Finite Group Databasebeta

← #10690750

Subgroups — #10690750

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84032C211yesmaximal, characteristic
2C22×#830C211yesmaximal, characteristic, complemented
3C3×C22⋊#95C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67612C212yesmaximal, complemented
6#9113—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C2×#953—21no—
10C2×#930C411yescharacteristic, complemented
11C2×#930C411yescharacteristic, complemented
12C2×#930C2211yescharacteristic
13C23×C3⋊C8—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
22C2×#830—21no—
23C2×#830—24no—
24C2×#830C2212yes—
25#4451—24no—
26C22×C22⋊C8—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—42no—
51#953—42no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—21no—
55#930—22no—
56#930—22no—
57#930—42no—
58#930—42no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×C4⋊Dic3C2×C411yescharacteristic
65C22×C3⋊C8—24no—
66C22×C3⋊C8—22no—
67C22×C3⋊C8—42no—
68C22×C3⋊C8—42no—
69C3×#95—22no—
70C3×#95—22no—
71C3×#95—42no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75#830—24no—
76#830—22no—
77#830D414yescomplemented
78C24×C4C2×D311yescharacteristic
79C23×C8—61no—
80C2×#121—31no—
81C2×#121—31no—
82C2×#121—31no—
83C2×#120—61no—
84C22×C22⋊C4—21no—
85C2×C22⋊C8—32no—
86C2×C22⋊C8—64no—
87C2×C22⋊C8—61no—
88#386—61no—
89#386—61no—
90#386—61no—
91#380—31no—
92#380—31no—
93#380—31no—
94C2×#95—21no—
95C2×#95C2×D312yes—
96#327—64no—
97C24×C3—22no—
98C24×C3—22no—
99C24×C3—22no—
100C24×C3—22no—
101C24×C3—41no—
102C24×C3—41no—
103C24×C3C22⋊C411yescharacteristic
104C24×C3C22⋊C411yescharacteristic
105C24×C3D812yes—
106C24×C3SD1612yes—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—24no—
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—41no—
129C22×C3×C4—48no—
130C22×C3×C4—48no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4C22⋊C411yescharacteristic
138C22×C3×C4C22⋊C411yescharacteristic
139C22×C3×C4C22×C411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4#4112yes—
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—
152C2×#67—42no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C2×D12—22no—
157C2×D12—41no—
158C2×D12—41no—
159C3×C22⋊C4—48no—
160#217—42no—
161#217—42no—
162#217—41no—
163#217—41no—
164(C2×D3)⋊C4—42no—
165(C2×D3)⋊C4—42no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168C4⋊Dic3—22no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171C2×C3⋊C8—24no—
172C2×C3⋊C8—48no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—42no—
175C25#6711yescharacteristic
176C22×D4—31no—
177C22×D4—61no—
178C22×D4—61no—
179C22×D4—61no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C4×D311yescharacteristic
186C23×C4C4×D311yescharacteristic
187C23×C4D1211yescharacteristic
188C23×C4D1211yescharacteristic
189C23×C4#6711yescharacteristic
190C23×C4C22×D311yescharacteristic
191C22×C8—64no—
192C22×C8—62no—
193C22×C8—122no—
194C22×C8—122no—
195#121—62no—
196#121—62no—
197#121—122no—
198#121—122no—
199#120—122no—
200#120—122no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C22⋊C8—34no—
214C22⋊C8—64no—
215C22⋊C8—62no—
216#95—22no—
217#95—22no—
218#95—42no—
219#95—61no—
220#95—61no—
221#95—61no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—24no—
231C23×C3—24no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—24no—
239C23×C3—24no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—44no—
243C23×C3—44no—
244C23×C3—44no—
245C23×C3—44no—
246C23×C3#9912yes—
247C23×C3#10212yes—
248C23×C3#10212yes—
249C23×C3#10412yes—
250C23×C3C2×C22⋊C411yescharacteristic
251C23×C3#12112yes—
252C23×C3C2×D811yescharacteristic
253C23×C3C2×SD1611yescharacteristic
254C23×C3#13512yes—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
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—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—82no—
284C2×C3×C4—82no—
285C2×C3×C4—82no—
286C2×C3×C4—82no—
287C2×C3×C4#9912yes—
288C2×C3×C4C4×D412yes—
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4#12311yescharacteristic
291#67—84no—
292#67—84no—
293#67—82no—
294#67—82no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304D12—82no—
305C3⋊C8—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24—61no—
313C24(C2×D3)⋊C411yescharacteristic
314C24(C2×D3)⋊C411yescharacteristic
315C24#21312yes—
316C24#21512yes—
317C24C2×#6711yescharacteristic
318C2×D4—62no—
319C2×D4—121no—
320C2×D4—121no—
321C2×D4—122no—
322C2×D4—122no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—121no—
328C2×D4—122no—
329C2×D4—121no—
330C2×D4—122no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4(C2×D3)⋊C411yescharacteristic
363C22×C4(C2×D3)⋊C411yescharacteristic
364C22×C4C2×C4×D311yescharacteristic
365C22×C4C2×D1211yescharacteristic
366C22×C4D3×D412yes—
367C22×C4#23712yes—
368C2×C8—64no—
369C2×C8—128no—
370C2×C8—122no—
371C2×C8—122no—
372C4⋊C4—62no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C22⋊C4—48no—
376C22⋊C4—122no—
377C22⋊C4—122no—
378C22⋊C4—122no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—122no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—48no—
402C22×C3—48no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3#32714yescomplemented
408C22×C3#38611yescharacteristic
409C22×C3C2×#9911yescharacteristic
410C22×C3C2×#9911yescharacteristic
411C22×C3C2×#10211yescharacteristic
412C22×C3#41611yescharacteristic
413C22×C3C2×#10411yescharacteristic
414C22×C3#42111yescharacteristic
415C22×C3#46512yes—
416C22×C3#46612yes—
417C2×D3—81no—
418C2×D3—81no—
419C2×D3—81no—
420C2×D3—81no—
421C2×D3—81no—
422C2×D3—81no—
423C2×D3—81no—
424C3×C4—21no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—84no—
430Dic3—81no—
431Dic3—81no—
432Dic3—81no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—24no—
446C23—24no—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—24no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23#80512yes—
465C23#80612yes—
466C23#80912yes—
467C23#80912yes—
468C23C2×(C2×D3)⋊C411yescharacteristic
469C23C2×#21311yescharacteristic
470C23#93912yes—
471C23C2×#21511yescharacteristic
472C23#95012yes—
473D4—242no—
474D4—244no—
475D4—244no—
476D4—242no—
477D4—242no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4#80612yes—
513C2×C4#67⋊C412yes—
514C2×C4#88211yescharacteristic
515C2×C4#88611yescharacteristic
516C8—124no—
517C2×C3—22no—
518C2×C3—24no—
519C2×C3—22no—
520C2×C3—42no—
521C2×C3C2×#32711yescharacteristic
522C2×C3#150111yescharacteristic
523C2×C3#151611yescharacteristic
524C2×C3#151911yescharacteristic
525C2×C3#187111yescharacteristic
526C2×C3#189311yescharacteristic
527C2×C3#189811yescharacteristic
528D3—161no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—21no—
538C22—22no—
539C22—21no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—48no—
547C22—48no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—241no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22#445114yescomplemented
560C22C2×#80511yescharacteristic
561C22C2×#80611yescharacteristic
562C22C2×#80611yescharacteristic
563C22#495811yescharacteristic
564C22C2×#80911yescharacteristic
565C22#500211yescharacteristic
566C22#500511yescharacteristic
567C22#503812yes—
568C22#504812yes—
569C4—21no—
570C4—22no—
571C4—21no—
572C4—22no—
573C4—41no—
574C4—84no—
575C4—241no—
576C4—241no—
577C4—241no—
578C3#911311yescharacteristic, complemented, Sylow
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—481no—
584C2C2×#445111yescharacteristic
585C2#6944311yescharacteristic
586C2#6945311yescharacteristic
587C2#6949611yescharacteristic
588C2#6950911yescharacteristic
589C2#6951511yescharacteristic
590C2#6953211yescharacteristic
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#9113
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. C22×#830C2semidirect
3. C3×C22⋊#95C2semidirect
4. C2×#4451C2semidirect
5. #67612C2semidirect
10. C2×#930C4semidirect
11. C2×#930C4semidirect
19. #5212C4semidirect
20. #5212C4semidirect
29. C22⋊#95D3semidirect
77. #830D4semidirect
407. C22×C3#327semidirect
559. C22#4451semidirect
578. C3#9113semidirect

Maximal subgroups

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

subgroupindex
#840322
C22×#8302
C3×C22⋊#952
C2×#44512
#676122
#91133