Finite Group Databasebeta

← #10822753

Subgroups — #10822753

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5833C211yesmaximal, characteristic
2C2×C3×#416C211yesmaximal, characteristic, complemented
3C2×#5232C211yesmaximal, characteristic, complemented
4C2×#5110C211yesmaximal, characteristic, complemented
5C22×#818C211yesmaximal, characteristic, complemented
6#82270C211yesmaximal, characteristic, complemented
7#71581C211yesmaximal, characteristic, complemented
8#70244C214yesmaximal, complemented
9#70175C214yesmaximal, complemented
10#23488—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5833—24no—
16#5833—24no—
17C2×#946C2212yes—
18C2×#939—22no—
19C2×#938C2212yes—
20C2×#930—21no—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#924—21no—
24C22×#208C2211yescharacteristic
25C3×#416—24no—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28#5232—24no—
29C2×#835C2212yes—
30C2×#832C2212yes—
31#5110—24no—
32C2×(C2×D3)⋊C8C2214yes—
33C2×#818—24no—
34C2×#818C2214yes—
35#4987C2214yes—
36#4958—22no—
37#4939—21no—
38#4929—21no—
39#4529C2214yes—
40C2×#537—31no—
41C22×#107—31no—
42C2×#417—31no—
43C2×#416D311yescharacteristic, complemented
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2033—34no—
48#1736—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41D414yes—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67D412yes—
62C23×Dic3—21no—
63C23×Dic3—21no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C22×D12D411yescharacteristic
71C22×C4×D3C2312yes—
72C22×Dic6D411yescharacteristic
73C22×C3×C8—22no—
74C22×C3×C8—22no—
75C22×C3×C8D414yescomplemented
76C22×C3×C8C2312yes—
77C3×#117—22no—
78C3×#117—22no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—21no—
82C2×#217—22no—
83#946—24no—
84#946—24no—
85#939—216no—
86#938—24no—
87#938—24no—
88#930—28no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—24no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3D414yes—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2312yes—
99#924—28no—
100C2×Dic3⋊C4—22no—
101C2×#208—22no—
102C2×#208—22no—
103C2×#208—22no—
104C22×C3⋊C8C2312yes—
105C3×#103—24no—
106C3×#102—24no—
107#835—24no—
108#832—24no—
109C3⋊#95—24no—
110(C2×D3)⋊C8—28no—
111#818—28no—
112C22×#41—31no—
113C22×#41C2×D311yescharacteristic
114C22×M16—31no—
115C23×C8C2×D311yescharacteristic
116#537—64no—
117#537—64no—
118C2×C22⋊Q8—32no—
119C2×C22⋊Q8—61no—
120C2×#121—32no—
121C2×#121—61no—
122C2×#121—62no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4—31no—
125C22×C4⋊C4—31no—
126C2×#107—34no—
127C2×#107—64no—
128#417—64no—
129#416—24no—
130C2×#103—32no—
131C2×#103C2×D312yes—
132C2×#102—32no—
133C2×#102C2×D312yes—
134#408—64no—
135C2×C22⋊C8—34no—
136#391—61no—
137#390—61no—
138#386—62no—
139#380—34no—
140#340—34no—
141C24×C3—21no—
142C24×C3C2×D411yescharacteristic
143C23×D3C2×D412yes—
144C3×#41—28no—
145C3×#41—28no—
146C3×#41—44no—
147C3×#41—44no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8C2×D412yes—
154C2×C3×D4—24no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4C2×D412yes—
163C22×C3×C4—21no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—24no—
177C2×#67—24no—
178C2×#67—24no—
179C2×#67—28no—
180C2×#67—84no—
181C22×Dic3—22no—
182C22×Dic3—24no—
183C22×Dic3—22no—
184C22×Dic3—42no—
185C22×Dic3—82no—
186C22×Dic3—82no—
187C22×Dic3C2×D412yes—
188#235—44no—
189#235—44no—
190#235—44no—
191#235—44no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×C3×C8—28no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8C2×D414yes—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C22⋊C4—42no—
211C3×C42—42no—
212#217—28no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—84no—
215C4⋊Dic3—24no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—42no—
220C4⋊Dic3C2×D414yes—
221Dic3⋊C4—28no—
222#208—44no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—62no—
232C2×#41—62no—
233C2×#41—62no—
234C2×#41—62no—
235C2×#41—62no—
236C2×#41—62no—
237C2×#41D1214yes—
238C22×Q8—31no—
239C22×Q8C22×D311yescharacteristic
240C22×D4—21no—
241C22×D4—32no—
242C22×D4—31no—
243C22×D4—62no—
244C22×D4C22×D311yescharacteristic
245C23×C4—21no—
246C23×C4—32no—
247C23×C4—61no—
248C23×C4—61no—
249C23×C4C22×D311yescharacteristic
250C2×M16—62no—
251C2×M16—62no—
252C2×M16—62no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—32no—
256C22×C8#6714yescomplemented
257C22×C8C22×D312yes—
258C22⋊Q8—64no—
259C22⋊Q8—68no—
260C22⋊Q8—64no—
261#121—64no—
262#121—68no—
263#121—616no—
264#121—64no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—64no—
281C2×C42—21no—
282#107—68no—
283#103—24no—
284#103—64no—
285#102—24no—
286#102—64no—
287C22⋊C8—68no—
288#95—64no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3#12012yes—
297C23×C3#12112yes—
298C23×C3#12312yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—24no—
304C22×D3—24no—
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—48no—
312C3×D4—41no—
313C3×D4—41no—
314C2×C3×C4—22no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—48no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12114yes—
323C2×C3×C4#12112yes—
324C2×C3×C4#12114yes—
325C2×C3×C4#12312yes—
326C2×C3×C4#12314yes—
327C2×C3×C4#12414yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334#67—48no—
335#67—88no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—44no—
341C2×Dic3—42no—
342C2×Dic3—44no—
343C2×Dic3—84no—
344C2×Dic3—84no—
345D12—44no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—42no—
350Dic6—44no—
351C3×C8—24no—
352C3⋊C8—42no—
353C24—21no—
354C24—32no—
355C24D3×D411yescharacteristic
356#41—28no—
357#41—28no—
358#41—44no—
359#41—44no—
360#41—124no—
361#41—124no—
362#41—124no—
363#41—124no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8C2×D1212yes—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—64no—
382C2×D4—64no—
383C2×D4—64no—
384C2×D4—68no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—244no—
389C2×D4C2×D1212yes—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—32no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—62no—
397C22×C4—64no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—122no—
406C22×C4—242no—
407C22×C4—242no—
408C22×C4C2×D1212yes—
409C22×C4#23512yes—
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D412yes—
412C22×C4#23712yes—
413C22×C4#23712yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416M16—124no—
417C2×C8—28no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8C2×#6714yes—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—34no—
427C4⋊C4—68no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—122no—
433C22⋊C4—42no—
434C22⋊C4—68no—
435C22⋊C4—68no—
436C22⋊C4—244no—
437C42—42no—
438C22×C3—24no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3#39414yes—
448C22×C3C2×#12011yescharacteristic
449C22×C3C2×#12111yescharacteristic
450C22×C3C2×#12311yescharacteristic
451C22×C3#57612yes—
452C22×C3#57812yes—
453C2×D3—44no—
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—44no—
458C3×C4—42no—
459C3×C4—82no—
460C3×C4#39414yes—
461C3×C4C2×#12111yescharacteristic
462C3×C4C2×#12111yescharacteristic
463C3×C4C2×#12311yescharacteristic
464C3×C4C2×#12411yescharacteristic
465Dic3—42no—
466Dic3—44no—
467Dic3—82no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—62no—
476C23—62no—
477C23—64no—
478C23—64no—
479C23—62no—
480C23#88212yes—
481C23#88312yes—
482C23#93812yes—
483C23C2×D3×D411yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—124no—
489D4—22no—
490D4—22no—
491D4—48no—
492D4—41no—
493D4—41no—
494D4—128no—
495D4—124no—
496D4—248no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—124no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—124no—
515C2×C4—244no—
516C2×C4—244no—
517C2×C4#88212yes—
518C2×C4#88312yes—
519C2×C4#88514yes—
520C2×C4#88614yes—
521C2×C4#93014yes—
522C2×C4#93812yes—
523C2×C4#93914yes—
524C2×C4C22×D1211yescharacteristic
525C2×C4C2×#23511yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×#23711yescharacteristic
528C2×C4C2×#23711yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—24no—
531C8—122no—
532C2×C3—24no—
533C2×C3—82no—
534C2×C3C2×#39411yescharacteristic
535C2×C3#300811yescharacteristic
536C2×C3#301111yescharacteristic
537C2×C3#306112yes—
538C2×C3#309812yes—
539D3—82no—
540C22—24no—
541C22—22no—
542C22—22no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—122no—
550C22—124no—
551C22—122no—
552C22—122no—
553C22—124no—
554C22#496114yes—
555C22C2×#88211yescharacteristic
556C22C2×#88311yescharacteristic
557C22#576912yes—
558C22#577112yes—
559C22C2×#93811yescharacteristic
560C4—42no—
561C4—82no—
562C4—122no—
563C4—124no—
564C4—242no—
565C4#496114yes—
566C4C2×#88511yescharacteristic
567C4C2×#88611yescharacteristic
568C4C2×#93011yescharacteristic
569C4C2×#93911yescharacteristic
570C3#2348811yescharacteristic, complemented, Sylow
571C2—24no—
572C2—82no—
573C2—242no—
574C2#7883411yescharacteristic
575C2#7883711yescharacteristic
576C2#7884312yes—
577C2C2×#496111yescharacteristic
578C2#8391712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#416C2
socleC23×C3C22×D4
prime radicalC2×#5833C2
hypercenterC2×#416D3
hypocenterC3#23488
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×C3×#416C2semidirect
3. C2×#5232C2semidirect
4. C2×#5110C2semidirect
5. C22×#818C2semidirect
6. #82270C2semidirect
7. #71581C2semidirect
8. #70244C2semidirect
9. #70175C2semidirect
43. C2×#416D3semidirect
75. C22×C3×C8D4semidirect
256. C22×C8#67semidirect
570. C3#23488semidirect

Maximal subgroups

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

subgroupindex
C2×#58332
C2×C3×#4162
C2×#52322
C2×#51102
C22×#8182
#822702
#715812
#702442
#701752
#234883