Finite Group Databasebeta

← #10945021

Subgroups — #10945021

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic
3#82263C211yesmaximal, characteristic, complemented
4C2×#4971C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic
6#71632C212yesmaximal, complemented
7#71629C212yesmaximal, complemented
8#71612C212yesmaximal, complemented
9#71607C212yesmaximal, complemented
10#71584C211yesmaximal, characteristic, complemented
11#71554C211yesmaximal, characteristic, complemented
12#33016—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×#520—24no—
15C3×#520—24no—
16C2×C3×C4×Q8C2212yes—
17C2×C3×C4×D4—22no—
18C2×C3×C4×D4C2212yes—
19C2×C3×#117—21no—
20C2×C3×#117C2211yescharacteristic
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C22×#214C2211yescharacteristic
25C2×#924C2211yescharacteristic
26#5239—24no—
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2211yescharacteristic
30C2×#832C2212yes—
31#5212C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#4985—21no—
35#4983—21no—
36#4971—24no—
37C2×#810C2212yes—
38C2×#808C2212yes—
39#4952—22no—
40#4929C2212yes—
41#4928C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#520D311yescharacteristic
47C2×#483—31no—
48C2×#452—31no—
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1865—32no—
53#1864—32no—
54#1824—31no—
55#1818—31no—
56#1788—31no—
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4—22no—
67C23×C3×C4C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×D4—216no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×#117—28no—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4—28no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D414yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—22no—
88C2×C3×C42C2312yes—
89C2×#217—21no—
90C2×#217—21no—
91C2×#216—22no—
92C2×#216—24no—
93C2×#214—22no—
94C2×#214—24no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97#924—22no—
98#924—22no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—22no—
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104#835—22no—
105#835—24no—
106#832—22no—
107#832—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112#830—22no—
113#830—22no—
114#810—24no—
115#808—24no—
116C22×#41C2×D311yescharacteristic
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8—31no—
120#520—24no—
121#520—24no—
122C2×C4×Q8C2×D312yes—
123C2×C4×D4—22no—
124C2×C4×D4C2×D312yes—
125C2×#117—21no—
126C2×#117C2×D311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2×D311yescharacteristic
129#483—64no—
130#452—64no—
131C2×#106—32no—
132C2×#103—31no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#102—31no—
136C2×#102—32no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—31no—
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C3×#41—216no—
152C3×#41—28no—
153C3×#41—28no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D412yes—
157C2×C3×Q8#4114yes—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4C2×D412yes—
163C2×C3×D4#4114yes—
164C22×C3×C4—28no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—28no—
171C22×C3×C4—24no—
172C22×C3×C4—84no—
173C22×C3×C4—82no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2411yescharacteristic
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D414yes—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—24no—
195C3×C42—28no—
196C3×C42—22no—
197C3×C42—22no—
198#217—42no—
199#217—81no—
200#216—44no—
201#214—44no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206Dic3⋊C4—42no—
207Dic3⋊C4—82no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—22no—
218C2×#41#6714yes—
219C22×Q8—31no—
220C22×Q8C22×D311yescharacteristic
221C22×D4—21no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4C22×D311yescharacteristic
227C2×Q16—64no—
228C2×Q16—62no—
229C2×SD16—64no—
230C2×SD16—62no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8—32no—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×D4—216no—
239C4×D4—24no—
240C4×D4—24no—
241#117—28no—
242#117—22no—
243#117—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4#6711yescharacteristic
253C2×C4⋊C4#6711yescharacteristic
254C2×C4⋊C4#6714yes—
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4#6712yes—
260C2×C42—22no—
261C2×C42C22×D312yes—
262#106—64no—
263#103—64no—
264#103—62no—
265#103—64no—
266#102—64no—
267#102—62no—
268C22⋊C8—62no—
269C22⋊C8—62no—
270#95—61no—
271#95—62no—
272#95—62no—
273#95—62no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—24no—
277C23×C3—21no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3#12012yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8#12314yes—
288C3×D4—216no—
289C3×D4—22no—
290C3×D4—24no—
291C3×D4#12314yes—
292C2×C3×C4—216no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—44no—
306C2×C3×C4—84no—
307C2×C3×C4—88no—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C4C2×#4112yes—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321Dic6—41no—
322Dic6—41no—
323Dic6—42no—
324C3⋊C8—41no—
325C3⋊C8—41no—
326C3⋊C8—42no—
327C24—21no—
328C24C2×#6711yescharacteristic
329#41—216no—
330#41—28no—
331#41—28no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8#23514yes—
339C2×Q8C2×#6712yes—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4#23514yes—
345C2×D4C2×#6712yes—
346C22×C4—28no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—84no—
358C22×C4—82no—
359C22×C4C2×#6712yes—
360C22×C4C2×#6712yes—
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6712yes—
363C22×C4C2×#6712yes—
364C22×C4C23×D311yescharacteristic
365Q16—124no—
366SD16—124no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—61no—
372C2×C8—62no—
373C2×C8—61no—
374C4⋊C4—28no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—122no—
384C4⋊C4—242no—
385C4⋊C4C2×#6714yes—
386C22⋊C4—28no—
387C22⋊C4—24no—
388C22⋊C4—122no—
389C22⋊C4—241no—
390C42—28no—
391C42—22no—
392C42—22no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#52512yes—
405C22×C3#57712yes—
406C22×C3#57711yescharacteristic
407C22×C3#57811yescharacteristic
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—42no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4C2×#12311yescharacteristic
414C3×C4C2×#12311yescharacteristic
415C3×C4#52512yes—
416Dic3—81no—
417Dic3—82no—
418Dic3—81no—
419C23—22no—
420C23—24no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23#95312yes—
429C23C22×#6711yescharacteristic
430Q8—22no—
431Q8—24no—
432Q8—121no—
433Q8—121no—
434Q8—122no—
435Q8#92914yes—
436D4—216no—
437D4—22no—
438D4—24no—
439D4#92914yes—
440C2×C4—216no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—44no—
454C2×C4—84no—
455C2×C4—88no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4#95312yes—
465C2×C4C2×#23512yes—
466C2×C4C2×#23512yes—
467C2×C4C22×#6711yescharacteristic
468C2×C4C22×#6711yescharacteristic
469C8—121no—
470C8—121no—
471C8—122no—
472C2×C3—42no—
473C2×C3—44no—
474C2×C3#284911yescharacteristic
475C2×C3#300911yescharacteristic
476C2×C3#301311yescharacteristic
477C2×C3#312012yes—
478C2×C3#312512yes—
479C22—22no—
480C22—24no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—44no—
488C22—42no—
489C22#579112yes—
490C22#583612yes—
491C22#583711yescharacteristic
492C22#583811yescharacteristic
493C22C2×#95311yescharacteristic
494C4—22no—
495C4—24no—
496C4—42no—
497C4—44no—
498C4—42no—
499C4—241no—
500C4—242no—
501C4—241no—
502C4C2×#92911yescharacteristic
503C4C2×#92911yescharacteristic
504C4#579112yes—
505C3#3301611yescharacteristic, complemented, Sylow
506C2—42no—
507C2—44no—
508C2#8257512yes—
509C2#8257612yes—
510C2#8403311yescharacteristic
511C2#8410511yescharacteristic
512C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#520C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#520D3
hypocenterC3#33016
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #82263C2semidirect
4. C2×#4971C2semidirect
6. #71632C2semidirect
7. #71629C2semidirect
8. #71612C2semidirect
9. #71607C2semidirect
10. #71584C2semidirect
11. #71554C2semidirect
505. C3#33016semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×#52392
#822632
C2×#49712
#721352
#716322
#716292
#716122
#716072
#715842
#715542
#330163