Finite Group Databasebeta

← #10845921

Subgroups — #10845921

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2537C211yesmaximal, characteristic
2C2×#5041C212yesmaximal, complemented
3#82379C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic, complemented
5C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
6#71685C212yesmaximal, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8#69579C211yesmaximal, characteristic, complemented
9#69568C211yesmaximal, characteristic, complemented
10#69526C212yesmaximal, complemented
11#69514C212yesmaximal, complemented
12#25385—31nomaximal, Sylow
13C2×C3×#121—21no—
14C2×C3×#121C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#214C2212yes—
18C2×#924C2212yes—
19C2×#924C2212yes—
20C23×C3⋊C8C2212yes—
21C2×#869C2211yescharacteristic
22C3×#394—22no—
23C3×#384C2211yescharacteristic
24C3×#380—21no—
25C3×#380C2212yes—
26#5221—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#830C2212yes—
30#5041—28no—
31#5019—24no—
32D4×C3⋊C8—24no—
33#4975—21no—
34C2×#810C2212yes—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#808C2212yes—
38#4949—22no—
39C4×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#465—32no—
46C2×#461—31no—
47C2×C8×D4—31no—
48#2542—31no—
49#2537D311yescharacteristic
50#2035—32no—
51#2000—32no—
52#1956—31no—
53#1948—31no—
54#1892—32no—
55C4×#102—31no—
56C22×C3×D4—22no—
57C22×C3×D4D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4C2312yes—
61C22×Dic6C2312yes—
62C3×#121—24no—
63C3×#121—42no—
64C3×#121—44no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80C2×#217—22no—
81C2×#214—24no—
82C2×#214—24no—
83C2×#214—42no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88#924—24no—
89#924—24no—
90#924—42no—
91#924—42no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8—24no—
97C22×C3⋊C8—42no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101#869—22no—
102#869—22no—
103C3×#95—22no—
104#832—22no—
105#832D414yescomplemented
106C3⋊#95—21no—
107#830—24no—
108#810—24no—
109#810—24no—
110#810—42no—
111#808—24no—
112C4⋊(C3⋊C8)D414yescomplemented
113C4×C3⋊C8—22no—
114C22×SD16—32no—
115C23×C8—32no—
116C2×C4⋊Q8—31no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×#121—21no—
120C2×#121C2×D312yes—
121C2×C4×D4C2×D311yescharacteristic
122C2×C4×D4C2×D311yescharacteristic
123#465—68no—
124#461—64no—
125C8×D4—64no—
126C2×#106—32no—
127C2×C4⋊C8—31no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#102—31no—
132C2×#102—31no—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#399—62no—
136#394—22no—
137#390—61no—
138#384—61no—
139#384C2×D311yescharacteristic
140#380—21no—
141#380C2×D312yes—
142C4×C4⋊C4—31no—
143#340—31no—
144#340—32no—
145#340—31no—
146C24×C3—21no—
147C24×C3C2×D412yes—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4—82no—
158C2×C3×D4SD1614yescomplemented
159C2×C3×D4C2×D412yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—81no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—42no—
195C3×C42C2×D412yes—
196C3×C42#4112yes—
197#217—44no—
198#217—44no—
199#214—48no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—42no—
203C4⋊Dic3—42no—
204C4⋊Dic3C2×D412yes—
205C4⋊Dic3#4112yes—
206Dic3⋊C4—44no—
207Dic3⋊C4—44no—
208C4×Dic3—42no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—48no—
217C2×C3⋊C8C2×D414yes—
218C2×C3⋊C8C2×D412yes—
219C22×Q8—32no—
220C22×D4—22no—
221C22×D4#6711yescharacteristic
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4C22×D312yes—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C22×C8—31no—
229C22×C8—32no—
230C22×C8—31no—
231C22×C8—64no—
232C22×C8—64no—
233C22×C8—122no—
234C4⋊Q8—62no—
235C4⋊Q8—62no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C22⋊Q8—122no—
240#121—24no—
241#121—42no—
242#121—44no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4#6712yes—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#106—64no—
267C4⋊C8—34no—
268#103—64no—
269#103—64no—
270#103—122no—
271#102—34no—
272#102—62no—
273C22⋊C8—64no—
274C4×C8—62no—
275#95—22no—
276#95—61no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3C2×SD1614yes—
291C23×C3C22×D411yescharacteristic
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—28no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—88no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12112yes—
315C2×C3×C4#12112yes—
316C2×C3×C4C2×SD1612yes—
317C2×C3×C4#13512yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333Dic6—42no—
334Dic6—42no—
335Dic6—42no—
336Dic6—42no—
337C3⋊C8—22no—
338C3⋊C8—24no—
339C3⋊C8—41no—
340C24—21no—
341C24C2×#6712yes—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—82no—
358C2×D4#21414yescomplemented
359C2×D4D3×D412yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4C2×#6712yes—
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379C2×C8—34no—
380C2×C8—32no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—61no—
384C2×C8—62no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—128no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—32no—
395C4⋊C4—32no—
396C4⋊C4—42no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—81no—
400C4⋊C4—124no—
401C4⋊C4—124no—
402C4⋊C4—122no—
403C4⋊C4—122no—
404C22⋊C4—24no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—124no—
409C22⋊C4—124no—
410C42—122no—
411C42D3×D412yes—
412C42#23712yes—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3C2×#12111yescharacteristic
428C22×C3D4211yescharacteristic
429C22×C3#54611yescharacteristic
430C22×C3C22×SD1611yescharacteristic
431C22×C3C2×#13511yescharacteristic
432C22×C3#57711yescharacteristic
433C22×C3#57811yescharacteristic
434C3×C4—22no—
435C3×C4—42no—
436C3×C4—82no—
437C3×C4—81no—
438C3×C4#46514yescomplemented
439C3×C4#53211yescharacteristic
440C3×C4#54011yescharacteristic
441C3×C4D4211yescharacteristic
442C3×C4#55011yescharacteristic
443Dic3—42no—
444Dic3—42no—
445Dic3—82no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23C2×#21414yes—
460C23C22×#6711yescharacteristic
461Q8—122no—
462Q8—122no—
463Q8—122no—
464Q8—122no—
465D4—28no—
466D4—22no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—88no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—84no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4C2×#21412yes—
497C2×C4#93912yes—
498C2×C4#93912yes—
499C2×C4#95012yes—
500C2×C4C2×D3×D411yescharacteristic
501C2×C4C2×D3×D411yescharacteristic
502C2×C4C2×#23711yescharacteristic
503C2×C4#100411yescharacteristic
504C2×C4D3×#4111yescharacteristic
505C2×C4#100911yescharacteristic
506C8—62no—
507C8—64no—
508C8—121no—
509C2×C3—44no—
510C2×C3—82no—
511C2×C3#242811yescharacteristic
512C2×C3C2×#46511yescharacteristic
513C2×C3#306311yescharacteristic
514C2×C3D4×SD1611yescharacteristic
515C2×C3#328311yescharacteristic
516C2×C3#329811yescharacteristic
517C2×C3#330311yescharacteristic
518C22—24no—
519C22—24no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22C22×#21411yescharacteristic
533C22D4×#6711yescharacteristic
534C22C2×#93911yescharacteristic
535C22C2×#95011yescharacteristic
536C22#582911yescharacteristic
537C22#583611yescharacteristic
538C22#583811yescharacteristic
539C4—22no—
540C4—42no—
541C4—82no—
542C4—81no—
543C4—122no—
544C4—122no—
545C4—242no—
546C4#504114yescomplemented
547C4#559611yescharacteristic
548C4#559911yescharacteristic
549C4#560911yescharacteristic
550C4#561911yescharacteristic
551C3#2538511yescharacteristic, complemented, Sylow
552C2—44no—
553C2—82no—
554C2C2×#504111yescharacteristic
555C2#8264911yescharacteristic
556C2#8342311yescharacteristic
557C2D4×#21411yescharacteristic
558C2#8352411yescharacteristic
559C2#8353111yescharacteristic
560C2#8365211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3C22×D4
hypercenter#2537D3
hypocenterC3#25385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. C2×#5041C2semidirect
3. #82379C2semidirect
4. C2×#5019C2semidirect
5. C2×D4×C3⋊C8C2semidirect
6. #71685C2semidirect
7. C4×#832C2semidirect
8. #69579C2semidirect
9. #69568C2semidirect
10. #69526C2semidirect
11. #69514C2semidirect
105. #832D4semidirect
112. C4⋊(C3⋊C8)D4semidirect
158. C2×C3×D4SD16semidirect
358. C2×D4#214semidirect
438. C3×C4#465semidirect
546. C4#5041semidirect
551. C3#25385semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
C2×#50412
#823792
C2×#50192
C2×D4×C3⋊C82
#716852
C4×#8322
#695792
#695682
#695262
#695142
#253853