Finite Group Databasebeta

← #10731974

Subgroups — #10731974

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic
2C2×#4985C211yesmaximal, characteristic
3C2×#4986C211yesmaximal, characteristic
4C2×#4976C211yesmaximal, characteristic
5C2×#4975C211yesmaximal, characteristic
6C2×#4659C211yesmaximal, characteristic
7#69421C212yesmaximal
8#69421C212yesmaximal
9Dic3⋊#95C212yesmaximal
10Dic3⋊#95C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15153—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C3×#400—24no—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37#4986—24no—
38#4985—24no—
39#4976—24no—
40#4975—24no—
41#4659—24no—
42C2×#400—31no—
43C2×#400—31no—
44C2×#400—31no—
45C2×#400D311yescharacteristic
46C2×#389—31no—
47C2×#384—31no—
48#1445—32no—
49#1445—32no—
50C4⋊#95—32no—
51C4⋊#95—32no—
52C4×#95—31no—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—28no—
83C2×Dic3⋊C4—28no—
84C2×Dic3⋊C4Q814yes—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3D414yes—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95D414yes—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95D414yes—
103C22×C4⋊C4—31no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C42—31no—
112C22×C42—32no—
113#400—24no—
114#400—64no—
115#400—64no—
116#400—64no—
117#389—64no—
118#384—64no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95C2×D312yes—
130C2×#95C2×D312yes—
131C2×#95C2×D311yescharacteristic
132C24×C3C2411yescharacteristic
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q812yes—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3C2×D414yes—
188C22×Dic3C2×D412yes—
189C22×Dic3C2×D412yes—
190C22×Dic3C2×Q812yes—
191C22×Dic3C2×Q812yes—
192C22×Dic3#4114yes—
193C22×Dic3#4112yes—
194C22×Dic3#4114yes—
195C22×Dic3#4114yes—
196C3×C4⋊C4—44no—
197C3×C4⋊C4—44no—
198C4⋊Dic3—44no—
199Dic3⋊C4—28no—
200Dic3⋊C4—44no—
201Dic3⋊C4—44no—
202Dic3⋊C4—48no—
203Dic3⋊C4—48no—
204C4×Dic3—28no—
205C4×Dic3—48no—
206C23×C4—31no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—32no—
210C23×C4—32no—
211C23×C4—31no—
212C23×C4C22×D312yes—
213C23×C4C22×D311yescharacteristic
214C23×C4C22×D311yescharacteristic
215C23×C4C22×D312yes—
216C23×C4C22×D311yescharacteristic
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—34no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—68no—
231C2×C4⋊C4—68no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C42—34no—
239C2×C42—64no—
240C2×C42—64no—
241C2×C42—64no—
242C2×C42—64no—
243#95—24no—
244#95—24no—
245#95—34no—
246#95—64no—
247#95—62no—
248#95—62no—
249#95—64no—
250#95—64no—
251#95—64no—
252#95—64no—
253#95—64no—
254#95—64no—
255#95#6714yes—
256C23×C3C22×D411yescharacteristic
257C23×C3C22×D411yescharacteristic
258C23×C3C22×D411yescharacteristic
259C23×C3C22×Q811yescharacteristic
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4112yes—
262C23×C3C2×#4112yes—
263C23×C3C2×#4112yes—
264C23×C32−1+411yescharacteristic
265C23×C32−1+412yes—
266C23×C32−1+411yescharacteristic
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—44no—
291C2×C3×C4—44no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4#12714yes—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—44no—
320C2×Dic3—44no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3C22⋊Q814yes—
325C2×Dic3#12314yes—
326C2×Dic3#12414yes—
327C2×Dic3#12414yes—
328C2×Dic3#12518yes—
329C2×Dic3#12618yes—
330C24C23×D311yescharacteristic
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—34no—
358C22×C4—32no—
359C22×C4—34no—
360C22×C4—32no—
361C22×C4—34no—
362C22×C4—34no—
363C22×C4—32no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—64no—
385C22×C4—64no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4D3×D412yes—
390C22×C4D3×D412yes—
391C22×C4D3×Q812yes—
392C22×C4C2×#6714yes—
393C22×C4C2×#6712yes—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—68no—
397C4⋊C4—124no—
398C4⋊C4—128no—
399C4⋊C4—128no—
400C4⋊C4—124no—
401C4⋊C4—124no—
402C42—68no—
403C42—128no—
404C22×C3C2×C22⋊Q811yescharacteristic
405C22×C3C2×#12311yescharacteristic
406C22×C3C2×#12411yescharacteristic
407C22×C3C2×#12411yescharacteristic
408C22×C3C2×#12512yes—
409C22×C3C2×#12612yes—
410C22×C3C2×#12711yescharacteristic
411C22×C3#53212yes—
412C22×C3#53212yes—
413C22×C3#53611yescharacteristic
414C22×C3#53611yescharacteristic
415C22×C3#54112yes—
416C22×C3#54112yes—
417C22×C3#54712yes—
418C22×C3#54711yescharacteristic
419C22×C3#54712yes—
420C22×C3#54711yescharacteristic
421C22×C3D4×Q811yescharacteristic
422C22×C3D4×Q811yescharacteristic
423C22×C3#55212yes—
424C22×C3#55212yes—
425C22×C3#55612yes—
426C22×C3#55712yes—
427C3×C4—24no—
428C3×C4—44no—
429C3×C4—42no—
430C3×C4—82no—
431C3×C4—81no—
432Dic3—24no—
433Dic3—44no—
434Dic3—42no—
435Dic3—82no—
436Dic3—81no—
437Dic3#40018yescomplemented
438C23C2×D3×D411yescharacteristic
439C23C2×D3×D411yescharacteristic
440C23C2×D3×Q811yescharacteristic
441C23#100711yescharacteristic
442C23D3×#4112yes—
443C23D3×#4112yes—
444C23D3×#4112yes—
445C23D3×#4111yescharacteristic
446C23#101011yescharacteristic
447C23#101012yes—
448C23C22×#6711yescharacteristic
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—34no—
460C2×C4—34no—
461C2×C4—38no—
462C2×C4—38no—
463C2×C4—34no—
464C2×C4—34no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—62no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—64no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—124no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—121no—
509C2×C4—122no—
510C2×C4—124no—
511C2×C4#94014yes—
512C2×C3C2×#40011yescharacteristic
513C2×C3#249612yes—
514C2×C3#249612yes—
515C2×C3#250111yescharacteristic
516C2×C3#250311yescharacteristic
517C2×C3#250612yes—
518C2×C3#250612yes—
519C2×C3#254211yescharacteristic
520C2×C3#256811yescharacteristic
521C2×C3#264211yescharacteristic
522C2×C3#285811yescharacteristic
523C22D3×C22⋊Q811yescharacteristic
524C22#562811yescharacteristic
525C22#563111yescharacteristic
526C22#563211yescharacteristic
527C22#564811yescharacteristic
528C22D3×#12311yescharacteristic
529C22#565512yes—
530C22#566812yes—
531C22D3×#12411yescharacteristic
532C22D3×#12411yescharacteristic
533C22#568512yes—
534C22D3×#12512yes—
535C22#569312yes—
536C22#569612yes—
537C22#570112yes—
538C22#570212yes—
539C22D3×#12612yes—
540C22#570812yes—
541C22C2×#94011yescharacteristic
542C22Q8×#6711yescharacteristic
543C22#582412yes—
544C22#582411yescharacteristic
545C22#583212yes—
546C4—24no—
547C4—38no—
548C4—42no—
549C4—44no—
550C4—64no—
551C4—82no—
552C4—81no—
553C4—124no—
554C4—122no—
555C4—242no—
556C4—241no—
557C3#1515311yescharacteristic, complemented, Sylow
558C2#7815012yes—
559C2#7815211yescharacteristic
560C2#7815312yes—
561C2#7818111yescharacteristic
562C2#7821411yescharacteristic
563C2D3×#40011yescharacteristic
564C2#8278011yescharacteristic
565C2#8285911yescharacteristic
566C2#8299311yescharacteristic
567C2#8303412yes—
568C2#8308312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#400C2
hypercenterC2×#400D3
hypocenterC3#15153
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
437. Dic3#400semidirect
557. C3#15153semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4002
C2×#49852
C2×#49862
C2×#49762
C2×#49752
C2×#46592
#694212
Dic3⋊#952
Dic3×#952
#151533