Finite Group Databasebeta

← #10822355

Subgroups — #10822355

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×C3×#137C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic
3C22×#944C211yesmaximal, characteristic, complemented
4#84086C211yesmaximal, characteristic
5C2×#5225C211yesmaximal, characteristic, complemented
6C2×#5120C211yesmaximal, characteristic, complemented
7#72073C214yesmaximal, complemented
8#72057C214yesmaximal, complemented
9#71979C211yesmaximal, characteristic, complemented
10#23432—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C2×C3×#137—24no—
15C2×C3×#137—24no—
16C2×C3×#137—24no—
17C22×C3×Q16C2212yes—
18C22×C3×SD16C2212yes—
19C22×C3×M16C2211yescharacteristic
20C2×#951—24no—
21C2×#951—24no—
22C2×#951—24no—
23C2×#944—24no—
24C2×#944—24no—
25C2×#944—22no—
26C2×#944C2214yes—
27C22×#216C2212yes—
28C22×#214C2212yes—
29C2×#926C2211yescharacteristic
30C22×Dic3⋊C4—22no—
31C22×C4×Dic3C2211yescharacteristic
32C22×#208C2211yescharacteristic
33#5229C2212yes—
34#5225—24no—
35C2×#835C2214yes—
36#5221C2212yes—
37#5220—22no—
38C4×#217—22no—
39#5120—24no—
40C2×#816C2214yes—
41#4550C2214yes—
42C22×#137—31no—
43C22×#137D311yescharacteristic
44C22×C4⋊Q8—31no—
45C2×#419—31no—
46C2×#419—31no—
47#2297—31no—
48#2016—34no—
49#2013—34no—
50#1728—31no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41D414yes—
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8D414yes—
64C22×C3×Q8C2312yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—21no—
72C23×Dic3D412yes—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—21no—
78C22×Dic6—22no—
79C22×Dic6D414yes—
80C22×Dic6C2312yes—
81C3×#137—232no—
82C2×C3×Q16—24no—
83C2×C3×Q16—24no—
84C2×C3×Q16—24no—
85C2×C3×SD16—24no—
86C2×C3×SD16—24no—
87C2×C3×SD16—24no—
88C2×C3×M16—22no—
89C2×C3×M16—22no—
90C2×C3×M16D414yes—
91C22×C3×C8C2312yes—
92C2×#217—21no—
93C2×#217—24no—
94#951—232no—
95#944—216no—
96#944—28no—
97#944—28no—
98C2×#216—24no—
99C2×#216—24no—
100C2×#216—24no—
101C2×#214—24no—
102C2×#214—24no—
103C2×#214—24no—
104#926—22no—
105#926—22no—
106C2×C4⋊Dic3C2312yes—
107C2×Dic3⋊C4—216no—
108C2×Dic3⋊C4—28no—
109C2×C4×Dic3—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3D412yes—
113C2×C4×Dic3C2312yes—
114C2×#208—22no—
115C2×#208—22no—
116C2×#208D414yes—
117C22×C3⋊C8C2312yes—
118#835—28no—
119C3⋊#95—24no—
120#816—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×Q8C2×D311yescharacteristic
124C2×#137—24no—
125C2×#137—24no—
126C2×#137—24no—
127C2×#137—64no—
128C2×#137—64no—
129C2×#137—64no—
130C22×Q16—32no—
131C22×Q16C2×D312yes—
132C22×SD16—32no—
133C22×SD16C2×D312yes—
134C22×M16—31no—
135C22×M16C2×D311yescharacteristic
136C2×C4⋊Q8—34no—
137C2×C4⋊Q8—62no—
138C2×C4⋊Q8—64no—
139C2×C4⋊Q8—64no—
140C2×#117—31no—
141C22×C4⋊C4—62no—
142C22×C42—31no—
143#419—64no—
144#419—64no—
145C2×#103—34no—
146C2×#103—34no—
147#391—32no—
148#390—32no—
149#386—62no—
150C4×C22⋊C4—62no—
151#336—34no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C3×#41—28no—
155C3×#41—28no—
156C3×#41—44no—
157C3×#41—44no—
158C2×C3×Q8—24no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—28no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8—42no—
174C2×C3×Q8—42no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×D414yes—
177C2×C3×D4—24no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—24no—
203C22×Dic3—28no—
204C22×Dic3—24no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—42no—
208C22×Dic3—42no—
209C22×Dic3C2×D414yes—
210C2×Dic6—28no—
211C2×Dic6—28no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—24no—
217C2×Dic6—22no—
218C2×Dic6—44no—
219C2×Dic6—44no—
220C2×Dic6—42no—
221C2×Dic6—42no—
222C2×Dic6C2×D414yes—
223C3×Q16—216no—
224C3×SD16—216no—
225C3×M16—28no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8C2×D414yes—
229#217—42no—
230#217—48no—
231#216—216no—
232#214—216no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235Dic3⋊C4—232no—
236C4×Dic3—24no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239C4×Dic3—42no—
240#208—28no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8C2×D414yes—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41#6714yes—
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—61no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8—62no—
263C22×Q8—62no—
264C22×Q8#6714yes—
265C22×Q8C22×D312yes—
266C22×Q8C22×D311yescharacteristic
267C22×D4—21no—
268C22×D4C22×D311yescharacteristic
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—32no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274#137—232no—
275#137—632no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—24no—
279C2×Q16—64no—
280C2×Q16—64no—
281C2×Q16—64no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—24no—
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—62no—
300#117—62no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—68no—
303C2×C4⋊C4—616no—
304C2×C22⋊C4—64no—
305C2×C22⋊C4—61no—
306C2×C42—32no—
307C2×C42—32no—
308C2×C42—62no—
309C2×C42—62no—
310C2×C42—61no—
311#103—68no—
312#103—68no—
313#95—64no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—44no—
319C23×C3#12012yes—
320C23×C3#12112yes—
321C23×C3#12712yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—22no—
324C3×Q8—24no—
325C3×Q8—22no—
326C3×Q8—24no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×Q8—48no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—48no—
335C3×D4—41no—
336C3×D4—41no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—48no—
344C2×C3×C4—48no—
345C2×C3×C4#12014yes—
346C2×C3×C4#12012yes—
347C2×C3×C4#12014yes—
348C2×C3×C4#12114yes—
349C2×C3×C4#12114yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12712yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×Dic3—216no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—48no—
365C2×Dic3—42no—
366C2×Dic3—44no—
367C2×Dic3#13718yescomplemented
368Dic6—24no—
369Dic6—24no—
370Dic6—42no—
371Dic6—42no—
372Dic6—48no—
373C3×C8—24no—
374C3⋊C8—24no—
375C24—21no—
376C24D3×D411yescharacteristic
377#41—28no—
378#41—28no—
379#41—44no—
380#41—44no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—34no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—64no—
400C2×Q8—62no—
401C2×Q8—68no—
402C2×Q8—68no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—124no—
408C2×Q8—122no—
409C2×Q8—122no—
410C2×Q8—124no—
411C2×Q8C2×#6714yes—
412C2×Q8C2×#6712yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×#6712yes—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—34no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—68no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4D3×D412yes—
439C22×C4D3×D411yescharacteristic
440C22×C4D3×D412yes—
441C22×C4#23712yes—
442C22×C4C2×#6712yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446Q16—216no—
447Q16—616no—
448SD16—216no—
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—632no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C22⋊C4—128no—
462C22⋊C4—122no—
463C42—64no—
464C42—62no—
465C42—62no—
466C42—122no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3#39214yes—
475C22×C3C2×#12011yescharacteristic
476C22×C3C2×#12111yescharacteristic
477C22×C3C2×#12711yescharacteristic
478C22×C3C2×#13714yes—
479C3×C4—42no—
480C3×C4—44no—
481C3×C4#39214yes—
482C3×C4C2×#12011yescharacteristic
483C3×C4C2×#12011yescharacteristic
484C3×C4C2×#12111yescharacteristic
485C3×C4C2×#12111yescharacteristic
486Dic3—28no—
487Dic3—44no—
488Dic3—82no—
489C23—24no—
490C23—22no—
491C23—44no—
492C23—41no—
493C23—41no—
494C23#93712yes—
495C23#93912yes—
496C23#94012yes—
497C23C2×D3×D411yescharacteristic
498Q8—22no—
499Q8—24no—
500Q8—24no—
501Q8—22no—
502Q8—48no—
503Q8—42no—
504Q8—42no—
505Q8—41no—
506Q8—41no—
507Q8—64no—
508Q8—64no—
509Q8—128no—
510Q8—122no—
511Q8—122no—
512D4—22no—
513D4—22no—
514D4—48no—
515D4—41no—
516D4—41no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—38no—
520C2×C4—48no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—616no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—128no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—124no—
534C2×C4—122no—
535C2×C4#93712yes—
536C2×C4#93714yes—
537C2×C4#93814yes—
538C2×C4#93912yes—
539C2×C4#93914yes—
540C2×C4#94012yes—
541C2×C4#95314yes—
542C2×C4C2×D3×D411yescharacteristic
543C2×C4C2×D3×D411yescharacteristic
544C2×C4C2×#23711yescharacteristic
545C2×C4C22×#6711yescharacteristic
546C2×C4C22×#6711yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—24no—
549C8—64no—
550C2×C3—24no—
551C2×C3—82no—
552C2×C3C2×#39211yescharacteristic
553C2×C3#300412yes—
554C2×C3#303912yes—
555C2×C3#315212yes—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—42no—
560C22—44no—
561C22—82no—
562C22—82no—
563C22#522314yes—
564C22D3×#13714yes—
565C22C2×#93711yescharacteristic
566C22C2×#93911yescharacteristic
567C22C2×#94011yescharacteristic
568C4—42no—
569C4—44no—
570C4—68no—
571C4—124no—
572C4—242no—
573C4#522314yes—
574C4C2×#93711yescharacteristic
575C4C2×#93811yescharacteristic
576C4C2×#93911yescharacteristic
577C4C2×#95311yescharacteristic
578C3#2343211yescharacteristic, complemented, Sylow
579C2—24no—
580C2—82no—
581C2#8396712yes—
582C2#8397212yes—
583C2#8397912yes—
584C2C2×#522311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#137C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC22×#137D3
hypocenterC3#23432
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C22×#944C2semidirect
5. C2×#5225C2semidirect
6. C2×#5120C2semidirect
7. #72073C2semidirect
8. #72057C2semidirect
9. #71979C2semidirect
367. 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×C3×#1372
C22×#9512
C22×#9442
#840862
C2×#52252
C2×#51202
#720732
#720572
#719792
#234323