Finite Group Databasebeta

← #10751714

Subgroups — #10751714

Order 768.

1598 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×#888C211yesmaximal, characteristic
2C2×#4980C211yesmaximal, characteristic
3C2×#4656C212yesmaximal
4C3×#1441C211yesmaximal, characteristic
5#71568C211yesmaximal, characteristic
6#71565C211yesmaximal, characteristic
7#69424C212yesmaximal
8#69420C212yesmaximal
9#69419C211yesmaximal, characteristic
10#69414C212yesmaximal
11#69403C211yesmaximal, characteristic
12#16603—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C2×#888—28no—
23C2×#888—24no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4980—24no—
39#4656—28no—
40C22×C4⋊Q8—31no—
41C2×#395—32no—
42C2×#387—31no—
43#1446—32no—
44#1444—31no—
45#1444—32no—
46#1444—31no—
47#1442—32no—
48#1441—31no—
49#1441D311yescharacteristic
50#1434—31no—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—42no—
65C22×Dic6—42no—
66C22×Dic6—42no—
67C22×Dic6—42no—
68C22×Dic6D418yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4Q814yescomplemented
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—42no—
80C2×C4⋊Dic3—42no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—42no—
86C2×Dic3⋊C4—42no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3D414yes—
89#888—232no—
90C3×#95—24no—
91C3×#95—22no—
92C3×#95—42no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—42no—
101C3⋊#95—42no—
102C3⋊#95—42no—
103C3⋊#95Q818yescomplemented
104C23×Q8—32no—
105C2×C4⋊Q8—64no—
106C2×C4⋊Q8—68no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—32no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C42—31no—
115#395—68no—
116#387—64no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—31no—
121C2×#95—31no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95C2×D312yes—
129C2×#95C2×D311yescharacteristic
130C2×#95C2×D312yes—
131C24×C3C2411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
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—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D414yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×Q812yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4C2×Q812yes—
161C22×C3×C4C2×Q814yes—
162C22×C3×C4#4112yes—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D414yes—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×Q814yes—
196C22×Dic3C2×Q814yes—
197C2×Dic6—216no—
198C2×Dic6—48no—
199C2×Dic6—48no—
200C2×Dic6—48no—
201C2×Dic6—48no—
202C2×Dic6—48no—
203C2×Dic6—48no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—28no—
206C4⋊Dic3—28no—
207C4⋊Dic3—48no—
208C4⋊Dic3—44no—
209Dic3⋊C4—216no—
210Dic3⋊C4—48no—
211C4×Dic3—28no—
212C22×Q8—38no—
213C22×Q8—64no—
214C22×Q8—64no—
215C22×Q8—122no—
216C22×Q8—122no—
217C22×Q8—122no—
218C22×Q8—122no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—32no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D312yes—
228C23×C4C22×D312yes—
229C23×C4C22×D311yescharacteristic
230C4⋊Q8—632no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—64no—
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—122no—
245C2×C4⋊C4—122no—
246C2×C4⋊C4—122no—
247C2×C4⋊C4—122no—
248C2×C4⋊C4Dic614yescomplemented
249C2×C42—34no—
250C2×C42—64no—
251#95—24no—
252#95—22no—
253#95—38no—
254#95—42no—
255#95—62no—
256#95—62no—
257#95—62no—
258#95—64no—
259#95—64no—
260#95—62no—
261#95—62no—
262#95—122no—
263#95—122no—
264#95—122no—
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D412yes—
267C23×C3C22×Q811yescharacteristic
268C23×C3C22×Q812yes—
269C23×C3C2×#4111yescharacteristic
270C23×C3H(5,𝔽2)11yescharacteristic
271C23×C3H(5,𝔽2)11yescharacteristic
272C23×C3H(5,𝔽2)12yes—
273C23×C32−1+411yescharacteristic
274C23×C32−1+411yescharacteristic
275C23×C32−1+412yes—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4C22⋊Q814yes—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3#12014yes—
340C2×Dic3C4⋊Q814yes—
341Dic6—432no—
342C24C23×D311yescharacteristic
343C2×Q8—616no—
344C2×Q8—128no—
345C2×Q8—128no—
346C2×Q8—128no—
347C2×Q8—128no—
348C2×Q8—128no—
349C2×Q8—128no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—32no—
375C22×C4—32no—
376C22×C4—34no—
377C22×C4—34no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—62no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4C2×Dic612yes—
409C22×C4C2×Dic612yes—
410C22×C4C2×Dic612yes—
411C22×C4D3×D414yes—
412C22×C4D3×D412yes—
413C22×C4#23712yes—
414C22×C4D3×Q814yes—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—616no—
418C4⋊C4—68no—
419C4⋊C4—128no—
420C4⋊C4—124no—
421C4⋊C4—128no—
422C42—68no—
423C22×C3C2×#12011yescharacteristic
424C22×C3C2×C22⋊Q811yescharacteristic
425C22×C3C2×C4⋊Q811yescharacteristic
426C22×C3#53412yes—
427C22×C3#53412yes—
428C22×C3#53511yescharacteristic
429C22×C3#53511yescharacteristic
430C22×C3#53611yescharacteristic
431C22×C3#53611yescharacteristic
432C22×C3#54211yescharacteristic
433C22×C3#54211yescharacteristic
434C22×C3#54412yes—
435C22×C3#54411yescharacteristic
436C22×C3#54411yescharacteristic
437C22×C3#54412yes—
438C22×C3#54812yes—
439C22×C3D4×Q812yes—
440C22×C3D4×Q812yes—
441C22×C3D4×Q812yes—
442C22×C3#55412yes—
443C22×C3Q8212yes—
444C22×C3#56312yes—
445C22×C3#56312yes—
446C3×C4—24no—
447C3×C4—44no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4—44no—
451Dic3—24no—
452Dic3—24no—
453Dic3—44no—
454Dic3—44no—
455Dic3—81no—
456Dic3—81no—
457C23C22×Dic611yescharacteristic
458C23C2×D3×D412yes—
459C23C2×D3×D411yescharacteristic
460C23C2×#23711yescharacteristic
461C23#100411yescharacteristic
462C23C2×D3×Q812yes—
463C23#100711yescharacteristic
464C23#100912yes—
465C23#100911yescharacteristic
466C23#101011yescharacteristic
467C23#101012yes—
468Q8—1232no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—34no—
485C2×C4—34no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—62no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—64no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4#87814yes—
534C2×C3#240911yescharacteristic
535C2×C3#243411yescharacteristic
536C2×C3#256711yescharacteristic
537C2×C3#261311yescharacteristic
538C2×C3#262811yescharacteristic
539C2×C3#266011yescharacteristic
540C2×C3#272912yes—
541C2×C3#272912yes—
542C2×C3#273412yes—
543C2×C3#273412yes—
544C2×C3#282111yescharacteristic
545C22C2×#87811yescharacteristic
546C22#550011yescharacteristic
547C22#551911yescharacteristic
548C22D4×Dic612yes—
549C22Q8×Dic612yes—
550C22D3×#12011yescharacteristic
551C22#559812yes—
552C22#560311yescharacteristic
553C22#560512yes—
554C22#560711yescharacteristic
555C22#562312yes—
556C22#562811yescharacteristic
557C22#563212yes—
558C22#563512yes—
559C22#564311yescharacteristic
560C22#564811yescharacteristic
561C22#565312yes—
562C22#567411yescharacteristic
563C22#568912yes—
564C22#569912yes—
565C22#571212yes—
566C22D3×C4⋊Q811yescharacteristic
567C22#572712yes—
568C4—24no—
569C4—44no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—64no—
574C4—64no—
575C4—124no—
576C4—124no—
577C4—241no—
578C4—241no—
579C3#1660311yescharacteristic, complemented, Sylow
580C2#7395411yescharacteristic
581C2#7468911yescharacteristic
582C2#7517912yes—
583C2#7613512yes—
584C2#7654311yescharacteristic
585C2#7712211yescharacteristic
586C2#7717311yescharacteristic
587C2#7736912yes—
588C2#7783711yescharacteristic
589C2#7798512yes—
590C2#7831711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1441C2
hypercenter#1441D3
hypocenterC3#16603
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
73. C2×C3×C4⋊C4Q8semidirect
103. C3⋊#95Q8semidirect
248. C2×C4⋊C4Dic6semidirect
579. C3#16603semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49802
C2×#46562
C3×#14412
#715682
#715652
#694242
#694202
#694192
#694142
#694032
#166033