Finite Group Databasebeta

← #10710159

Subgroups — #10710159

Order 768.

1026 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#99C211yesmaximal, characteristic
2#84044C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4#67695C214yesmaximal, complemented
5#12582—31nomaximal, Sylow
6C23×C3×D4C2211yescharacteristic
7C22×C3×C22⋊C4—21no—
8C22×#217—21no—
9C2×#926C411yescharacteristic, complemented
10C2×#926C411yescharacteristic, complemented
11C2×C3×#99—21no—
12C2×C3×#99—22no—
13C2×C3×#99C414yescomplemented
14C2×C3×#99C2212yes—
15#5247—21no—
16#5247—21no—
17#5221—21no—
18#5221—21no—
19#5221C412yescomplemented
20#5221C2212yes—
21C2×#834—21no—
22C2×#834—22no—
23C2×#834C414yescomplemented
24C2×#834C2212yes—
25C22×#99—31no—
26C22×#99D311yescharacteristic
27#2309—31no—
28#1393—34no—
29C25×C3D411yescharacteristic
30C25×C3D411yescharacteristic
31C22×C3×D4—41no—
32C22×C3×D4—41no—
33C22×C3×D4—42no—
34C22×C3×D4C2×C412yes—
35C22×C3×D4C2×C411yescharacteristic
36C22×C3×D4C2×C412yes—
37C22×C3×D4D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C22×C3×D4D412yes—
40C22×C3×D4Q812yes—
41C22×C3×D4C2311yescharacteristic
42C23×C3×C4—41no—
43C23×C3×C4C2×C411yescharacteristic
44C23×Dic3—41no—
45C2×C3×C22⋊C4—24no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—41no—
48C2×C3×C22⋊C4—42no—
49C2×#217—24no—
50C2×#217—22no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—41no—
58C2×#217—42no—
59#926—41no—
60#926—41no—
61#926—42no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2×C412yescomplemented
64C2×C4×Dic3—21no—
65C2×C4×Dic3C2×C412yescomplemented
66C3×#99—24no—
67C3×#99—22no—
68C3×#99C2×C414yescomplemented
69#834—24no—
70#834—22no—
71#834C2×C414yescomplemented
72C23×D4C2×D311yescharacteristic
73C2×#117—31no—
74C2×#117—31no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—61no—
77C2×#99—21no—
78C2×#99—22no—
79C2×#99—34no—
80C2×#99—32no—
81C2×#99—61no—
82C2×#99—62no—
83C2×#99Dic314yescomplemented
84C2×#99C2×D312yes—
85#390—32no—
86#390—32no—
87#390—61no—
88#390—61no—
89#379—61no—
90#379—61no—
91C24×C3—21no—
92C24×C3—21no—
93C24×C3—22no—
94C24×C3—21no—
95C24×C3—22no—
96C24×C3—21no—
97C24×C3—82no—
98C24×C3—82no—
99C24×C3C22⋊C411yescharacteristic
100C24×C3C22⋊C411yescharacteristic
101C24×C3C22⋊C412yes—
102C24×C3C22⋊C411yescharacteristic
103C24×C3C22⋊C412yes—
104C24×C3C4⋊C412yes—
105C24×C3C4⋊C412yes—
106C24×C3C2×D411yescharacteristic
107C24×C3C2×D411yescharacteristic
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4C4212yes—
125C2×C3×D4C22⋊C412yes—
126C2×C3×D4C22⋊C412yes—
127C2×C3×D4C4⋊C412yes—
128C2×C3×D4C4⋊C412yes—
129C2×C3×D4C4⋊C412yes—
130C2×C3×D4C22×C411yescharacteristic
131C2×C3×D4C22×C411yescharacteristic
132C2×C3×D4C2×D411yescharacteristic
133C2×C3×D4C2×Q811yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—44no—
138C22×C3×C4—42no—
139C22×C3×C4—84no—
140C22×C3×C4C4212yes—
141C22×C3×C4C22⋊C411yescharacteristic
142C22×C3×C4C4⋊C412yes—
143C22×C3×C4C22×C411yescharacteristic
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—44no—
147C22×Dic3—42no—
148C22×Dic3—84no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—44no—
151C3×C22⋊C4—42no—
152#217—24no—
153#217—44no—
154#217—42no—
155#217—41no—
156#217—41no—
157#217—42no—
158#217—82no—
159#217—82no—
160#217—82no—
161#217—82no—
162C4⋊Dic3—42no—
163C4⋊Dic3—42no—
164C4×Dic3—42no—
165C4×Dic3—42no—
166C25D1211yescharacteristic
167C25#6711yescharacteristic
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—42no—
171C22×D4Dic612yes—
172C22×D4C4×D312yes—
173C22×D4C4×D311yescharacteristic
174C22×D4D1211yescharacteristic
175C22×D4C2×Dic312yes—
176C22×D4#6712yes—
177C22×D4#6711yescharacteristic
178C22×D4C22×D311yescharacteristic
179C23×C4—41no—
180C23×C4—121no—
181C23×C4C4×D311yescharacteristic
182#117—122no—
183#117—121no—
184#117—121no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—61no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—122no—
200C2×C22⋊C4—121no—
201C2×C42—32no—
202C2×C42—61no—
203#99—24no—
204#99—22no—
205#99—34no—
206#99—64no—
207#99—62no—
208#99C2×Dic314yescomplemented
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—22no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—82no—
233C23×C3—82no—
234C23×C3—82no—
235C23×C3—82no—
236C23×C3—82no—
237C23×C3—82no—
238C23×C3—84no—
239C23×C3—84no—
240C23×C3#9512yes—
241C23×C3#9512yes—
242C23×C3#9512yes—
243C23×C3#9911yescharacteristic
244C23×C3#9911yescharacteristic
245C23×C3#9912yes—
246C23×C3C2×C22⋊C411yescharacteristic
247C23×C3C2×C22⋊C411yescharacteristic
248C23×C3C2×C22⋊C411yescharacteristic
249C23×C3C2×C4⋊C411yescharacteristic
250C23×C3C2×C4⋊C411yescharacteristic
251C3×D4—42no—
252C3×D4—42no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—44no—
270C2×C3×C4—88no—
271C2×C3×C4—84no—
272C2×C3×C4#9512yes—
273C2×C3×C4C2×C4211yescharacteristic
274C2×C3×C4C2×C4⋊C411yescharacteristic
275C2×Dic3—44no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—88no—
283C2×Dic3—84no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—21no—
289C24—22no—
290C24—82no—
291C24—82no—
292C24Dic3⋊C412yes—
293C24C4⋊Dic312yes—
294C24(C2×D3)⋊C411yescharacteristic
295C24(C2×D3)⋊C411yescharacteristic
296C24(C2×D3)⋊C412yes—
297C24(C2×D3)⋊C411yescharacteristic
298C24#21712yes—
299C24C2×D1211yescharacteristic
300C24C2×#6711yescharacteristic
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4C4×Dic312yes—
318C2×D4Dic3⋊C412yes—
319C2×D4Dic3⋊C412yes—
320C2×D4C4⋊Dic312yes—
321C2×D4(C2×D3)⋊C412yes—
322C2×D4#21712yes—
323C2×D4C2×Dic611yescharacteristic
324C2×D4C2×C4×D311yescharacteristic
325C2×D4C22×Dic311yescharacteristic
326C2×D4C2×#6711yescharacteristic
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—84no—
335C22×C4—122no—
336C22×C4—124no—
337C22×C4—244no—
338C22×C4C4×Dic312yes—
339C22×C4Dic3⋊C412yes—
340C22×C4(C2×D3)⋊C411yescharacteristic
341C22×C4C2×C4×D311yescharacteristic
342C4⋊C4—122no—
343C4⋊C4—122no—
344C22⋊C4—24no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—64no—
348C22⋊C4—122no—
349C22⋊C4—124no—
350C22⋊C4—122no—
351C22⋊C4—121no—
352C22⋊C4—121no—
353C22⋊C4—242no—
354C22⋊C4—242no—
355C22⋊C4—242no—
356C22⋊C4—242no—
357C42—122no—
358C42—122no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3#34212yes—
393C22×C3#34212yes—
394C22×C3C2×#9511yescharacteristic
395C22×C3C2×#9911yescharacteristic
396C22×C3C2×#9911yescharacteristic
397C3×C4—22no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—84no—
401Dic3—82no—
402Dic3—84no—
403Dic3—82no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—84no—
434C23—84no—
435C23#80611yescharacteristic
436C23#80611yescharacteristic
437C23#80612yes—
438C23C3⋊#9512yes—
439C23C3⋊#9512yes—
440C23C3⋊#9512yes—
441C23C2×Dic3⋊C411yescharacteristic
442C23C2×C4⋊Dic311yescharacteristic
443C23C2×(C2×D3)⋊C411yescharacteristic
444C23C2×(C2×D3)⋊C411yescharacteristic
445C23C2×#21711yescharacteristic
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—88no—
466C2×C4—84no—
467C2×C4—124no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—248no—
475C2×C4—244no—
476C2×C4C3⋊#9512yes—
477C2×C4C2×C4×Dic311yescharacteristic
478C2×C4C2×Dic3⋊C411yescharacteristic
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—42no—
486C2×C3—42no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3C2×#34211yescharacteristic
490C2×C3#212712yes—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—21no—
499C22—21no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22#452712yes—
525C22#452712yes—
526C22C2×#80611yescharacteristic
527C22C2×#80611yescharacteristic
528C22C2×C3⋊#9511yescharacteristic
529C4—22no—
530C4—21no—
531C4—21no—
532C4—84no—
533C4—242no—
534C4—244no—
535C4—242no—
536C3#1258211yescharacteristic, complemented, Sylow
537C2—21no—
538C2—21no—
539C2—22no—
540C2—22no—
541C2—41no—
542C2—41no—
543C2—42no—
544C2—42no—
545C2—41no—
546C2—41no—
547C2#7001912yes—
548C2C2×#452711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC2×C3×C4C2×C42
FrattiniC22×D4C22×D3
FittingC22×C3×#99C2
socleC22×C3C2×#99
prime radicalC23×C3×D4C22
cosocleC22×C3×D4C23
hypercenterC22×#99D3
hypocenterC3#12582
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
4. #67695C2semidirect
9. C2×#926C4semidirect
10. C2×#926C4semidirect
13. C2×C3×#99C4semidirect
19. #5221C4semidirect
23. C2×#834C4semidirect
63. C2×C4⋊Dic3C2×C4semidirect
65. C2×C4×Dic3C2×C4semidirect
68. C3×#99C2×C4semidirect
71. #834C2×C4semidirect
83. C2×#99Dic3semidirect
208. #99C2×Dic3semidirect
536. C3#12582semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#992
#840442
C22×#8342
#676952
#125823