Finite Group Databasebeta

← #10900976

Subgroups — #10900976

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#4888C212yesmaximal
3#76287C211yesmaximal, characteristic
4C3×#2057C211yesmaximal, characteristic, complemented
5#71794C211yesmaximal, characteristic, complemented
6#70618C212yesmaximal, complemented
7#69605C211yesmaximal, characteristic, complemented
8#68911C212yesmaximal, complemented
9#68873C212yesmaximal
10#68689C211yesmaximal, characteristic, complemented
11#68664C211yesmaximal, characteristic
12#29645—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—21no—
15C2×Q8×Dic3—21no—
16#5728—28no—
17C2×#897—21no—
18C2×#897C2211yescharacteristic
19C2×#892C2212yes—
20C2×#890C2212yes—
21C2×#887—21no—
22C2×C42×D3C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C3×#107C2212yes—
25C3×#389C2211yescharacteristic
26C2×(C2×D3)⋊C8C2212yes—
27C2×#816C2212yes—
28#4989—21no—
29#4987C2211yescharacteristic
30#4980—21no—
31#4973C2211yescharacteristic
32#4972C2211yescharacteristic
33C2×#810C2212yes—
34C2×#808C2212yes—
35#4929C2211yescharacteristic
36#4888—28no—
37#4664—21no—
38#4655—21no—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#485—32no—
48#2855—31no—
49#2059—32no—
50#2058—31no—
51#2057D311yescharacteristic, complemented
52#2026—32no—
53#2024—31no—
54#2024—31no—
55#1825—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×Dic6—21no—
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71#946—28no—
72Q8×Dic3—28no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—21no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88#897—28no—
89#897—24no—
90#892—24no—
91#892D418yes—
92#890—24no—
93#890—24no—
94#887—28no—
95C42×D3—24no—
96#869—24no—
97C3×#107—24no—
98C3×#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105(C2×D3)⋊C8—24no—
106#816—24no—
107#810—24no—
108#808—24no—
109#533—68no—
110C2×C4⋊Q8—31no—
111C2×C4⋊Q8C2×D311yescharacteristic
112C2×#125—32no—
113C2×C22⋊Q8—31no—
114C2×C22⋊Q8—61no—
115C2×C22⋊Q8—61no—
116C2×C4×Q8—61no—
117C2×C4×Q8—61no—
118C2×#117—32no—
119C22×C42—31no—
120#485—68no—
121C2×#107C2×D312yes—
122C2×#106—32no—
123C2×#103—32no—
124C2×#103—32no—
125C2×C22⋊C8—32no—
126#391—31no—
127#391—31no—
128#389—31no—
129#389C2×D311yescharacteristic
130#387—61no—
131#387—61no—
132#385—61no—
133#381—61no—
134#380—31no—
135#340—32no—
136#340—32no—
137#340—32no—
138#340C2×D312yes—
139#336—32no—
140#336—31no—
141#336C2×D311yescharacteristic
142C23×D3C2×D411yescharacteristic
143C2×C3×Q8—24no—
144C2×C3×Q8—42no—
145C2×C3×Q8—42no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—22no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3C2×D411yescharacteristic
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158C2×C4×D3—21no—
159C2×C4×D3—22no—
160C2×C4×D3—21no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—41no—
166C2×C4×D3—41no—
167C2×C4×D3C2×D414yes—
168C2×Dic6—24no—
169C2×Dic6—22no—
170C2×Dic6—21no—
171C2×Dic6—22no—
172C2×Dic6—21no—
173C2×Dic6—42no—
174C2×Dic6—42no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×D414yes—
185C3×C42—22no—
186(C2×D3)⋊C4—28no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—24no—
190C4⋊Dic3—28no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3C2×D414yes—
195Dic3⋊C4—28no—
196Dic3⋊C4—42no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—28no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—21no—
205C22×Q8—31no—
206C22×Q8—61no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—31no—
210C22×C8—32no—
211C22×C8C22×D312yes—
212C4⋊Q8—24no—
213C4⋊Q8—64no—
214#125—64no—
215#125—64no—
216C22⋊Q8—68no—
217C22⋊Q8—68no—
218C22⋊Q8—64no—
219C4×Q8—68no—
220C4×Q8—68no—
221#117—38no—
222#117—64no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—32no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C4⋊C4C22×D312yes—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C42—31no—
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—64no—
242C2×C42—62no—
243C2×C42C22×D311yescharacteristic
244#107—24no—
245#106—64no—
246#103—64no—
247#103—64no—
248C22⋊C8—64no—
249#95—21no—
250#95—61no—
251#95—61no—
252#95—61no—
253#95—61no—
254#95—61no—
255#95—61no—
256C23×C3C22×D411yescharacteristic
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—21no—
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—21no—
263C22×D3#12012yes—
264C3×Q8—48no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—24no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4#13512yes—
279C2×C3×C4#13712yes—
280C2×C3×C4C22×D412yes—
281C2×C3×C42−1+411yescharacteristic
282C2×C3×C42−1+412yes—
283C2×C3×C42−1+411yescharacteristic
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—24no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3#12012yes—
302C4×D3—24no—
303C4×D3—24no—
304C4×D3—44no—
305C4×D3—44no—
306Dic6—48no—
307Dic6—42no—
308Dic6—42no—
309C3×C8—42no—
310C3⋊C8—42no—
311C24—31no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—64no—
316C2×Q8—62no—
317C2×Q8—62no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8—122no—
321C2×Q8—122no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—31no—
326C22×C4—34no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4C23×D311yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—34no—
356C4⋊C4—42no—
357C4⋊C4—68no—
358C4⋊C4—68no—
359C4⋊C4—64no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—122no—
363C4⋊C4—122no—
364C4⋊C4D3×D414yes—
365C22⋊C4—68no—
366C22⋊C4—122no—
367C22⋊C4—122no—
368C42—22no—
369C42—62no—
370C42—62no—
371C42—68no—
372C42—62no—
373C22×C3C2×#12011yescharacteristic
374C22×C3#53611yescharacteristic
375C22×C3#53611yescharacteristic
376C22×C3C2×#13511yescharacteristic
377C22×C3C2×#13711yescharacteristic
378C22×C3#57511yescharacteristic
379C22×C3#57511yescharacteristic
380C2×D3—22no—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—41no—
384C2×D3—41no—
385C2×D3—41no—
386C2×D3—41no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—44no—
390C3×C4—81no—
391C3×C4—81no—
392C3×C4#53612yes—
393C3×C4#53612yes—
394Dic3—22no—
395Dic3—41no—
396Dic3—44no—
397Dic3—41no—
398Dic3—81no—
399Dic3—81no—
400C23—32no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23C2×D3×D411yescharacteristic
408Q8—48no—
409Q8—128no—
410Q8—122no—
411Q8—122no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—32no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—64no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4#91212yes—
448C2×C4#91812yes—
449C2×C4C2×D3×D412yes—
450C2×C4#100711yescharacteristic
451C2×C4#100712yes—
452C2×C4#101011yescharacteristic
453C8—42no—
454C8—122no—
455C2×C3#261411yescharacteristic
456C2×C3#300211yescharacteristic
457C2×C3#300711yescharacteristic
458C2×C3#323111yescharacteristic
459C2×C3#323311yescharacteristic
460C2×C3#323711yescharacteristic
461C2×C3#324011yescharacteristic
462D3—42no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22D3×#12011yescharacteristic
471C22#562811yescharacteristic
472C22#572411yescharacteristic
473C22C2×#91211yescharacteristic
474C22C2×#91811yescharacteristic
475C22#577311yescharacteristic
476C22#577711yescharacteristic
477C4—22no—
478C4—22no—
479C4—44no—
480C4—62no—
481C4—81no—
482C4—81no—
483C4—121no—
484C4—124no—
485C4—121no—
486C4—241no—
487C4—241no—
488C4#562812yes—
489C4#572412yes—
490C3#2964511yescharacteristic, complemented, Sylow
491C2—122no—
492C2#7725611yescharacteristic
493C2#8000111yescharacteristic
494C2#8018411yescharacteristic
495C2#8097011yescharacteristic
496C2#8111411yescharacteristic
497C2#8194411yescharacteristic
498C2#8198411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2057C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2057D3
hypocenterC3#29645
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
4. C3×#2057C2semidirect
5. #71794C2semidirect
6. #70618C2semidirect
7. #69605C2semidirect
8. #68911C2semidirect
10. #68689C2semidirect
51. #2057D3semidirect
490. C3#29645semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#48882
#762872
C3×#20572
#717942
#706182
#696052
#689112
#688732
#686892
#686642
#296453