Finite Group Databasebeta

← #10702265

Subgroups — #10702265

Order 768.

2538 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#379C211yesmaximal, characteristic
2C2×#5247C212yesmaximal
3#82094C211yesmaximal, characteristic, complemented
4C2×#4527C212yesmaximal, complemented
5C2×#4526C211yesmaximal, characteristic, complemented
6#11069—31nomaximal, Sylow
7C26×C3C2211yescharacteristic
8C22×C3×C22⋊C4—22no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217—24no—
12C22×#217C2212yes—
13C22×#217C2212yes—
14C3×#379—22no—
15C3×#379C414yescomplemented
16#5247—24no—
17#5247—24no—
18#5212—22no—
19#4950—24no—
20#4527—24no—
21#4526—22no—
22#2315—31no—
23C2×#379—32no—
24C2×#379D311yescharacteristic
25C2×#342—31no—
26C2×#342—32no—
27C25×C3—22no—
28C25×C3—21no—
29C25×C3—22no—
30C25×C3—21no—
31C25×C3—21no—
32C25×C3—21no—
33C25×C3—22no—
34C25×C3—22no—
35C25×C3—48no—
36C25×C3C2×C412yes—
37C25×C3D412yes—
38C25×C3Q812yes—
39C25×C3C2311yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4D412yes—
42C23×Dic3—22no—
43C23×Dic3D414yes—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—28no—
50C2×C3×C22⋊C4—48no—
51C2×C3×C22⋊C4C2×C412yes—
52C2×C3×C22⋊C4C2×C412yes—
53C2×C3×C22⋊C4D412yes—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217—216no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—22no—
65C2×#217—22no—
66C2×#217—416no—
67C2×#217D414yes—
68C2×C4⋊Dic3—24no—
69C3⋊#95—24no—
70C26C2×D311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—32no—
73C22×C22⋊C4—32no—
74C22×C22⋊C4—64no—
75C22×C22⋊C4C2×D311yescharacteristic
76C22×C22⋊C4C2×D311yescharacteristic
77#385—64no—
78#380—62no—
79#379—22no—
80#379—64no—
81#379—64no—
82#379Dic314yescomplemented
83#342—64no—
84#342—62no—
85C24×C3—216no—
86C24×C3—28no—
87C24×C3—22no—
88C24×C3—24no—
89C24×C3—24no—
90C24×C3—22no—
91C24×C3—24no—
92C24×C3—22no—
93C24×C3—22no—
94C24×C3—24no—
95C24×C3—22no—
96C24×C3—24no—
97C24×C3—22no—
98C24×C3—24no—
99C24×C3—48no—
100C24×C3—48no—
101C24×C3—48no—
102C24×C3—44no—
103C24×C3—48no—
104C24×C3—44no—
105C24×C3—416no—
106C24×C3—48no—
107C24×C3—48no—
108C24×C3—48no—
109C24×C3—416no—
110C24×C3—416no—
111C24×C3—41no—
112C24×C3—42no—
113C24×C3—42no—
114C24×C3—42no—
115C24×C3—42no—
116C24×C3—41no—
117C24×C3—42no—
118C24×C3—41no—
119C24×C3—42no—
120C24×C3—41no—
121C24×C3C4⋊C414yescomplemented
122C24×C3C22×C411yescharacteristic
123C24×C3C2×D411yescharacteristic
124C24×C3C2×D412yes—
125C24×C3C2×D412yes—
126C24×C3C2×D411yescharacteristic
127C24×C3C2×D411yescharacteristic
128C24×C3C2×Q811yescharacteristic
129C24×C3#4111yescharacteristic
130C24×C3#4111yescharacteristic
131C24×C3#4112yes—
132C24×C3#4112yes—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4C22⋊C414yescomplemented
141C22×Dic3—24no—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—24no—
145C22×Dic3—48no—
146C22×Dic3—48no—
147C22×Dic3—42no—
148C22×Dic3—42no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—416no—
153C3×C22⋊C4—42no—
154#217—24no—
155#217—24no—
156#217—432no—
157#217—42no—
158#217—42no—
159#217—42no—
160#217—42no—
161#217—42no—
162#217—42no—
163C4⋊Dic3—84no—
164C25—22no—
165C25—21no—
166C25—21no—
167C25—21no—
168C25—22no—
169C25—22no—
170C25—22no—
171C25—21no—
172C25—48no—
173C25Dic612yes—
174C25D1212yes—
175C25C2×Dic312yes—
176C25C22×D311yescharacteristic
177C23×C4—21no—
178C23×C4—34no—
179C23×C4—62no—
180C23×C4#6712yes—
181C2×C4⋊C4—64no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—616no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—1216no—
203C2×C22⋊C4C2×Dic312yes—
204C2×C22⋊C4C2×Dic312yes—
205C2×C22⋊C4#6712yes—
206#95—64no—
207C23×C3—28no—
208C23×C3—24no—
209C23×C3—216no—
210C23×C3—28no—
211C23×C3—216no—
212C23×C3—28no—
213C23×C3—216no—
214C23×C3—28no—
215C23×C3—24no—
216C23×C3—28no—
217C23×C3—48no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—44no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—44no—
226C23×C3—44no—
227C23×C3—416no—
228C23×C3—44no—
229C23×C3—48no—
230C23×C3—48no—
231C23×C3—48no—
232C23×C3—416no—
233C23×C3—48no—
234C23×C3—44no—
235C23×C3—416no—
236C23×C3—48no—
237C23×C3—48no—
238C23×C3—44no—
239C23×C3—48no—
240C23×C3—416no—
241C23×C3—48no—
242C23×C3—416no—
243C23×C3—48no—
244C23×C3—416no—
245C23×C3—416no—
246C23×C3—416no—
247C23×C3—416no—
248C23×C3—42no—
249C23×C3—44no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—44no—
254C23×C3—44no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—48no—
258C23×C3C2×C22⋊C411yescharacteristic
259C23×C3C2×C4⋊C411yescharacteristic
260C23×C3#11711yescharacteristic
261C23×C3C4×D412yes—
262C23×C3C4×D412yes—
263C23×C3#12012yes—
264C23×C3#12011yescharacteristic
265C23×C3#12112yes—
266C23×C3#12111yescharacteristic
267C23×C3#12111yescharacteristic
268C23×C3#12112yes—
269C23×C3#12112yes—
270C23×C3C22⋊Q812yes—
271C23×C3C22⋊Q812yes—
272C23×C3C22⋊Q811yescharacteristic
273C23×C3C22⋊Q811yescharacteristic
274C23×C3#12312yes—
275C23×C3#12312yes—
276C23×C3#12311yescharacteristic
277C23×C3#12412yes—
278C23×C3#12411yescharacteristic
279C23×C3#12612yes—
280C23×C3#12711yescharacteristic
281C2×C3×C4—24no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4—42no—
286C2×C3×C4—82no—
287C2×C3×C4—82no—
288C2×C3×C4—82no—
289C2×Dic3—48no—
290C2×Dic3—48no—
291C2×Dic3—44no—
292C2×Dic3—44no—
293C2×Dic3—48no—
294C2×Dic3—84no—
295C2×Dic3—84no—
296C2×Dic3—84no—
297C24—28no—
298C24—216no—
299C24—24no—
300C24—24no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—44no—
315C24—44no—
316C24—48no—
317C24—48no—
318C24—416no—
319C24—48no—
320C24—48no—
321C24—416no—
322C24—416no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—41no—
328C24—41no—
329C24—41no—
330C24—42no—
331C24—42no—
332C24—41no—
333C24C4⋊Dic314yescomplemented
334C24C2×Dic611yescharacteristic
335C24C2×D1211yescharacteristic
336C24#23511yescharacteristic
337C24#23511yescharacteristic
338C24D3×D412yes—
339C24D3×D412yes—
340C24#23712yes—
341C24#23712yes—
342C24C22×Dic311yescharacteristic
343C24C2×#6711yescharacteristic
344C24C2×#6711yescharacteristic
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—64no—
353C22×C4—64no—
354C22×C4—64no—
355C22×C4—64no—
356C22×C4—128no—
357C22×C4—128no—
358C22×C4—122no—
359C22×C4—122no—
360C22×C4#21714yescomplemented
361C4⋊C4—244no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—416no—
366C22⋊C4—42no—
367C22⋊C4—64no—
368C22⋊C4—64no—
369C22⋊C4—1232no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—122no—
374C22⋊C4—122no—
375C22⋊C4—122no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—22no—
388C22×C3—28no—
389C22×C3—216no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—42no—
393C22×C3—41no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—416no—
401C22×C3—48no—
402C22×C3—44no—
403C22×C3—48no—
404C22×C3—416no—
405C22×C3—416no—
406C22×C3—48no—
407C22×C3—48no—
408C22×C3—48no—
409C22×C3—44no—
410C22×C3—48no—
411C22×C3—48no—
412C22×C3#38011yescharacteristic
413C22×C3#38512yes—
414C22×C3#38611yescharacteristic
415C22×C3#38812yes—
416C22×C3#39011yescharacteristic
417C22×C3#39211yescharacteristic
418C22×C3#39311yescharacteristic
419C22×C3#39412yes—
420C22×C3#39612yes—
421C22×C3#39711yescharacteristic
422C22×C3#39911yescharacteristic
423C22×C3B(4,𝔽2)14yes—
424C3×C4—44no—
425C3×C4—82no—
426Dic3—84no—
427Dic3—84no—
428C23—24no—
429C23—28no—
430C23—28no—
431C23—216no—
432C23—216no—
433C23—28no—
434C23—28no—
435C23—216no—
436C23—28no—
437C23—24no—
438C23—48no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—416no—
450C23—48no—
451C23—44no—
452C23—48no—
453C23—48no—
454C23—416no—
455C23—44no—
456C23—44no—
457C23—48no—
458C23—416no—
459C23—48no—
460C23—416no—
461C23—416no—
462C23—48no—
463C23—48no—
464C23—48no—
465C23—416no—
466C23—416no—
467C23—416no—
468C23—416no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—48no—
479C23#87812yes—
480C23#87812yes—
481C23#87912yes—
482C23#88212yes—
483C23#88312yes—
484C23#88412yes—
485C23#88512yes—
486C23#88612yes—
487C23#92411yescharacteristic
488C23#92411yescharacteristic
489C23C2×C4⋊Dic311yescharacteristic
490C23#92611yescharacteristic
491C23#93011yescharacteristic
492C23#93011yescharacteristic
493C23D4×Dic312yes—
494C23D4×Dic312yes—
495C23#93511yescharacteristic
496C23#93611yescharacteristic
497C23#93711yescharacteristic
498C23#93812yes—
499C23#93912yes—
500C23#94011yescharacteristic
501C23C2×#21711yescharacteristic
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—128no—
511C2×C4—128no—
512C2×C4—124no—
513C2×C4—124no—
514C2×C4—128no—
515C2×C4—244no—
516C2×C4—244no—
517C2×C4—244no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—48no—
527C2×C3#143911yescharacteristic
528C2×C3#188912yes—
529C2×C3#202112yes—
530C2×C3#202912yes—
531C22—22no—
532C22—24no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—24no—
542C22—24no—
543C22—216no—
544C22—28no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—42no—
555C22—48no—
556C22—416no—
557C22—48no—
558C22—44no—
559C22—416no—
560C22—48no—
561C22—416no—
562C22—48no—
563C22—48no—
564C22—44no—
565C22—48no—
566C22—48no—
567C22#494711yescharacteristic
568C22#494812yes—
569C22#494911yescharacteristic
570C22#495012yes—
571C22#495112yes—
572C22#495311yescharacteristic
573C22#496011yescharacteristic
574C22#496112yes—
575C22#503214yes—
576C22#521211yescharacteristic
577C22#522011yescharacteristic
578C22#522111yescharacteristic
579C4—44no—
580C4—82no—
581C4—244no—
582C4—244no—
583C3#1106911yescharacteristic, complemented, Sylow
584C2—21no—
585C2—22no—
586C2—22no—
587C2—21no—
588C2—22no—
589C2—21no—
590C2—22no—
591C2—21no—
592C2—48no—
593C2#6949712yes—
594C2#6950012yes—
595C2#7154611yescharacteristic
596C2#7166912yes—
subgroupG/H
centerC23#937
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC2×C3×#379C2
socleC23×C3#120
prime radicalC26×C3C22
cosocleC25×C3C23
hypercenterC2×#379D3
hypocenterC3#11069
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
3. #82094C2semidirect
4. C2×#4527C2semidirect
5. C2×#4526C2semidirect
15. C3×#379C4semidirect
82. #379Dic3semidirect
121. C24×C3C4⋊C4semidirect
140. C22×C3×C4C22⋊C4semidirect
333. C24C4⋊Dic3semidirect
360. C22×C4#217semidirect
583. C3#11069semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3792
C2×#52472
#820942
C2×#45272
C2×#45262
#110693