Finite Group Databasebeta

← #10732635

Subgroups — #10732635

Order 768.

1678 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×#4976C211yesmaximal, characteristic
3C2×#4650C212yesmaximal
4C3×#1441C211yesmaximal, characteristic
5#71566C211yesmaximal, characteristic
6#71565C211yesmaximal, characteristic
7#69424C212yesmaximal
8#69424C212yesmaximal
9#69410C211yesmaximal, characteristic
10Dic3⋊#95C212yesmaximal
11Dic3×#95C211yesmaximal, characteristic
12#15230—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2212yes—
23C2×#888—28no—
24C2×#888—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4976—24no—
39#4650—28no—
40C22×C4⋊Q8—31no—
41C2×#389—31no—
42C2×#382—32no—
43#1446—32no—
44#1446—32no—
45#1441—31no—
46#1441—31no—
47#1441—31no—
48#1441D311yescharacteristic
49C4⋊#95—32no—
50C4×#95—31no—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6—24no—
67C22×Dic6—42no—
68C22×Dic6—42no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—24no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4D418yes—
86C2×Dic3⋊C4Q814yescomplemented
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3D414yes—
91C2×C4×Dic3Q818yescomplemented
92#888—232no—
93C3×#95—24no—
94C3×#95—22no—
95C3×#95—42no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C23×Q8—32no—
107C2×C4⋊Q8—64no—
108C2×C4⋊Q8—68no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C42—31no—
117C22×C42—32no—
118#389—64no—
119#382—68no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95C2×D311yescharacteristic
131C2×#95C2×D312yes—
132C2×#95C2×D312yes—
133C24×C3C2411yescharacteristic
134C22×C3×C4—22no—
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—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q814yes—
162C22×C3×C4C2×Q812yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D414yes—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×Q814yes—
196C22×Dic3C2×Q812yes—
197C22×Dic3C2×Q814yes—
198C22×Dic3C2×Q812yes—
199C22×Dic3#4114yes—
200C22×Dic3#4112yes—
201C2×Dic6—216no—
202C2×Dic6—48no—
203C2×Dic6—48no—
204C2×Dic6—48no—
205C2×Dic6—48no—
206C2×Dic6—48no—
207C2×Dic6—48no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—44no—
210C4⋊Dic3—28no—
211Dic3⋊C4—216no—
212Dic3⋊C4—28no—
213Dic3⋊C4—44no—
214Dic3⋊C4—44no—
215C4×Dic3—28no—
216C4×Dic3—216no—
217C22×Q8—64no—
218C22×Q8—64no—
219C22×Q8—64no—
220C22×Q8—64no—
221C22×Q8—64no—
222C22×Q8—122no—
223C22×Q8—122no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—31no—
228C23×C4—32no—
229C23×C4—31no—
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D312yes—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235C4⋊Q8—632no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—38no—
242C2×C4⋊C4—34no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C42—34no—
255C2×C42—38no—
256C2×C42—64no—
257C2×C42—64no—
258C2×C42—64no—
259#95—24no—
260#95—22no—
261#95—42no—
262#95—64no—
263#95—64no—
264#95—64no—
265#95—62no—
266#95—62no—
267#95—62no—
268#95—64no—
269#95—122no—
270#95—122no—
271#95—122no—
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D412yes—
274C23×C3C22×Q812yes—
275C23×C3C22×Q811yescharacteristic
276C23×C3C2×#4111yescharacteristic
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4112yes—
280C23×C3H(5,𝔽2)11yescharacteristic
281C23×C3H(5,𝔽2)11yescharacteristic
282C23×C32−1+412yes—
283C2×C3×C4—22no—
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—22no—
295C2×C3×C4—22no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12414yes—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—22no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3#12014yes—
345C2×Dic3C22⋊Q818yes—
346C2×Dic3C22⋊Q814yes—
347C2×Dic3#12414yes—
348C2×Dic3C4⋊Q814yes—
349C2×Dic3C4⋊Q818yes—
350Dic6—432no—
351C24C23×D311yescharacteristic
352C2×Q8—616no—
353C2×Q8—128no—
354C2×Q8—128no—
355C2×Q8—128no—
356C2×Q8—128no—
357C2×Q8—128no—
358C2×Q8—128no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—32no—
385C22×C4—34no—
386C22×C4—34no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—34no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—32no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4#23512yes—
421C22×C4D3×D414yes—
422C22×C4D3×D412yes—
423C22×C4#23712yes—
424C22×C4D3×Q814yes—
425C22×C4D3×Q812yes—
426C4⋊C4—28no—
427C4⋊C4—44no—
428C4⋊C4—616no—
429C4⋊C4—68no—
430C4⋊C4—68no—
431C4⋊C4—124no—
432C4⋊C4—124no—
433C42—68no—
434C42—616no—
435C22×C3C2×#12011yescharacteristic
436C22×C3C2×C22⋊Q811yescharacteristic
437C22×C3C2×C22⋊Q812yes—
438C22×C3C2×#12411yescharacteristic
439C22×C3C2×#12411yescharacteristic
440C22×C3C2×C4⋊Q811yescharacteristic
441C22×C3C2×C4⋊Q812yes—
442C22×C3#53312yes—
443C22×C3#53312yes—
444C22×C3#53511yescharacteristic
445C22×C3#53611yescharacteristic
446C22×C3#54412yes—
447C22×C3#54412yes—
448C22×C3#54612yes—
449C22×C3#54612yes—
450C22×C3#54812yes—
451C22×C3#54812yes—
452C22×C3#55411yescharacteristic
453C22×C3#55411yescharacteristic
454C22×C3#55511yescharacteristic
455C22×C3#55511yescharacteristic
456C22×C3#55612yes—
457C22×C3#55712yes—
458C3×C4—24no—
459C3×C4—42no—
460C3×C4—44no—
461C3×C4—44no—
462C3×C4—81no—
463Dic3—24no—
464Dic3—42no—
465Dic3—44no—
466Dic3—44no—
467Dic3—81no—
468Dic3#39518yes—
469C23C2×#23511yescharacteristic
470C23C2×D3×D411yescharacteristic
471C23C2×D3×D412yes—
472C23C2×#23711yescharacteristic
473C23#100411yescharacteristic
474C23C2×D3×Q811yescharacteristic
475C23C2×D3×Q812yes—
476C23D3×#4112yes—
477C23D3×#4111yescharacteristic
478C23#100911yescharacteristic
479C23#101012yes—
480Q8—1232no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—34no—
495C2×C4—38no—
496C2×C4—34no—
497C2×C4—38no—
498C2×C4—34no—
499C2×C4—34no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—64no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—62no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4#88514yes—
548C2×C3C2×#39511yescharacteristic
549C2×C3#243511yescharacteristic
550C2×C3#243611yescharacteristic
551C2×C3#243912yes—
552C2×C3#243912yes—
553C2×C3#255012yes—
554C2×C3#255012yes—
555C2×C3#256311yescharacteristic
556C2×C3#256811yescharacteristic
557C2×C3#266011yescharacteristic
558C2×C3#286211yescharacteristic
559C22C2×#88511yescharacteristic
560C22#551111yescharacteristic
561C22#553211yescharacteristic
562C22#555212yes—
563C22#557812yes—
564C22D3×#12011yescharacteristic
565C22#559812yes—
566C22#560711yescharacteristic
567C22#561012yes—
568C22#561611yescharacteristic
569C22#562212yes—
570C22#562312yes—
571C22D3×C22⋊Q811yescharacteristic
572C22D3×C22⋊Q812yes—
573C22#563511yescharacteristic
574C22#564811yescharacteristic
575C22#565412yes—
576C22D3×#12411yescharacteristic
577C22#568512yes—
578C22#568912yes—
579C22#570212yes—
580C22D3×C4⋊Q811yescharacteristic
581C22D3×C4⋊Q812yes—
582C4—24no—
583C4—38no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—64no—
588C4—81no—
589C4—124no—
590C4—122no—
591C4—124no—
592C4—241no—
593C3#1523011yescharacteristic, complemented, Sylow
594C2#7394211yescharacteristic
595C2#7470111yescharacteristic
596C2#7498312yes—
597C2#7606312yes—
598C2#7660711yescharacteristic
599C2#7690511yescharacteristic
600C2D3×#39511yescharacteristic
601C2#7728812yes—
602C2#7778612yes—
603C2#7783711yescharacteristic
604C2#7818111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1441C2
hypercenter#1441D3
hypocenterC3#15230
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
86. C2×Dic3⋊C4Q8semidirect
91. C2×C4×Dic3Q8semidirect
593. C3#15230semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49762
C2×#46502
C3×#14412
#715662
#715652
#694242
#694102
Dic3⋊#952
Dic3×#952
#152303