Finite Group Databasebeta

← #10828032

Subgroups — #10828032

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3487C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic, complemented
3#82914C212yesmaximal, complemented
4#72143C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71852C214yesmaximal, complemented
7#71712C212yesmaximal, complemented
8#71658C212yesmaximal, complemented
9#24024—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C3×#550—22no—
13C3×#550—22no—
14C3×#548—22no—
15C3×#548—22no—
16C3×#535—22no—
17C3×#535C2212yes—
18C3×#525—22no—
19C3×#525C2212yes—
20C3×#519C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#837C2214yes—
24#5232C2214yes—
25#5229C2212yes—
26C2×#836C2212yes—
27#5221C2212yes—
28#5220—22no—
29C2×#833C2212yes—
30#5212—22no—
31#5207C2212yes—
32#5044—22no—
33#5043—22no—
34#5034—22no—
35#5033—22no—
36#4995C2214yes—
37#4532C2214yes—
38#3487D311yescharacteristic
39#3092—32no—
40#2311—31no—
41#2037—34no—
42#1897—32no—
43#1857—32no—
44#1793—32no—
45#1785—31no—
46C3×2−1+4—24no—
47C3×2−1+4—42no—
48C2×C3×#41—24no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—42no—
54C2×C3×#41C2312yes—
55C22×C3×Q8—22no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4D412yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4D412yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C3×#127—24no—
63C3×#127—22no—
64C3×#124—24no—
65C3×#124—22no—
66C3×#123—22no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8D412yes—
70C3×C22⋊Q8D412yes—
71C3×#121—22no—
72C3×#121—22no—
73C3×#121—24no—
74C3×#121—24no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#121D412yes—
78C3×#121D412yes—
79C3×#120—24no—
80C3×#120—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×#117—22no—
88C3×#117C2312yes—
89C2×#217—21no—
90C2×#217—24no—
91#926—42no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×C4×Dic3C2312yes—
98C2×#208C2314yes—
99#837—44no—
100#836—24no—
101#835—24no—
102#833—42no—
103#832—24no—
104C3⋊#95—24no—
105#830—22no—
106#830—22no—
107#808—24no—
108#807—24no—
109C2×2−1+4C2×D311yescharacteristic
110C22×#41C2×D311yescharacteristic
111#550—22no—
112#550—22no—
113#548—22no—
114#548—22no—
115#535—22no—
116#535C2×D312yes—
117#525—22no—
118#525C2×D312yes—
119#519C2×D311yescharacteristic
120C2×#117—31no—
121C22×C4⋊C4—31no—
122#484—62no—
123#483—62no—
124#481—62no—
125#480—62no—
126#428—34no—
127C2×#104—34no—
128#417—34no—
129C2×#101—32no—
130C2×#100—32no—
131#407—32no—
132#391—32no—
133#390—32no—
134#386—62no—
135#380—62no—
136#339—34no—
137C24×C3C2×D412yes—
138C24×C3C2×D411yescharacteristic
139C3×#41—24no—
140C3×#41—24no—
141C3×#41—48no—
142C3×#41—48no—
143C3×#41—44no—
144C3×#41—48no—
145C3×#41—44no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—48no—
150C2×C3×Q8C2×D412yes—
151C2×C3×Q8#4114yes—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—42no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—42no—
181C22×Dic3—82no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C42—24no—
196C3×C42—22no—
197#217—42no—
198#217—84no—
199C4⋊Dic3—24no—
200C4⋊Dic3—42no—
201C4⋊Dic3—41no—
202C4⋊Dic3—41no—
203C4×Dic3—24no—
204#208—24no—
205#208—24no—
206C2×C3⋊C8—24no—
2072−1+4—24no—
2082−1+4—42no—
209C2×#41—24no—
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41—42no—
215C2×#41C22×D312yes—
216C22×Q8—22no—
217C22×Q8C22×D311yescharacteristic
218C22×D4#6712yes—
219C22×D4C22×D311yescharacteristic
220C23×C4—61no—
221C23×C4#6712yes—
222C23×C4C22×D311yescharacteristic
223C2×M16—34no—
224#127—24no—
225#127—22no—
226#124—24no—
227#124—22no—
228#123—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8#6712yes—
232C22⋊Q8#6712yes—
233#121—22no—
234#121—22no—
235#121—24no—
236#121—24no—
237#121—22no—
238#121—22no—
239#121#6712yes—
240#121#6712yes—
241#120—24no—
242#120—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—22no—
248C4×D4—22no—
249#117—22no—
250#117—122no—
251#117C22×D312yes—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—64no—
259C2×C42—32no—
260#107—64no—
261#106—64no—
262#104—124no—
263#103—64no—
264#102—64no—
265#101—64no—
266#100—122no—
267C22⋊C8—62no—
268C22⋊C8—62no—
269#95—64no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3#12012yes—
281C23×C3C22×D411yescharacteristic
282C3×Q8—28no—
283C3×Q8—24no—
284C3×Q8—44no—
285C3×D4—24no—
286C3×D4—48no—
287C3×D4—44no—
288C3×D4—42no—
289C3×D4—42no—
290C3×D4—44no—
291C3×D4—82no—
292C3×D4—82no—
293C3×D4—84no—
294C3×D4—84no—
295C2×C3×C4—22no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—44no—
305C2×C3×C4—44no—
306C2×C3×C4—48no—
307C2×C3×C4—44no—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12314yes—
311C2×C3×C4C22×D412yes—
312C2×C3×C4C2×#4112yes—
313C2×C3×C4H(5,𝔽2)12yes—
314C2×Dic3—42no—
315C2×Dic3—44no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—84no—
320C3⋊C8—44no—
321C24C2×#6712yes—
322C24C2×#6711yescharacteristic
323#41—24no—
324#41—24no—
325#41—48no—
326#41—48no—
327#41—44no—
328#41—48no—
329#41—44no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—48no—
334C2×Q8#23714yes—
335C2×Q8C2×#6712yes—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—42no—
348C2×D4C2×#6712yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—122no—
360C22×C4—242no—
361C22×C4C2×#6712yes—
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6712yes—
364C22×C4C2×#6712yes—
365C22×C4C23×D311yescharacteristic
366M16—64no—
367M16—64no—
368C2×C8—64no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—64no—
374C4⋊C4—122no—
375C4⋊C4—121no—
376C4⋊C4—121no—
377C4⋊C4C2×#6712yes—
378C4⋊C4C2×#6712yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—122no—
387C22⋊C4—244no—
388C42—24no—
389C42—22no—
390C42—64no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—82no—
406C22×C3C2×#12011yescharacteristic
407C22×C3C2×#12311yescharacteristic
408C22×C3#53411yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—24no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4#54614yes—
416Dic3—82no—
417Dic3—82no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23#95312yes—
429C23C22×#6711yescharacteristic
430Q8—28no—
431Q8—24no—
432Q8—44no—
433D4—24no—
434D4—48no—
435D4—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—82no—
440D4—82no—
441D4—84no—
442D4—84no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—122no—
457C2×C4—124no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—244no—
462C2×C4#93514yes—
463C2×C4#95311yescharacteristic
464C2×C4#95311yescharacteristic
465C2×C4C2×#23712yes—
466C2×C4#100412yes—
467C2×C4C22×#6712yes—
468C8—124no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—42no—
472C2×C3—82no—
473C2×C3—82no—
474C2×C3#240511yescharacteristic
475C2×C3#302111yescharacteristic
476C2×C3#302211yescharacteristic
477C22—21no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22C2×#93511yescharacteristic
493C22#580411yescharacteristic
494C22C2×#95311yescharacteristic
495C4—21no—
496C4—21no—
497C4—24no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—242no—
502C4—242no—
503C4#580614yes—
504C3#2402411yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—42no—
508C2—82no—
509C2—82no—
510C2#8404711yescharacteristic
511C2#8412211yescharacteristic
512C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3487C2
socleC22×C3C2×#120
hypercenter#3487D3
hypocenterC3#24024
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3487

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. #82914C2semidirect
4. #72143C2semidirect
5. #72137C2semidirect
6. #71852C2semidirect
7. #71712C2semidirect
8. #71658C2semidirect
504. C3#24024semidirect

Maximal subgroups

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

subgroupindex
C3×#34872
#840872
#829142
#721432
#721372
#718522
#717122
#716582
#240243