Finite Group Databasebeta

← #10823010

Subgroups — #10823010

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3C22×#836C211yesmaximal, characteristic, complemented
4C2×#5132C211yesmaximal, characteristic
5#82213C211yesmaximal, characteristic
6#82204C211yesmaximal, characteristic, complemented
7#71994C211yesmaximal, characteristic, complemented
8#70340C214yesmaximal, complemented
9#70339C214yesmaximal, complemented
10#23522—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×#216C2212yes—
16C2×#942—24no—
17C2×#942—24no—
18C2×#942—22no—
19C2×#942—22no—
20C22×#215C2212yes—
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#419—22no—
25C3×#419C2214yes—
26C2×C3×#103—22no—
27C2×#836—24no—
28C2×#836C2214yes—
29C2×#824C2212yes—
30#5132—24no—
31C2×#823C2212yes—
32#5004C2212yes—
33#5002—22no—
34#5001C2212yes—
35#4999—22no—
36#4999C2214yes—
37#4966—22no—
38#4924—22no—
39#4558C2214yes—
40C22×#137—31no—
41#2905—31no—
42#2896—31no—
43C2×#419D311yescharacteristic, complemented
44C2×#413—31no—
45C22×#101—31no—
46#2047—34no—
47#2046—34no—
48#1738—31no—
49C22×C3×Q8—22no—
50C22×C3×Q8—21no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—21no—
54C22×C3×Q8—21no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8D414yes—
58C22×C3×Q8C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C2×#235—22no—
63C2×#235—21no—
64C2×#235—21no—
65C2×#235—22no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—22no—
69C2×#235D414yes—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6C2311yescharacteristic
73C2×C3×M16—21no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16C2312yes—
77C22×C3×C8—21no—
78C3×#117—21no—
79C3×#117—21no—
80C3×#117D412yes—
81C3×#117C2312yes—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—24no—
88#942—216no—
89#942—216no—
90C2×#215—24no—
91C2×#215—24no—
92C2×#215—24no—
93C2×#208—22no—
94C2×#208—22no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—22no—
101C2×#208D412yes—
102C2×#208D412yes—
103C2×#208C2312yes—
104C2×#208C2312yes—
105C22×C3⋊C8—21no—
106C22×C3⋊C8—21no—
107C22×C3⋊C8C2312yes—
108C3×#103—28no—
109#836—28no—
110#824—24no—
111#823—24no—
112#810—24no—
113#809—24no—
114C4⋊(C3⋊C8)—28no—
115C22×#41—31no—
116C23×Q8C2×D311yescharacteristic
117C2×#137—62no—
118C2×#137—62no—
119C2×#137—64no—
120C2×#137—64no—
121C22×Q16—32no—
122C22×SD16—32no—
123C22×M16—31no—
124C22×M16—31no—
125C22×M16—31no—
126C22×M16C2×D311yescharacteristic
127C2×#117C2×D311yescharacteristic
128#424—34no—
129#424—62no—
130#423—62no—
131#419—22no—
132#419—32no—
133#419C2×D314yes—
134#418—32no—
135#417—62no—
136#416—62no—
137C2×#103—22no—
138#413—64no—
139C2×#101—34no—
140C2×#101—32no—
141C2×#101—64no—
142C2×#100—32no—
143#341—34no—
144C24×C3#4111yescharacteristic
145C23×D3—21no—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—24no—
150C2×C3×Q8—24no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×Q8—41no—
159C2×C3×Q8—41no—
160C2×C3×Q8—42no—
161C2×C3×Q8—42no—
162C2×C3×Q8—42no—
163C2×C3×Q8C2×D414yes—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—42no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4#4112yes—
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—41no—
182C2×#67—41no—
183C2×#67—42no—
184C22×Dic3—21no—
185#235—24no—
186#235—24no—
187#235—24no—
188#235—24no—
189#235—42no—
190#235—42no—
191#235—42no—
192#235—42no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12C2×D412yes—
199C2×C4×D3—24no—
200C2×C4×D3—41no—
201C2×C4×D3—41no—
202C2×C4×D3—42no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6C2×D412yes—
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—42no—
212C2×C3×C8—22no—
213C2×C3×C8—42no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4C2×D414yes—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—22no—
219C3×C42—22no—
220#216—216no—
221#215—216no—
222#208—24no—
223#208—24no—
224#208—22no—
225#208—22no—
226#208—22no—
227#208—22no—
228#208—42no—
229#208—42no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—42no—
235C2×C3⋊C8—42no—
236C2×C3⋊C8C2×D414yes—
237C2×#41—34no—
238C2×#41—62no—
239C2×#41—62no—
240C2×#41—61no—
241C2×#41—61no—
242C2×#41—62no—
243C2×#41—61no—
244C2×#41—61no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—31no—
254C22×Q8D1214yes—
255C22×Q8C22×D312yes—
256C22×D4—31no—
257C22×D4—61no—
258C23×C4—21no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261#137—616no—
262#137—616no—
263C2×Q16—64no—
264C2×Q16—64no—
265C2×Q16—64no—
266C2×SD16—64no—
267C2×SD16—64no—
268C2×SD16—64no—
269C2×M16—21no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—32no—
273C2×M16—32no—
274C2×M16—32no—
275C2×M16—32no—
276C2×M16—62no—
277C2×M16—62no—
278C2×M16—61no—
279C2×M16—61no—
280C2×M16—62no—
281C2×M16—62no—
282C2×M16—62no—
283C2×M16—62no—
284C2×M16C22×D312yes—
285C22×C8—21no—
286C22×C8—32no—
287C22×C8—61no—
288C22×C8—61no—
289#117—21no—
290#117—21no—
291#117#6712yes—
292#117C22×D312yes—
293C2×C4⋊C4#6712yes—
294C2×C22⋊C4—21no—
295C2×C42—21no—
296C4⋊C8—68no—
297#103—28no—
298#103—64no—
299#102—64no—
300#101—64no—
301#101—68no—
302#100—64no—
303C23×C3—42no—
304C23×C3#12111yescharacteristic
305C23×C3#12111yescharacteristic
306C23×C3#12312yes—
307C23×C3#12312yes—
308C23×C3C2×#4111yescharacteristic
309C22×D3—22no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—82no—
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—44no—
318C3×Q8—44no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—44no—
325C2×C3×C4—82no—
326C2×C3×C4#12012yes—
327C2×C3×C4#12012yes—
328C2×C3×C4#12112yes—
329C2×C3×C4#12112yes—
330C2×C3×C4#12112yes—
331C2×C3×C4#12112yes—
332C2×C3×C4#12111yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12312yes—
335C2×C3×C4#12312yes—
336C2×C3×C4#12412yes—
337C2×C3×C4#12412yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344#67—44no—
345#67—44no—
346C2×Dic3—22no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349D12—22no—
350D12—22no—
351D12—41no—
352D12—41no—
353C4×D3—44no—
354C4×D3—44no—
355Dic6—22no—
356Dic6—22no—
357Dic6—41no—
358Dic6—41no—
359C3×C8—42no—
360C3⋊C8—24no—
361C3⋊C8—42no—
362C3⋊C8—42no—
363C24—61no—
364C24#23711yescharacteristic
365#41—64no—
366#41—64no—
367#41—64no—
368#41—64no—
369#41—122no—
370#41—122no—
371#41—122no—
372#41—122no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—32no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8C2×D1214yes—
397C2×D4—32no—
398C2×D4—64no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—122no—
405C2×D4—121no—
406C2×D4—121no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—61no—
414C22×C4—64no—
415C22×C4—122no—
416C22×C4—121no—
417C22×C4—121no—
418C22×C4C2×D1212yes—
419C22×C4#23512yes—
420C22×C4D3×D412yes—
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4#23711yescharacteristic
424C22×C4#23712yes—
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C23×D311yescharacteristic
428Q16—616no—
429SD16—616no—
430M16—22no—
431M16—22no—
432M16—42no—
433M16—64no—
434M16—64no—
435M16—62no—
436M16—62no—
437M16—62no—
438M16—62no—
439M16—122no—
440M16—122no—
441C2×C8—22no—
442C2×C8—34no—
443C2×C8—42no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—122no—
449C2×C8—122no—
450C4⋊C4—22no—
451C4⋊C4C2×#6714yes—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C42—22no—
455C42—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3#39412yes—
462C22×C3#39412yes—
463C22×C3C2×#12111yescharacteristic
464C22×C3C2×#12311yescharacteristic
465C22×C3C2×#12311yescharacteristic
466C2×D3—42no—
467C2×D3—82no—
468C2×D3—82no—
469C2×D3—82no—
470C3×C4—44no—
471C3×C4—44no—
472C3×C4#39412yes—
473C3×C4#39412yes—
474C3×C4C2×#12011yescharacteristic
475C3×C4C2×#12111yescharacteristic
476C3×C4C2×#12111yescharacteristic
477C3×C4C2×#12411yescharacteristic
478Dic3—42no—
479C23—42no—
480C23—62no—
481C23—121no—
482C23—121no—
483C23—242no—
484C23#88312yes—
485C23#88612yes—
486C23#93911yescharacteristic
487C23#93911yescharacteristic
488C23C2×#23711yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8—62no—
496Q8—62no—
497Q8—121no—
498Q8—121no—
499D4—62no—
500D4—62no—
501D4—124no—
502D4—124no—
503D4—121no—
504D4—121no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—62no—
512C2×C4—82no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—124no—
516C2×C4—124no—
517C2×C4#88212yes—
518C2×C4#88212yes—
519C2×C4#88312yes—
520C2×C4#88512yes—
521C2×C4#88512yes—
522C2×C4#88612yes—
523C2×C4#93012yes—
524C2×C4#93012yes—
525C2×C4#93812yes—
526C2×C4#93812yes—
527C2×C4#93911yescharacteristic
528C2×C4#93911yescharacteristic
529C2×C4C22×D1211yescharacteristic
530C2×C4C2×#23511yescharacteristic
531C2×C4C2×D3×D411yescharacteristic
532C2×C4C2×D3×D411yescharacteristic
533C2×C4C2×#23711yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C8—42no—
536C8—64no—
537C8—122no—
538C8—122no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3C2×#39411yescharacteristic
544C2×C3#306912yes—
545D3—82no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—122no—
552C22—242no—
553C22—242no—
554C22—242no—
555C22#496112yes—
556C22#496112yes—
557C22C2×#88311yescharacteristic
558C22C2×#88611yescharacteristic
559C22C2×#93911yescharacteristic
560C4—44no—
561C4—44no—
562C4—122no—
563C4#496112yes—
564C4#496112yes—
565C4C2×#88211yescharacteristic
566C4C2×#88511yescharacteristic
567C4C2×#93011yescharacteristic
568C4C2×#93811yescharacteristic
569C3#2352211yescharacteristic, complemented, Sylow
570C2—21no—
571C2—21no—
572C2—22no—
573C2—22no—
574C2—242no—
575C2C2×#496111yescharacteristic
576C2#8397812yes—
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC22×C3C2×#121
prime radicalC22×#235C22
hypercenterC2×#419D3
hypocenterC3#23522
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
3. C22×#836C2semidirect
6. #82204C2semidirect
7. #71994C2semidirect
8. #70340C2semidirect
9. #70339C2semidirect
43. C2×#419D3semidirect
569. C3#23522semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C2×C3×#4192
C22×#8362
C2×#51322
#822132
#822042
#719942
#703402
#703392
#235223