Finite Group Databasebeta

← #10690579

Subgroups — #10690579

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#386C211yesmaximal, characteristic
2C2×#5207C211yesmaximal, characteristic, complemented
3#72126C211yesmaximal, characteristic, complemented
4C2×#4525C212yesmaximal, complemented
5#67613C212yesmaximal, complemented
6#9083—31nomaximal, Sylow
7C23×C3×D4C411yescharacteristic
8C24×C3×C4C2211yescharacteristic
9C22×C3×C22⋊C4C411yescharacteristic
10C22×C3×C22⋊C4C411yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#208—21no—
13C3×#386—21no—
14C3×#386—21no—
15C3×#386—21no—
16C3×#386—21no—
17C2×C3×#95C411yescharacteristic
18C2×C3×#95C2211yescharacteristic
19#5207—24no—
20C2×#830—21no—
21C2×#830—21no—
22C2×#830—21no—
23C2×#830C2212yes—
24C2×#830C2212yes—
25#4525—24no—
26C2×#407—31no—
27C2×#386D311yescharacteristic
28#1779—31no—
29C2×#323—32no—
30#1280—32no—
31C25×C3C2×C411yescharacteristic
32C25×C3C2×C411yescharacteristic
33C22×C3×D4—21no—
34C22×C3×D4—21no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—42no—
38C22×C3×D4—42no—
39C23×C3×C4—22no—
40C23×C3×C4—22no—
41C23×C3×C4—42no—
42C23×C3×C4—42no—
43C23×C3×C4C2×C411yescharacteristic
44C23×C3×C4C2×C411yescharacteristic
45C23×C3×C4C2×C411yescharacteristic
46C23×C3×C4C2×C411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—42no—
61C2×C3×C22⋊C4—42no—
62C2×#208—24no—
63C2×#208—42no—
64C2×#208—42no—
65C22×C3⋊C8—22no—
66C22×C3⋊C8—22no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71#830—24no—
72#830—22no—
73#830—22no—
74#830—22no—
75#830—42no—
76#830—42no—
77#830D414yescomplemented
78C23×D4Dic311yescharacteristic
79C24×C4C2×D311yescharacteristic
80C22×M16—61no—
81C22×C22⋊C4Dic311yescharacteristic
82C22×C22⋊C4Dic311yescharacteristic
83C22×C22⋊C4C2×D311yescharacteristic
84#407—64no—
85C2×C22⋊C8—32no—
86C2×C22⋊C8—32no—
87C2×C22⋊C8—61no—
88C2×C22⋊C8—61no—
89C2×C22⋊C8—61no—
90#386—21no—
91#386—21no—
92#386—21no—
93#386—21no—
94C2×#95Dic311yescharacteristic
95C2×#95C2×D311yescharacteristic
96#323—64no—
97C24×C3—22no—
98C24×C3—22no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—21no—
102C24×C3—22no—
103C24×C3—21no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—41no—
108C24×C3—41no—
109C24×C3—41no—
110C24×C3—41no—
111C24×C3—41no—
112C24×C3—41no—
113C24×C3C22⋊C411yescharacteristic
114C24×C3C22⋊C411yescharacteristic
115C24×C3M1612yescomplemented
116C24×C3C22×C411yescharacteristic
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—48no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—48no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4—48no—
151C22×C3×C4—48no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4C22⋊C411yescharacteristic
159C22×C3×C4C22⋊C411yescharacteristic
160C22×C3×C4M1612yescomplemented
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4112yes—
165C3×C22⋊C4—42no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—42no—
168C3×C22⋊C4—42no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—41no—
172#208—48no—
173C2×C3⋊C8—24no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—42no—
176C2×C3⋊C8—42no—
177C2×C3⋊C8—42no—
178C2×C3⋊C8—42no—
179C25C2×Dic311yescharacteristic
180C25C2×Dic311yescharacteristic
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4—42no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C2×Dic311yescharacteristic
192C23×C4C2×Dic311yescharacteristic
193C23×C4C2×Dic311yescharacteristic
194C23×C4C2×Dic311yescharacteristic
195C23×C4#6711yescharacteristic
196C23×C4#6711yescharacteristic
197C23×C4#6711yescharacteristic
198C23×C4#6711yescharacteristic
199C23×C4C22×D311yescharacteristic
200C2×M16—64no—
201C2×M16—122no—
202C2×M16—122no—
203C22×C8—62no—
204C22×C8—62no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C22⋊C8—34no—
216C22⋊C8—64no—
217C22⋊C8—62no—
218C22⋊C8—62no—
219C22⋊C8—62no—
220C22⋊C8—122no—
221C22⋊C8—122no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—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—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—22no—
245C23×C3—24no—
246C23×C3—44no—
247C23×C3—44no—
248C23×C3—44no—
249C23×C3—44no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—82no—
266C23×C3—82no—
267C23×C3—82no—
268C23×C3—82no—
269C23×C3#9912yes—
270C23×C3#10012yes—
271C23×C3C2×C22⋊C411yescharacteristic
272C23×C3C4×D412yes—
273C23×C3C2×M1611yescharacteristic
274C23×C3C2×M1611yescharacteristic
275C23×C3#13112yes—
276C3×D4—48no—
277C3×D4—48no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—44no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—82no—
308C2×C3×C4—82no—
309C2×C3×C4#9912yes—
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12112yes—
312C2×C3×C4#12311yescharacteristic
313C3⋊C8—44no—
314C3⋊C8—82no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—41no—
326C24—41no—
327C24—41no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24#20812yescomplemented
332C24#21711yescharacteristic
333C24#21711yescharacteristic
334C24C22×Dic311yescharacteristic
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—48no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—48no—
341C2×D4—42no—
342C2×D4—42no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4#20812yescomplemented
377C22×C4#21711yescharacteristic
378C22×C4#21711yescharacteristic
379C22×C4D3×D412yes—
380C22×C4#23712yes—
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6711yescharacteristic
383M16—128no—
384C2×C8—64no—
385C2×C8—122no—
386C2×C8—122no—
387C2×C8—122no—
388C2×C8—122no—
389C2×C8—122no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—42no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—48no—
412C22×C3—48no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—84no—
425C22×C3—84no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3#38611yescharacteristic
431C22×C3#40711yescharacteristic
432C22×C3#40811yescharacteristic
433C22×C3C2×#9911yescharacteristic
434C22×C3C2×#9911yescharacteristic
435C22×C3C2×#10011yescharacteristic
436C22×C3#41311yescharacteristic
437C22×C3#43512yes—
438C22×C3#43612yes—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—42no—
445C3×C4—81no—
446C3×C4—81no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—24no—
452C23—24no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—24no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23#83312yes—
491C23#83412yes—
492C23C2×#20811yescharacteristic
493C23C2×#20811yescharacteristic
494C23D4×Dic312yes—
495C23#94812yes—
496C23C2×#21711yescharacteristic
497D4—48no—
498D4—48no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#83412yes—
531C2×C4#93511yescharacteristic
532C2×C4#93711yescharacteristic
533C2×C4#93912yes—
534C8—124no—
535C8—242no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—42no—
539C2×C3—42no—
540C2×C3—82no—
541C2×C3#146211yescharacteristic
542C2×C3#146511yescharacteristic
543C2×C3#146811yescharacteristic
544C2×C3#147111yescharacteristic
545C2×C3#187011yescharacteristic
546C2×C3#189811yescharacteristic
547C2×C3#190011yescharacteristic
548C22—22no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—48no—
563C22—48no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—84no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#501112yes—
582C22#501212yes—
583C22#520711yescharacteristic
584C22C2×#83311yescharacteristic
585C22C2×#83411yescharacteristic
586C22C2×#83411yescharacteristic
587C22#522011yescharacteristic
588C22#523411yescharacteristic
589C22#523711yescharacteristic
590C4—21no—
591C4—22no—
592C4—22no—
593C4—21no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—81no—
598C3#908311yescharacteristic, complemented, Sylow
599C2—22no—
600C2—22no—
601C2—42no—
602C2—42no—
603C2—82no—
604C2#7153911yescharacteristic
605C2#7160211yescharacteristic
606C2#7161111yescharacteristic
607C2#7167011yescharacteristic
608C2#7167611yescharacteristic
609C2#7168311yescharacteristic
610C2#7170211yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#386C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C4
cosocleC23×C3×C4C23
hypercenterC2×#386D3
hypocenterC3#9083
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C2×#5207C2semidirect
3. #72126C2semidirect
4. C2×#4525C2semidirect
5. #67613C2semidirect
77. #830D4semidirect
115. C24×C3M16semidirect
160. C22×C3×C4M16semidirect
331. C24#208semidirect
376. C22×C4#208semidirect
598. C3#9083semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3862
C2×#52072
#721262
C2×#45252
#676132
#90833