Finite Group Databasebeta

← #10718088

Subgroups — #10718088

Order 768.

2358 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D12C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C2×#4988C211yesmaximal, characteristic
4C2×#4988C211yesmaximal, characteristic
5C2×#4977C211yesmaximal, characteristic, complemented
6#71570C214yesmaximal, complemented
7#71563C211yesmaximal, characteristic, complemented
8C4×C3⋊#95C211yesmaximal, characteristic, complemented
9(C2×D3)⋊#95C214yesmaximal, complemented
10#13979—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C23×C4×D3C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2214yes—
19C22×(C2×D3)⋊C4C2214yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C2×C4×D12—24no—
24C2×C4×D12—24no—
25C2×C4×D12—24no—
26C3×#389—24no—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2214yes—
32#4988—24no—
33#4988C418yescomplemented
34#4977—24no—
35C22×C4×D4—31no—
36C2×#390—31no—
37C2×#390—31no—
38C2×#389D311yescharacteristic, complemented
39C2×#382—31no—
40#1439—34no—
41C4⋊#95—31no—
42C22⋊#95—34no—
43C4×#95—31no—
44C24×D3C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C22×D12—22no—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12—22no—
57C22×D12C2×C414yes—
58C22×D12D414yes—
59C22×D12Q814yes—
60C22×C4×D3—24no—
61C22×C4×D3—24no—
62C22×C4×D3—24no—
63C22×C4×D3—28no—
64C22×C4×D3—44no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—44no—
72C2×C3×C4⋊C4C2×C414yescomplemented
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42D414yes—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—28no—
80C2×(C2×D3)⋊C4—28no—
81C2×(C2×D3)⋊C4—44no—
82C2×(C2×D3)⋊C4—44no—
83C2×(C2×D3)⋊C4—44no—
84C2×(C2×D3)⋊C4C2×C4116yescomplemented
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D414yes—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3C2×C414yescomplemented
94C4×D12—232no—
95C3×#95—24no—
96C3⋊#95—24no—
97C3⋊#95—28no—
98C3⋊#95—24no—
99C3⋊#95—44no—
100C23×D4—31no—
101C24×C4—32no—
102C2×C4×D4—64no—
103C2×C4×D4—64no—
104C2×C4×D4—64no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C4⋊C4C2×D312yes—
109C22×C22⋊C4—34no—
110C22×C22⋊C4—32no—
111C22×C22⋊C4—34no—
112C22×C42—31no—
113C22×C42—31no—
114C22×C42C2×D311yescharacteristic
115#390—38no—
116#390—64no—
117#389—24no—
118#382—64no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—34no—
122C2×#95—34no—
123C2×#95C2×D312yes—
124C24×C3C2411yescharacteristic
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—24no—
128C23×D3—24no—
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—24no—
132C23×D3—24no—
133C23×D3C22×C414yes—
134C23×D3C2×D414yes—
135C23×D3C2×Q814yes—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4C22×C414yes—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4#4112yes—
161C22×C3×C4#4114yes—
162C22×C3×C4#4112yes—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—22no—
175C22×Dic3—24no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3C22×C414yes—
180C22×Dic3C2×D414yes—
181C22×Dic3#4114yes—
182C2×D12—28no—
183C2×D12—28no—
184C2×D12—28no—
185C2×D12—28no—
186C2×D12—28no—
187C2×D12—28no—
188C2×D12C4⋊C4116yescomplemented
189C2×C4×D3—216no—
190C2×C4×D3—416no—
191C2×C4×D3—416no—
192C2×C4×D3—48no—
193C2×C4×D3—48no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—48no—
196C3×C4⋊C4—44no—
197C3×C42—28no—
198(C2×D3)⋊C4—216no—
199(C2×D3)⋊C4—416no—
200(C2×D3)⋊C4—416no—
201C4⋊Dic3—28no—
202C4×Dic3—28no—
203C4×Dic3—44no—
204C25—32no—
205C22×D4—34no—
206C22×D4—34no—
207C22×D4—34no—
208C22×D4—62no—
209C22×D4—62no—
210C22×D4—62no—
211C22×D4—62no—
212C23×C4—32no—
213C23×C4—32no—
214C23×C4—32no—
215C23×C4—64no—
216C23×C4—64no—
217C23×C4—64no—
218C23×C4—68no—
219C23×C4—124no—
220C23×C4C22×D312yes—
221C23×C4C22×D312yes—
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C4×D4—632no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4C4×D314yescomplemented
237C2×C22⋊C4—316no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—68no—
242C2×C22⋊C4—68no—
243C2×C22⋊C4—124no—
244C2×C22⋊C4—124no—
245C2×C22⋊C4—124no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—34no—
249C2×C42—62no—
250C2×C42—62no—
251C2×C42—62no—
252C2×C42—62no—
253C2×C42—62no—
254C2×C42#6714yes—
255#95—24no—
256#95—64no—
257#95—68no—
258#95—64no—
259#95—124no—
260C23×C3C23×C411yescharacteristic
261C23×C3C22×D411yescharacteristic
262C23×C3C22×D411yescharacteristic
263C23×C3C22×D411yescharacteristic
264C23×C3C22×Q811yescharacteristic
265C23×C3C2×#4111yescharacteristic
266C23×C3C2×#4111yescharacteristic
267C23×C3C2×#4112yes—
268C23×C3C2×#4112yes—
269C23×C3H(5,𝔽2)12yes—
270C23×C3H(5,𝔽2)12yes—
271C22×D3—24no—
272C22×D3—24no—
273C22×D3—28no—
274C22×D3—28no—
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—24no—
278C22×D3—24no—
279C22×D3—28no—
280C22×D3—28no—
281C22×D3—28no—
282C22×D3—28no—
283C22×D3—42no—
284C22×D3—44no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—44no—
289C22×D3—44no—
290C22×D3—44no—
291C22×D3—44no—
292C22×D3—44no—
293C22×D3C2×C4⋊C418yes—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—28no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—44no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4C2×C4⋊C414yes—
314C2×C3×C4C4×D414yes—
315C2×C3×C4C4×D414yes—
316C2×C3×C4#12014yes—
317C2×C3×C4C22⋊Q814yes—
318C2×C3×C4C22⋊Q814yes—
319C2×C3×C4#12314yes—
320C2×C3×C4#12414yes—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—28no—
328C2×Dic3—24no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—44no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340D12—232no—
341C4×D3—432no—
342C24—34no—
343C24—34no—
344C24—34no—
345C24—62no—
346C24—62no—
347C24—64no—
348C24—64no—
349C24—62no—
350C24—64no—
351C24—64no—
352C24—62no—
353C24C23×D311yescharacteristic
354C2×D4—316no—
355C2×D4—68no—
356C2×D4—68no—
357C2×D4—68no—
358C2×D4—68no—
359C2×D4—68no—
360C2×D4—68no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—34no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—616no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—1216no—
398C22×C4—1216no—
399C22×C4—128no—
400C22×C4—128no—
401C22×C4C2×C4×D314yes—
402C22×C4C2×C4×D312yes—
403C22×C4#23512yes—
404C22×C4D3×D412yes—
405C22×C4D3×D412yes—
406C22×C4D3×Q812yes—
407C22×C4#23914yes—
408C22×C4#23912yes—
409C22×C4C2×#6712yes—
410C22×C4C2×#6712yes—
411C22×C4C2×#6712yes—
412C4⋊C4—28no—
413C4⋊C4—48no—
414C4⋊C4—44no—
415C4⋊C4—68no—
416C22⋊C4—616no—
417C22⋊C4—1216no—
418C22⋊C4—1216no—
419C42—28no—
420C42—68no—
421C42—124no—
422C22×C3C22×C4⋊C411yescharacteristic
423C22×C3C2×C4×D411yescharacteristic
424C22×C3C2×C4×D411yescharacteristic
425C22×C3C4×#4112yes—
426C22×C3#51812yes—
427C22×C3C2×#12011yescharacteristic
428C22×C3C2×C22⋊Q811yescharacteristic
429C22×C3C2×C22⋊Q811yescharacteristic
430C22×C3C2×#12311yescharacteristic
431C22×C3C2×#12411yescharacteristic
432C22×C3#52912yes—
433C22×C3#53211yescharacteristic
434C22×C3#53511yescharacteristic
435C22×C3#53511yescharacteristic
436C22×C3#54012yes—
437C22×C3#54612yes—
438C22×C3#54612yes—
439C22×C3#54612yes—
440C22×C3#55012yes—
441C22×C3#55312yes—
442C22×C3#55412yes—
443C22×C3#55412yes—
444C22×C3#55512yes—
445C2×D3—216no—
446C2×D3—28no—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—48no—
451C2×D3—48no—
452C2×D3—44no—
453C2×D3—44no—
454C2×D3—48no—
455C2×D3—44no—
456C3×C4—24no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—82no—
460C3×C4#38518yes—
461Dic3—44no—
462Dic3—44no—
463Dic3—44no—
464C23—38no—
465C23—68no—
466C23—68no—
467C23—64no—
468C23—64no—
469C23—64no—
470C23—64no—
471C23—64no—
472C23—68no—
473C23—64no—
474C23—68no—
475C23—68no—
476C23—68no—
477C23—124no—
478C23—124no—
479C23—122no—
480C23—122no—
481C23—124no—
482C23—122no—
483C23—122no—
484C23—124no—
485C23—124no—
486C23—124no—
487C23C22×C4×D311yescharacteristic
488C23C2×#23511yescharacteristic
489C23C2×D3×D411yescharacteristic
490C23C2×D3×D411yescharacteristic
491C23#100412yes—
492C23C2×D3×Q811yescharacteristic
493C23C2×#23911yescharacteristic
494C23C2×#23912yes—
495C23D3×#4112yes—
496C23#100912yes—
497C23C22×#6711yescharacteristic
498D4—632no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—68no—
526C2×C4—1232no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—124no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4D3×C4⋊C414yes—
539C2×C4#89314yes—
540C2×C4#89414yes—
541C2×C4#89614yes—
542C2×C4C4×#6714yes—
543C2×C4#93714yes—
544C2×C4#94614yes—
545C2×C4#94714yes—
546C2×C3C2×#38511yescharacteristic
547C2×C3C4×#12411yescharacteristic
548C2×C3#232811yescharacteristic
549C2×C3#234912yes—
550C2×C3#236312yes—
551C2×C3#242012yes—
552C2×C3#243111yescharacteristic
553C2×C3#243511yescharacteristic
554C2×C3#245412yes—
555C2×C3#256311yescharacteristic
556C2×C3#256611yescharacteristic
557D3—48no—
558C22—68no—
559C22—616no—
560C22—124no—
561C22—124no—
562C22—124no—
563C22—124no—
564C22—124no—
565C22—128no—
566C22—128no—
567C22—124no—
568C22—128no—
569C22C2×D3×C4⋊C411yescharacteristic
570C22C2×#89311yescharacteristic
571C22C2×#89411yescharacteristic
572C22C2×#89611yescharacteristic
573C22#554512yes—
574C22#556212yes—
575C22C4×#23912yes—
576C22#558012yes—
577C22#559912yes—
578C22#561212yes—
579C22#563012yes—
580C22#563012yes—
581C22#569012yes—
582C22#569512yes—
583C22#573012yes—
584C22#573412yes—
585C22C2×C4×#6711yescharacteristic
586C22C2×#93711yescharacteristic
587C22#580511yescharacteristic
588C22C2×#94611yescharacteristic
589C22C2×#94711yescharacteristic
590C22#583011yescharacteristic
591C22#583211yescharacteristic
592C4—24no—
593C4—44no—
594C4—42no—
595C4—82no—
596C4—124no—
597C4—124no—
598C4—124no—
599C4#498918yes—
600C3#1397911yescharacteristic, complemented, Sylow
601C2—128no—
602C2#7598512yes—
603C2#7601512yes—
604C2#7620312yes—
605C2#7621312yes—
606C2C2×#498911yescharacteristic
607C2#8232911yescharacteristic
608C2C4×#94711yescharacteristic
609C2#8265211yescharacteristic
610C2#8276111yescharacteristic
611C2#8304011yescharacteristic
612C2#8326011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#389C2
socle, cosocleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#389D3
hypocenterC3#13979
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
5. C2×#4977C2semidirect
6. #71570C2semidirect
7. #71563C2semidirect
8. C4×C3⋊#95C2semidirect
9. (C2×D3)⋊#95C2semidirect
33. #4988C4semidirect
38. C2×#389D3semidirect
72. C2×C3×C4⋊C4C2×C4semidirect
84. C2×(C2×D3)⋊C4C2×C4semidirect
93. C2×C4×Dic3C2×C4semidirect
188. C2×D12C4⋊C4semidirect
236. C2×C4⋊C4C4×D3semidirect
600. C3#13979semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
C2×C3×#3892
C2×#49882
C2×#49772
#715702
#715632
C4×C3⋊#952
(C2×D3)⋊#952
#139793