Finite Group Databasebeta

← #10827619

Subgroups — #10827619

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#552C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic
4C2×#5004C211yesmaximal, characteristic, complemented
5#71744C212yesmaximal, complemented
6#71700C212yesmaximal, complemented
7#71697C211yesmaximal, characteristic, complemented
8#71632C212yesmaximal, complemented
9#71611C211yesmaximal, characteristic
10#71575C211yesmaximal, characteristic, complemented
11#71559C212yesmaximal, complemented
12#23983—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C3×#552—24no—
15C3×#552—24no—
16C2×C3×#126—21no—
17C2×C3×#125—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×#216C2212yes—
28C2×#924C2212yes—
29C22×#208C2211yescharacteristic
30C2×#835C2212yes—
31#5220—21no—
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35#5050—24no—
36#5004—24no—
37#4979—21no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41#4947—21no—
42#4946—21no—
43#4929C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#552D311yescharacteristic
49C2×#451—31no—
50C2×#419—31no—
51#2409—31no—
52#2026—32no—
53#2001—32no—
54#1894—31no—
55#1878—32no—
56#1872—32no—
57#1870—31no—
58#1825—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4D411yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—21no—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—22no—
68C22×Dic6—21no—
69C22×Dic6—42no—
70C22×Dic6C2312yes—
71C3×#126—28no—
72C3×#125—28no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#123—28no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8D414yes—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×#117—28no—
85C3×#117—22no—
86C3×#117D414yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C42—21no—
105C2×C3×C42—21no—
106C2×C3×C42C2311yescharacteristic
107C2×#217—21no—
108C2×#217—22no—
109C2×#216—24no—
110C2×#216—24no—
111C2×#216—42no—
112C2×C4⋊Dic3C2312yes—
113#924—24no—
114#924—42no—
115C2×Dic3⋊C4—22no—
116C2×#208—22no—
117C2×#208—42no—
118C22×C3⋊C8C2312yes—
119C22×C3⋊C8C2312yes—
120#835—42no—
121C3⋊#95—22no—
122C3⋊#95—22no—
123C3⋊#95—21no—
124#830—22no—
125#830—42no—
126#810—24no—
127#810—24no—
128#810—24no—
129C23×Q8—31no—
130C22×Q16—32no—
131C22×M16—31no—
132#552—24no—
133#552—24no—
134C2×#126—21no—
135C2×#125—21no—
136C2×#125C2×D311yescharacteristic
137C2×#123—21no—
138C2×C22⋊Q8—32no—
139C2×C22⋊Q8C2×D311yescharacteristic
140C2×C22⋊Q8C2×D311yescharacteristic
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×D4C2×D311yescharacteristic
143C2×#117—21no—
144C2×#117C2×D311yescharacteristic
145C22×C4⋊C4C2×D311yescharacteristic
146#451—64no—
147#419—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×C22⋊C8—31no—
153C2×C22⋊C8—32no—
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—41no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—44no—
196C22×Dic3—42no—
197C2×Dic6—28no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—48no—
204C2×Dic6—44no—
205C2×Dic6—44no—
206C2×Dic6—42no—
207C2×Dic6#4114yes—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4C2×D412yes—
219C3×C4⋊C4C2×D412yes—
220C3×C4⋊C4C2×D412yes—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C42—24no—
227C3×C42—24no—
228C3×C42—21no—
229C3×C42—21no—
230#217—44no—
231#217—42no—
232#216—48no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235Dic3⋊C4—44no—
236#208—44no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C22×Q8—32no—
244C22×Q8—64no—
245C22×Q8—61no—
246C22×Q8—62no—
247C22×Q8—62no—
248C22×Q8—122no—
249C22×Q8C22×D311yescharacteristic
250C22×D4#6711yescharacteristic
251C23×C4—21no—
252C23×C4—61no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C2×Q16—64no—
256C2×Q16—64no—
257C2×Q16—122no—
258C2×M16—62no—
259C2×M16—122no—
260C22×C8—32no—
261C22×C8—32no—
262#126—28no—
263#125—28no—
264#125—22no—
265#125—22no—
266#123—28no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—64no—
271C22⋊Q8—122no—
272C22⋊Q8#6714yes—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—22no—
276C4×D4—22no—
277#117—28no—
278#117—22no—
279#117#6714yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4#6711yescharacteristic
300C2×C22⋊C4#6711yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—64no—
307#103—122no—
308C22⋊C8—62no—
309C22⋊C8—122no—
310#95—61no—
311#95—62no—
312#95—62no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3#12012yes—
322C23×C3#13712yes—
323C23×C3C22×D411yescharacteristic
324C3×Q8—24no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×D4—44no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—48no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4#12012yes—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4112yes—
357C2×C3×C4H(5,𝔽2)12yes—
358C2×Dic3—24no—
359C2×Dic3—48no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365Dic6—24no—
366Dic6—24no—
367Dic6—48no—
368Dic6—42no—
369Dic6—42no—
370C3⋊C8—42no—
371C3⋊C8—42no—
372C24C2×#6711yescharacteristic
373C24C2×#6711yescharacteristic
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—34no—
378C2×Q8—68no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—62no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×Q8—124no—
385C2×Q8—128no—
386C2×Q8—124no—
387C2×Q8—122no—
388C2×Q8C2×#6712yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—124no—
408C22×C4—122no—
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6712yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416Q16—128no—
417M16—124no—
418C2×C8—62no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—124no—
437C4⋊C4C2×#6712yes—
438C4⋊C4C2×#6712yes—
439C4⋊C4C2×#6712yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—124no—
446C22⋊C4—122no—
447C42—24no—
448C42—24no—
449C42—21no—
450C42—21no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#54612yes—
463C22×C3C2×#13711yescharacteristic
464C22×C3#57511yescharacteristic
465C22×C3#57812yes—
466C3×C4—22no—
467C3×C4—41no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4#52511yescharacteristic
475C3×C4#53511yescharacteristic
476C3×C4#54612yes—
477Dic3—44no—
478Dic3—82no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23#95112yes—
488C23#95312yes—
489C23C22×#6711yescharacteristic
490Q8—24no—
491Q8—41no—
492Q8—41no—
493Q8—64no—
494Q8—64no—
495Q8—128no—
496Q8—122no—
497Q8—122no—
498D4—44no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—64no—
525C2×C4—128no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#95312yes—
532C2×C4D3×#4112yes—
533C2×C4#100912yes—
534C2×C4C22×#6711yescharacteristic
535C2×C4C22×#6711yescharacteristic
536C8—122no—
537C8—122no—
538C2×C3—42no—
539C2×C3—42no—
540C2×C3#240411yescharacteristic
541C2×C3#300711yescharacteristic
542C2×C3#301111yescharacteristic
543C2×C3#331212yes—
544C2×C3#332712yes—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—22no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—41no—
555C22#582211yescharacteristic
556C22C2×#95111yescharacteristic
557C22#582912yes—
558C22#583812yes—
559C22C2×#95311yescharacteristic
560C4—22no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—124no—
569C4—242no—
570C4#582811yescharacteristic
571C4#582912yes—
572C4#583011yescharacteristic
573C3#2398311yescharacteristic, complemented, Sylow
574C2—42no—
575C2—42no—
576C2#8360312yes—
577C2#8360512yes—
578C2#8408411yescharacteristic
579C2#8409611yescharacteristic
580C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#552C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#552D3
hypocenterC3#23983
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×#5050C2semidirect
4. C2×#5004C2semidirect
5. #71744C2semidirect
6. #71700C2semidirect
7. #71697C2semidirect
8. #71632C2semidirect
10. #71575C2semidirect
11. #71559C2semidirect
573. C3#23983semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×#50502
#823492
C2×#50042
#717442
#717002
#716972
#716322
#716112
#715752
#715592
#239833