Finite Group Databasebeta

← #10740333

Subgroups — #10740333

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4979C212yesmaximal
2C2×#4976C212yesmaximal
3C3×#1444C211yesmaximal, characteristic
4#71564C212yesmaximal
5#69424C211yesmaximal, characteristic
6#69420C211yesmaximal, characteristic
7#69414C212yesmaximal
8#69411C212yesmaximal
9#69405C212yesmaximal
10#15779—31nomaximal, Sylow
11C23×Dic6C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4C2212yes—
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2212yes—
19C2×C3×#95C2211yescharacteristic
20C2×C3×#95C2212yes—
21C2×C3×#95C2211yescharacteristic
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32#4979—28no—
33#4976—28no—
34C2×#395—32no—
35C2×#389—32no—
36#1446—31no—
37#1444—31no—
38#1444D311yescharacteristic
39#1442—32no—
40#1441—32no—
41#1436—32no—
42#1436—32no—
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C22×Dic6—22no—
53C22×Dic6—24no—
54C22×Dic6—41no—
55C22×Dic6—42no—
56C22×Dic6—41no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4—41no—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3—41no—
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4—24no—
66C2×Dic3⋊C4—24no—
67C2×Dic3⋊C4—24no—
68C2×Dic3⋊C4—24no—
69C2×Dic3⋊C4—42no—
70C2×Dic3⋊C4—42no—
71C2×Dic3⋊C4—42no—
72C2×Dic3⋊C4D418yes—
73C2×C4×Dic3—24no—
74C2×C4×Dic3—42no—
75C2×C4×Dic3—42no—
76C3×#95—22no—
77C3×#95—42no—
78C3×#95—41no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—41no—
86C3⋊#95—42no—
87C3⋊#95Q818yescomplemented
88C3⋊#95Q814yescomplemented
89C23×Q8—31no—
90C22×C4⋊C4—32no—
91C22×C4⋊C4—32no—
92C22×C4⋊C4—32no—
93C22×C4⋊C4—31no—
94C22×C4⋊C4C2×D312yes—
95C22×C4⋊C4C2×D311yescharacteristic
96C22×C42—32no—
97#395—68no—
98#389—68no—
99C2×#95—32no—
100C2×#95—31no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—31no—
108C2×#95—32no—
109C2×#95C2×D311yescharacteristic
110C2×#95C2×D312yes—
111C2×#95C2×D311yescharacteristic
112C24×C3C2411yescharacteristic
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D414yes—
138C22×C3×C4C2×Q814yes—
139C22×C3×C4C2×Q812yes—
140C22×C3×C4#4112yes—
141C22×C3×C4#4112yes—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3C2×D414yes—
165C22×Dic3C2×D414yes—
166C22×Dic3C2×Q814yes—
167C22×Dic3C2×Q814yes—
168C22×Dic3C2×Q814yes—
169C22×Dic3#4114yes—
170C2×Dic6—44no—
171C2×Dic6—48no—
172C2×Dic6—48no—
173C2×Dic6—44no—
174C2×Dic6—44no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—48no—
177C4⋊Dic3—44no—
178Dic3⋊C4—216no—
179Dic3⋊C4—48no—
180Dic3⋊C4—48no—
181C4×Dic3—48no—
182C22×Q8—64no—
183C22×Q8—62no—
184C22×Q8—121no—
185C22×Q8—122no—
186C22×Q8—121no—
187C23×C4—32no—
188C23×C4—32no—
189C23×C4—32no—
190C23×C4—32no—
191C23×C4C22×D311yescharacteristic
192C23×C4C22×D312yes—
193C23×C4C22×D311yescharacteristic
194C23×C4C22×D311yescharacteristic
195C23×C4C22×D312yes—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—38no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—64no—
203C2×C4⋊C4—64no—
204C2×C4⋊C4—64no—
205C2×C4⋊C4—64no—
206C2×C4⋊C4—64no—
207C2×C4⋊C4—64no—
208C2×C4⋊C4—122no—
209C2×C4⋊C4—122no—
210C2×C4⋊C4—122no—
211C2×C4⋊C4—121no—
212C2×C42—64no—
213C2×C42—122no—
214C2×C42—122no—
215#95—22no—
216#95—34no—
217#95—38no—
218#95—42no—
219#95—41no—
220#95—64no—
221#95—64no—
222#95—64no—
223#95—64no—
224#95—64no—
225#95—64no—
226#95—122no—
227#95—121no—
228C23×C3C22×D412yes—
229C23×C3C22×Q811yescharacteristic
230C23×C3C22×Q812yes—
231C23×C3C2×#4111yescharacteristic
232C23×C3C2×#4111yescharacteristic
233C23×C3C2×#4112yes—
234C23×C3H(5,𝔽2)12yes—
235C23×C3H(5,𝔽2)11yescharacteristic
236C23×C3H(5,𝔽2)11yescharacteristic
237C23×C32−1+411yescharacteristic
238C23×C32−1+411yescharacteristic
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—24no—
271C2×Dic3—24no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274C2×Dic3—24no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—24no—
278C2×Dic3—24no—
279C2×Dic3—24no—
280C2×Dic3—24no—
281C2×Dic3—24no—
282C2×Dic3—24no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3C4⋊Q818yes—
296Dic6—416no—
297C24C23×D311yescharacteristic
298C2×Q8—128no—
299C2×Q8—124no—
300C2×Q8—124no—
301C2×Q8—128no—
302C2×Q8—124no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—34no—
328C22×C4—34no—
329C22×C4—34no—
330C22×C4—34no—
331C22×C4—34no—
332C22×C4—34no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4#23512yes—
356C22×C4D3×D414yes—
357C22×C4#23712yes—
358C22×C4D3×Q814yes—
359C22×C4D3×Q812yes—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—616no—
363C4⋊C4—128no—
364C4⋊C4—128no—
365C4⋊C4—124no—
366C42—128no—
367C22×C3#52512yes—
368C22×C3C2×C4⋊Q812yes—
369C22×C3#53312yes—
370C22×C3#53412yes—
371C22×C3#53811yescharacteristic
372C22×C3#53911yescharacteristic
373C22×C3#54212yes—
374C22×C3#54412yes—
375C22×C3#54411yescharacteristic
376C22×C3D4211yescharacteristic
377C22×C3#54612yes—
378C22×C3#54812yes—
379C22×C3D4×Q812yes—
380C22×C3#55111yescharacteristic
381C22×C3#55412yes—
382C22×C3#55412yes—
383C22×C3#55412yes—
384C22×C3#55411yescharacteristic
385C22×C3#55611yescharacteristic
386C22×C3#55711yescharacteristic
387C22×C3Q8211yescharacteristic
388C22×C3#56011yescharacteristic
389C22×C3#56311yescharacteristic
390C3×C4—42no—
391C3×C4—44no—
392C3×C4—42no—
393C3×C4—44no—
394C3×C4—42no—
395Dic3—28no—
396Dic3—44no—
397Dic3—44no—
398Dic3—44no—
399C23C2×#23511yescharacteristic
400C23C2×D3×D412yes—
401C23C2×#23711yescharacteristic
402C23#100412yes—
403C23#100411yescharacteristic
404C23C2×D3×Q812yes—
405C23C2×D3×Q811yescharacteristic
406C23#100711yescharacteristic
407C23D3×#4112yes—
408C23#100911yescharacteristic
409C23#101011yescharacteristic
410Q8—1216no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—38no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—64no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—64no—
449C2×C4—64no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C3#246311yescharacteristic
469C2×C3#246512yes—
470C2×C3#255012yes—
471C2×C3#255111yescharacteristic
472C2×C3#258312yes—
473C2×C3#258611yescharacteristic
474C2×C3#266811yescharacteristic
475C2×C3#272912yes—
476C2×C3#274211yescharacteristic
477C2×C3#279211yescharacteristic
478C2×C3#280511yescharacteristic
479C22#551112yes—
480C22#551511yescharacteristic
481C22#554211yescharacteristic
482C22#554712yes—
483C22#557911yescharacteristic
484C22#559011yescharacteristic
485C22#559112yes—
486C22#560512yes—
487C22#561711yescharacteristic
488C22#562212yes—
489C22#562311yescharacteristic
490C22#563212yes—
491C22#563312yes—
492C22#563612yes—
493C22#565911yescharacteristic
494C22#566212yes—
495C22#569012yes—
496C22#569911yescharacteristic
497C22#572011yescharacteristic
498C22#572111yescharacteristic
499C22#572211yescharacteristic
500C22D3×C4⋊Q812yes—
501C22#572912yes—
502C4—42no—
503C4—44no—
504C4—42no—
505C4—44no—
506C4—42no—
507C4—68no—
508C4—124no—
509C4—124no—
510C4—124no—
511C3#1577911yescharacteristic, complemented, Sylow
512C2#7435011yescharacteristic
513C2#7443312yes—
514C2#7461311yescharacteristic
515C2#7618111yescharacteristic
516C2#7624712yes—
517C2#7676311yescharacteristic
518C2#7718512yes—
519C2#7718611yescharacteristic
520C2#7780111yescharacteristic
521C2#7787112yes—
522C2#7807611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1444C2
hypercenter#1444D3
hypocenterC3#15779
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
87. C3⋊#95Q8semidirect
88. C3⋊#95Q8semidirect
511. C3#15779semidirect

Maximal subgroups

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

subgroupindex
C2×#49792
C2×#49762
C3×#14442
#715642
#694242
#694202
#694142
#694112
#694052
#157793