Finite Group Databasebeta

← #10848750

Subgroups — #10848750

Order 768.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2C3×#2850C211yesmaximal, characteristic
3C2×#5060C211yesmaximal, characteristic, complemented
4C2×#5055C211yesmaximal, characteristic, complemented
5#71700C212yesmaximal, complemented
6#71685C212yesmaximal
7#71529C211yesmaximal, characteristic, complemented
8#69690C212yesmaximal, complemented
9#69660C212yesmaximal, complemented
10#69555C211yesmaximal, characteristic, complemented
11#69552C211yesmaximal, characteristic, complemented
12#25603—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#216C2212yes—
18C22×#214C2212yes—
19C2×#926—21no—
20C2×#924—21no—
21C2×#924C2212yes—
22#5470—24no—
23#5470—24no—
24C2×#870—21no—
25C2×#869C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C4×Dic6—21no—
28C2×C4×Dic6C2212yes—
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#388—21no—
32C3×#388—21no—
33C3×#386—21no—
34C3×#386—21no—
35C3×#382C2211yescharacteristic
36C2×#835C2212yes—
37C2×#832C2212yes—
38C2×#830C2212yes—
39#5060—24no—
40#5055—24no—
41C2×#810C2212yes—
42C2×#810C2212yes—
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#533—31no—
49C2×#497—31no—
50C2×#489—31no—
51#2850D311yescharacteristic
52#2001—32no—
53#2000—32no—
54#1993—31no—
55#1960—31no—
56#1880—32no—
57#1878—32no—
58#1749—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×Dic6—21no—
66C22×Dic6C2312yes—
67C3×#124—22no—
68C3×#124—22no—
69C3×C22⋊Q8—42no—
70C3×#121—42no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—22no—
87C2×#216—24no—
88C2×#216—24no—
89C2×#216—42no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—42no—
93#926—28no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2312yes—
98#924—28no—
99#924—24no—
100#924—24no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×C4×Dic3—22no—
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106#870—28no—
107#869—22no—
108#869—22no—
109#869D414yescomplemented
110#869D414yescomplemented
111C4×Dic6—28no—
112C4×Dic6—24no—
113C4×Dic6—24no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118#835—24no—
119#832—24no—
120#830—24no—
121#810—42no—
122#810—42no—
123#803—22no—
124#803—22no—
125C22×Q16—32no—
126C22×SD16—32no—
127#533—64no—
128#533—64no—
129C2×C4⋊Q8—31no—
130C2×C4⋊Q8—31no—
131C2×#125—61no—
132C2×#124C2×D311yescharacteristic
133C2×C22⋊Q8—32no—
134C2×C22⋊Q8—61no—
135C2×C22⋊Q8C2×D311yescharacteristic
136C2×#121C2×D311yescharacteristic
137C2×C4×Q8—32no—
138C2×C4×Q8—61no—
139C2×#117—61no—
140C22×C42C2×D311yescharacteristic
141#497—64no—
142#489—64no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#102—32no—
147C2×C22⋊C8—32no—
148C2×#97—31no—
149C2×#97—31no—
150#391C2×D311yescharacteristic
151#390C2×D311yescharacteristic
152#388—21no—
153#388—21no—
154#386—21no—
155#386—21no—
156#382C2×D311yescharacteristic
157#340—32no—
158#340—32no—
159#336—32no—
160C24×C3—21no—
161C24×C3C2×D411yescharacteristic
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—41no—
185C22×C3×C4—41no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—22no—
193C22×Dic3—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—24no—
199C2×Dic6—22no—
200C2×Dic6—42no—
201C2×Dic6—42no—
202C2×Dic6C2×D414yes—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—42no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—42no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#217—28no—
219#216—48no—
220#214—48no—
221C4⋊Dic3—24no—
222C4⋊Dic3—24no—
223C4⋊Dic3—24no—
224C4⋊Dic3—22no—
225C4⋊Dic3C2×D414yes—
226Dic3⋊C4—28no—
227Dic3⋊C4—28no—
228C4×Dic3—28no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8#4114yes—
235C22×Q8—32no—
236C22×Q8—61no—
237C22×Q8C22×D311yescharacteristic
238C22×D4—21no—
239C22×D4C22×D311yescharacteristic
240C23×C4#6711yescharacteristic
241C23×C4#6711yescharacteristic
242C23×C4C22×D311yescharacteristic
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8—32no—
250C22×C8—32no—
251C4⋊Q8—34no—
252C4⋊Q8—34no—
253C4⋊Q8—62no—
254C4⋊Q8—62no—
255#125—68no—
256#124—22no—
257#124—22no—
258C22⋊Q8—42no—
259C22⋊Q8—64no—
260C22⋊Q8—64no—
261C22⋊Q8—68no—
262#121—42no—
263C4×Q8—64no—
264C4×Q8—64no—
265C4×Q8—68no—
266#117—68no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—62no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—62no—
286C2×C42#6711yescharacteristic
287C2×C42#6711yescharacteristic
288C2×C42C22×D311yescharacteristic
289C2×C42C22×D311yescharacteristic
290#103—64no—
291#103—122no—
292#103—122no—
293#102—64no—
294C22⋊C8—64no—
295#97—62no—
296#97—62no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3#13712yes—
315C23×C3C22×D411yescharacteristic
316C3×Q8—42no—
317C3×Q8—42no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—84no—
321C2×C3×C4—28no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4#12012yes—
347C2×C3×C4#12012yes—
348C2×C3×C4#13712yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4112yes—
352C2×C3×C4H(5,𝔽2)12yes—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359Dic6—24no—
360Dic6—24no—
361Dic6—48no—
362Dic6—42no—
363Dic6—42no—
364C3⋊C8—24no—
365C3⋊C8—42no—
366C24—21no—
367C24C2×#6711yescharacteristic
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—34no—
373C2×Q8—64no—
374C2×Q8—62no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—122no—
380C2×Q8—122no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C2×#6711yescharacteristic
408C22×C4C23×D311yescharacteristic
409Q16—128no—
410SD16—128no—
411C2×C8—34no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—62no—
416C2×C8—62no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—34no—
420C4⋊C4—42no—
421C4⋊C4—68no—
422C4⋊C4—68no—
423C4⋊C4—64no—
424C4⋊C4—64no—
425C4⋊C4—64no—
426C4⋊C4—62no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—68no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—68no—
439C42D3×D412yes—
440C42D3×D412yes—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3C2×#12011yescharacteristic
456C22×C3#53211yescharacteristic
457C22×C3#53511yescharacteristic
458C22×C3C2×#13711yescharacteristic
459C22×C3C2×#13711yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57511yescharacteristic
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4#54612yes—
471C3×C4#55012yes—
472Dic3—44no—
473Dic3—44no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23#95112yes—
488C23C22×#6711yescharacteristic
489Q8—42no—
490Q8—42no—
491Q8—64no—
492Q8—64no—
493Q8—128no—
494Q8—122no—
495Q8—122no—
496D4—42no—
497D4—42no—
498D4—84no—
499C2×C4—28no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4#93712yes—
531C2×C4#93712yes—
532C2×C4#95112yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4#100412yes—
536C2×C4D3×#4112yes—
537C8—64no—
538C8—122no—
539C2×C3—42no—
540C2×C3—161no—
541C2×C3#243111yescharacteristic
542C2×C3#300411yescharacteristic
543C2×C3#300511yescharacteristic
544C2×C3#317211yescharacteristic
545C2×C3#318111yescharacteristic
546C2×C3#321511yescharacteristic
547C2×C3#321611yescharacteristic
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22C2×#93711yescharacteristic
563C22#580511yescharacteristic
564C22#582211yescharacteristic
565C22#582211yescharacteristic
566C22C2×#95111yescharacteristic
567C22C2×#95111yescharacteristic
568C22#583211yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—124no—
578C4—124no—
579C4#559912yes—
580C4#561212yes—
581C3#2560311yescharacteristic, complemented, Sylow
582C2—42no—
583C2—161no—
584C2#8265211yescharacteristic
585C2#8292811yescharacteristic
586C2#8293311yescharacteristic
587C2#8294911yescharacteristic
588C2#8313611yescharacteristic
589C2#8313811yescharacteristic
590C2#8331911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2850C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2850D3
hypocenterC3#25603
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
3. C2×#5060C2semidirect
4. C2×#5055C2semidirect
5. #71700C2semidirect
7. #71529C2semidirect
8. #69690C2semidirect
9. #69660C2semidirect
10. #69555C2semidirect
11. #69552C2semidirect
109. #869D4semidirect
110. #869D4semidirect
581. C3#25603semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×#28502
C2×#50602
C2×#50552
#717002
#716852
#715292
#696902
#696602
#695552
#695522
#256033