Finite Group Databasebeta

← #10822359

Subgroups — #10822359

Order 768.

2018 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic, complemented
2C22×#942C211yesmaximal, characteristic
3C22×#909C211yesmaximal, characteristic
4C2×C3×#419C211yesmaximal, characteristic, complemented
5#82228C211yesmaximal, characteristic
6C2×#5004C211yesmaximal, characteristic, complemented
7#71979C211yesmaximal, characteristic, complemented
8#70283C214yesmaximal, complemented
9#70282C214yesmaximal, complemented
10#23432—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#944—24no—
17C2×#944—24no—
18C2×#944—22no—
19C2×#944C2214yes—
20C22×#216C2212yes—
21C2×#942—24no—
22C2×#942—24no—
23C2×#942—24no—
24C22×#215C2212yes—
25C22×Dic3⋊C4—22no—
26C22×C4×Dic3C2211yescharacteristic
27C22×#208C2211yescharacteristic
28C2×#909—24no—
29C2×#909—24no—
30C2×#909—24no—
31C22×Dic12C2212yes—
32C22×#204C2212yes—
33C3×#419—24no—
34C2×C3×#103C2214yes—
35#5004—24no—
36#4988C2212yes—
37#4973C2212yes—
38C2×#810C2214yes—
39#4958—22no—
40C4×(C2×D3)⋊C4—22no—
41#4550C2214yes—
42C22×#137—31no—
43C22×#137—31no—
44C22×C4⋊Q8—31no—
45C2×#419—31no—
46C2×#419D311yescharacteristic, complemented
47#2297—31no—
48#2016—34no—
49#2013—34no—
50#1728—31no—
51C22×C3×Q8—22no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8D414yes—
58C22×C3×Q8C2312yes—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C23×Dic3D412yes—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235D414yes—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6D414yes—
79C22×Dic6C2312yes—
80C22×Dic6C2311yescharacteristic
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C2×C3×M16D414yes—
84C22×C3×C8C2312yes—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90#944—216no—
91#944—28no—
92#944—28no—
93C2×#216—24no—
94C2×#216—24no—
95C2×#216—24no—
96#942—232no—
97C2×#215—24no—
98C2×#215—24no—
99C2×#215—24no—
100C2×(C2×D3)⋊C4—24no—
101C2×Dic3⋊C4—216no—
102C2×Dic3⋊C4—28no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3D412yes—
106C2×C4×Dic3C2312yes—
107C2×#208—22no—
108C2×#208—22no—
109C2×#208D414yes—
110C22×C3⋊C8C2312yes—
111#909—232no—
112C2×Dic12—24no—
113C2×Dic12—24no—
114C2×Dic12—24no—
115C2×#204—24no—
116C2×#204—24no—
117C2×#204—24no—
118C3×#103—28no—
119C3⋊#95—24no—
120#810—28no—
121C22×#41—31no—
122C23×Q8—31no—
123C23×Q8C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C2×#137—64no—
128C2×#137—64no—
129C2×#137—64no—
130C22×Q16—32no—
131C22×Q16—32no—
132C22×SD16—32no—
133C22×SD16—32no—
134C22×M16—31no—
135C22×M16C2×D311yescharacteristic
136C2×C4⋊Q8—34no—
137C2×C4⋊Q8—62no—
138C2×C4⋊Q8—64no—
139C2×C4⋊Q8—64no—
140C2×#117C2×D311yescharacteristic
141C22×C4⋊C4—62no—
142C22×C42—31no—
143#419—24no—
144#419—64no—
145C2×#103—34no—
146C2×#103C2×D314yes—
147#391—32no—
148#390—32no—
149#386—62no—
150C4×C22⋊C4—62no—
151#336—34no—
152C24×C3C2×D411yescharacteristic
153C23×D3—21no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—28no—
157C2×C3×Q8—28no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—44no—
163C2×C3×Q8—42no—
164C2×C3×Q8—44no—
165C2×C3×Q8—42no—
166C2×C3×Q8C2×D414yes—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C22×Dic3—24no—
184C22×Dic3—28no—
185C22×Dic3—24no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—42no—
189C22×Dic3—42no—
190C22×Dic3C2×D414yes—
191#235—28no—
192#235—28no—
193#235—44no—
194#235—44no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12C2×D412yes—
201C2×C4×D3—24no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×Dic6—28no—
205C2×Dic6—28no—
206C2×Dic6—22no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—22no—
214C2×Dic6—24no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6—44no—
218C2×Dic6—42no—
219C2×Dic6—44no—
220C2×Dic6—42no—
221C2×Dic6C2×D414yes—
222C2×Dic6C2×D412yes—
223C3×M16—28no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8C2×D414yes—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C22⋊C4—42no—
230C3×C42—42no—
231#216—216no—
232#215—216no—
233(C2×D3)⋊C4—48no—
234Dic3⋊C4—232no—
235C4×Dic3—24no—
236C4×Dic3—22no—
237C4×Dic3—22no—
238#208—28no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8C2×D414yes—
242Dic12—216no—
243#204—216no—
244C2×#41—34no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C22×Q8—22no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—34no—
257C22×Q8—32no—
258C22×Q8—31no—
259C22×Q8—62no—
260C22×Q8—61no—
261C22×Q8—62no—
262C22×Q8—62no—
263C22×Q8—62no—
264C22×Q8—62no—
265C22×Q8D1214yes—
266C22×Q8C22×D312yes—
267C22×D4—31no—
268C22×D4—61no—
269C23×C4—21no—
270C23×C4—32no—
271C23×C4—61no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274#137—632no—
275#137—632no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×Q16—64no—
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C2×SD16—64no—
286C2×SD16—64no—
287C2×SD16—64no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—34no—
291C2×M16—62no—
292C2×M16—62no—
293C2×M16#6714yes—
294C22×C8—32no—
295C22×C8C22×D312yes—
296C4⋊Q8—68no—
297C4⋊Q8—616no—
298C4⋊Q8—68no—
299#117—22no—
300#117—22no—
301C2×C4⋊C4—68no—
302C2×C4⋊C4—616no—
303C2×C4⋊C4C22×D312yes—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—64no—
306C2×C42—21no—
307C2×C42—32no—
308C2×C42—32no—
309C2×C42—62no—
310C2×C42—62no—
311#103—28no—
312#103—68no—
313#95—64no—
314C23×C3—24no—
315C23×C3#12012yes—
316C23×C3#12112yes—
317C23×C3#12712yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—44no—
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—48no—
328C2×C3×C4—24no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—48no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4#12014yes—
335C2×C3×C4#12012yes—
336C2×C3×C4#12014yes—
337C2×C3×C4#12114yes—
338C2×C3×C4#12112yes—
339C2×C3×C4#12114yes—
340C2×C3×C4#12712yes—
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347#67—48no—
348C2×Dic3—216no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—41no—
357C2×Dic3—48no—
358C2×Dic3#13718yescomplemented
359D12—22no—
360D12—22no—
361D12—41no—
362D12—41no—
363C4×D3—48no—
364Dic6—24no—
365Dic6—22no—
366Dic6—22no—
367Dic6—24no—
368Dic6—42no—
369Dic6—42no—
370Dic6—41no—
371Dic6—41no—
372Dic6—48no—
373C3×C8—24no—
374C3⋊C8—24no—
375C24—61no—
376C24D3×D411yescharacteristic
377#41—68no—
378#41—68no—
379#41—124no—
380#41—124no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—34no—
390C2×Q8—32no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—64no—
396C2×Q8—62no—
397C2×Q8—61no—
398C2×Q8—68no—
399C2×Q8—68no—
400C2×Q8—61no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—62no—
407C2×Q8—61no—
408C2×Q8—124no—
409C2×Q8—122no—
410C2×Q8—124no—
411C2×Q8—122no—
412C2×Q8C2×D1214yes—
413C2×D4—32no—
414C2×D4—64no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—122no—
421C2×D4—122no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—34no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—68no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4C2×D1212yes—
439C22×C4#23512yes—
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D412yes—
442C22×C4D3×D412yes—
443C22×C4D3×D412yes—
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446Q16—616no—
447Q16—616no—
448SD16—616no—
449SD16—616no—
450M16—28no—
451M16—68no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—34no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8C2×#6714yes—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—632no—
461C22⋊C4—42no—
462C22⋊C4—128no—
463C42—42no—
464C42—64no—
465C42—62no—
466C42—62no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3#39214yes—
471C22×C3C2×#12011yescharacteristic
472C22×C3C2×#12111yescharacteristic
473C22×C3C2×#12711yescharacteristic
474C22×C3C2×#13714yes—
475C2×D3—42no—
476C2×D3—44no—
477C2×D3—82no—
478C2×D3—82no—
479C3×C4—44no—
480C3×C4—82no—
481C3×C4#39214yes—
482C3×C4C2×#12011yescharacteristic
483C3×C4C2×#12011yescharacteristic
484C3×C4C2×#12111yescharacteristic
485C3×C4C2×#12111yescharacteristic
486Dic3—28no—
487Dic3—44no—
488Dic3—42no—
489C23—24no—
490C23—62no—
491C23—121no—
492C23—124no—
493C23—121no—
494C23#88212yes—
495C23#88412yes—
496C23#94012yes—
497C23C2×D3×D411yescharacteristic
498Q8—24no—
499Q8—24no—
500Q8—48no—
501Q8—42no—
502Q8—42no—
503Q8—62no—
504Q8—64no—
505Q8—64no—
506Q8—62no—
507Q8—128no—
508Q8—122no—
509Q8—121no—
510Q8—122no—
511Q8—121no—
512D4—62no—
513D4—62no—
514D4—128no—
515D4—121no—
516D4—121no—
517C2×C4—24no—
518C2×C4—38no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—616no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—62no—
529C2×C4—128no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—128no—
535C2×C4#88212yes—
536C2×C4#88214yes—
537C2×C4#88412yes—
538C2×C4#88414yes—
539C2×C4#93014yes—
540C2×C4#93714yes—
541C2×C4#94012yes—
542C2×C4C22×D1211yescharacteristic
543C2×C4C2×#23511yescharacteristic
544C2×C4C2×D3×D411yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—24no—
549C8—64no—
550C2×C3—24no—
551C2×C3C2×#39211yescharacteristic
552C2×C3#300412yes—
553C2×C3#303912yes—
554C2×C3#315212yes—
555D3—82no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—124no—
560C22—122no—
561C22—242no—
562C22—242no—
563C22#496014yes—
564C22C2×#88211yescharacteristic
565C22C2×#88411yescharacteristic
566C22D3×#13714yes—
567C22C2×#94011yescharacteristic
568C4—44no—
569C4—68no—
570C4—82no—
571C4—122no—
572C4—124no—
573C4#496014yes—
574C4C2×#88211yescharacteristic
575C4C2×#88411yescharacteristic
576C4C2×#93011yescharacteristic
577C4C2×#93711yescharacteristic
578C3#2343211yescharacteristic, complemented, Sylow
579C2—24no—
580C2—242no—
581C2#7895812yes—
582C2#7897412yes—
583C2C2×#496011yescharacteristic
584C2#8397212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#419D3
hypocenterC3#23432
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
1. C22×#944C2semidirect
4. C2×C3×#419C2semidirect
6. C2×#5004C2semidirect
7. #71979C2semidirect
8. #70283C2semidirect
9. #70282C2semidirect
46. C2×#419D3semidirect
358. C2×Dic3#137semidirect
578. C3#23432semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C22×#9422
C22×#9092
C2×C3×#4192
#822282
C2×#50042
#719792
#702832
#702822
#234323