Finite Group Databasebeta

← #10804452

Subgroups — #10804452

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5597C211yesmaximal, characteristic
2#73867C211yesmaximal, characteristic, complemented
3#73445C211yesmaximal, characteristic, complemented
4C3×#1964C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic, complemented
6#70030C214yesmaximal, complemented
7#70027C214yesmaximal, complemented
8#69528C211yesmaximal, characteristic, complemented
9#69456C211yesmaximal, characteristic, complemented
10#21796—31nomaximal, Sylow
11C22×#237C2212yes—
12C2×C3×#121C2211yescharacteristic
13C2×#939C412yescomplemented
14C2×#939C2212yes—
15C2×#930C2211yescharacteristic
16C22×Dic3⋊C4C2211yescharacteristic
17#5597—22no—
18#5597—22no—
19#5597—22no—
20#5597—22no—
21C2×#896C412yescomplemented
22C2×#884C412yescomplemented
23C2×#878C412yescomplemented
24C2×C3×#99C2214yes—
25C3×#380C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27C2×#834C2214yes—
28#5212C2211yescharacteristic
29C4×#217C2211yescharacteristic
30#4978—22no—
31C2×#806C2214yes—
32C2×#806C2214yes—
33#4943C2212yes—
34#4943C2212yes—
35#4942C2212yes—
36Dic3×C22⋊C4C2212yes—
37#4933—22no—
38#4655—22no—
39#4652—22no—
40C2×#537—31no—
41#2313—31no—
42#2313—31no—
43#1964—34no—
44#1964—31no—
45#1964—31no—
46#1964—31no—
47#1964D311yescharacteristic, complemented
48#1782—34no—
49C22×C3×D4C2×C411yescharacteristic
50C22×C3×D4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C22×#67C2×C412yes—
53C22×#67C2×C412yes—
54C22×#67C2312yes—
55C23×Dic3C2×C412yes—
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C2×#237—24no—
59C2×#237—24no—
60C2×#237—44no—
61C2×#237—44no—
62C22×D12C2×C411yescharacteristic
63C22×C4×D3C2×C412yes—
64C22×Dic6C2×C412yes—
65C3×#121—21no—
66C3×#121—21no—
67C3×#121—21no—
68C3×#121—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2×C411yescharacteristic
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C22⋊C4C2312yes—
73C2×C3×C22⋊C4C2312yes—
74C2×C3×C42—21no—
75C2×#217C2×C412yes—
76C2×#217C2312yes—
77C2×#217C2312yes—
78#939—24no—
79#939—24no—
80#939—24no—
81#939—24no—
82#930—21no—
83#930—21no—
84#930—21no—
85#930—21no—
86C2×(C2×D3)⋊C4C2×C412yes—
87C2×(C2×D3)⋊C4C2×C412yes—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2×C411yescharacteristic
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4C2×C412yes—
92C2×Dic3⋊C4C2×C412yes—
93C2×Dic3⋊C4C2×C412yes—
94C2×Dic3⋊C4D412yes—
95C2×Dic3⋊C4D412yes—
96C2×Dic3⋊C4D412yes—
97C2×Dic3⋊C4D412yes—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—22no—
100#896—24no—
101#896—24no—
102#884—24no—
103#884—24no—
104#878—24no—
105#878—24no—
106C3×#99—44no—
107C3×#95—21no—
108C3×#95—21no—
109#834—44no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113C3⋊#95—24no—
114C3⋊#95—22no—
115C3⋊#95—24no—
116#806—44no—
117#806—44no—
118C22×#41—32no—
119#537—62no—
120#537—62no—
121#537—62no—
122#537—62no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—32no—
125C2×#121—32no—
126C2×#121—32no—
127C2×#121—31no—
128C2×#121—32no—
129C2×#121C2×D311yescharacteristic
130C22×C4⋊C4—31no—
131C2×#99—34no—
132C2×#99—34no—
133C2×#99—34no—
134C2×#99C2×D314yes—
135#387—62no—
136#387—62no—
137#387—62no—
138#387—62no—
139#381—32no—
140#381—32no—
141#380—31no—
142#380—32no—
143#380C2×D311yescharacteristic
144C4×C22⋊C4—31no—
145C4×C22⋊C4—32no—
146C4×C22⋊C4C2×D311yescharacteristic
147C24×C3C22×C412yes—
148C24×C3C2411yescharacteristic
149C23×D3C22×C412yes—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—81no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—81no—
160C22×C3×C4C22×C412yes—
161C22×C3×C4C22×C411yescharacteristic
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C2×#67—22no—
168C2×#67—22no—
169C2×#67—24no—
170C2×#67—24no—
171C2×#67—24no—
172C2×#67—24no—
173C2×#67—44no—
174C2×#67—82no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—24no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—44no—
181C22×Dic3—81no—
182C22×Dic3C22×C412yes—
183C22×Dic3C22×C412yes—
184C22×Dic3C22×C412yes—
185C22×Dic3C2×D412yes—
186C22×Dic3C2×D412yes—
187C22×Dic3C2×D412yes—
188C22×Dic3C2×D412yes—
189#237—48no—
190#237—48no—
191#237—416no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—81no—
195C2×C4×D3—24no—
196C2×C4×D3—44no—
197C2×Dic6—24no—
198C2×Dic6—44no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—81no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—22no—
207C3×C42—81no—
208#217—24no—
209#217—22no—
210#217—24no—
211#217—22no—
212(C2×D3)⋊C4—22no—
213(C2×D3)⋊C4—22no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—81no—
218Dic3⋊C4—24no—
219Dic3⋊C4—82no—
220Dic3⋊C4C22⋊C414yes—
221Dic3⋊C4C22⋊C414yes—
222Dic3⋊C4C22⋊C414yes—
223Dic3⋊C4C22⋊C414yes—
224C4×Dic3—82no—
225C4×Dic3—81no—
226C2×#41—64no—
227C2×#41—64no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—32no—
231C22×D4—32no—
232C22×D4—32no—
233C22×D4—32no—
234C22×D4—31no—
235C22×D4C4×D311yescharacteristic
236C22×D4C22×D312yes—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—32no—
240C23×C4—32no—
241C23×C4C22×D311yescharacteristic
242C22⋊Q8—64no—
243C22⋊Q8—64no—
244C22⋊Q8—64no—
245C22⋊Q8—64no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—64no—
251#121—64no—
252#121—64no—
253#121—64no—
254#121—64no—
255#121—64no—
256#121—61no—
257#121—61no—
258#121—61no—
259#121—61no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C4×D311yescharacteristic
272C2×C22⋊C4—32no—
273C2×C22⋊C4—32no—
274C2×C22⋊C4—32no—
275C2×C22⋊C4—32no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4C4×D312yes—
278C2×C22⋊C4C22×D312yes—
279C2×C22⋊C4C22×D312yes—
280C2×C42—21no—
281C2×C42—62no—
282C2×C42—61no—
283#99—44no—
284#99—124no—
285#99—124no—
286#99—124no—
287#95—21no—
288#95—21no—
289#95—62no—
290#95—62no—
291#95—64no—
292#95—64no—
293#95—61no—
294#95—61no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3C23×C411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3H(5,𝔽2)12yes—
305C23×C3H(5,𝔽2)12yes—
306C22×D3—22no—
307C22×D3—24no—
308C22×D3—42no—
309C22×D3—42no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—44no—
313C3×D4—81no—
314C3×D4—81no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4C2×C22⋊C411yescharacteristic
330C2×C3×C4C2×C22⋊C411yescharacteristic
331C2×C3×C4C2×C22⋊C411yescharacteristic
332C2×C3×C4C2×C22⋊C411yescharacteristic
333#67—48no—
334#67—42no—
335#67—42no—
336#67—44no—
337#67—84no—
338C2×Dic3—24no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—48no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3C2×C22⋊C414yes—
355C2×Dic3C2×C22⋊C414yes—
356D12—81no—
357D12—81no—
358C4×D3—48no—
359Dic6—48no—
360C24—32no—
361C24C2×C4×D312yes—
362C24C23×D311yescharacteristic
363#41—128no—
364#41—128no—
365#41—1216no—
366C2×Q8—64no—
367C2×Q8—124no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—64no—
375C2×D4—64no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—64no—
379C2×D4—64no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—81no—
383C2×D4—124no—
384C2×D4—241no—
385C2×D4—242no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—81no—
402C22×C4—124no—
403C22×C4—124no—
404C22×C4—241no—
405C22×C4C2×C4×D312yes—
406C22×C4C2×C4×D311yescharacteristic
407C22×C4C2×C4×D311yescharacteristic
408C22×C4D3×D411yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—34no—
415C4⋊C4—34no—
416C4⋊C4—34no—
417C4⋊C4—34no—
418C4⋊C4—64no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—81no—
422C4⋊C4—242no—
423C4⋊C4—241no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—64no—
430C22⋊C4—64no—
431C22⋊C4—62no—
432C22⋊C4—62no—
433C22⋊C4—62no—
434C22⋊C4—64no—
435C22⋊C4—62no—
436C42—81no—
437C42—241no—
438C42—242no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—84no—
451C22×C3C22×C22⋊C411yescharacteristic
452C22×C3#51812yes—
453C22×C3#53511yescharacteristic
454C22×C3#53511yescharacteristic
455C22×C3#53512yes—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—84no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—42no—
465C3×C4—82no—
466C3×C4—82no—
467Dic3—24no—
468Dic3—24no—
469Dic3—42no—
470Dic3—82no—
471Dic3—82no—
472C23—22no—
473C23—24no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—62no—
479C23—64no—
480C23—122no—
481C23—122no—
482C23C22×C4×D311yescharacteristic
483C23C2×D3×D411yescharacteristic
484C23C2×D3×D411yescharacteristic
485C23#100412yes—
486C23#100912yes—
487Q8—128no—
488D4—42no—
489D4—42no—
490D4—44no—
491D4—81no—
492D4—81no—
493D4—128no—
494D4—122no—
495D4—122no—
496D4—124no—
497D4—241no—
498D4—241no—
499D4—244no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—128no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—128no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4D3×C22⋊C411yescharacteristic
534C2×C4D3×C22⋊C411yescharacteristic
535C2×C4D3×C22⋊C411yescharacteristic
536C2×C4D3×C22⋊C411yescharacteristic
537C2×C3—21no—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—82no—
542C2×C3#230911yescharacteristic
543C2×C3#288311yescharacteristic
544C2×C3#288411yescharacteristic
545D3—82no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—84no—
558C22—122no—
559C22—122no—
560C22—122no—
561C22—244no—
562C22C2×D3×C22⋊C411yescharacteristic
563C22#554512yes—
564C22#560712yes—
565C22#567411yescharacteristic
566C22#571511yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—42no—
572C4—64no—
573C4—64no—
574C4—82no—
575C4—82no—
576C4—122no—
577C4—242no—
578C4—242no—
579C3#2179611yescharacteristic, complemented, Sylow
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—82no—
585C2—242no—
586C2#7642811yescharacteristic
587C2#7868811yescharacteristic
588C2#7868911yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1964C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5597C2
cosocleC24×C3C24
hypercenter#1964D3
hypocenterC3#21796
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #73867C2semidirect
3. #73445C2semidirect
4. C3×#1964C2semidirect
5. #71686C2semidirect
6. #70030C2semidirect
7. #70027C2semidirect
8. #69528C2semidirect
9. #69456C2semidirect
13. C2×#939C4semidirect
21. C2×#896C4semidirect
22. C2×#884C4semidirect
23. C2×#878C4semidirect
47. #1964D3semidirect
579. C3#21796semidirect

Maximal subgroups

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

subgroupindex
C2×#55972
#738672
#734452
C3×#19642
#716862
#700302
#700272
#695282
#694562
#217963