Finite Group Databasebeta

← #10727915

Subgroups — #10727915

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4983C211yesmaximal, characteristic
3C2×#4976C212yesmaximal
4C2×#4933C211yesmaximal, characteristic
5C2×C4×C4⋊Dic3C211yesmaximal, characteristic
6#71569C211yesmaximal, characteristic
7#71568C211yesmaximal, characteristic
8#71566C211yesmaximal, characteristic
9#71564C212yesmaximal
10#69411C212yesmaximal
11#69411C212yesmaximal
12#14818—31nomaximal, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×C4×Dic3C2212yes—
23C3×#389—24no—
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37#4983—24no—
38#4976—28no—
39C4×C4⋊Dic3—24no—
40#4933—24no—
41C2×#398—31no—
42C2×#389—32no—
43C2×#389D311yescharacteristic
44C2×#387—31no—
45C2×C4×C4⋊C4—31no—
46#1445—31no—
47#1444—31no—
48#1441—32no—
49#1441—32no—
50#1441—31no—
51#1436—32no—
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42D414yes—
74C2×C4⋊Dic3—24no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3Q818yes—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—42no—
88C2×Dic3⋊C4D414yes—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—24no—
91C2×C4×Dic3—42no—
92C3×#95—24no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—42no—
104C3⋊#95D414yes—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C4⋊C4C2×D312yes—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42—32no—
114C22×C42C2×D311yescharacteristic
115#398—64no—
116#389—24no—
117#389—68no—
118#387—64no—
119C4×C4⋊C4—64no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—31no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95C2×D312yes—
133C24×C3C2411yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
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—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q814yes—
162C22×C3×C4#4114yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D414yes—
198C22×Dic3C2×Q814yes—
199C22×Dic3C2×Q814yes—
200C22×Dic3#4114yes—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—48no—
203C3×C4⋊C4—44no—
204C3×C42—28no—
205C4⋊Dic3—216no—
206Dic3⋊C4—216no—
207Dic3⋊C4—28no—
208Dic3⋊C4—48no—
209Dic3⋊C4—44no—
210C4×Dic3—48no—
211C23×C4—32no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—32no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4C22×D312yes—
218C23×C4C22×D312yes—
219C23×C4C22×D311yescharacteristic
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—38no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—122no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—64no—
248C2×C42—64no—
249C2×C42—122no—
250C2×C42#6714yes—
251#95—24no—
252#95—34no—
253#95—64no—
254#95—64no—
255#95—64no—
256#95—64no—
257#95—64no—
258#95—62no—
259#95—62no—
260#95—62no—
261#95—62no—
262#95—62no—
263#95—122no—
264C23×C3C22×D411yescharacteristic
265C23×C3C22×D411yescharacteristic
266C23×C3C22×D411yescharacteristic
267C23×C3C22×Q812yes—
268C23×C3C2×#4112yes—
269C23×C3C2×#4111yescharacteristic
270C23×C3C2×#4111yescharacteristic
271C23×C3C2×#4112yes—
272C23×C3H(5,𝔽2)11yescharacteristic
273C23×C32−1+412yes—
274C23×C32−1+411yescharacteristic
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4#12114yes—
304C2×C3×C4#12114yes—
305C2×C3×C4C4⋊Q814yes—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3C22⋊Q818yes—
342C24C23×D311yescharacteristic
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—34no—
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—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4#23514yes—
402C22×C4D3×D412yes—
403C22×C4D3×D412yes—
404C22×C4#23712yes—
405C22×C4#23712yes—
406C22×C4D3×Q814yes—
407C22×C4C2×#6712yes—
408C22×C4C2×#6712yes—
409C22×C4C2×#6712yes—
410C4⋊C4—28no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—616no—
414C4⋊C4—616no—
415C4⋊C4—68no—
416C4⋊C4—128no—
417C4⋊C4—124no—
418C42—28no—
419C42—128no—
420C22×C3C2×#12111yescharacteristic
421C22×C3C2×#12111yescharacteristic
422C22×C3C2×C22⋊Q812yes—
423C22×C3#52511yescharacteristic
424C22×C3#52512yes—
425C22×C3#52912yes—
426C22×C3C2×C4⋊Q811yescharacteristic
427C22×C3#53211yescharacteristic
428C22×C3#53312yes—
429C22×C3#53611yescharacteristic
430C22×C3#53711yescharacteristic
431C22×C3#53912yes—
432C22×C3#54112yes—
433C22×C3D4211yescharacteristic
434C22×C3#54712yes—
435C22×C3#54711yescharacteristic
436C22×C3#54811yescharacteristic
437C22×C3D4×Q812yes—
438C22×C3#55212yes—
439C22×C3#55311yescharacteristic
440C22×C3#55412yes—
441C22×C3#55712yes—
442C22×C3#55712yes—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—24no—
446C3×C4—44no—
447C3×C4—82no—
448Dic3—28no—
449Dic3—42no—
450Dic3—42no—
451Dic3—44no—
452Dic3—81no—
453Dic3—81no—
454C23C2×#23512yes—
455C23C2×D3×D411yescharacteristic
456C23C2×D3×D411yescharacteristic
457C23C2×#23711yescharacteristic
458C23C2×#23711yescharacteristic
459C23C2×D3×Q812yes—
460C23#100712yes—
461C23D3×#4112yes—
462C23#100911yescharacteristic
463C23#101011yescharacteristic
464C23C22×#6711yescharacteristic
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—38no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—121no—
511C2×C4—122no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—122no—
518C2×C4—121no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4#93814yes—
530C2×C4#93914yes—
531C2×C4#94414yes—
532C2×C3#239211yescharacteristic
533C2×C3#241311yescharacteristic
534C2×C3#241611yescharacteristic
535C2×C3#244912yes—
536C2×C3#245312yes—
537C2×C3#246212yes—
538C2×C3#254611yescharacteristic
539C2×C3#258411yescharacteristic
540C2×C3#258512yes—
541C2×C3#266411yescharacteristic
542C2×C3#285711yescharacteristic
543C22#550712yes—
544C22#551312yes—
545C22#553812yes—
546C22#554712yes—
547C22#556712yes—
548C22#557912yes—
549C22#559811yescharacteristic
550C22#560911yescharacteristic
551C22#561111yescharacteristic
552C22#561811yescharacteristic
553C22#562212yes—
554C22D3×C22⋊Q812yes—
555C22#569212yes—
556C22#569612yes—
557C22#572112yes—
558C22#572712yes—
559C22#579111yescharacteristic
560C22C2×#93811yescharacteristic
561C22C2×#93911yescharacteristic
562C22C2×#94411yescharacteristic
563C22#581611yescharacteristic
564C22#583211yescharacteristic
565C22#583311yescharacteristic
566C4—24no—
567C4—24no—
568C4—24no—
569C4—44no—
570C4—68no—
571C4—82no—
572C4—122no—
573C4—124no—
574C4—122no—
575C4—241no—
576C4—241no—
577C3#1481811yescharacteristic, complemented, Sylow
578C2#7601412yes—
579C2#7601712yes—
580C2#7608212yes—
581C2#7622512yes—
582C2#8217011yescharacteristic
583C2#8240111yescharacteristic
584C2#8248311yescharacteristic
585C2#8270511yescharacteristic
586C2#8281111yescharacteristic
587C2#8304511yescharacteristic
588C2#8323911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#389C2
hypercenterC2×#389D3
hypocenterC3#14818
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
577. C3#14818semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49832
C2×#49762
C2×#49332
C2×C4×C4⋊Dic32
#715692
#715682
#715662
#715642
#694112
#148183