Finite Group Databasebeta

← #10900963

Subgroups — #10900963

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×#4784C212yesmaximal
3#76475C211yesmaximal, characteristic
4C3×#2023C211yesmaximal, characteristic, complemented
5#71783C211yesmaximal, characteristic, complemented
6#70164C212yesmaximal, complemented
7#69638C211yesmaximal, characteristic, complemented
8#68873C212yesmaximal, complemented
9#68829C212yesmaximal, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68632C211yesmaximal, characteristic, complemented
12#29644—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—21no—
15C2×Q8×Dic3—21no—
16C2×#938C2211yescharacteristic
17#5728—28no—
18C2×#897—21no—
19C2×#892C2212yes—
20C2×#890C2212yes—
21C2×#887—21no—
22C2×#883—21no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#102C2212yes—
26C3×#390C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#818C2212yes—
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33C2×#808C2212yes—
34#4956—21no—
35#4951—21no—
36#4945—21no—
37#4928C2211yescharacteristic
38#4784—28no—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#482—32no—
48#2854—31no—
49#2059—32no—
50#2058—31no—
51#2057—31no—
52#2025—32no—
53#2023—31no—
54#2023D311yescharacteristic, complemented
55#1825—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C22×#67—21no—
60C22×C4×D3D411yescharacteristic
61C22×C4×D3D411yescharacteristic
62C22×C4×D3C2311yescharacteristic
63C22×Dic6—21no—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—21no—
73C2×#217—21no—
74C2×#217—21no—
75#946—28no—
76Q8×Dic3—28no—
77#938—24no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—21no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#897—28no—
94#892—24no—
95#892D418yes—
96#890—24no—
97#890—24no—
98#887—28no—
99#883—44no—
100C42×D3—24no—
101#869—24no—
102C3×#102—24no—
103#832—24no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107(C2×D3)⋊C8—24no—
108#818—24no—
109#808—24no—
110#533—68no—
111C2×C4⋊Q8—31no—
112C2×C4⋊Q8C2×D311yescharacteristic
113C2×#125—32no—
114C2×#123—61no—
115C2×C22⋊Q8—61no—
116C2×C22⋊Q8—61no—
117C2×#121—31no—
118C2×C4×Q8—61no—
119C2×C4×Q8—61no—
120C2×#117—32no—
121C22×C42—31no—
122#482—68no—
123C2×#107—32no—
124C2×#106—32no—
125C2×#102—32no—
126C2×#102C2×D312yes—
127C2×C22⋊C8—32no—
128#390—31no—
129#390C2×D311yescharacteristic
130#389—31no—
131#389—31no—
132#388—61no—
133#388—61no—
134#386—61no—
135#380—31no—
136#340—32no—
137#340—32no—
138#340—32no—
139#340C2×D312yes—
140#336—32no—
141#336—31no—
142#336C2×D311yescharacteristic
143C24×C3—21no—
144C23×D3C2×D411yescharacteristic
145C2×C3×Q8—24no—
146C2×C3×Q8—42no—
147C2×C3×Q8—42no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C2×#67—42no—
158C2×#67—42no—
159C2×#67—42no—
160C22×Dic3—22no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166C2×C4×D3—21no—
167C2×C4×D3—22no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—41no—
174C2×C4×D3—41no—
175C2×C4×D3C2×D414yes—
176C2×Dic6—24no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×D414yes—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C42—22no—
189#217—42no—
190#217—42no—
191#217—42no—
192(C2×D3)⋊C4—28no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—24no—
195C4⋊Dic3—28no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—42no—
201C4⋊Dic3C2×D414yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—42no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C4×Dic3—28no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—21no—
212C22×Q8—61no—
213C22×D4—61no—
214C22×D4C22×D311yescharacteristic
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—31no—
218C22×C8—32no—
219C22×C8C22×D312yes—
220C4⋊Q8—24no—
221C4⋊Q8—64no—
222#125—64no—
223#125—64no—
224#123—124no—
225C22⋊Q8—68no—
226C22⋊Q8—68no—
227#121—64no—
228C4×Q8—68no—
229C4×Q8—68no—
230#117—38no—
231#117—64no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C42—31no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—64no—
254C2×C42—62no—
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#102—24no—
259#102—64no—
260C22⋊C8—64no—
261#95—61no—
262#95—61no—
263#95—61no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3C22×D411yescharacteristic
272C22×D3—21no—
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—21no—
278C22×D3#12012yes—
279C3×Q8—48no—
280C3×D4—42no—
281C3×D4—42no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4#13512yes—
293C2×C3×C4AGL(1,ℤ/8)12yes—
294C2×C3×C4C22×D412yes—
295C2×C3×C42−1+411yescharacteristic
296C2×C3×C42−1+412yes—
297C2×C3×C42−1+411yescharacteristic
298#67—84no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—24no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3#12012yes—
317C4×D3—24no—
318C4×D3—24no—
319C4×D3—44no—
320C4×D3—44no—
321Dic6—48no—
322C3×C8—42no—
323C3⋊C8—42no—
324C24—21no—
325C24—31no—
326C2×Q8—24no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—64no—
330C2×Q8—122no—
331C2×Q8—122no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—122no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—31no—
342C22×C4—34no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—34no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—64no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—122no—
378C4⋊C4—122no—
379C4⋊C4D3×D414yes—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—68no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C42—22no—
388C42—62no—
389C42—62no—
390C42—68no—
391C42—62no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#53611yescharacteristic
401C22×C3#53611yescharacteristic
402C22×C3C2×#13511yescharacteristic
403C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C3#57511yescharacteristic
405C22×C3#57511yescharacteristic
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—41no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—41no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—44no—
416C3×C4—81no—
417C3×C4#53612yes—
418C3×C4#53612yes—
419Dic3—22no—
420Dic3—41no—
421Dic3—44no—
422Dic3—41no—
423Dic3—81no—
424Dic3—81no—
425C23—32no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—61no—
434C23—61no—
435C23—61no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23C2×D3×D411yescharacteristic
440Q8—48no—
441Q8—128no—
442D4—42no—
443D4—42no—
444D4—244no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—32no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—64no—
467C2×C4—124no—
468C2×C4—124no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4#91212yes—
478C2×C4#91412yes—
479C2×C4C2×D3×D412yes—
480C2×C4#100711yescharacteristic
481C2×C4#101011yescharacteristic
482C2×C4#101012yes—
483C8—42no—
484C8—122no—
485C2×C3—161no—
486C2×C3#261411yescharacteristic
487C2×C3#300211yescharacteristic
488C2×C3#300611yescharacteristic
489C2×C3#322811yescharacteristic
490C2×C3#323311yescharacteristic
491C2×C3#323511yescharacteristic
492C2×C3#324011yescharacteristic
493D3—42no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22D3×#12011yescharacteristic
509C22#562811yescharacteristic
510C22#567511yescharacteristic
511C22C2×#91211yescharacteristic
512C22C2×#91411yescharacteristic
513C22#577311yescharacteristic
514C22#577711yescharacteristic
515C4—22no—
516C4—22no—
517C4—44no—
518C4—62no—
519C4—81no—
520C4—121no—
521C4—121no—
522C4—124no—
523C4—241no—
524C4—241no—
525C4#562812yes—
526C4#567512yes—
527C3#2964411yescharacteristic, complemented, Sylow
528C2—122no—
529C2—161no—
530C2#7687011yescharacteristic
531C2#8000111yescharacteristic
532C2#8004011yescharacteristic
533C2#8097011yescharacteristic
534C2#8106611yescharacteristic
535C2#8157311yescharacteristic
536C2#8169211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2023D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#5728C2semidirect
4. C3×#2023C2semidirect
5. #71783C2semidirect
6. #70164C2semidirect
7. #69638C2semidirect
8. #68873C2semidirect
9. #68829C2semidirect
10. #68664C2semidirect
11. #68632C2semidirect
54. #2023D3semidirect
527. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#47842
#764752
C3×#20232
#717832
#701642
#696382
#688732
#688292
#686642
#686322
#296443