
207C4⋊Dic3—24no—
208C4⋊Dic3—24no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3C2×D412yes—
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—22no—
219Dic3⋊C4—82no—
220C4×Dic3—24no—
221C4×Dic3—24no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C2×C3⋊C8—28no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8#4114yes—
228C22×Q8C22×D311yescharacteristic
229C22×D4—31no—
230C23×C4—21no—
231C23×C4—31no—
232C23×C4—61no—
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—64no—
236C22×C8—62no—
237C22×C8C22×D312yes—
238#126—68no—
239#125—62no—
240#125—68no—
241#125—62no—
242#123—68no—
243C22⋊Q8—22no—
244C22⋊Q8—62no—
245C22⋊Q8—62no—
246C22⋊Q8#6714yes—
247C4×Q8—62no—
248C4×Q8—62no—
249C4×D4—62no—
250C4×D4—62no—
251#117—68no—
252#117—62no—
253#117—62no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4C22×D312yes—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—31no—
274C2×C22⋊C4—31no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279C2×C42—31no—
280C2×C42—61no—
281C2×C42—61no—
282#107—24no—
283#106—64no—
284#106—64no—
285#103—24no—
286#103—64no—
287C22⋊C8—22no—
288C22⋊C8—64no—
289#95—62no—
290#95—62no—
291#95—61no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#12012yes—
295C23×C3#13512yes—
296C23×C3#13712yes—
297C23×C3C22×D411yescharacteristic
298C22×D3—21no—
299C22×D3—22no—
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—22no—
303C3×Q8—24no—
304C3×Q8—41no—
305C3×Q8—41no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—84no—
318C2×C3×C4#12012yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4112yes—
322C2×C3×C42−1+412yes—
323#67—44no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—22no—
327C2×Dic3—21no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—22no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—48no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C4×D3—41no—
342C4×D3—41no—
343C4×D3—42no—
344C3×C8—42no—
345C3⋊C8—24no—
346C24—31no—
347C24D3×D411yescharacteristic
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8C2×#6712yes—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—62no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—31no—
363C22×C4—42no—
364C22×C4—64no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4—82no—
373C22×C4—122no—
374C22×C4—122no—
375C22×C4D3×D411yescharacteristic
376C22×C4D3×D412yes—
377C22×C4C2×#6712yes—
378C22×C4C23×D311yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—34no—
383C2×C8—68no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—32no—
391C4⋊C4—32no—
392C4⋊C4—42no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—242no—
404C4⋊C4C2×#6712yes—
405C22⋊C4—24no—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—122no—
411C22⋊C4—242no—
412C42—64no—
413C42—64no—
414C42—61no—
415C42—61no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3#45214yes—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54712yes—
423C22×C3C2×#13511yescharacteristic
424C22×C3C2×#13711yescharacteristic
425C22×C3#57712yes—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—41no—
429C2×D3—41no—
430C2×D3—41no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—82no—
435C3×C4#52511yescharacteristic
436C3×C4#53611yescharacteristic
437C3×C4#54712yes—
438Dic3—41no—
439Dic3—41no—
440Dic3—42no—
441Dic3—42no—
442Dic3—41no—
443Dic3—82no—
444C23—22no—
445C23—22no—
446C23—62no—
447C23—61no—
448C23—61no—
449C23—62no—
450C23—61no—
451C23#91812yes—
452C23#91912yes—
453C23#93712yes—
454C23C2×D3×D411yescharacteristic
455Q8—24no—
456Q8—41no—
457Q8—41no—
458D4—124no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—62no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—62no—
474C2×C4—61no—
475C2×C4—62no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—62no—
479C2×C4—84no—
480C2×C4—128no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4#93712yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4D3×#4112yes—
494C2×C4#101012yes—
495C2×C4C22×#6711yescharacteristic
496C8—42no—
497C8—64no—
498C2×C3—24no—
499C2×C3#240611yescharacteristic
500C2×C3C2×#45211yescharacteristic
501C2×C3#301011yescharacteristic
502C2×C3#333712yes—
503C2×C3#334412yes—
504D3—81no—
505C22—24no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—122no—
510C22—122no—
511C22—121no—
512C22—121no—
513C22—121no—
514C22#518714yes—
515C22#565512yes—
516C22C2×#91811yescharacteristic
517C22C2×#91911yescharacteristic
518C22#577512yes—
519C22C2×#93711yescharacteristic
520C4—22no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—121no—
525C4—121no—
526C4—122no—
527C4—122no—
528C4—121no—
529C4—242no—
530C4#564811yescharacteristic
531C4#565611yescharacteristic
532C4#582412yes—
533C3#2409511yescharacteristic, complemented, Sylow
534C2—24no—
535C2—241no—
536C2#8146412yes—
537C2#8146512yes—
538C2#8284111yescharacteristic
539C2C2×#518711yescharacteristic
540C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#484C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenterC2×#484D3
hypocenterC3#24095
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C2×C3×#484C2semidirect
4. C22×#808C2semidirect
5. #71696C2semidirect
6. #70633C2semidirect
7. #70618C2semidirect
8. #70274C2semidirect
9. #70237C2semidirect
11. #69924C2semidirect
49. C2×#484D3semidirect
533. C3#24095semidirect

Maximal subgroups

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

subgroupindex
C2×#56422
C2×C3×#4842
#821972
C22×#8082
#716962
#706332
#706182
#702742
#702372
#699522
#699242
#240953