Finite Group Databasebeta

← #10690752

Subgroups — #10690752

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
1C22×C3×C22⋊C8C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#72127C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9113—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×C3×C8—21no—
9C2×#953—21no—
10C22×#217—21no—
11C2×#930C411yescharacteristic, complemented
12C2×#930C411yescharacteristic, complemented
13C2×#930C2211yescharacteristic
14C2×C3×C22⋊C8—21no—
15C2×C3×C22⋊C8—24no—
16C2×C3×C22⋊C8C2212yes—
17#5213—21no—
18#5213—21no—
19#5213—21no—
20#5212C411yescharacteristic, complemented
21#5212C411yescharacteristic, complemented
22#5212C2211yescharacteristic
23C2×C3⋊#95—21no—
24C2×C3⋊#95C2212yes—
25#4470—24no—
26C22×C22⋊C8D311yescharacteristic, complemented
27#2847—31no—
28C2×#327—32no—
29C22⋊#95—31no—
30#1294—32no—
31C25×C3D411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—42no—
35C23×C3×C4—42no—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67—21no—
44C22×#67—21no—
45C23×Dic3—22no—
46C22×D12C2×C411yescharacteristic
47C22×C3×C8—24no—
48C22×C3×C8—22no—
49C22×C3×C8—42no—
50C22×C3×C8—42no—
51#953—42no—
52#953—42no—
53C2×#217—24no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—42no—
58C2×#217—42no—
59#930—22no—
60#930—22no—
61#930—42no—
62#930—42no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×C4⋊Dic3C2×C411yescharacteristic
67C2×C4⋊Dic3C2×C411yescharacteristic
68C2×C4⋊Dic3C2×C411yescharacteristic
69C3×C22⋊C8—24no—
70C3×C22⋊C8—22no—
71C3×C22⋊C8D414yescomplemented
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—42no—
78C24×C4C2×D311yescharacteristic
79C23×C8—21no—
80C2×#121—31no—
81C2×#121—31no—
82C2×#121—31no—
83C2×#120—61no—
84C22×C22⋊C4—61no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8C2×D312yes—
88#386—61no—
89#386—61no—
90#386—61no—
91#380—31no—
92#380—31no—
93#380—31no—
94C2×#95—32no—
95C2×#95—61no—
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—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×C4C22⋊C411yescharacteristic
131C22×C3×C4C22⋊C411yescharacteristic
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4#4112yes—
136C2×#67—42no—
137C2×#67—42no—
138C2×#67—42no—
139C2×#67—42no—
140C2×#67—42no—
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—42no—
146C22×Dic3—24no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—42no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3—42no—
156C2×D12—22no—
157C2×D12—41no—
158C2×D12—41no—
159C2×C3×C8—24no—
160C2×C3×C8—48no—
161C2×C3×C8—42no—
162C2×C3×C8—42no—
163#217—48no—
164#217—42no—
165#217—42no—
166#217—41no—
167#217—41no—
168(C2×D3)⋊C4—42no—
169(C2×D3)⋊C4—42no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172C4⋊Dic3—22no—
173C4⋊Dic3—41no—
174C4⋊Dic3—41no—
175C25D1211yescharacteristic
176C22×D4—31no—
177C22×D4—61no—
178C22×D4—61no—
179C22×D4—61no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—62no—
185C23×C4C4×D311yescharacteristic
186C23×C4C4×D311yescharacteristic
187C23×C4D1211yescharacteristic
188C23×C4#6711yescharacteristic
189C23×C4#6711yescharacteristic
190C23×C4C22×D311yescharacteristic
191C22×C8—24no—
192C22×C8—22no—
193C22×C8—42no—
194C22×C8—42no—
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—64no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—122no—
212C2×C22⋊C4—122no—
213C22⋊C8—24no—
214C22⋊C8—22no—
215C22⋊C8#6714yescomplemented
216#95—61no—
217#95—61no—
218#95—61no—
219#95—62no—
220#95—62no—
221#95—122no—
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#9912yes—
281C2×C3×C4C4×D412yes—
282C2×C3×C4#12011yescharacteristic
283C2×C3×C4#12311yescharacteristic
284#67—84no—
285#67—84no—
286#67—82no—
287#67—82no—
288C2×Dic3—44no—
289C2×Dic3—44no—
290C2×Dic3—44no—
291C2×Dic3—41no—
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—82no—
301C2×Dic3—82no—
302C2×Dic3—82no—
303C2×Dic3—82no—
304D12—82no—
305C3×C8—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24—61no—
313C24#20412yes—
314C24D2412yes—
315C24(C2×D3)⋊C411yescharacteristic
316C24(C2×D3)⋊C411yescharacteristic
317C24C2×D1211yescharacteristic
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—121no—
329C2×D4—122no—
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—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—64no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—122no—
357C22×C4—122no—
358C22×C4—122no—
359C22×C4—122no—
360C22×C4—122no—
361C22×C4—122no—
362C22×C4(C2×D3)⋊C411yescharacteristic
363C22×C4(C2×D3)⋊C411yescharacteristic
364C22×C4C2×C4×D311yescharacteristic
365C22×C4#23512yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6712yes—
368C2×C8—24no—
369C2×C8—48no—
370C2×C8—42no—
371C2×C8—42no—
372C4⋊C4—62no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C22⋊C4—122no—
376C22⋊C4—128no—
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—
429Dic3—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—21no—
446C23—24no—
447C23—24no—
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#80612yes—
465C23#82112yes—
466C23#82112yes—
467C23#82512yes—
468C23C2×#20411yescharacteristic
469C23C2×D2411yescharacteristic
470C23#90412yes—
471C23C2×(C2×D3)⋊C411yescharacteristic
472C23#93012yes—
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—124no—
497C2×C4—124no—
498C2×C4—124no—
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—242no—
509C2×C4—242no—
510C2×C4—242no—
511C2×C4—242no—
512C2×C4#80612yes—
513C2×C4C4×#6712yes—
514C2×C4#92911yescharacteristic
515C2×C4#95311yescharacteristic
516C8—44no—
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—22no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—21no—
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#447014yescomplemented
560C22C2×#80611yescharacteristic
561C22C2×#80611yescharacteristic
562C22C2×#82111yescharacteristic
563C22#511311yescharacteristic
564C22#511412yes—
565C22#511512yes—
566C22C2×#82511yescharacteristic
567C22#513911yescharacteristic
568C22#521311yescharacteristic
569C4—21no—
570C4—22no—
571C4—21no—
572C4—22no—
573C4—41no—
574C4—244no—
575C4—241no—
576C4—241no—
577C4—241no—
578C3#911311yescharacteristic, complemented, Sylow
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—481no—
584C2C2×#447011yescharacteristic
585C2#6994211yescharacteristic
586C2#6994311yescharacteristic
587C2#6994411yescharacteristic
588C2#7154311yescharacteristic
589C2#7197311yescharacteristic
590C2#7198411yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC22×C3×C22⋊C8C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenterC22×C22⋊C8D3
hypocenterC3#9113
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊C8C2semidirect
3. #72127C2semidirect
4. C2×#4470C2semidirect
5. #67675C2semidirect
11. C2×#930C4semidirect
12. C2×#930C4semidirect
20. #5212C4semidirect
21. #5212C4semidirect
26. C22×C22⋊C8D3semidirect
71. C3×C22⋊C8D4semidirect
215. C22⋊C8#67semidirect
407. C22×C3#327semidirect
559. C22#4470semidirect
578. C3#9113semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C82
#840322
#721272
C2×#44702
#676752
#91133