Finite Group Databasebeta

← #10700577

Subgroups — #10700577

Order 768.

1522 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5798C211yesmaximal, characteristic
2C22×C3×#99C211yesmaximal, characteristic
3#84043C211yesmaximal, characteristic
4C2×#4527C212yesmaximal, complemented
5C2×#4526C212yesmaximal, complemented
6#10754—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4C2212yes—
9C22×#217—21no—
10C22×#217C2211yescharacteristic
11C22×#217C2211yescharacteristic
12C22×#217C2212yes—
13#5798—22no—
14#5798—24no—
15#5798C414yescomplemented
16C2×D4×Dic3—24no—
17C2×#926—21no—
18C2×C3×#99—22no—
19C2×C3×#99—22no—
20C2×C3×#99C414yescomplemented
21#5247—21no—
22#5220—22no—
23#5220C414yescomplemented
24#5211—21no—
25#4527—24no—
26#4526—24no—
27C2×#518—31no—
28C22×#99D311yescharacteristic
29#2308—31no—
30C2×#342—32no—
31C2×#342—32no—
32C25×C3D411yescharacteristic
33C25×C3C2311yescharacteristic
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—22no—
38C22×C3×D4C2×C412yes—
39C22×C3×D4C2×C412yes—
40C22×C3×D4C2×C412yes—
41C22×C3×D4D412yes—
42C22×C3×D4D412yes—
43C22×C3×D4Q812yes—
44C23×C3×C4—22no—
45C23×C3×C4D411yescharacteristic
46C23×Dic3—22no—
47C23×Dic3—22no—
48C23×Dic3—21no—
49C23×Dic3C2×C412yescomplemented
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4C2×C414yescomplemented
56C2×#217—22no—
57C2×#217—24no—
58C2×#217—24no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217—22no—
68C2×#217—22no—
69C2×#217—24no—
70C2×#217C2×C414yes—
71C2×#217C2×C412yescomplemented
72C2×#217D414yes—
73C2×#217D414yes—
74D4×Dic3—216no—
75D4×Dic3—216no—
76#926—24no—
77#926—24no—
78C2×C4⋊Dic3—22no—
79C2×C4×Dic3—22no—
80C3×#99—28no—
81C3⋊#95—24no—
82C23×D4C2×D311yescharacteristic
83#518—34no—
84#518—64no—
85#518—62no—
86C2×C4×D4—64no—
87C2×#117—61no—
88C22×C22⋊C4—32no—
89C22×C22⋊C4—31no—
90C22×C22⋊C4—31no—
91C22×C22⋊C4—61no—
92C22×C22⋊C4C2×D312yes—
93C2×#99—22no—
94C2×#99—22no—
95C2×#99Dic314yescomplemented
96#386—34no—
97#386—62no—
98#381—61no—
99#379—61no—
100#342—64no—
101#342—64no—
102C24×C3—21no—
103C24×C3—21no—
104C24×C3—21no—
105C24×C3—21no—
106C24×C3—82no—
107C24×C3—82no—
108C24×C3C22⋊C412yes—
109C24×C3C22⋊C412yes—
110C24×C3C4⋊C412yes—
111C24×C3C22×C411yescharacteristic
112C24×C3C22×C411yescharacteristic
113C24×C3C22×C411yescharacteristic
114C24×C3C2×D411yescharacteristic
115C24×C3C2×D412yes—
116C24×C3C2×D412yes—
117C24×C3C2×D411yescharacteristic
118C24×C3C2×D411yescharacteristic
119C24×C3C2×Q811yescharacteristic
120C24×C3#4112yes—
121C24×C3#4112yes—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4C4214yes—
135C2×C3×D4C22⋊C414yes—
136C2×C3×D4C4⋊C414yes—
137C2×C3×D4C4⋊C414yes—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—24no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—84no—
146C22×C3×C4C22⋊C412yes—
147C22×C3×C4C22⋊C412yes—
148C22×C3×C4C4⋊C412yes—
149C22×Dic3—24no—
150C22×Dic3—28no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—24no—
156C22×Dic3—42no—
157C22×Dic3—84no—
158C22×Dic3—84no—
159C22×Dic3—82no—
160C22×Dic3—82no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—42no—
164C3×C22⋊C4—42no—
165#217—28no—
166#217—24no—
167#217—24no—
168#217—24no—
169#217—24no—
170#217—41no—
171#217—41no—
172#217—42no—
173#217—41no—
174#217—41no—
175#217—84no—
176#217C22⋊C418yescomplemented
177C4⋊Dic3—28no—
178C4×Dic3—28no—
179C25D1211yescharacteristic
180C25C22×D311yescharacteristic
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4Dic612yes—
186C22×D4C4×D312yes—
187C22×D4C4×D312yes—
188C22×D4C2×Dic312yes—
189C22×D4#6712yes—
190C22×D4#6712yes—
191C23×C4—22no—
192C23×C4—32no—
193C23×C4—62no—
194C23×C4—62no—
195C23×C4—61no—
196C23×C4D1211yescharacteristic
197C4×D4—616no—
198C4×D4—616no—
199#117—64no—
200#117—64no—
201C2×C4⋊C4—62no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—34no—
208C2×C22⋊C4—34no—
209C2×C22⋊C4—34no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—64no—
214C2×C22⋊C4—62no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4C2×Dic314yescomplemented
226C2×C42—62no—
227#99—28no—
228#95—64no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—42no—
236C23×C3—44no—
237C23×C3—42no—
238C23×C3—44no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3—44no—
242C23×C3—82no—
243C23×C3—81no—
244C23×C3—81no—
245C23×C3—82no—
246C23×C3—81no—
247C23×C3—81no—
248C23×C3—82no—
249C23×C3—82no—
250C23×C3—82no—
251C23×C3—82no—
252C23×C3—82no—
253C23×C3#9514yes—
254C23×C3C2×C4211yescharacteristic
255C23×C3C2×C22⋊C412yes—
256C23×C3C2×C22⋊C411yescharacteristic
257C23×C3C2×C22⋊C411yescharacteristic
258C23×C3C2×C22⋊C411yescharacteristic
259C23×C3C2×C4⋊C411yescharacteristic
260C23×C3C2×C4⋊C411yescharacteristic
261C23×C3C2×C4⋊C411yescharacteristic
262C23×C3#11712yes—
263C23×C3C4×D412yes—
264C23×C3C4×D412yes—
265C23×C3C4×D412yes—
266C23×C3C4×D412yes—
267C23×C3#12011yescharacteristic
268C23×C3#12011yescharacteristic
269C23×C3#12012yes—
270C23×C3#12112yes—
271C23×C3#12112yes—
272C23×C3C22⋊Q812yes—
273C23×C3C22⋊Q812yes—
274C23×C3#12311yescharacteristic
275C23×C3#12311yescharacteristic
276C23×C3#12312yes—
277C3×D4—28no—
278C3×D4—28no—
279C3×D4—28no—
280C3×D4—28no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—44no—
288C2×C3×C4—84no—
289C2×C3×C4—84no—
290C2×C3×C4—84no—
291C2×C3×C4#9514yes—
292C2×Dic3—28no—
293C2×Dic3—48no—
294C2×Dic3—44no—
295C2×Dic3—44no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—84no—
299C2×Dic3—84no—
300C2×Dic3—88no—
301C2×Dic3—84no—
302C2×Dic3—84no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—82no—
308C24—82no—
309C24C4⋊Dic312yes—
310C24(C2×D3)⋊C412yes—
311C24(C2×D3)⋊C412yes—
312C24C2×Dic611yescharacteristic
313C24C2×C4×D311yescharacteristic
314C24C2×C4×D311yescharacteristic
315C24C2×D1211yescharacteristic
316C24D3×D412yes—
317C24D3×D412yes—
318C24#23712yes—
319C24#23712yes—
320C24C22×Dic311yescharacteristic
321C24C2×#6711yescharacteristic
322C24C2×#6711yescharacteristic
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4C4×Dic314yes—
336C2×D4Dic3⋊C414yes—
337C2×D4Dic3⋊C414yes—
338C2×D4#21714yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—64no—
347C22×C4—68no—
348C22×C4—64no—
349C22×C4—64no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—64no—
353C22×C4—84no—
354C22×C4—122no—
355C22×C4—244no—
356C22×C4—244no—
357C22×C4—242no—
358C22×C4—242no—
359C22×C4C4⋊Dic312yes—
360C22×C4(C2×D3)⋊C412yes—
361C22×C4(C2×D3)⋊C412yes—
362C4⋊C4—68no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—38no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—68no—
369C22⋊C4—64no—
370C22⋊C4—64no—
371C22⋊C4—64no—
372C22⋊C4—64no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—122no—
378C22⋊C4—244no—
379C42—68no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—24no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3C2×#9511yescharacteristic
412C22×C3C4×C22⋊C412yes—
413C22×C3#37911yescharacteristic
414C22×C3#38012yes—
415C22×C3#38111yescharacteristic
416C22×C3#38512yes—
417C22×C3#38511yescharacteristic
418C22×C3#38511yescharacteristic
419C22×C3#38611yescharacteristic
420C22×C3#38611yescharacteristic
421C22×C3#38612yes—
422C3×C4—24no—
423C3×C4—84no—
424Dic3—48no—
425Dic3—82no—
426Dic3—82no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—44no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—82no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—81no—
446C23—81no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23C3⋊#9514yes—
452C23#87712yes—
453C23#87812yes—
454C23#87812yes—
455C23D3×C22⋊C412yes—
456C23#67⋊C412yes—
457C23#67⋊C412yes—
458C23#88211yescharacteristic
459C23#88211yescharacteristic
460C23#88611yescharacteristic
461C23#88611yescharacteristic
462C23C2×C4×Dic311yescharacteristic
463C23C2×Dic3⋊C411yescharacteristic
464C23C2×Dic3⋊C411yescharacteristic
465C23C2×C4⋊Dic311yescharacteristic
466C23C2×(C2×D3)⋊C411yescharacteristic
467C23C2×(C2×D3)⋊C411yescharacteristic
468C23D4×Dic312yes—
469C23D4×Dic312yes—
470C23#93512yes—
471C23#93712yes—
472C23#93912yes—
473C23#93912yes—
474C23C2×#21711yescharacteristic
475D4—28no—
476D4—28no—
477D4—28no—
478D4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—44no—
486C2×C4—68no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—128no—
491C2×C4—124no—
492C2×C4—124no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—244no—
496C2×C4—244no—
497C2×C4—248no—
498C2×C4—244no—
499C2×C4—244no—
500C2×C4C3⋊#9514yes—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—22no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—44no—
512C2×C3C22⋊#9511yescharacteristic
513C2×C3#191312yes—
514C22—24no—
515C22—22no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22Dic3×C22⋊C412yes—
546C22#494212yes—
547C22#494311yescharacteristic
548C22#494612yes—
549C22#495011yescharacteristic
550C22#495011yescharacteristic
551C22#495511yescharacteristic
552C22#495811yescharacteristic
553C22#495811yescharacteristic
554C22C2×C3⋊#9511yescharacteristic
555C22#522012yes—
556C4—24no—
557C4—84no—
558C4—128no—
559C4—242no—
560C4—242no—
561C3#1075411yescharacteristic, complemented, Sylow
562C2—22no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—22no—
571C2—22no—
572C2—44no—
573C2#7002412yes—
574C2#7154411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC22×C3×#99C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenterC22×#99D3
hypocenterC3#10754
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
4. C2×#4527C2semidirect
5. C2×#4526C2semidirect
15. #5798C4semidirect
20. C2×C3×#99C4semidirect
23. #5220C4semidirect
49. C23×Dic3C2×C4semidirect
55. C2×C3×C22⋊C4C2×C4semidirect
71. C2×#217C2×C4semidirect
95. C2×#99Dic3semidirect
176. #217C22⋊C4semidirect
225. C2×C22⋊C4C2×Dic3semidirect
561. C3#10754semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
C22×C3×#992
#840432
C2×#45272
C2×#45262
#107543