Finite Group Databasebeta

← #10827626

Subgroups — #10827626

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×#5541C211yesmaximal, characteristic
2C3×#2409C211yesmaximal, characteristic
3C2×#5228C211yesmaximal, characteristic, complemented
4C2×#5225C211yesmaximal, characteristic, complemented
5#71700C212yesmaximal, complemented
6#71692C212yesmaximal, complemented
7#71611C211yesmaximal, characteristic
8#69690C212yesmaximal, complemented
9#69669C212yesmaximal, complemented
10#69516C211yesmaximal, characteristic, complemented
11#69503C211yesmaximal, characteristic, complemented
12#23983—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C22×#216C2212yes—
17C2×#935—21no—
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C22×#208C2211yescharacteristic
23#5541—24no—
24#5541—24no—
25C2×#889—21no—
26C2×#879—21no—
27C2×#877—21no—
28C2×#870C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#395—21no—
31C3×#393—21no—
32C3×#391C2212yes—
33C3×#386—21no—
34C3×#385—21no—
35C3×#380C2211yescharacteristic
36#5228—24no—
37#5225—24no—
38C2×#835C2212yes—
39C2×#835C2212yes—
40C2×#835C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2212yes—
43C2×#810C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#552—31no—
49C2×#451—31no—
50C2×#419—31no—
51#2409D311yescharacteristic
52#2026—32no—
53#2001—32no—
54#1894—31no—
55#1878—32no—
56#1872—32no—
57#1870—31no—
58#1825—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3—21no—
70C22×Dic6C2311yescharacteristic
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—42no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84C2×#217—21no—
85C2×#217—21no—
86C2×#216—24no—
87C2×#216—24no—
88C2×#216—42no—
89#935—28no—
90#926—22no—
91#926D414yes—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98#924—22no—
99#924—22no—
100#924—22no—
101#924D414yes—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×#208—22no—
110C2×#208—42no—
111C22×C3⋊C8C2312yes—
112C22×C3⋊C8C2312yes—
113#889—28no—
114#879—28no—
115#877—28no—
116#870—22no—
117#870—22no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×#95—21no—
121C3×#95—22no—
122C3×#95—22no—
123#835—24no—
124#835—24no—
125#835—24no—
126#830—22no—
127#830—42no—
128#810—42no—
129C23×Q8C2×D311yescharacteristic
130C22×Q16—32no—
131C22×M16—31no—
132#552—64no—
133#552—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#125—61no—
137C2×#123—61no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8C2×D312yes—
141C2×C4×Q8—31no—
142C2×C4×D4C2×D311yescharacteristic
143C2×#117—31no—
144C2×#117—61no—
145C22×C4⋊C4—31no—
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—21no—
155#393—21no—
156#391C2×D312yes—
157#386—21no—
158#385—21no—
159#380C2×D311yescharacteristic
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C2×C3×Q8—22no—
167C2×C3×Q8—28no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—44no—
173C2×C3×Q8—48no—
174C2×C3×Q8—44no—
175C2×C3×Q8—42no—
176C2×C3×Q8#4114yes—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—41no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—44no—
189C22×C3×C4—42no—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—24no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—21no—
207C2×Dic6C2×D412yes—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—44no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—44no—
215C3×C22⋊C4—42no—
216C3×C42—21no—
217C3×C42—21no—
218#217—24no—
219#217—24no—
220#217—24no—
221#216—48no—
222C4⋊Dic3—24no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3C2×D412yes—
227C4⋊Dic3C2×D412yes—
228C4⋊Dic3C2×D412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234C4×Dic3—24no—
235C4×Dic3—24no—
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—21no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—31no—
248C22×Q8—42no—
249C22×Q8C22×D312yes—
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—68no—
263#125—62no—
264#125—62no—
265#125—68no—
266#123—68no—
267C22⋊Q8—24no—
268C22⋊Q8—34no—
269C22⋊Q8—42no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272C22⋊Q8—62no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—22no—
276C4×D4—22no—
277#117—34no—
278#117—68no—
279#117—62no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4#6711yescharacteristic
293C2×C4⋊C4C22×D312yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4#6711yescharacteristic
300C2×C22⋊C4#6711yescharacteristic
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—64no—
307#103—122no—
308C22⋊C8—62no—
309C22⋊C8—122no—
310#95—21no—
311#95—22no—
312#95—22no—
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—24no—
326C3×Q8—48no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×D4—44no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—48no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4#12012yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C2×#4112yes—
355C2×C3×C4H(5,𝔽2)12yes—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—48no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367Dic6—24no—
368Dic6—41no—
369Dic6—41no—
370C3⋊C8—42no—
371C3⋊C8—42no—
372C24C2×#6711yescharacteristic
373C24C2×#6711yescharacteristic
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—32no—
381C2×Q8—44no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—61no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8#23714yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—64no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4D3×D412yes—
410C22×C4D3×D412yes—
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
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—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—32no—
428C4⋊C4—32no—
429C4⋊C4—32no—
430C4⋊C4—44no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—64no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—64no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C42—21no—
448C42—21no—
449C42—64no—
450C42—64no—
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—44no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—82no—
472C3×C4#52511yescharacteristic
473C3×C4#53511yescharacteristic
474C3×C4#54612yes—
475Dic3—42no—
476Dic3—42no—
477Dic3—42no—
478Dic3—42no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23#93712yes—
488C23#94212yes—
489C23C22×#6711yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—48no—
493Q8—42no—
494Q8—42no—
495Q8—64no—
496Q8—121no—
497Q8—121no—
498D4—44no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—128no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4#93712yes—
532C2×C4C2×D3×D411yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×#23712yes—
535C2×C4#100412yes—
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—24no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22C2×#93711yescharacteristic
556C22#580612yes—
557C22C2×#94211yescharacteristic
558C22#582211yescharacteristic
559C22#583812yes—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—44no—
564C4—41no—
565C4—82no—
566C4—122no—
567C4—122no—
568C4—122no—
569C4—122no—
570C4#558611yescharacteristic
571C4#559112yes—
572C4#559211yescharacteristic
573C3#2398311yescharacteristic, complemented, Sylow
574C2—42no—
575C2—42no—
576C2#8258111yescharacteristic
577C2#8273511yescharacteristic
578C2#8341111yescharacteristic
579C2#8354112yes—
580C2#8355212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2409C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2409D3
hypocenterC3#23983
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. C2×#5228C2semidirect
4. C2×#5225C2semidirect
5. #71700C2semidirect
6. #71692C2semidirect
8. #69690C2semidirect
9. #69669C2semidirect
10. #69516C2semidirect
11. #69503C2semidirect
573. C3#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C3×#24092
C2×#52282
C2×#52252
#717002
#716922
#716112
#696902
#696692
#695162
#695032
#239833