Finite Group Databasebeta

← #10828831

Subgroups — #10828831

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×C3×#552C211yesmaximal, characteristic
2C2×#5044C211yesmaximal, characteristic, complemented
3#82374C211yesmaximal, characteristic
4C22×#808C211yesmaximal, characteristic, complemented
5#71739C212yesmaximal, complemented
6#71696C211yesmaximal, characteristic, complemented
7#71694C212yesmaximal, complemented
8#71629C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic
10#71576C212yesmaximal, complemented
11#71553C211yesmaximal, characteristic, complemented
12#24095—31nomaximal, Sylow
13C3×#552—24no—
14C3×#552—24no—
15C2×C3×#126—21no—
16C2×C3×#125—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117—21no—
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30#5220—21no—
31#5212C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2211yescharacteristic
35#5044—24no—
36#4986—21no—
37C2×#808—24no—
38C2×#808C2212yes—
39C2×#808C2212yes—
40C2×#807C2212yes—
41#4949—21no—
42#4946—21no—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#552D311yescharacteristic
49C2×#484—31no—
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
60C22×C3×D4D411yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—21no—
65C3×#126—28no—
66C3×#125—28no—
67C3×#125—22no—
68C3×#125—22no—
69C3×#123—28no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8D414yes—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×#117—28no—
79C3×#117—22no—
80C3×#117—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D414yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2312yes—
108C2×Dic3⋊C4—22no—
109C22×C3⋊C8—22no—
110C22×C3⋊C8—24no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#835—24no—
114#835—24no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—21no—
118#830—22no—
119#830—24no—
120#808—24no—
121#808—24no—
122#807—24no—
123C23×C8—31no—
124#552—24no—
125#552—24no—
126C2×#126—21no—
127C2×#125—21no—
128C2×#125C2×D311yescharacteristic
129C2×#123—21no—
130C2×C22⋊Q8C2×D311yescharacteristic
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C4×Q8C2×D311yescharacteristic
133C2×C4×D4C2×D311yescharacteristic
134C2×#117—21no—
135C2×#117C2×D311yescharacteristic
136C22×C4⋊C4—31no—
137C22×C4⋊C4C2×D311yescharacteristic
138#484—64no—
139C2×#107—32no—
140C2×#106—32no—
141C2×#106—32no—
142C2×#106—64no—
143C2×#103—32no—
144C2×#103—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—32no—
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
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—42no—
190C22×Dic3—82no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D412yes—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C42—24no—
210C3×C42—24no—
211C3×C42—21no—
212C3×C42—21no—
213#217—42no—
214#217—82no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220Dic3⋊C4—82no—
221C2×C3⋊C8—28no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8#4114yes—
228C22×Q8C22×D311yescharacteristic
229C22×D4#6711yescharacteristic
230C23×C4—21no—
231C23×C4—61no—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—32no—
236C22×C8—64no—
237C22×C8—62no—
238#126—28no—
239#125—28no—
240#125—22no—
241#125—22no—
242#123—28no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8#6714yes—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×D4—22no—
250C4×D4—22no—
251#117—28no—
252#117—22no—
253#117—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4#6711yescharacteristic
268C2×C4⋊C4#6714yes—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4#6711yescharacteristic
278C2×C22⋊C4#6711yescharacteristic
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C22×D311yescharacteristic
282#107—64no—
283#106—64no—
284#106—64no—
285#103—64no—
286#103—64no—
287C22⋊C8—64no—
288C22⋊C8—62no—
289#95—62no—
290#95—61no—
291#95—62no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3#12012yes—
300C23×C3#13512yes—
301C23×C3#13712yes—
302C23×C3C22×D411yescharacteristic
303C3×Q8—24no—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×D4—44no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—48no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4#12012yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C2×#4112yes—
336C2×C3×C42−1+412yes—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—84no—
344C3⋊C8—24no—
345C3⋊C8—42no—
346C24C2×#6711yescharacteristic
347C24C2×#6711yescharacteristic
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8C2×#6712yes—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—122no—
371C22×C4—242no—
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6712yes—
377C22×C4C2×#6712yes—
378C22×C4C23×D311yescharacteristic
379C2×C8—34no—
380C2×C8—68no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—122no—
401C4⋊C4—242no—
402C4⋊C4C2×#6712yes—
403C4⋊C4C2×#6712yes—
404C4⋊C4C2×#6712yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—122no—
411C22⋊C4—242no—
412C42—24no—
413C42—24no—
414C42—21no—
415C42—21no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3#45214yes—
426C22×C3C2×#12011yescharacteristic
427C22×C3#54712yes—
428C22×C3C2×#13511yescharacteristic
429C22×C3C2×#13711yescharacteristic
430C22×C3#57712yes—
431C3×C4—22no—
432C3×C4—41no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—42no—
439C3×C4#52511yescharacteristic
440C3×C4#53611yescharacteristic
441C3×C4#54712yes—
442Dic3—82no—
443Dic3—82no—
444C23—22no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—21no—
451C23#95012yes—
452C23#95112yes—
453C23#95312yes—
454C23C22×#6711yescharacteristic
455Q8—24no—
456Q8—41no—
457Q8—41no—
458D4—44no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—48no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—244no—
491C2×C4#95312yes—
492C2×C4D3×#4112yes—
493C2×C4#101012yes—
494C2×C4C22×#6711yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—64no—
497C8—122no—
498C2×C3—24no—
499C2×C3—81no—
500C2×C3#240611yescharacteristic
501C2×C3C2×#45211yescharacteristic
502C2×C3#301011yescharacteristic
503C2×C3#333712yes—
504C2×C3#334412yes—
505C22—22no—
506C22—24no—
507C22—24no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22—42no—
513C22—41no—
514C22#523914yes—
515C22C2×#95011yescharacteristic
516C22C2×#95111yescharacteristic
517C22#582412yes—
518C22#583712yes—
519C22C2×#95311yescharacteristic
520C4—22no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—242no—
529C4—242no—
530C4#582412yes—
531C4#582511yescharacteristic
532C4#582811yescharacteristic
533C3#2409511yescharacteristic, complemented, Sylow
534C2—24no—
535C2—81no—
536C2#8359112yes—
537C2#8359312yes—
538C2C2×#523911yescharacteristic
539C2#8409011yescharacteristic
540C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#552C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#552D3
hypocenterC3#24095
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#5044C2semidirect
4. C22×#808C2semidirect
5. #71739C2semidirect
6. #71696C2semidirect
7. #71694C2semidirect
8. #71629C2semidirect
10. #71576C2semidirect
11. #71553C2semidirect
533. C3#24095semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×#50442
#823742
C22×#8082
#717392
#716962
#716942
#716292
#716102
#715762
#715532
#240953