Finite Group Databasebeta

← #11235190

Subgroups — #11235190

Order 768.

1514 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2568C211yesmaximal, characteristic
2#83238C212yesmaximal
3#82479C211yesmaximal, characteristic
4#82467C211yesmaximal, characteristic
5#78273C212yesmaximal
6#78272C214yesmaximal
7#78207C212yesmaximal
8#76558C214yesmaximal
9#76545C212yesmaximal
10#76535C212yesmaximal
11#76490C212yesmaximal
12#76485C212yesmaximal
13#74209C212yesmaximal
14#74192C212yesmaximal
15#73644C212yesmaximal
16#45436—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2212yes—
20C2×#944C2212yes—
21C2×#944C2212yes—
22C2×#887C2212yes—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#400C2212yes—
26C3×#400C2212yes—
27C3×#391C2212yes—
28C3×#391C2212yes—
29C3×#391C2212yes—
30C3×#382C2212yes—
31C3×C4×C4⋊C4C2211yescharacteristic
32#5230C2212yes—
33#5229C2212yes—
34#5229C2212yes—
35#4986C2212yes—
36#4986C2212yes—
37#4983C2212yes—
38#4982C2214yes—
39#4982C2214yes—
40#4981C2211yescharacteristic
41#4981C2211yescharacteristic
42#4979C2211yescharacteristic
43#4979C2211yescharacteristic
44#4977C2212yes—
45#4977C2212yes—
46#4976C2214yes—
47#4976C2214yes—
48Dic3×C4⋊C4C2212yes—
49#4973C2212yes—
50C4×C4⋊Dic3C2211yescharacteristic
51#4933C2214yes—
52#4932C2214yes—
53#4932C2211yescharacteristic
54C4×Dic3⋊C4C2212yes—
55C4×Dic3⋊C4C2212yes—
56#4929C2212yes—
57#4929C2212yes—
58#4660C2214yes—
59#4658C2214yes—
60#4657C2214yes—
61#4657C2214yes—
62#4655C2214yes—
63#4655C2214yes—
64#4654C2214yes—
65#4653C2214yes—
66#4652C2214yes—
67#4652C2214yes—
68#4652C2214yes—
69#4651C2214yes—
70#4651C2214yes—
71#4650C2214yes—
72#4648C2214yes—
73#4647C2214yes—
74#4647C2214yes—
75#4646C2214yes—
76#2769—34no—
77#2769—32no—
78#2769—32no—
79#2612—32no—
80#2612—34no—
81#2612—32no—
82#2568—32no—
83#2568—31no—
84#2568D311yescharacteristic
85#2546—32no—
86#2546—32no—
87#2399—31no—
88#2356—32no—
89#2356—32no—
90#2327—32no—
91C22×C3×Q8C2312yes—
92C22×C3×Q8C2311yescharacteristic
93C22×Dic6C2312yes—
94C22×Dic6C2311yescharacteristic
95C3×C4⋊Q8—42no—
96C3×C4×Q8—42no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2312yes—
107Q8×Dic3—28no—
108#944—28no—
109#944—44no—
110C2×C4⋊Dic3C2314yes—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2312yes—
113C2×C4⋊Dic3C2312yes—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4C2314yes—
116C2×Dic3⋊C4C2312yes—
117C2×Dic3⋊C4C2312yes—
118C2×Dic3⋊C4C2314yes—
119C2×Dic3⋊C4C2312yes—
120C2×Dic3⋊C4C2314yes—
121C2×Dic3⋊C4C2312yes—
122C2×Dic3⋊C4C2314yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2312yes—
125C2×C4×Dic3C2314yes—
126C2×C4×Dic3C2314yes—
127C2×C4×Dic3C2312yes—
128C2×C4×Dic3C2314yes—
129C2×C4×Dic3C2312yes—
130C2×C4×Dic3C2312yes—
131#887—28no—
132#869—42no—
133C4×Dic6—42no—
134C3×#95C2314yes—
135C3×#95C2314yes—
136C3×#95C2314yes—
137C3×#95C2314yes—
138C3×#95C2312yes—
139C3⋊#95C2314yes—
140C3⋊#95C2314yes—
141C3⋊#95C2314yes—
142C3⋊#95C2314yes—
143C3⋊#95C2312yes—
144C3⋊#95C2314yes—
145C3⋊#95C2314yes—
146C3⋊#95C2314yes—
147C3⋊#95C2314yes—
148C3⋊#95C2312yes—
149C3⋊#95C2312yes—
150C3⋊#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2312yes—
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C2×C4⋊Q8—32no—
159C2×C4⋊Q8—32no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8C2×D311yescharacteristic
162C2×C4×Q8—32no—
163C2×C4×Q8—32no—
164C2×C4×Q8—31no—
165C2×C4×Q8C2×D311yescharacteristic
166#400—32no—
167#400—32no—
168#400—34no—
169#400—34no—
170#400C2×D312yes—
171#400C2×D312yes—
172#398—34no—
173#398—31no—
174#398—31no—
175#398—32no—
176#398—34no—
177#398—34no—
178#398—34no—
179#395—32no—
180#395—31no—
181#395—31no—
182#391—32no—
183#391—32no—
184#391—32no—
185#391—32no—
186#391—32no—
187#391—34no—
188#391C2×D312yes—
189#391C2×D312yes—
190#391C2×D312yes—
191#389—34no—
192#389—34no—
193#389—34no—
194#389—34no—
195#387—34no—
196#387—34no—
197#387—34no—
198#387—34no—
199#387—34no—
200#387—34no—
201#387—34no—
202#387—34no—
203#383—34no—
204#383—34no—
205#382—32no—
206#382—32no—
207#382—34no—
208#382C2×D312yes—
209C4×C4⋊C4—32no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4—32no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4C2×D311yescharacteristic
214#376—31no—
215#376—34no—
216#376—34no—
217C2×C3×Q8—24no—
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—41no—
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2414yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2414yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C22×Dic3C2414yes—
233C22×Dic3C2412yes—
234C22×Dic3C2414yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C22×Dic3C2412yes—
238C2×Dic6—24no—
239C2×Dic6—24no—
240C2×Dic6—24no—
241C2×Dic6—24no—
242C2×Dic6—41no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C42—22no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—24no—
253C4⋊Dic3—24no—
254C4⋊Dic3—28no—
255C4⋊Dic3—41no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258Dic3⋊C4—24no—
259Dic3⋊C4—24no—
260Dic3⋊C4—28no—
261Dic3⋊C4—28no—
262Dic3⋊C4—28no—
263Dic3⋊C4—24no—
264Dic3⋊C4—24no—
265Dic3⋊C4—42no—
266Dic3⋊C4—44no—
267Dic3⋊C4—44no—
268C4×Dic3—24no—
269C4×Dic3—24no—
270C4×Dic3—28no—
271C4×Dic3—28no—
272C4×Dic3—28no—
273C4×Dic3#4118yes—
274C22×Q8—32no—
275C22×Q8—31no—
276C22×Q8C22×D312yes—
277C22×Q8C22×D311yescharacteristic
278C4⋊Q8—42no—
279C4⋊Q8—68no—
280C4⋊Q8—122no—
281C4⋊Q8—124no—
282C4×Q8—42no—
283C4×Q8—68no—
284C4×Q8—68no—
285C4×Q8—122no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—34no—
288C2×C4⋊C4—34no—
289C2×C4⋊C4—34no—
290C2×C4⋊C4—34no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—32no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—34no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—34no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4C22×D312yes—
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D312yes—
306C2×C4⋊C4C22×D312yes—
307C2×C42—32no—
308C2×C42—32no—
309C2×C42—34no—
310C2×C42—34no—
311C2×C42—34no—
312C2×C42—32no—
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D312yes—
317#95—32no—
318#95—32no—
319#95—34no—
320#95—34no—
321#95—32no—
322#95—31no—
323#95—31no—
324#95—34no—
325#95—34no—
326#95—34no—
327#95—32no—
328#95—34no—
329#95—32no—
330#95—32no—
331#95—34no—
332#95—34no—
333#95—34no—
334#95—34no—
335#95—34no—
336#95C22×D314yes—
337#95C22×D312yes—
338#95C22×D314yes—
339#95C22×D314yes—
340#95C22×D314yes—
341C23×C3C2511yescharacteristic
342C3×Q8—44no—
343C3×Q8—48no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—24no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4C2×#4114yes—
364C2×C3×C4H(5,𝔽2)12yes—
365C2×C3×C42−1+412yes—
366C2×C3×C42−1+412yes—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3C2×#4114yes—
383C2×Dic3C2×#4114yes—
384C2×Dic32−1+414yes—
385Dic6—44no—
386Dic6—48no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—41no—
393C2×Q8—64no—
394C2×Q8—64no—
395C2×Q8—64no—
396C2×Q8—64no—
397C2×Q8—121no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—34no—
402C22×C4—32no—
403C22×C4—34no—
404C22×C4C23×D311yescharacteristic
405C22×C4C23×D311yescharacteristic
406C22×C4C23×D311yescharacteristic
407C22×C4C23×D314yes—
408C22×C4C23×D311yescharacteristic
409C22×C4C23×D314yes—
410C22×C4C23×D311yescharacteristic
411C22×C4C23×D311yescharacteristic
412C22×C4C23×D311yescharacteristic
413C4⋊C4—24no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—68no—
420C4⋊C4—68no—
421C4⋊C4—68no—
422C4⋊C4—68no—
423C4⋊C4—64no—
424C4⋊C4—64no—
425C4⋊C4—64no—
426C4⋊C4—64no—
427C4⋊C4—64no—
428C4⋊C4—122no—
429C4⋊C4—121no—
430C4⋊C4—124no—
431C4⋊C4—124no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—38no—
439C42—68no—
440C42—68no—
441C42—68no—
442C42—64no—
443C42—64no—
444C22×C3C22×#4112yes—
445C22×C3C2×H(5,𝔽2)11yescharacteristic
446C22×C3C2×2−1+411yescharacteristic
447C22×C3C2×2−1+411yescharacteristic
448C22×C3C2×2−1+412yes—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—44no—
452C3×C4—44no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4#55514yes—
458Dic3—24no—
459Dic3—24no—
460Dic3—24no—
461Dic3—44no—
462Dic3—44no—
463Dic3—42no—
464C23C24×D311yescharacteristic
465Q8—44no—
466Q8—48no—
467Q8—124no—
468Q8—128no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—34no—
489C2×C4—34no—
490C2×C4—34no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—62no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4#100412yes—
507C2×C4#100712yes—
508C2×C4D3×#4114yes—
509C2×C4#101012yes—
510C2×C3C2×#55511yescharacteristic
511C2×C3#350912yes—
512C2×C3#350912yes—
513C2×C3#353811yescharacteristic
514C2×C3#353811yescharacteristic
515C22C2×#100411yescharacteristic
516C22C2×#100711yescharacteristic
517C22C2×D3×#4112yes—
518C22C2×#101011yescharacteristic
519C22D3×2−1+412yes—
520C4—22no—
521C4—22no—
522C4—44no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—44no—
527C4—41no—
528C4—64no—
529C4—64no—
530C4—64no—
531C4—124no—
532C4—122no—
533C4—124no—
534C4#567014yes—
535C3#4543611yescharacteristic, complemented, Sylow
536C2#8597311yescharacteristic
537C2C2×#567011yescharacteristic
538C2#8635911yescharacteristic
539C2#8698312yes—
540C2#8702012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2568C2
hypercenter#2568D3
hypocenterC3#45436
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2568

Complements

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

NH ≅ G/Ntype
535. C3#45436semidirect

Maximal subgroups

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

subgroupindex
C3×#25682
#832382
#824792
#824672
#782732
#782722
#782072
#765582
#765452
#765352
#764902
#764852
#742092
#741922
#736442
#454363