Finite Group Databasebeta

← #10695439

Subgroups — #10695439

Order 768.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#392C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic, complemented
3#82117C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic, complemented
6C2×#4452C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9922—31nomaximal, Sylow
9C23×C3×D4—21no—
10C23×C3×D4C2211yescharacteristic
11C22×C3×C22⋊C4—21no—
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×#214—21no—
14C2×#924C2211yescharacteristic
15C3×#392—22no—
16C3×#392—22no—
17C2×C3×#95C2211yescharacteristic
18#5247—21no—
19#5218—22no—
20#5218—22no—
21C2×#832—21no—
22C2×#832—22no—
23#5212C2211yescharacteristic
24#5210—21no—
25C2×#830C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4949—21no—
28#4948—22no—
29#4947—21no—
30#4525—22no—
31#4452—22no—
32C2×#450—31no—
33#2626—31no—
34C2×#392D311yescharacteristic
35#1786—31no—
36C2×#328—31no—
37C2×#323—31no—
38#1294—31no—
39C25×C3—21no—
40C25×C3D411yescharacteristic
41C22×C3×D4—24no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—44no—
48C22×C3×D4—44no—
49C22×C3×D4—44no—
50C22×C3×D4—42no—
51C22×C3×D4—42no—
52C22×C3×D4D412yes—
53C23×C3×C4—21no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×Dic6D411yescharacteristic, complemented
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—44no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217—22no—
68C2×#217—22no—
69C2×#214—24no—
70C2×#214—44no—
71C2×C4⋊Dic3D411yescharacteristic, complemented
72C2×C4⋊Dic3D412yescomplemented
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—22no—
79C22×C3⋊C8—21no—
80C22×C3⋊C8—21no—
81C3×#95—21no—
82C3×#95D412yes—
83#832—24no—
84#832—44no—
85C3⋊#95—22no—
86C3⋊#95—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830D412yescomplemented
91C23×D4—21no—
92C23×D4C2×D311yescharacteristic
93C22×SD16—61no—
94C2×C22⋊Q8—31no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4C2×D311yescharacteristic
97#450—62no—
98#450—62no—
99C2×#102—61no—
100C2×#102—62no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#399—61no—
104#396—62no—
105#393—61no—
106#392—22no—
107#392—22no—
108#385—61no—
109#380—31no—
110#379—61no—
111C2×#95C2×D311yescharacteristic
112#328—62no—
113#323—62no—
114C24×C3—24no—
115C24×C3—22no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—42no—
123C24×C3—42no—
124C24×C3—42no—
125C24×C3—42no—
126C24×C3—42no—
127C24×C3—41no—
128C24×C3—41no—
129C24×C3—41no—
130C24×C3—41no—
131C24×C3—41no—
132C24×C3—41no—
133C24×C3SD1612yescomplemented
134C24×C3C2×D411yescharacteristic
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—48no—
142C2×C3×D4—44no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4—44no—
146C2×C3×D4—48no—
147C2×C3×D4—48no—
148C2×C3×D4—416no—
149C2×C3×D4—416no—
150C2×C3×D4—41no—
151C2×C3×D4—41no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—42no—
174C3×C22⋊C4—48no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179#217—41no—
180#217—41no—
181#217—41no—
182#217—41no—
183#217—82no—
184#217—82no—
185#214—48no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—42no—
189Dic3⋊C4—41no—
190Dic3⋊C4—41no—
191Dic3⋊C4—82no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—42no—
194C2×C3⋊C8—42no—
195C2×C3⋊C8—41no—
196C25—21no—
197C25#6711yescharacteristic
198C22×Q8—31no—
199C22×D4—24no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4#6712yes—
211C23×C4—21no—
212C23×C4#6711yescharacteristic
213C23×C4C22×D311yescharacteristic
214C2×SD16—64no—
215C2×SD16—124no—
216C22×C8—61no—
217C22×C8—61no—
218C22⋊Q8—61no—
219C22⋊Q8—61no—
220C22⋊Q8—61no—
221C22⋊Q8—61no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—62no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238#102—64no—
239#102—124no—
240C22⋊C8—32no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244#95—21no—
245#95—62no—
246#95—61no—
247#95#6712yes—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—24no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—44no—
255C23×C3—42no—
256C23×C3—44no—
257C23×C3—44no—
258C23×C3—44no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3—44no—
265C23×C3—44no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3—41no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—44no—
275C23×C3—41no—
276C23×C3—82no—
277C23×C3—82no—
278C23×C3—81no—
279C23×C3—81no—
280C23×C3—82no—
281C23×C3—82no—
282C23×C3—82no—
283C23×C3—82no—
284C23×C3—82no—
285C23×C3—81no—
286C23×C3—81no—
287C23×C3—82no—
288C23×C3—81no—
289C23×C3—81no—
290C23×C3#12011yescharacteristic
291C23×C3#12011yescharacteristic
292C23×C3#12111yescharacteristic
293C23×C3C2×SD1611yescharacteristic
294C23×C3#13511yescharacteristic
295C23×C3AGL(1,ℤ/8)11yescharacteristic
296C23×C3#13711yescharacteristic
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—42no—
300C3×D4—44no—
301C3×D4—44no—
302C3×D4—42no—
303C3×D4—432no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—81no—
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4#12711yescharacteristic
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330Dic6—42no—
331Dic6—81no—
332C3⋊C8—42no—
333C3⋊C8—81no—
334C24—24no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24—41no—
351C24—41no—
352C24—41no—
353C24#21412yescomplemented
354C24C2×#6711yescharacteristic
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—122no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—41no—
374C2×D4—41no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4D3×D411yescharacteristic
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4#23711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394SD16—128no—
395C2×C8—62no—
396C2×C8—122no—
397C2×C8—122no—
398C2×C8—121no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—122no—
402C4⋊C4—121no—
403C4⋊C4—121no—
404C4⋊C4—242no—
405C22⋊C4—48no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—121no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C22⋊C4—242no—
415C22⋊C4—242no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—44no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—84no—
435C22×C3—82no—
436C22×C3—84no—
437C22×C3—84no—
438C22×C3—82no—
439C22×C3—84no—
440C22×C3—82no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—81no—
444C22×C3#39211yescharacteristic
445C22×C3#44811yescharacteristic
446C22×C3#44911yescharacteristic
447C22×C3#45011yescharacteristic
448C22×C3#45211yescharacteristic
449C22×C3#45312yes—
450C22×C3B(4,𝔽2)12yes—
451C22×C3#46511yescharacteristic
452C22×C3#46811yescharacteristic
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—44no—
458C3×C4—81no—
459Dic3—82no—
460Dic3—81no—
461Dic3—81no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—44no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—82no—
491C23—82no—
492C23—81no—
493C23—81no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—82no—
504C23#93211yescharacteristic
505C23C2×#21411yescharacteristic
506C23#93711yescharacteristic
507C23#93911yescharacteristic
508C23#95011yescharacteristic
509C23#95111yescharacteristic
510C23#95311yescharacteristic
511Q8—122no—
512Q8—241no—
513D4—24no—
514D4—24no—
515D4—42no—
516D4—44no—
517D4—44no—
518D4—42no—
519D4—432no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—81no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4#93711yescharacteristic
543C2×C4#93811yescharacteristic
544C2×C4#93911yescharacteristic
545C2×C4#94011yescharacteristic
546C8—122no—
547C8—241no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—82no—
553C2×C3—84no—
554C2×C3#159612yes—
555C2×C3#160012yes—
556C2×C3#200011yescharacteristic
557C2×C3#200211yescharacteristic
558C2×C3#202111yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—84no—
578C22—82no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22—81no—
585C22—82no—
586C22—81no—
587C22#503212yes—
588C22#503711yescharacteristic
589C22#504011yescharacteristic
590C22#504111yescharacteristic
591C22#504211yescharacteristic
592C22#507712yes—
593C22#521811yescharacteristic
594C22#522311yescharacteristic
595C22#523911yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—44no—
601C4—81no—
602C4—242no—
603C4—241no—
604C4—241no—
605C3#992211yescharacteristic, complemented, Sylow
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—84no—
612C2#6969712yes—
613C2#6969812yes—
614C2#7167111yescharacteristic
615C2#7168511yescharacteristic
616C2#7176911yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#392C2
socleC23×C3#120
hypercenterC2×#392D3
hypocenterC3#9922
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
2. C2×#5218C2semidirect
3. #82117C2semidirect
4. #72128C2semidirect
5. C2×#4525C2semidirect
6. C2×#4452C2semidirect
7. #67612C2semidirect
56. C22×Dic6D4semidirect
71. C2×C4⋊Dic3D4semidirect
72. C2×C4⋊Dic3D4semidirect
90. #830D4semidirect
133. C24×C3SD16semidirect
353. C24#214semidirect
605. C3#9922semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3922
C2×#52182
#821172
#721282
C2×#45252
C2×#44522
#676122
#99223