Finite Group Databasebeta

← #10728699

Subgroups — #10728699

Order 768.

1742 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic
2C2×#4656C212yesmaximal
3C2×#4646C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic
5#72134C211yesmaximal, characteristic
6#69420C211yesmaximal, characteristic
7#69419C211yesmaximal, characteristic
8#69411C212yesmaximal
9#69407C212yesmaximal
10#69406C212yesmaximal
11#69403C211yesmaximal, characteristic
12#14881—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×Q8×Dic3—28no—
16C2×Q8×Dic3—24no—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C22×C4⋊Dic3C2212yes—
20C22×Dic3⋊C4C2212yes—
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⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4656—28no—
39#4646—24no—
40C22×C4×Q8—31no—
41C2×#395—32no—
42C2×#391—31no—
43#1446—31no—
44#1446D311yescharacteristic
45#1444—31no—
46#1444—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×Q8Q814yescomplemented
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—24no—
71C22×Dic6—42no—
72Q8×Dic3—232no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—42no—
80C2×C4⋊Dic3D418yes—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—42no—
85C2×Dic3⋊C4—42no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3D414yes—
91C3×#95—22no—
92C3×#95—24no—
93C3×#95—22no—
94C3×#95—24no—
95C3⋊#95—22no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95Q818yescomplemented
105C23×Q8—31no—
106C23×Q8C2×D311yescharacteristic
107C2×C4×Q8—64no—
108C2×C4×Q8—68no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C42—31no—
115C22×C42—32no—
116#395—68no—
117#391—64no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—31no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95C2×D312yes—
129C2×#95C2×D311yescharacteristic
130C2×#95C2×D312yes—
131C2×#95C2×D311yescharacteristic
132C24×C3C2411yescharacteristic
133C2×C3×Q8—28no—
134C2×C3×Q8—28no—
135C2×C3×Q8—216no—
136C2×C3×Q8—28no—
137C2×C3×Q8—216no—
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—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q814yes—
162C22×C3×C4C2×Q812yes—
163C22×C3×C4C2×Q814yes—
164C22×C3×C4#4114yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×Dic3—22no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×D414yes—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D412yes—
198C22×Dic3C2×Q814yes—
199C22×Dic3C2×Q814yes—
200C22×Dic3#4112yes—
201C2×Dic6—28no—
202C2×Dic6—48no—
203C2×Dic6—48no—
204C2×Dic6—44no—
205C2×Dic6—44no—
206C4⋊Dic3—28no—
207C4⋊Dic3—216no—
208C4⋊Dic3—48no—
209Dic3⋊C4—48no—
210Dic3⋊C4—44no—
211C4×Dic3—216no—
212C4×Dic3—28no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—22no—
216C22×Q8—24no—
217C22×Q8—62no—
218C22×Q8—62no—
219C22×Q8—64no—
220C22×Q8—62no—
221C22×Q8—122no—
222C22×Q8Dic614yescomplemented
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—32no—
226C23×C4—31no—
227C23×C4—32no—
228C23×C4—31no—
229C23×C4C22×D311yescharacteristic
230C23×C4C22×D312yes—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D312yes—
234C4×Q8—632no—
235C2×C4⋊C4—38no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—64no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—122no—
246C2×C4⋊C4—122no—
247C2×C4⋊C4—122no—
248C2×C42—34no—
249C2×C42—64no—
250C2×C42—64no—
251C2×C42—64no—
252C2×C42—64no—
253#95—22no—
254#95—24no—
255#95—22no—
256#95—24no—
257#95—38no—
258#95—62no—
259#95—62no—
260#95—64no—
261#95—62no—
262#95—64no—
263#95—62no—
264#95—64no—
265#95—122no—
266#95—122no—
267C23×C3C22×D411yescharacteristic
268C23×C3C22×D412yes—
269C23×C3C22×Q811yescharacteristic
270C23×C3C22×Q812yes—
271C23×C3C2×#4112yes—
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4111yescharacteristic
274C23×C3C2×#4111yescharacteristic
275C23×C3H(5,𝔽2)12yes—
276C23×C32−1+411yescharacteristic
277C23×C32−1+411yescharacteristic
278C3×Q8—232no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4C22⋊Q814yes—
307C2×C3×C4C22⋊Q818yes—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3#12014yes—
341C2×Dic3C4⋊Q814yes—
342Dic6—416no—
343C24C23×D311yescharacteristic
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—216no—
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—68no—
350C2×Q8—128no—
351C2×Q8—124no—
352C2×Q8—128no—
353C2×Q8—124no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—32no—
376C22×C4—34no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—32no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4C2×Dic614yes—
410C22×C4C2×Dic612yes—
411C22×C4D3×D412yes—
412C22×C4D3×D414yes—
413C22×C4#23712yes—
414C22×C4#23714yes—
415C22×C4D3×Q814yes—
416C22×C4#23912yes—
417C4⋊C4—616no—
418C4⋊C4—68no—
419C4⋊C4—128no—
420C4⋊C4—128no—
421C4⋊C4—124no—
422C42—68no—
423C42—616no—
424C22×C3C2×#12011yescharacteristic
425C22×C3C2×C22⋊Q811yescharacteristic
426C22×C3C2×C22⋊Q812yes—
427C22×C3#52511yescharacteristic
428C22×C3#52512yes—
429C22×C3#52511yescharacteristic
430C22×C3C2×C4⋊Q811yescharacteristic
431C22×C3#53312yes—
432C22×C3#53312yes—
433C22×C3#53511yescharacteristic
434C22×C3#53611yescharacteristic
435C22×C3#54212yes—
436C22×C3#54212yes—
437C22×C3#54612yes—
438C22×C3#54612yes—
439C22×C3D4×Q812yes—
440C22×C3D4×Q812yes—
441C22×C3#55412yes—
442C22×C3#55611yescharacteristic
443C22×C3#55611yescharacteristic
444C22×C3#55711yescharacteristic
445C22×C3#55711yescharacteristic
446C22×C3Q8212yes—
447C3×C4—28no—
448C3×C4—24no—
449C3×C4—42no—
450C3×C4—44no—
451C3×C4—81no—
452Dic3—24no—
453Dic3—24no—
454Dic3—42no—
455Dic3—44no—
456Dic3—44no—
457Dic3—81no—
458C23C22×Dic611yescharacteristic
459C23C2×D3×D412yes—
460C23C2×D3×D411yescharacteristic
461C23C2×#23711yescharacteristic
462C23C2×#23712yes—
463C23C2×D3×Q812yes—
464C23C2×#23911yescharacteristic
465C23#100711yescharacteristic
466C23D3×#4111yescharacteristic
467C23#100912yes—
468C23#101011yescharacteristic
469Q8—232no—
470Q8—1216no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—34no—
487C2×C4—34no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—122no—
517C2×C4—121no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—121no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4#87814yes—
533C2×C4#87818yes—
534C2×C3#239911yescharacteristic
535C2×C3#240411yescharacteristic
536C2×C3#243411yescharacteristic
537C2×C3#245112yes—
538C2×C3#245112yes—
539C2×C3#246512yes—
540C2×C3#246512yes—
541C2×C3#256711yescharacteristic
542C2×C3#258111yescharacteristic
543C2×C3#263211yescharacteristic
544C2×C3#284811yescharacteristic
545C22C2×#87811yescharacteristic
546C22C2×#87812yes—
547C22Q8×Dic612yes—
548C22#556911yescharacteristic
549C22#557011yescharacteristic
550C22#558611yescharacteristic
551C22#558612yes—
552C22D3×#12011yescharacteristic
553C22#562212yes—
554C22#562711yescharacteristic
555C22#562811yescharacteristic
556C22#563012yes—
557C22#563212yes—
558C22#563512yes—
559C22#564312yes—
560C22#567111yescharacteristic
561C22#567411yescharacteristic
562C22#567612yes—
563C22#568212yes—
564C22D3×C4⋊Q811yescharacteristic
565C22#572712yes—
566C22#572812yes—
567C22#573211yescharacteristic
568C4—28no—
569C4—24no—
570C4—42no—
571C4—44no—
572C4—64no—
573C4—64no—
574C4—81no—
575C4—122no—
576C4—124no—
577C4—124no—
578C4—241no—
579C3#1488111yescharacteristic, complemented, Sylow
580C2#7384611yescharacteristic
581C2#7650812yes—
582C2#7650912yes—
583C2#7654311yescharacteristic
584C2#7654711yescharacteristic
585C2#7678812yes—
586C2#7681211yescharacteristic
587C2#7686311yescharacteristic
588C2#7717111yescharacteristic
589C2#7717311yescharacteristic
590C2#7717912yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1446C2
hypercenter#1446D3
hypocenterC3#14881
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
55. C22×C3×Q8Q8semidirect
104. C3⋊#95Q8semidirect
222. C22×Q8Dic6semidirect
579. C3#14881semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×#46562
C2×#46462
C3×#14462
#721342
#694202
#694192
#694112
#694072
#694062
#694032
#148813