Finite Group Databasebeta

← #10728726

Subgroups — #10728726

Order 768.

1590 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic
2C2×#4660C212yesmaximal
3C2×#4647C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic
5#72134C211yesmaximal, characteristic
6#69415C211yesmaximal, characteristic
7#69412C211yesmaximal, characteristic
8#69411C212yesmaximal
9#69407C212yesmaximal
10#69406C212yesmaximal
11#69403C211yesmaximal, characteristic
12#14884—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×Q8×Dic3—28no—
15C2×Q8×Dic3—24no—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×C4×Dic3C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4660—28no—
39#4647—24no—
40C22×C4×Q8—31no—
41C2×#400—32no—
42C2×#389—31no—
43#1446—31no—
44#1446D311yescharacteristic
45#1442—31no—
46#1442—31no—
47#1441—32no—
48#1436—32no—
49#1436—32no—
50#1434—31no—
51C22×C3×Q8—24no—
52C22×C3×Q8—22no—
53C22×C3×Q8—24no—
54C22×C3×Q8—22no—
55C22×C3×Q8Q814yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67Q8×Dic3—232no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3—24no—
71C2×C4⋊Dic3—24no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3D418yes—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×C4×Dic3—24no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—24no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3D414yes—
87C3×#95—22no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—22no—
91C3⋊#95—22no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C23×Q8C2×D311yescharacteristic
102C2×C4×Q8—64no—
103C2×C4×Q8—68no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C42—31no—
111C22×C42—32no—
112#400—68no—
113#389—64no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—31no—
124C2×#95C2×D312yes—
125C2×#95C2×D311yescharacteristic
126C2×#95C2×D312yes—
127C2×#95C2×D311yescharacteristic
128C24×C3C2411yescharacteristic
129C2×C3×Q8—28no—
130C2×C3×Q8—28no—
131C2×C3×Q8—216no—
132C2×C3×Q8—28no—
133C2×C3×Q8—216no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D414yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×Q814yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4#4114yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D414yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195C22×Dic3#4112yes—
196C22×Dic3#4114yes—
197C4⋊Dic3—28no—
198C4⋊Dic3—216no—
199C4⋊Dic3—44no—
200Dic3⋊C4—28no—
201Dic3⋊C4—44no—
202Dic3⋊C4—44no—
203C4×Dic3—216no—
204C4×Dic3—28no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8Dic614yes—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—32no—
214C23×C4—31no—
215C23×C4—32no—
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D312yes—
218C23×C4C22×D311yescharacteristic
219C23×C4C22×D311yescharacteristic
220C23×C4C22×D312yes—
221C4×Q8—632no—
222C2×C4⋊C4—38no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C42—34no—
237C2×C42—64no—
238C2×C42—64no—
239C2×C42—64no—
240C2×C42—64no—
241#95—22no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—64no—
246#95—64no—
247#95—64no—
248#95—64no—
249#95—64no—
250#95—62no—
251#95—64no—
252#95—64no—
253#95—62no—
254#95—64no—
255C23×C3C22×D411yescharacteristic
256C23×C3C22×D412yes—
257C23×C3C22×Q811yescharacteristic
258C23×C3C2×#4112yes—
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4111yescharacteristic
262C23×C3C2×#4112yes—
263C23×C32−1+412yes—
264C23×C32−1+411yescharacteristic
265C23×C32−1+411yescharacteristic
266C3×Q8—232no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4C22⋊Q814yes—
295C2×C3×C4C22⋊Q818yes—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—22no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3#12014yes—
329C2×Dic3#12414yes—
330C24C23×D311yescharacteristic
331C2×Q8—28no—
332C2×Q8—216no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—216no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—34no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—34no—
362C22×C4—34no—
363C22×C4—32no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4C2×Dic614yes—
393C22×C4C2×Dic612yes—
394C22×C4D3×D412yes—
395C22×C4D3×D414yes—
396C22×C4#23712yes—
397C22×C4#23714yes—
398C22×C4#23912yes—
399C4⋊C4—616no—
400C4⋊C4—68no—
401C4⋊C4—68no—
402C4⋊C4—124no—
403C4⋊C4—124no—
404C4⋊C4—124no—
405C42—68no—
406C42—616no—
407C22×C3C2×#12011yescharacteristic
408C22×C3C2×C22⋊Q811yescharacteristic
409C22×C3C2×C22⋊Q812yes—
410C22×C3#52511yescharacteristic
411C22×C3#52512yes—
412C22×C3#52511yescharacteristic
413C22×C3C2×#12411yescharacteristic
414C22×C3#52912yes—
415C22×C3#52912yes—
416C22×C3#53611yescharacteristic
417C22×C3#53611yescharacteristic
418C22×C3#54112yes—
419C22×C3#54112yes—
420C22×C3#54712yes—
421C22×C3#54712yes—
422C22×C3#54812yes—
423C22×C3#54812yes—
424C22×C3#55612yes—
425C22×C3#55611yescharacteristic
426C22×C3#55611yescharacteristic
427C22×C3#55712yes—
428C22×C3#55711yescharacteristic
429C22×C3#55711yescharacteristic
430C3×C4—28no—
431C3×C4—24no—
432C3×C4—42no—
433C3×C4—81no—
434C3×C4—82no—
435Dic3—24no—
436Dic3—24no—
437Dic3—42no—
438Dic3—44no—
439Dic3—81no—
440Dic3—82no—
441C23C22×Dic611yescharacteristic
442C23C2×D3×D412yes—
443C23C2×D3×D411yescharacteristic
444C23C2×#23711yescharacteristic
445C23C2×#23712yes—
446C23C2×#23911yescharacteristic
447C23#100711yescharacteristic
448C23#100712yes—
449C23D3×#4112yes—
450C23D3×#4111yescharacteristic
451C23#101011yescharacteristic
452Q8—232no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—34no—
466C2×C4—34no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—64no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4#87814yes—
515C2×C4#87818yes—
516C2×C3#239811yescharacteristic
517C2×C3#240611yescharacteristic
518C2×C3#243611yescharacteristic
519C2×C3#245312yes—
520C2×C3#245312yes—
521C2×C3#246912yes—
522C2×C3#246912yes—
523C2×C3#256811yescharacteristic
524C2×C3#258111yescharacteristic
525C2×C3#263411yescharacteristic
526C2×C3#284811yescharacteristic
527C22C2×#87811yescharacteristic
528C22C2×#87812yes—
529C22#556912yes—
530C22#556911yescharacteristic
531C22#557011yescharacteristic
532C22#558612yes—
533C22#558611yescharacteristic
534C22D3×#12011yescharacteristic
535C22#562412yes—
536C22#562512yes—
537C22#562711yescharacteristic
538C22#562811yescharacteristic
539C22#563112yes—
540C22#563312yes—
541C22#564112yes—
542C22#567111yescharacteristic
543C22D3×#12411yescharacteristic
544C22#567712yes—
545C22#568012yes—
546C22#572411yescharacteristic
547C22#572612yes—
548C22#573112yes—
549C22#573211yescharacteristic
550C4—28no—
551C4—24no—
552C4—42no—
553C4—64no—
554C4—64no—
555C4—81no—
556C4—82no—
557C4—124no—
558C4—122no—
559C4—242no—
560C4—241no—
561C3#1488411yescharacteristic, complemented, Sylow
562C2#7384611yescharacteristic
563C2#7651112yes—
564C2#7651212yes—
565C2#7654411yescharacteristic
566C2#7654711yescharacteristic
567C2#7677911yescharacteristic
568C2#7678111yescharacteristic
569C2#7678912yes—
570C2#7718112yes—
571C2#7720411yescharacteristic
572C2#7724911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1446C2
hypercenter#1446D3
hypocenterC3#14884
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
561. C3#14884semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×#46602
C2×#46472
C3×#14462
#721342
#694152
#694122
#694112
#694072
#694062
#694032
#148843