Finite Group Databasebeta

← #10690770

Subgroups — #10690770

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84032C211yesmaximal, characteristic
2C2×#5207C211yesmaximal, characteristic, complemented
3C3×C22⋊#95C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67612C212yesmaximal, complemented
6#9117—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C2×#953—21no—
10C2×#930C411yescharacteristic, complemented
11C2×#930C411yescharacteristic, complemented
12C2×#930C2211yescharacteristic
13C22×#208—21no—
14C2×C3×#95—21no—
15C2×C3×#95C2212yes—
16#5213—21no—
17#5213—21no—
18#5213—21no—
19#5212C411yescharacteristic, complemented
20#5212C411yescharacteristic, complemented
21#5212C2211yescharacteristic
22#5207—24no—
23C2×#830—21no—
24C2×#830C2212yes—
25#4451—24no—
26C2×#407—31no—
27#2847—31no—
28C2×#327—32no—
29C22⋊#95D311yescharacteristic, complemented
30#1294—32no—
31C25×C3D411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—22no—
35C23×C3×C4—42no—
36C23×C3×C4—42no—
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4C2×C411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—21no—
45C22×#67—21no—
46C22×D12C2×C411yescharacteristic
47C2×C3×C22⋊C4—24no—
48C2×C3×C22⋊C4—42no—
49C2×C3×C22⋊C4—42no—
50#953—44no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54#930—22no—
55#930—22no—
56#930—42no—
57#930—42no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×#208—24no—
65C2×#208—42no—
66C2×#208—42no—
67C22×C3⋊C8—22no—
68C3×#95—22no—
69C3×#95—22no—
70C3×#95—42no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74#830—24no—
75#830—22no—
76#830D414yescomplemented
77C24×C4C2×D311yescharacteristic
78C22×M16—61no—
79C2×#121—31no—
80C2×#121—31no—
81C2×#121—31no—
82C2×#120—61no—
83C22×C22⋊C4—21no—
84#407—64no—
85C2×C22⋊C8—32no—
86C2×C22⋊C8—61no—
87#386—61no—
88#386—61no—
89#386—61no—
90#380—31no—
91#380—31no—
92#380—31no—
93C2×#95—21no—
94C2×#95C2×D312yes—
95#327—64no—
96C24×C3—22no—
97C24×C3—22no—
98C24×C3—21no—
99C24×C3—21no—
100C24×C3—22no—
101C24×C3—22no—
102C24×C3—41no—
103C24×C3—41no—
104C24×C3C22⋊C411yescharacteristic
105C24×C3C22⋊C411yescharacteristic
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—24no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—48no—
129C22×C3×C4—48no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4C22⋊C411yescharacteristic
137C22×C3×C4C22⋊C411yescharacteristic
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D412yes—
141C22×C3×C4#4112yes—
142C2×#67—42no—
143C2×#67—42no—
144C2×#67—42no—
145C2×#67—42no—
146C2×#67—42no—
147C2×#67—42no—
148C2×#67—42no—
149C2×#67—42no—
150C2×#67—41no—
151C2×#67—41no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C2×D12—22no—
156C2×D12—41no—
157C2×D12—41no—
158C3×C22⋊C4—48no—
159#217—42no—
160#217—42no—
161#217—42no—
162(C2×D3)⋊C4—42no—
163(C2×D3)⋊C4—42no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166C4⋊Dic3—22no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169#208—48no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—42no—
172C2×C3⋊C8—42no—
173C25#6711yescharacteristic
174C22×D4—31no—
175C22×D4—61no—
176C22×D4—61no—
177C22×D4—61no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C4×D311yescharacteristic
184C23×C4C4×D311yescharacteristic
185C23×C4D1211yescharacteristic
186C23×C4D1211yescharacteristic
187C23×C4#6711yescharacteristic
188C23×C4C22×D311yescharacteristic
189C2×M16—64no—
190C2×M16—122no—
191C2×M16—122no—
192C22×C8—62no—
193#121—62no—
194#121—62no—
195#121—122no—
196#121—122no—
197#120—124no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C22⋊C8—34no—
211C22⋊C8—64no—
212C22⋊C8—62no—
213#95—22no—
214#95—22no—
215#95—42no—
216#95—61no—
217#95—61no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—44no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3#9912yes—
246C23×C3#10412yes—
247C23×C3C2×C22⋊C411yescharacteristic
248C23×C3#12112yes—
249C23×C3AGL(1,ℤ/8)11yescharacteristic
250C23×C3AGL(1,ℤ/8)12yes—
251C23×C3#13711yescharacteristic
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—82no—
281C2×C3×C4—82no—
282C2×C3×C4—82no—
283C2×C3×C4—82no—
284C2×C3×C4#9912yes—
285C2×C3×C4C4×D412yes—
286C2×C3×C4#12011yescharacteristic
287C2×C3×C4#12311yescharacteristic
288#67—84no—
289#67—84no—
290#67—82no—
291#67—82no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301D12—82no—
302C3⋊C8—44no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—41no—
310C24—41no—
311C24—61no—
312C24(C2×D3)⋊C411yescharacteristic
313C24(C2×D3)⋊C411yescharacteristic
314C24C2×#6711yescharacteristic
315C2×D4—62no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—122no—
319C2×D4—122no—
320C2×D4—122no—
321C2×D4—122no—
322C2×D4—122no—
323C2×D4—122no—
324C2×D4—121no—
325C2×D4—122no—
326C2×D4—121no—
327C2×D4—122no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4(C2×D3)⋊C411yescharacteristic
360C22×C4(C2×D3)⋊C411yescharacteristic
361C22×C4C2×C4×D311yescharacteristic
362C22×C4C2×D1211yescharacteristic
363C22×C4D3×D412yes—
364C22×C4#23712yes—
365M16—128no—
366C2×C8—64no—
367C2×C8—122no—
368C2×C8—122no—
369C4⋊C4—62no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C22⋊C4—48no—
373C22⋊C4—122no—
374C22⋊C4—122no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C22⋊C4—122no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—48no—
395C22×C3—48no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3#38611yescharacteristic
405C22×C3C2×#9911yescharacteristic
406C22×C3C2×#9911yescharacteristic
407C22×C3#41711yescharacteristic
408C22×C3#41811yescharacteristic
409C22×C3C2×#10411yescharacteristic
410C22×C3#42111yescharacteristic
411C22×C3#46812yes—
412C22×C3#46912yes—
413C2×D3—81no—
414C2×D3—81no—
415C2×D3—81no—
416C2×D3—81no—
417C2×D3—81no—
418C2×D3—81no—
419C2×D3—81no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—41no—
425C3×C4—84no—
426Dic3—81no—
427Dic3—81no—
428Dic3—81no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—24no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23#80512yes—
463C23#80612yes—
464C23C2×(C2×D3)⋊C411yescharacteristic
465C23#93211yescharacteristic
466C23#93912yes—
467C23#94211yescharacteristic
468C23#94912yes—
469D4—242no—
470D4—244no—
471D4—244no—
472D4—242no—
473D4—242no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4#80612yes—
509C2×C4#67⋊C412yes—
510C2×C4#88211yescharacteristic
511C2×C4#88611yescharacteristic
512C8—124no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—42no—
516C2×C3—42no—
517C2×C3#150411yescharacteristic
518C2×C3#150711yescharacteristic
519C2×C3#151011yescharacteristic
520C2×C3#151311yescharacteristic
521C2×C3#187111yescharacteristic
522C2×C3#189511yescharacteristic
523C2×C3#189811yescharacteristic
524D3—161no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—48no—
540C22—48no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—241no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22C2×#80511yescharacteristic
557C22C2×#80611yescharacteristic
558C22C2×#80611yescharacteristic
559C22#495811yescharacteristic
560C22#496611yescharacteristic
561C22#500111yescharacteristic
562C22#500511yescharacteristic
563C22#503912yes—
564C22#504912yes—
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—41no—
570C4—84no—
571C4—241no—
572C4—241no—
573C4—241no—
574C3#911711yescharacteristic, complemented, Sylow
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—481no—
580C2#6939811yescharacteristic
581C2#6944411yescharacteristic
582C2#6945311yescharacteristic
583C2#6949611yescharacteristic
584C2#6950811yescharacteristic
585C2#6951711yescharacteristic
586C2#6953111yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×C22⋊#95C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenterC22⋊#95D3
hypocenterC3#9117
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#5207C2semidirect
3. C3×C22⋊#95C2semidirect
4. C2×#4451C2semidirect
5. #67612C2semidirect
10. C2×#930C4semidirect
11. C2×#930C4semidirect
19. #5212C4semidirect
20. #5212C4semidirect
29. C22⋊#95D3semidirect
76. #830D4semidirect
574. C3#9117semidirect

Maximal subgroups

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

subgroupindex
#840322
C2×#52072
C3×C22⋊#952
C2×#44512
#676122
#91173