Finite Group Databasebeta

← #10847367

Subgroups — #10847367

Order 768.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5041C212yesmaximal, complemented
3#82075C211yesmaximal, characteristic, complemented
4#71794C211yesmaximal, characteristic, complemented
5#71783C211yesmaximal, characteristic, complemented
6#71768C211yesmaximal, characteristic, complemented
7#71762C211yesmaximal, characteristic, complemented
8#71681C212yesmaximal, complemented
9#71612C212yesmaximal, complemented
10#71607C212yesmaximal, complemented
11C2×#4524C211yesmaximal, characteristic, complemented
12#25495—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#532—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×#214C2212yes—
24C2×#924C2211yescharacteristic
25C23×C3⋊C8C2212yes—
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212C2211yescharacteristic
30#5211—21no—
31#5210—21no—
32#5041—28no—
33C2×#810C2212yes—
34C2×#808C2212yes—
35#4933—21no—
36#4933—21no—
37#4929C2211yescharacteristic
38#4929C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#532D311yescharacteristic
47C2×#465—32no—
48#2855—31no—
49#2058—31no—
50#2041—31no—
51#2024—31no—
52#2014—31no—
53#1893—32no—
54#1877—32no—
55#1864—32no—
56C2×#336—31no—
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C3×C4⋊Q8—24no—
71C3×#127—24no—
72C3×#124—28no—
73C3×#121—28no—
74C3×#121—24no—
75C3×#121D418yes—
76C3×C4×D4—28no—
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—24no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—42no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—24no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C22×C3⋊C8—24no—
101C22×C3⋊C8—24no—
102C22×C3⋊C8—42no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#832—24no—
106#832—24no—
107#832—42no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115#810—24no—
116#808—24no—
117C22×#41—21no—
118C22×SD16—32no—
119C23×C8—32no—
120#532—28no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#127C2×D311yescharacteristic
123C2×#124—21no—
124C2×C22⋊Q8—31no—
125C2×#121—21no—
126C2×#121C2×D312yes—
127C2×C4×D4—21no—
128C2×C4×D4C2×D312yes—
129C22×C42C2×D311yescharacteristic
130#465—68no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136#391—31no—
137#391—31no—
138#389—31no—
139#389—31no—
140#387—61no—
141#387—61no—
142#385—61no—
143#381—61no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336—31no—
148#336—32no—
149#336—31no—
150C24×C3—22no—
151C24×C3C2×D411yescharacteristic
152C3×#41—432no—
153C2×C3×Q8—24no—
154C2×C3×Q8—42no—
155C2×C3×Q8—42no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—28no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—44no—
167C2×C3×D4—44no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4C2×D414yes—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—28no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—41no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4SD1614yescomplemented
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4C2×D414yes—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—28no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—22no—
208#217—42no—
209#217—42no—
210#214—48no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215Dic3⋊C4—42no—
216Dic3⋊C4—42no—
217Dic3⋊C4—42no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—48no—
224C2×C3⋊C8#4114yescomplemented
225C2×#41—28no—
226C2×#41—44no—
227C2×#41—44no—
228C22×Q8—21no—
229C22×Q8—31no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C22×D312yes—
234C23×C4—22no—
235C23×C4#6711yescharacteristic
236C23×C4#6711yescharacteristic
237C23×C4C22×D311yescharacteristic
238C2×SD16—64no—
239C2×SD16—64no—
240C2×SD16—122no—
241C22×C8—32no—
242C22×C8—32no—
243C22×C8—64no—
244C22×C8—64no—
245C22×C8—122no—
246C4⋊Q8—24no—
247#127—24no—
248#124—28no—
249C22⋊Q8—64no—
250#121—28no—
251#121—24no—
252#121#6718yes—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4C22×D312yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—61no—
268C2×C42—24no—
269C2×C42#6711yescharacteristic
270C2×C42#6711yescharacteristic
271C2×C42C22×D311yescharacteristic
272C2×C42C22×D311yescharacteristic
273#106—64no—
274#103—64no—
275#102—64no—
276#102—64no—
277#102—122no—
278#95—61no—
279#95—61no—
280#95—61no—
281#95—61no—
282#95—61no—
283#95—61no—
284#95—61no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—24no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3#12012yes—
297C23×C3C2×SD1612yes—
298C23×C3AGL(1,ℤ/8)12yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—48no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—48no—
304C3×D4—48no—
305C3×D4—416no—
306C3×D4—42no—
307C3×D4—42no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—28no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—44no—
324C2×C3×C4—416no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12012yes—
328C2×C3×C4C2×SD1612yes—
329C2×C3×C4C2×SD1612yes—
330C2×C3×C4#13512yes—
331C2×C3×C4C22×D412yes—
332C2×C3×C4C2×#4112yes—
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347Dic6—42no—
348Dic6—42no—
349C3⋊C8—24no—
350C3⋊C8—42no—
351C24—22no—
352C24C2×#6711yescharacteristic
353#41—432no—
354C2×Q8—24no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—62no—
358C2×Q8—61no—
359C2×Q8—61no—
360C2×Q8—62no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4C2×#6714yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4#21414yescomplemented
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6714yes—
399C22×C4C23×D311yescharacteristic
400SD16—128no—
401C2×C8—34no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—128no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—62no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—62no—
414C4⋊C4—122no—
415C4⋊C4—122no—
416C4⋊C4—122no—
417C4⋊C4C2×#6714yes—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—122no—
421C22⋊C4—122no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3#45014yes—
440C22×C3C2×#12011yescharacteristic
441C22×C3#53211yescharacteristic
442C22×C3#53711yescharacteristic
443C22×C3C22×SD1611yescharacteristic
444C22×C3C2×#13511yescharacteristic
445C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
446C22×C3#57511yescharacteristic
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—44no—
452C3×C4—41no—
453C3×C4#54612yes—
454C3×C4#54712yes—
455Dic3—81no—
456Dic3—81no—
457Dic3—81no—
458Dic3—81no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#93212yes—
471C23C2×#21412yes—
472C23#95312yes—
473C23C22×#6711yescharacteristic
474Q8—48no—
475Q8—122no—
476Q8—122no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—48no—
481D4—416no—
482D4—42no—
483D4—42no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—44no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4C2×#21412yes—
516C2×C4C2×#21412yes—
517C2×C4#95012yes—
518C2×C4#95312yes—
519C2×C4D3×#4112yes—
520C2×C4#100911yescharacteristic
521C2×C4#101011yescharacteristic
522C2×C4C22×#6712yes—
523C8—64no—
524C8—122no—
525C2×C3—24no—
526C2×C3—84no—
527C2×C3#243011yescharacteristic
528C2×C3C2×#45011yescharacteristic
529C2×C3#300211yescharacteristic
530C2×C3#316711yescharacteristic
531C2×C3#318011yescharacteristic
532C2×C3#324611yescharacteristic
533C2×C3#324711yescharacteristic
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22#521814yes—
548C22C2×#93211yescharacteristic
549C22C22×#21411yescharacteristic
550C22C2×#95011yescharacteristic
551C22#582211yescharacteristic
552C22#583211yescharacteristic
553C22#583311yescharacteristic
554C22C2×#95311yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—44no—
560C4—41no—
561C4—241no—
562C4—241no—
563C4—241no—
564C4—241no—
565C4#582412yes—
566C4#582912yes—
567C3#2549511yescharacteristic, complemented, Sylow
568C2—24no—
569C2—84no—
570C2#8329011yescharacteristic
571C2#8329311yescharacteristic
572C2#8331111yescharacteristic
573C2#8332011yescharacteristic
574C2C2×#521811yescharacteristic
575C2#8408011yescharacteristic
576C2#8409111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#25495
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5041C2semidirect
3. #82075C2semidirect
4. #71794C2semidirect
5. #71783C2semidirect
6. #71768C2semidirect
7. #71762C2semidirect
8. #71681C2semidirect
9. #71612C2semidirect
10. #71607C2semidirect
11. C2×#4524C2semidirect
183. C22×C3×C4SD16semidirect
224. C2×C3⋊C8#41semidirect
392. C22×C4#214semidirect
567. C3#25495semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50412
#820752
#717942
#717832
#717682
#717622
#716812
#716122
#716072
C2×#45242
#254953