Finite Group Databasebeta

← #10828505

Subgroups — #10828505

Order 768.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5600C211yesmaximal, characteristic
2C2×C3×#480C211yesmaximal, characteristic
3C2×#4972C211yesmaximal, characteristic, complemented
4C2×#4971C211yesmaximal, characteristic, complemented
5#71677C211yesmaximal, characteristic
6#70642C212yesmaximal, complemented
7#70618C212yesmaximal, complemented
8#70216C212yesmaximal, complemented
9#70169C212yesmaximal, complemented
10#69930C211yesmaximal, characteristic, complemented
11#69910C211yesmaximal, characteristic, complemented
12#24066—31nomaximal, Sylow
13C2×C3×#121C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#936C2211yescharacteristic
16C2×#935—21no—
17C2×D4×Dic3—21no—
18C2×D4×Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×Dic3⋊C4—21no—
22C22×C4×Dic3C2211yescharacteristic
23#5600—24no—
24#5600—24no—
25C2×#890C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#879—21no—
28C2×#877—21no—
29C3×#480—24no—
30C2×C3×#107C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×#832C2212yes—
34C2×C3⋊#95—21no—
35C2×#830C2211yescharacteristic
36C2×#816C2212yes—
37#4972—21no—
38#4972—21no—
39#4972C2212yes—
40#4972C2212yes—
41#4971—24no—
42C2×#810C2212yes—
43C2×#808C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#529—31no—
49C2×#483—31no—
50C2×#480D311yescharacteristic
51C2×#389—31no—
52#2059—32no—
53#2042—32no—
54#1874—31no—
55#1872—31no—
56#1864—32no—
57#1863—32no—
58#1817—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C23×Dic3—21no—
65C23×Dic3D411yescharacteristic
66C23×Dic3D411yescharacteristic
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C3×#121—22no—
70C3×#121D414yes—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—21no—
77C2×#217—21no—
78C2×#217—21no—
79C2×#217—21no—
80C2×#217—21no—
81#936—22no—
82#936—22no—
83#935—28no—
84D4×Dic3—28no—
85D4×Dic3—22no—
86D4×Dic3—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2312yes—
92#924—22no—
93#924D414yes—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—44no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108#890—22no—
109#890—22no—
110#887—22no—
111#887—22no—
112#879—28no—
113#877—28no—
114C3×#107—24no—
115C3×#102—24no—
116C3×C22⋊C8—22no—
117#832—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#830—22no—
121#816—24no—
122#810—24no—
123#808—24no—
124#529—64no—
125#529—64no—
126C2×#126—61no—
127C2×#125—31no—
128C2×#124—31no—
129C2×#123—61no—
130C2×C22⋊Q8—31no—
131C2×#121C2×D311yescharacteristic
132C2×C4×Q8—31no—
133C2×C4×D4—31no—
134C2×C4×D4—61no—
135C2×#117—61no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—61no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C42—31no—
140#483—64no—
141#480—24no—
142C2×#107C2×D312yes—
143C2×#106—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102C2×D312yes—
148C2×C22⋊C8—31no—
149C2×C22⋊C8C2×D311yescharacteristic
150#389—32no—
151#389—32no—
152#389—61no—
153#389—61no—
154C2×#95—61no—
155#340—32no—
156#340—32no—
157#336—32no—
158#336—32no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×D4—24no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C22×Dic3—24no—
180C22×Dic3—24no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—41no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C22×Dic3—42no—
193C22×Dic3—42no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—21no—
201C2×Dic6C2×D412yes—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211#217—24no—
212#217—24no—
213#217—24no—
214#217—24no—
215C4⋊Dic3—24no—
216C4⋊Dic3—24no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4⋊Dic3C2×D412yes—
220C4⋊Dic3#4114yes—
221Dic3⋊C4—24no—
222Dic3⋊C4—24no—
223Dic3⋊C4—24no—
224Dic3⋊C4—44no—
225Dic3⋊C4—44no—
226C4×Dic3—24no—
227C4×Dic3—22no—
228C4×Dic3—22no—
229C4×Dic3—21no—
230C4×Dic3—21no—
231C4×Dic3—21no—
232C4×Dic3—21no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C22×Q8—31no—
237C22×D4—21no—
238C22×D4C22×D311yescharacteristic
239C23×C4—21no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—61no—
243C23×C4—61no—
244C23×C4C22×D311yescharacteristic
245C22×C8—32no—
246C22×C8C22×D312yes—
247#126—68no—
248#125—62no—
249#125—62no—
250#124—62no—
251#124—62no—
252#123—68no—
253C22⋊Q8—34no—
254C22⋊Q8—62no—
255#121—22no—
256#121#6714yes—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—62no—
260C4×D4—68no—
261C4×D4—62no—
262#117—68no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—124no—
277C2×C4⋊C4C22×D312yes—
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—31no—
288C2×C42—61no—
289C2×C42—62no—
290C2×C42—62no—
291#107—24no—
292#106—64no—
293#103—64no—
294#103—64no—
295#102—24no—
296#102—64no—
297C22⋊C8—22no—
298C22⋊C8—62no—
299#95—62no—
300#95—62no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3#12012yes—
313C23×C3#13512yes—
314C23×C3AGL(1,ℤ/8)12yes—
315C23×C3C22×D411yescharacteristic
316C3×D4—24no—
317C3×D4—48no—
318C3×D4—42no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—84no—
330C2×C3×C4#12012yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C42−1+412yes—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—44no—
350C2×Dic3—48no—
351C2×Dic3—48no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—44no—
359C2×Dic3#13712yes—
360Dic6—24no—
361Dic6—42no—
362C3×C8—42no—
363C3⋊C8—42no—
364C24—21no—
365C24D3×D411yescharacteristic
366C2×Q8—32no—
367C2×Q8—61no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4C2×#6712yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—62no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—122no—
399C22×C4—121no—
400C22×C4—122no—
401C22×C4—122no—
402C22×C4—122no—
403C22×C4D3×D411yescharacteristic
404C22×C4D3×D412yes—
405C22×C4C2×#6712yes—
406C22×C4C23×D311yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—32no—
417C4⋊C4—34no—
418C4⋊C4—42no—
419C4⋊C4—64no—
420C4⋊C4—64no—
421C4⋊C4—64no—
422C4⋊C4—64no—
423C4⋊C4—64no—
424C4⋊C4—62no—
425C4⋊C4—61no—
426C4⋊C4—124no—
427C4⋊C4—124no—
428C4⋊C4C2×#6712yes—
429C22⋊C4—24no—
430C22⋊C4—64no—
431C22⋊C4—64no—
432C22⋊C4—64no—
433C22⋊C4—64no—
434C42—64no—
435C42—62no—
436C42—62no—
437C42—61no—
438C42—61no—
439C42—61no—
440C42—61no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3#44914yes—
455C22×C3C2×#12011yescharacteristic
456C22×C3#52511yescharacteristic
457C22×C3#53611yescharacteristic
458C22×C3C2×#13511yescharacteristic
459C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
460C22×C3C2×#13711yescharacteristic
461C22×C3#57511yescharacteristic
462C3×C4—22no—
463C3×C4—42no—
464C3×C4—82no—
465C3×C4#54712yes—
466C3×C4#54712yes—
467Dic3—22no—
468Dic3—44no—
469Dic3—42no—
470Dic3—41no—
471Dic3—41no—
472Dic3—41no—
473Dic3—42no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23#91112yes—
486C23#91212yes—
487C23#93712yes—
488C23C2×D3×D411yescharacteristic
489Q8—64no—
490Q8—122no—
491D4—24no—
492D4—48no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—32no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—64no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—84no—
520C2×C4—124no—
521C2×C4—128no—
522C2×C4—128no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—124no—
526C2×C4—122no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4#93712yes—
531C2×C4C2×D3×D411yescharacteristic
532C2×C4D3×#4112yes—
533C2×C4#101012yes—
534C2×C4C22×#6711yescharacteristic
535C8—42no—
536C8—122no—
537C2×C3—24no—
538C2×C3—82no—
539C2×C3#240611yescharacteristic
540C2×C3C2×#44911yescharacteristic
541C2×C3#300711yescharacteristic
542C2×C3#310811yescharacteristic
543C2×C3#311411yescharacteristic
544C2×C3#323211yescharacteristic
545C2×C3#323811yescharacteristic
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#515814yes—
560C22#564811yescharacteristic
561C22#565611yescharacteristic
562C22C2×#91111yescharacteristic
563C22C2×#91211yescharacteristic
564C22D3×#13711yescharacteristic
565C22#577711yescharacteristic
566C22C2×#93711yescharacteristic
567C4—22no—
568C4—42no—
569C4—62no—
570C4—82no—
571C4—121no—
572C4—122no—
573C4—124no—
574C4—121no—
575C4—121no—
576C4—122no—
577C4#565512yes—
578C4#582412yes—
579C3#2406611yescharacteristic, complemented, Sylow
580C2—24no—
581C2—82no—
582C2#8126911yescharacteristic
583C2#8127111yescharacteristic
584C2#8127511yescharacteristic
585C2#8127711yescharacteristic
586C2#8284111yescharacteristic
587C2C2×#515811yescharacteristic
588C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#480C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#480D3
hypocenterC3#24066
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
3. C2×#4972C2semidirect
4. C2×#4971C2semidirect
6. #70642C2semidirect
7. #70618C2semidirect
8. #70216C2semidirect
9. #70169C2semidirect
10. #69930C2semidirect
11. #69910C2semidirect
579. C3#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#56002
C2×C3×#4802
C2×#49722
C2×#49712
#716772
#706422
#706182
#702162
#701692
#699302
#699102
#240663