Finite Group Databasebeta

← #10900865

Subgroups — #10900865

Order 768.

1302 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×#5526C211yesmaximal, characteristic, complemented
2C3×#2624C211yesmaximal, characteristic, complemented
3C2×#5040C211yesmaximal, characteristic, complemented
4C2×#5037C211yesmaximal, characteristic
5#71584C211yesmaximal, characteristic, complemented
6#69661C212yesmaximal, complemented
7#69659C212yesmaximal, complemented
8#69633C212yesmaximal, complemented
9#69628C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic, complemented
12#29631—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×#214C2212yes—
18C22×#214C2211yescharacteristic
19C22×#213C2211yescharacteristic
20#5526—24no—
21#5526—24no—
22C2×#897—21no—
23C2×#897—21no—
24C2×#886—22no—
25C2×#875C2212yes—
26C2×#869C2212yes—
27C3×#399—21no—
28C3×#397—21no—
29C3×#394—22no—
30C3×#390C2212yes—
31C3×#389C2212yes—
32C3×#380C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5040—24no—
36#5037—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#810C2212yes—
42C2×#809C2211yescharacteristic
43C2×#809C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#536—31no—
47C2×#449—31no—
48C2×#448—31no—
49#2624D311yescharacteristic, complemented
50#2043—32no—
51#2042—32no—
52#2013—32no—
53#2012—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C23×Dic3—21no—
62C2×#235—22no—
63C2×#235—42no—
64C2×#235—42no—
65C2×#235D414yes—
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C22×Dic6—21no—
69C22×Dic6—22no—
70C22×Dic6—42no—
71C22×Dic6—42no—
72C22×Dic6D414yes—
73C22×Dic6C2312yes—
74C22×Dic6C2311yescharacteristic
75C3×#121—22no—
76C3×#121—22no—
77C3×#117—22no—
78C3×#117—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42C2312yes—
90C2×#214—22no—
91C2×#214—24no—
92C2×#214—42no—
93C2×#214—42no—
94C2×#214—42no—
95C2×#213—22no—
96C2×#213—42no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2311yescharacteristic
104#897—28no—
105#897—28no—
106#886—216no—
107#875—24no—
108#875—24no—
109#869—24no—
110#869—24no—
111C3×#95—21no—
112C3×#95—22no—
113#830—22no—
114#830—22no—
115#810—22no—
116#810—24no—
117#810—42no—
118#810—42no—
119#810—42no—
120#809—22no—
121#809—42no—
122C22×#41—31no—
123C23×Q8—31no—
124C22×SD16—32no—
125C22×SD16—31no—
126C22×D8—31no—
127#536—64no—
128#536—64no—
129C2×C4⋊Q8—32no—
130C2×#124—32no—
131C2×#123—62no—
132C2×C22⋊Q8—61no—
133C2×C22⋊Q8—61no—
134C2×#121C2×D311yescharacteristic
135C2×#117C2×D311yescharacteristic
136#449—64no—
137#448—64no—
138C2×#103—32no—
139C2×#103—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—31no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8—31no—
147#399—21no—
148#397—21no—
149#394—22no—
150#390C2×D312yes—
151#389C2×D312yes—
152#380C2×D311yescharacteristic
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C23×D3—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—42no—
163C2×C3×D4—82no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—24no—
175C2×#67—44no—
176C22×Dic3—24no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—44no—
180#235—44no—
181#235—44no—
182#235—48no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12C2×D412yes—
187C2×C4×D3—24no—
188C2×C4×D3—44no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—24no—
196C2×Dic6—21no—
197C2×Dic6—22no—
198C2×Dic6—48no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C2×Dic6—44no—
202C2×Dic6C2×D412yes—
203C2×Dic6C2×D414yes—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—82no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—82no—
214C3×C22⋊C4—82no—
215C3×C22⋊C4—81no—
216C3×C42—22no—
217C3×C42—22no—
218#214—48no—
219#214—44no—
220#213—44no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—28no—
223C4⋊Dic3—28no—
224C4⋊Dic3—28no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41—34no—
233C2×#41—62no—
234C2×#41—122no—
235C2×#41—122no—
236C22×Q8—34no—
237C22×Q8—32no—
238C22×Q8—31no—
239C22×Q8—62no—
240C22×Q8—61no—
241C22×Q8—122no—
242C22×Q8—122no—
243C22×D4—21no—
244C22×D4—31no—
245C22×D4—61no—
246C22×D4C22×D311yescharacteristic
247C23×C4—61no—
248C23×C4—61no—
249C23×C4C22×D311yescharacteristic
250C2×SD16—64no—
251C2×SD16—62no—
252C2×SD16—122no—
253C2×SD16—122no—
254C2×SD16—122no—
255C2×D8—62no—
256C2×D8—122no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—32no—
260C4⋊Q8—64no—
261C4⋊Q8—64no—
262#124—64no—
263#124—64no—
264#123—616no—
265C22⋊Q8—68no—
266C22⋊Q8—68no—
267#121—22no—
268#121—22no—
269#117—22no—
270#117—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4#6711yescharacteristic
275C2×C4⋊C4#6711yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D312yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4#6712yes—
285C2×C42C22×D312yes—
286#103—64no—
287#103—62no—
288#103—122no—
289#103—122no—
290#103—122no—
291#102—62no—
292#102—122no—
293C22⋊C8—62no—
294C22⋊C8—62no—
295#95—21no—
296#95—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3#12012yes—
307C23×C3C22×D411yescharacteristic
308C22×D3—22no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4#12012yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4H(5,𝔽2)12yes—
336C2×C3×C4H(5,𝔽2)12yes—
337#67—48no—
338C2×Dic3—22no—
339C2×Dic3—24no—
340C2×Dic3—48no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344D12—22no—
345D12—22no—
346D12—42no—
347C4×D3—48no—
348Dic6—24no—
349Dic6—22no—
350Dic6—24no—
351Dic6—22no—
352Dic6—48no—
353Dic6—42no—
354Dic6—42no—
355Dic6—42no—
356C3⋊C8—41no—
357C3⋊C8—42no—
358C3⋊C8—41no—
359C24—21no—
360C24—61no—
361C24C2×#6711yescharacteristic
362#41—124no—
363#41—124no—
364#41—128no—
365C2×Q8—34no—
366C2×Q8—32no—
367C2×Q8—64no—
368C2×Q8—62no—
369C2×Q8—61no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—62no—
373C2×Q8—62no—
374C2×Q8—62no—
375C2×Q8—62no—
376C2×Q8—128no—
377C2×Q8—124no—
378C2×Q8—124no—
379C2×Q8—124no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—32no—
385C2×D4—42no—
386C2×D4—64no—
387C2×D4—61no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—82no—
391C2×D4—124no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—64no—
400C22×C4—124no—
401C22×C4—124no—
402C22×C4D3×D412yes—
403C22×C4D3×D412yes—
404C22×C4C2×#6712yes—
405C22×C4C2×#6712yes—
406C22×C4C2×#6711yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—128no—
409SD16—124no—
410D8—124no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—62no—
416C2×C8—62no—
417C2×C8—62no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—68no—
425C4⋊C4—68no—
426C4⋊C4—82no—
427C22⋊C4—24no—
428C22⋊C4—42no—
429C22⋊C4—68no—
430C22⋊C4—68no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—81no—
434C42—22no—
435C42—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#53512yes—
448C22×C3#57711yescharacteristic
449C22×C3#57712yes—
450C22×C3#57811yescharacteristic
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—82no—
455C2×D3—82no—
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—82no—
460C3×C4—81no—
461C3×C4#53412yes—
462C3×C4#53411yescharacteristic
463C3×C4#53411yescharacteristic
464Dic3—44no—
465Dic3—42no—
466C23—22no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—62no—
476C23—122no—
477C23—122no—
478C23—121no—
479C23—121no—
480C23#93712yes—
481C23C22×#6711yescharacteristic
482Q8—64no—
483Q8—62no—
484Q8—64no—
485Q8—62no—
486Q8—128no—
487Q8—122no—
488Q8—122no—
489Q8—122no—
490D4—41no—
491D4—41no—
492D4—42no—
493D4—62no—
494D4—62no—
495D4—84no—
496D4—128no—
497D4—122no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—64no—
514C2×C4—62no—
515C2×C4—128no—
516C2×C4—128no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4#93712yes—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4#100912yes—
524C2×C4#100912yes—
525C8—121no—
526C8—121no—
527C8—122no—
528C2×C3—42no—
529C2×C3—161no—
530C2×C3#261311yescharacteristic
531C2×C3#301011yescharacteristic
532C2×C3#301311yescharacteristic
533C2×C3#322112yes—
534C2×C3#322512yes—
535D3—82no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22—82no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—82no—
546C22—122no—
547C22—122no—
548C22—122no—
549C22—242no—
550C22—242no—
551C22C2×#93711yescharacteristic
552C22#583012yes—
553C22#583612yes—
554C22#583611yescharacteristic
555C22#583811yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—81no—
560C4—82no—
561C4—122no—
562C4—124no—
563C4#565311yescharacteristic
564C4#565311yescharacteristic
565C4#565312yes—
566C3#2963111yescharacteristic, complemented, Sylow
567C2—42no—
568C2—161no—
569C2—242no—
570C2#8290311yescharacteristic
571C2#8339011yescharacteristic
572C2#8342711yescharacteristic
573C2#8344012yes—
574C2#8346412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2624C2
socleC23×C3C22×D4
prime radicalC2×#5037C2
hypercenter#2624D3
hypocenterC3#29631
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2624

Complements

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

NH ≅ G/Ntype
1. C2×#5526C2semidirect
2. C3×#2624C2semidirect
3. C2×#5040C2semidirect
5. #71584C2semidirect
6. #69661C2semidirect
7. #69659C2semidirect
8. #69633C2semidirect
9. #69628C2semidirect
10. #69525C2semidirect
11. #69522C2semidirect
49. #2624D3semidirect
566. C3#29631semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
C3×#26242
C2×#50402
C2×#50372
#715842
#696612
#696592
#696332
#696282
#695252
#695222
#296313