Finite Group Databasebeta

← #10725608

Subgroups — #10725608

Order 768.

1290 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4979C211yesmaximal, characteristic
2C3×C4⋊#95C211yesmaximal, characteristic
3#71565C211yesmaximal, characteristic
4#71564C212yesmaximal
5#71536C212yesmaximal
6#69424C212yesmaximal
7#69410C212yesmaximal
8#69406C212yesmaximal
9Dic3⋊#95C212yesmaximal
10#14601—31nomaximal, Sylow
11C23×Dic6C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2212yes—
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32#4979C418yescomplemented
33C2×#395—31no—
34#1446—32no—
35#1441—32no—
36#1441—31no—
37#1436—32no—
38#1436—32no—
39#1436—32no—
40C4⋊#95—32no—
41C4⋊#95D311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C22×Dic6—41no—
52C22×Dic6—41no—
53C22×Dic6—42no—
54C22×Dic6—42no—
55C22×Dic6C2×C414yes—
56C2×C3×C4⋊C4—41no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4—41no—
59C2×C3×C4⋊C4—41no—
60C2×C3×C4⋊C4C2×C414yes—
61C2×C3×C4⋊C4Q814yescomplemented
62C2×C3×C42—22no—
63C2×C3×C42—41no—
64C2×C3×C42—41no—
65C2×C4⋊Dic3—42no—
66C2×C4⋊Dic3—41no—
67C2×Dic3⋊C4—42no—
68C2×Dic3⋊C4—42no—
69C2×Dic3⋊C4—42no—
70C2×Dic3⋊C4—42no—
71C2×Dic3⋊C4C2×C418yes—
72C2×Dic3⋊C4Q818yescomplemented
73C2×C4×Dic3—24no—
74C2×C4×Dic3—42no—
75C2×C4×Dic3—42no—
76C3×#95—42no—
77C3×#95—42no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—41no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84C3⋊#95—42no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87C3⋊#95C2×C418yes—
88C3⋊#95C2×C414yes—
89C23×Q8—31no—
90C22×C4⋊C4—32no—
91C22×C4⋊C4—32no—
92C22×C4⋊C4—31no—
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C42—32no—
96C22×C42C2×D311yescharacteristic
97#395—38no—
98C2×#95—31no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—31no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95C2×D312yes—
110C2×#95C2×D312yes—
111C24×C3C2411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
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—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C22×C412yes—
138C22×C3×C4C22×C412yes—
139C22×C3×C4C2×Q814yes—
140C22×C3×C4C2×Q812yes—
141C22×C3×C4C2×Q814yes—
142C22×C3×C4#4112yes—
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×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3C22×C414yes—
167C22×Dic3C22×C414yes—
168C22×Dic3C2×Q814yes—
169C22×Dic3C2×Q814yes—
170C22×Dic3#4114yes—
171C2×Dic6—48no—
172C2×Dic6—44no—
173C2×Dic6—44no—
174C2×Dic6—48no—
175C2×Dic6—44no—
176C3×C4⋊C4—44no—
177C3×C4⋊C4—44no—
178C3×C42—44no—
179C4⋊Dic3—44no—
180Dic3⋊C4—48no—
181Dic3⋊C4—48no—
182C4×Dic3—48no—
183C22×Q8—34no—
184C22×Q8—122no—
185C22×Q8—121no—
186C22×Q8—121no—
187C22×Q8—122no—
188C23×C4—32no—
189C23×C4—32no—
190C23×C4—32no—
191C23×C4—32no—
192C23×C4C22×D312yes—
193C23×C4C22×D311yescharacteristic
194C23×C4C22×D311yescharacteristic
195C23×C4C22×D312yes—
196C23×C4C22×D311yescharacteristic
197C2×C4⋊C4—38no—
198C2×C4⋊C4—38no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—122no—
204C2×C4⋊C4—122no—
205C2×C4⋊C4—122no—
206C2×C4⋊C4—122no—
207C2×C4⋊C4—121no—
208C2×C4⋊C4—122no—
209C2×C4⋊C4Dic614yescomplemented
210C2×C4⋊C4C4×D314yes—
211C2×C42—22no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—64no—
215C2×C42—122no—
216C2×C42—122no—
217#95—34no—
218#95—38no—
219#95—42no—
220#95—42no—
221#95—64no—
222#95—64no—
223#95—64no—
224#95—122no—
225#95—122no—
226#95—121no—
227#95—122no—
228#95—122no—
229#95—122no—
230C23×C3C23×C411yescharacteristic
231C23×C3C22×Q811yescharacteristic
232C23×C3C22×Q812yes—
233C23×C3C2×#4111yescharacteristic
234C23×C3C2×#4112yes—
235C23×C3H(5,𝔽2)11yescharacteristic
236C23×C3H(5,𝔽2)12yes—
237C23×C3H(5,𝔽2)11yescharacteristic
238C23×C3H(5,𝔽2)11yescharacteristic
239C23×C3H(5,𝔽2)12yes—
240C23×C32−1+411yescharacteristic
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4C4×Q814yes—
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—24no—
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×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3C4×Q818yes—
300Dic6—416no—
301C24C23×D311yescharacteristic
302C2×Q8—124no—
303C2×Q8—128no—
304C2×Q8—124no—
305C2×Q8—124no—
306C2×Q8—128no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—34no—
332C22×C4—34no—
333C22×C4—34no—
334C22×C4—34no—
335C22×C4—34no—
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—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4C2×Dic614yes—
360C22×C4C2×Dic612yes—
361C22×C4C2×C4×D312yes—
362C22×C4C2×C4×D312yes—
363C22×C4C2×C4×D312yes—
364C22×C4#23512yes—
365C22×C4D3×Q814yes—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—128no—
369C4⋊C4—128no—
370C4⋊C4—124no—
371C42—44no—
372C42—128no—
373C22×C3C2×C4×Q811yescharacteristic
374C22×C3C2×C4×Q812yes—
375C22×C3#51811yescharacteristic
376C22×C3#51812yes—
377C22×C3#52011yescharacteristic
378C22×C3#53811yescharacteristic
379C22×C3#53812yes—
380C22×C3#53912yes—
381C22×C3#53911yescharacteristic
382C22×C3#54312yes—
383C22×C3#54311yescharacteristic
384C22×C3#54412yes—
385C22×C3#54411yescharacteristic
386C22×C3#55412yes—
387C22×C3#55412yes—
388C22×C3#55412yes—
389C22×C3#55511yescharacteristic
390C22×C3#55512yes—
391C22×C3Q8211yescharacteristic
392C22×C3Q8212yes—
393C22×C3#56011yescharacteristic
394C22×C3#56012yes—
395C22×C3H(3,𝔽4)11yescharacteristic
396C3×C4—24no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—44no—
401Dic3—28no—
402Dic3—44no—
403Dic3—44no—
404Dic3—82no—
405C23C22×Dic611yescharacteristic
406C23C22×C4×D311yescharacteristic
407C23C2×#23511yescharacteristic
408C23#100411yescharacteristic
409C23#100411yescharacteristic
410C23#100411yescharacteristic
411C23#100412yes—
412C23C2×D3×Q812yes—
413C23#100711yescharacteristic
414C23D3×#4112yes—
415C23#100912yes—
416Q8—1216no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—38no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
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—64no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—64no—
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×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4C4×Dic614yes—
476C2×C3#232911yescharacteristic
477C2×C3#232912yes—
478C2×C3#236912yes—
479C2×C3#236911yescharacteristic
480C2×C3#237511yescharacteristic
481C2×C3#278412yes—
482C2×C3#278411yescharacteristic
483C2×C3#278511yescharacteristic
484C2×C3#279211yescharacteristic
485C2×C3#279212yes—
486C2×C3#281111yescharacteristic
487C22C2×C4×Dic611yescharacteristic
488C22#548311yescharacteristic
489C22#548611yescharacteristic
490C22#549411yescharacteristic
491C22#550011yescharacteristic
492C22#550411yescharacteristic
493C22#551112yes—
494C22#551511yescharacteristic
495C22#553212yes—
496C22#553912yes—
497C22#554512yes—
498C22Q8×Dic612yes—
499C22C4×D3×Q812yes—
500C22#561312yes—
501C22#562012yes—
502C22#563412yes—
503C22#563612yes—
504C22#567011yescharacteristic
505C22#568311yescharacteristic
506C22#569212yes—
507C22#572011yescharacteristic
508C22#572211yescharacteristic
509C22#572912yes—
510C4—24no—
511C4—44no—
512C4—44no—
513C4—42no—
514C4—42no—
515C4—68no—
516C4—124no—
517C4—124no—
518C4—242no—
519C3#1460111yescharacteristic, complemented, Sylow
520C2#7333411yescharacteristic
521C2#7365712yes—
522C2#7399312yes—
523C2#7421511yescharacteristic
524C2#7435011yescharacteristic
525C2#7616912yes—
526C2#7619912yes—
527C2#7641411yescharacteristic
528C2#7643411yescharacteristic
529C2#7649411yescharacteristic
530C2#7660411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4⋊#95C2
socle, prime radical, cosocleC24×C3C24
hypercenterC4⋊#95D3
hypocenterC3#14601
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
32. #4979C4semidirect
61. C2×C3×C4⋊C4Q8semidirect
72. C2×Dic3⋊C4Q8semidirect
209. C2×C4⋊C4Dic6semidirect
519. C3#14601semidirect

Maximal subgroups

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

subgroupindex
C2×#49792
C3×C4⋊#952
#715652
#715642
#715362
#694242
#694102
#694062
Dic3⋊#952
#146013