Finite Group Databasebeta

← #10732465

Subgroups — #10732465

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4973C211yesmaximal, characteristic
2C2×#4647C211yesmaximal, characteristic
3C3×#1445C211yesmaximal, characteristic
4#71569C211yesmaximal, characteristic
5#69424C211yesmaximal, characteristic
6#69424C211yesmaximal, characteristic
7#69414C212yesmaximal
8#69413C212yesmaximal
9#69407C212yesmaximal
10#69406C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15209—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×C4×Dic3C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36#4973—24no—
37#4647—24no—
38C2×#391—31no—
39C2×#389—31no—
40#1446—31no—
41#1446—31no—
42#1445—31no—
43#1445D311yescharacteristic
44#1442—32no—
45#1442—32no—
46#1436—32no—
47#1436—32no—
48C4×#95—31no—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3C2311yescharacteristic
55C23×Dic3C2312yes—
56C23×Dic3C2312yes—
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C22×Dic6—22no—
61C22×Dic6—24no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—42no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—24no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—42no—
77C2×C4×Dic3—24no—
78C2×C4×Dic3—42no—
79C2×C4×Dic3—42no—
80C2×C4×Dic3—42no—
81C2×C4×Dic3D414yes—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—24no—
85C3×#95—42no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—42no—
94C3⋊#95—42no—
95C3⋊#95—42no—
96C3⋊#95Q818yescomplemented
97C23×Q8—31no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4C2×D311yescharacteristic
103C22×C42—31no—
104C22×C42—32no—
105#391—64no—
106#389—64no—
107C2×#95—32no—
108C2×#95—31no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—31no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95C2×D311yescharacteristic
119C2×#95C2×D312yes—
120C2×#95C2×D311yescharacteristic
121C2×#95C2×D312yes—
122C24×C3C2411yescharacteristic
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×Q814yes—
149C22×C3×C4#4112yes—
150C22×C3×C4#4112yes—
151C22×C3×C4#4112yes—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3C2×D412yes—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×Q814yes—
182C22×Dic3C2×Q814yes—
183C22×Dic3#4112yes—
184C22×Dic3#4114yes—
185C22×Dic3#4114yes—
186C2×Dic6—28no—
187C2×Dic6—48no—
188C2×Dic6—48no—
189C2×Dic6—44no—
190C2×Dic6—44no—
191C3×C4⋊C4—44no—
192C4⋊Dic3—44no—
193Dic3⋊C4—28no—
194Dic3⋊C4—44no—
195Dic3⋊C4—44no—
196C4×Dic3—44no—
197C4×Dic3—48no—
198C22×Q8—64no—
199C22×Q8—62no—
200C22×Q8—62no—
201C22×Q8—62no—
202C22×Q8—122no—
203C23×C4—31no—
204C23×C4—31no—
205C23×C4—31no—
206C23×C4—32no—
207C23×C4—32no—
208C23×C4—31no—
209C23×C4C22×D311yescharacteristic
210C23×C4C22×D312yes—
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D312yes—
213C23×C4C22×D311yescharacteristic
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4—62no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—122no—
226C2×C42—34no—
227C2×C42—64no—
228C2×C42—122no—
229C2×C42—122no—
230C2×C42—122no—
231#95—22no—
232#95—22no—
233#95—24no—
234#95—38no—
235#95—42no—
236#95—64no—
237#95—64no—
238#95—64no—
239#95—64no—
240#95—64no—
241#95—62no—
242#95—62no—
243#95—122no—
244#95—122no—
245#95—122no—
246C23×C3C22×D411yescharacteristic
247C23×C3C22×Q812yes—
248C23×C3C2×#4111yescharacteristic
249C23×C3C2×#4111yescharacteristic
250C23×C3C2×#4111yescharacteristic
251C23×C3C2×#4112yes—
252C23×C3C2×#4111yescharacteristic
253C23×C3C2×#4112yes—
254C23×C3H(5,𝔽2)11yescharacteristic
255C23×C3H(5,𝔽2)12yes—
256C23×C32−1+411yescharacteristic
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4#12414yes—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—22no—
295C2×Dic3—24no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3#12414yes—
319C2×Dic3C4⋊Q814yes—
320Dic6—416no—
321C24C23×D311yescharacteristic
322C2×Q8—68no—
323C2×Q8—128no—
324C2×Q8—128no—
325C2×Q8—124no—
326C2×Q8—124no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—34no—
354C22×C4—34no—
355C22×C4—34no—
356C22×C4—32no—
357C22×C4—34no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4#23512yes—
386C22×C4D3×D412yes—
387C22×C4#23712yes—
388C22×C4#23712yes—
389C22×C4D3×Q814yes—
390C4⋊C4—44no—
391C4⋊C4—68no—
392C4⋊C4—124no—
393C4⋊C4—124no—
394C4⋊C4—124no—
395C42—124no—
396C42—128no—
397C22×C3#52511yescharacteristic
398C22×C3#52511yescharacteristic
399C22×C3C2×#12411yescharacteristic
400C22×C3C2×#12411yescharacteristic
401C22×C3#52912yes—
402C22×C3#52912yes—
403C22×C3C2×C4⋊Q811yescharacteristic
404C22×C3#53312yes—
405C22×C3#53312yes—
406C22×C3#53411yescharacteristic
407C22×C3#53711yescharacteristic
408C22×C3#53812yes—
409C22×C3#53912yes—
410C22×C3#55112yes—
411C22×C3#55112yes—
412C22×C3#55412yes—
413C22×C3#55412yes—
414C22×C3#55412yes—
415C22×C3#55611yescharacteristic
416C22×C3#55611yescharacteristic
417C22×C3#55712yes—
418C22×C3#55911yescharacteristic
419C22×C3#55911yescharacteristic
420C3×C4—24no—
421C3×C4—44no—
422C3×C4—42no—
423C3×C4—82no—
424C3×C4—81no—
425Dic3—24no—
426Dic3—24no—
427Dic3—44no—
428Dic3—44no—
429Dic3—42no—
430Dic3—81no—
431C23C2×#23511yescharacteristic
432C23C2×D3×D411yescharacteristic
433C23C2×#23711yescharacteristic
434C23C2×#23711yescharacteristic
435C23#100411yescharacteristic
436C23C2×D3×Q812yes—
437C23D3×#4112yes—
438C23D3×#4112yes—
439C23D3×#4111yescharacteristic
440C23#100912yes—
441C23#101011yescharacteristic
442Q8—1216no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—34no—
453C2×C4—34no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—64no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—121no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4#88514yes—
506C2×C3#239811yescharacteristic
507C2×C3#256111yescharacteristic
508C2×C3#258311yescharacteristic
509C2×C3#258411yescharacteristic
510C2×C3#258812yes—
511C2×C3#258812yes—
512C2×C3#258912yes—
513C2×C3#258912yes—
514C2×C3#259211yescharacteristic
515C2×C3#263911yescharacteristic
516C2×C3#286211yescharacteristic
517C22C2×#88511yescharacteristic
518C22#552712yes—
519C22#554211yescharacteristic
520C22#555911yescharacteristic
521C22#557812yes—
522C22#560611yescharacteristic
523C22#560811yescharacteristic
524C22#562212yes—
525C22#563512yes—
526C22#563612yes—
527C22#564912yes—
528C22#565311yescharacteristic
529C22#565611yescharacteristic
530C22#565712yes—
531C22#565812yes—
532C22#567111yescharacteristic
533C22D3×#12411yescharacteristic
534C22#568512yes—
535C22#569012yes—
536C22#570612yes—
537C22#570712yes—
538C22#571311yescharacteristic
539C22D3×C4⋊Q811yescharacteristic
540C4—24no—
541C4—42no—
542C4—44no—
543C4—64no—
544C4—64no—
545C4—82no—
546C4—81no—
547C4—124no—
548C4—124no—
549C4—122no—
550C4—241no—
551C3#1520911yescharacteristic, complemented, Sylow
552C2#7413211yescharacteristic
553C2#7510212yes—
554C2#7510812yes—
555C2#7636511yescharacteristic
556C2#7660711yescharacteristic
557C2#7746011yescharacteristic
558C2#7768711yescharacteristic
559C2#7779711yescharacteristic
560C2#7796111yescharacteristic
561C2#7804212yes—
562C2#7819412yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1445C2
hypercenter#1445D3
hypocenterC3#15209
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
96. C3⋊#95Q8semidirect
551. C3#15209semidirect

Maximal subgroups

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

subgroupindex
C2×#49732
C2×#46472
C3×#14452
#715692
#694242
#694142
#694132
#694072
#694062
Dic3×#952
#152093