Finite Group Databasebeta

← #10944013

Subgroups — #10944013

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic
3C2×#5228C211yesmaximal, characteristic
4C2×#5182C212yesmaximal, complemented
5#82811C211yesmaximal, characteristic
6#72065C212yesmaximal, complemented
7#71692C211yesmaximal, characteristic
8#70239C212yesmaximal, complemented
9#70221C212yesmaximal, complemented
10#69930C211yesmaximal, characteristic
11#69908C211yesmaximal, characteristic
12#32906—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×Q8×Dic3—28no—
17C2×Q8×Dic3—21no—
18C2×Q8×Dic3—21no—
19C2×Q8×Dic3C2212yes—
20C2×Q8×Dic3C2212yes—
21C2×#944C2211yescharacteristic
22C22×#216C2212yes—
23C22×C4⋊Dic3—21no—
24C22×C4⋊Dic3C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C22×C4×Dic3—21no—
27C22×C4×Dic3C2211yescharacteristic
28C3×#451—24no—
29C2×C3×#103C2212yes—
30C2×C3×C22⋊C8C2211yescharacteristic
31#5229C2211yescharacteristic
32#5228—24no—
33C2×#835C2212yes—
34C2×#835C2212yes—
35C2×#830C2211yescharacteristic
36#5182—28no—
37C2×#816C2212yes—
38C2×Dic3⋊C8C2212yes—
39#4976—21no—
40#4976—21no—
41#4975C2211yescharacteristic
42#4973C2211yescharacteristic
43#4942—21no—
44#4942—21no—
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C22×C4×Q8—31no—
49C2×#462—32no—
50C2×#451—31no—
51C2×#451D311yescharacteristic
52#2857—31no—
53#1878—32no—
54#1878—32no—
55#1872—31no—
56#1872—31no—
57#1803—32no—
58#1797—31no—
59C22×C3×Q8—24no—
60C22×C3×Q8—24no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8D414yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3—21no—
69C23×Dic3D411yescharacteristic
70C23×Dic3D411yescharacteristic
71C22×Dic6C2311yescharacteristic
72C2×C3×Q16—24no—
73C2×C3×Q16—24no—
74C2×C3×Q16—24no—
75C22×C3×C8C2312yes—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×#217—21no—
82Q8×Dic3—28no—
83Q8×Dic3—28no—
84Q8×Dic3—24no—
85Q8×Dic3—24no—
86Q8×Dic3—24no—
87Q8×Dic3D418yes—
88#944—24no—
89C2×#216—24no—
90C2×#216—24no—
91C2×#216—24no—
92C2×C4⋊Dic3—28no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99#924—22no—
100#924—22no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—28no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3—22no—
109C2×C4×Dic3—24no—
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2312yes—
115C3×#103—24no—
116C3×C22⋊C8—22no—
117#835—24no—
118#835—24no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123#830—22no—
124#816—24no—
125Dic3⋊C8—24no—
126C23×Q8C2×D311yescharacteristic
127C22×Q16—32no—
128C22×Q16C2×D312yes—
129C2×C4⋊Q8—31no—
130C2×C22⋊Q8—31no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C4×Q8—32no—
133C2×C4×Q8—32no—
134C2×C4×Q8—61no—
135C2×C4×Q8—61no—
136C2×C4×Q8—68no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4—61no—
139C22×C42—31no—
140C22×C42—61no—
141#462—68no—
142#451—24no—
143#451—64no—
144C2×C4⋊C8—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#103—32no—
148C2×#103C2×D312yes—
149C2×C22⋊C8—31no—
150C2×C22⋊C8C2×D311yescharacteristic
151#391—31no—
152#391—31no—
153#389—61no—
154#389—61no—
155#384—31no—
156#380—61no—
157#380—61no—
158#340—32no—
159#340—32no—
160#336—32no—
161C24×C3C2×D411yescharacteristic
162C2×C3×Q8—24no—
163C2×C3×Q8—216no—
164C2×C3×Q8—24no—
165C2×C3×Q8—216no—
166C2×C3×Q8—24no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8C2×D414yes—
175C2×C3×Q8C2×D414yes—
176C2×C3×Q8#4114yes—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—24no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—41no—
198C22×Dic3—42no—
199C22×Dic3—42no—
200C22×Dic3Q1614yescomplemented
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C2×Dic6—22no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—22no—
209C3×Q16—48no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—42no—
216C3×C22⋊C4—42no—
217#217—42no—
218#216—48no—
219C4⋊Dic3—28no—
220C4⋊Dic3—24no—
221C4⋊Dic3—22no—
222C4⋊Dic3—28no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3C2×D414yes—
226Dic3⋊C4—42no—
227Dic3⋊C4—42no—
228Dic3⋊C4—42no—
229Dic3⋊C4—42no—
230Dic3⋊C4—42no—
231C4×Dic3—28no—
232C4×Dic3—22no—
233C4×Dic3—28no—
234C4×Dic3—22no—
235C4×Dic3—22no—
236C4×Dic3C2×D414yes—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—31no—
245C22×Q8#6714yes—
246C22×Q8C22×D311yescharacteristic
247C22×Q8C22×D312yes—
248C23×C4—21no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4—61no—
252C23×C4C22×D311yescharacteristic
253C2×Q16—24no—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—64no—
257C2×Q16—64no—
258C2×Q16—64no—
259C22×C8—32no—
260C22×C8C22×D312yes—
261C4⋊Q8—64no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266C4×Q8—38no—
267C4×Q8—64no—
268C4×Q8—64no—
269C4×Q8—68no—
270C4×Q8—68no—
271C4×Q8—64no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—68no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—61no—
288C2×C42—31no—
289C2×C42—31no—
290C2×C42—31no—
291C2×C42—31no—
292C2×C42—64no—
293C2×C42—62no—
294C2×C42—62no—
295C2×C42—68no—
296C4⋊C8—64no—
297#103—24no—
298#103—64no—
299#103—64no—
300#103—64no—
301C22⋊C8—22no—
302C22⋊C8—62no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3#12112yes—
312C23×C3C2×Q1612yes—
313C23×C3#13712yes—
314C23×C3C22×D411yescharacteristic
315C3×Q8—216no—
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—24no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8#12118yes—
322C2×C3×C4—216no—
323C2×C3×C4—24no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—21no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4#12112yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4112yes—
340C2×C3×C4C2×#4111yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—28no—
346C2×Dic3—21no—
347C2×Dic3—24no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—44no—
351C2×Dic3—48no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3#12012yes—
361C2×Dic3#12012yes—
362C2×Dic3C2×Q1612yes—
363C2×Dic3C2×Q1612yes—
364C2×Dic3#13712yes—
365Dic6—42no—
366Dic6—42no—
367C3×C8—42no—
368C3⋊C8—42no—
369C24D3×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—216no—
372C2×Q8—24no—
373C2×Q8—216no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—62no—
383C2×Q8—61no—
384C2×Q8—62no—
385C2×Q8—61no—
386C2×Q8D3×D414yes—
387C2×Q8#23714yes—
388C2×Q8C2×#6714yes—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—34no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—121no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4D3×D411yescharacteristic
414C22×C4#23712yes—
415C22×C4C2×#6712yes—
416C22×C4C23×D311yescharacteristic
417Q16—48no—
418Q16—128no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—34no—
428C4⋊C4—42no—
429C4⋊C4—68no—
430C4⋊C4—68no—
431C4⋊C4—64no—
432C4⋊C4—62no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—122no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C22⋊C4—42no—
441C22⋊C4—122no—
442C42—34no—
443C42—62no—
444C42—62no—
445C42—68no—
446C42—68no—
447C42—62no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3#46214yescomplemented
455C22×C3C2×#12011yescharacteristic
456C22×C3C2×#12111yescharacteristic
457C22×C3#52511yescharacteristic
458C22×C3C22×Q1611yescharacteristic
459C22×C3C2×#13711yescharacteristic
460C22×C3C2×#13711yescharacteristic
461C22×C3#57511yescharacteristic
462C3×C4—28no—
463C3×C4—22no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4C2×#12112yes—
467C3×C4#52512yes—
468Dic3—22no—
469Dic3—22no—
470Dic3—41no—
471Dic3—44no—
472Dic3—81no—
473Dic3—81no—
474Dic3#45114yescomplemented
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23D3×Q1612yes—
480C23#91812yes—
481C23#93912yes—
482C23C2×D3×D411yescharacteristic
483Q8—216no—
484Q8—24no—
485Q8—24no—
486Q8—24no—
487Q8—42no—
488Q8—42no—
489Q8—122no—
490Q8—122no—
491Q8#93918yes—
492C2×C4—216no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—32no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—68no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—64no—
519C2×C4—61no—
520C2×C4—124no—
521C2×C4—128no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4#93912yes—
531C2×C4C2×D3×D412yes—
532C2×C4C2×#23712yes—
533C2×C4C2×#23711yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C8—42no—
536C8—122no—
537C2×C3—24no—
538C2×C3#284711yescharacteristic
539C2×C3C2×#45111yescharacteristic
540C2×C3#300711yescharacteristic
541C2×C3C2×#46211yescharacteristic
542C2×C3#303711yescharacteristic
543C2×C3#310411yescharacteristic
544C2×C3#311311yescharacteristic
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22#518214yescomplemented
552C22#558611yescharacteristic
553C22D3×#12011yescharacteristic
554C22C2×D3×Q1611yescharacteristic
555C22C2×#91811yescharacteristic
556C22D3×#13711yescharacteristic
557C22#577711yescharacteristic
558C22C2×#93911yescharacteristic
559C4—28no—
560C4—22no—
561C4—34no—
562C4—62no—
563C4—62no—
564C4—81no—
565C4—81no—
566C4—121no—
567C4—124no—
568C4—241no—
569C4—241no—
570C4#558612yes—
571C4C2×#93912yes—
572C3#3290611yescharacteristic, complemented, Sylow
573C2—24no—
574C2#8017211yescharacteristic
575C2#8017411yescharacteristic
576C2D3×#45111yescharacteristic
577C2#8018411yescharacteristic
578C2#8260111yescharacteristic
579C2C2×#518211yescharacteristic
580C2#8396611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#451D3
hypocenterC3#32906
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
4. C2×#5182C2semidirect
6. #72065C2semidirect
8. #70239C2semidirect
9. #70221C2semidirect
200. C22×Dic3Q16semidirect
454. C22×C3#462semidirect
474. Dic3#451semidirect
551. C22#5182semidirect
572. C3#32906semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×C3×#4512
C2×#52282
C2×#51822
#828112
#720652
#716922
#702392
#702212
#699302
#699082
#329063