Finite Group Databasebeta

← #10901359

Subgroups — #10901359

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#4791C212yesmaximal, complemented
3#76288C211yesmaximal, characteristic
4C3×#2043C211yesmaximal, characteristic, complemented
5#71781C211yesmaximal, characteristic, complemented
6#70274C212yesmaximal
7#70243C212yesmaximal
8#70221C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#68662C211yesmaximal, characteristic
11#68633C211yesmaximal, characteristic, complemented
12#29688—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16#5728—28no—
17C2×#897—21no—
18C2×#896—21no—
19C2×#892—21no—
20C2×#890—21no—
21C2×#888C2211yescharacteristic
22C2×#887—21no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C3×#389C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#817C2212yes—
32C2×Dic3⋊C8C2212yes—
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4976—21no—
36#4975C2211yescharacteristic
37C2×#808C2212yes—
38#4928C2211yescharacteristic
39#4791—28no—
40#4667—21no—
41#4646—21no—
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#475—32no—
48#2857—31no—
49#2058—31no—
50#2058—31no—
51#2043—31no—
52#2043D311yescharacteristic, complemented
53#2035—32no—
54#2035—32no—
55#1803—32no—
56#1797—31no—
57C22×C3×Q8C2312yes—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×Dic6—21no—
62C22×Dic6—21no—
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71#946—24no—
72#946—24no—
73Q8×Dic3—24no—
74Q8×Dic3D418yes—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93#897—28no—
94#896—44no—
95#892—28no—
96#890—28no—
97#888—24no—
98#887—28no—
99C42×D3—24no—
100#869—24no—
101C3×#103—24no—
102C3×#103—42no—
103C3×#95—21no—
104#835—24no—
105#835—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109(C2×D3)⋊C8—42no—
110#817—24no—
111Dic3⋊C8—24no—
112#808—24no—
113#533—68no—
114C2×C4⋊Q8—31no—
115C2×C4⋊Q8—31no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125—61no—
118C2×C22⋊Q8—32no—
119C2×C22⋊Q8—61no—
120C2×C22⋊Q8—61no—
121C2×C4×Q8—32no—
122C2×C4×Q8—61no—
123C2×#117—61no—
124C22×C42—31no—
125#475—68no—
126C2×#106—32no—
127C2×#106—32no—
128C2×C4⋊C8—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#103C2×D312yes—
132C2×#103C2×D312yes—
133C2×C22⋊C8—32no—
134#391—61no—
135#389—31no—
136#389—61no—
137#389C2×D311yescharacteristic
138#384—31no—
139#380—31no—
140#380—31no—
141#380—61no—
142#340—32no—
143#340—32no—
144#336—31no—
145#336C2×D311yescharacteristic
146C23×D3C2×D411yescharacteristic
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D414yes—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—41no—
174C2×C4×D3SD1614yes—
175C2×Dic6—24no—
176C2×Dic6—42no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C2×Dic6—42no—
180C2×Dic6—42no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—42no—
190C3×C42—22no—
191(C2×D3)⋊C4—28no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—24no—
195C4⋊Dic3—22no—
196C4⋊Dic3—28no—
197C4⋊Dic3—22no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3C2×D414yes—
201Dic3⋊C4—28no—
202Dic3⋊C4—42no—
203Dic3⋊C4—42no—
204Dic3⋊C4—42no—
205Dic3⋊C4—42no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C4×Dic3—28no—
209C4×Dic3C2×D414yes—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C22×Q8—61no—
214C22×Q8—61no—
215C22×Q8C22×D312yes—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—31no—
219C22×C8—32no—
220C22×C8C22×D312yes—
221C4⋊Q8—24no—
222C4⋊Q8—64no—
223C4⋊Q8—64no—
224#125—68no—
225C22⋊Q8—68no—
226C22⋊Q8—64no—
227C22⋊Q8—64no—
228C22⋊Q8—124no—
229C4×Q8—38no—
230C4×Q8—64no—
231C4×Q8—68no—
232#117—68no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4C22×D311yescharacteristic
247C2×C4⋊C4C22×D311yescharacteristic
248C2×C22⋊C4—62no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—61no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—31no—
254C2×C42—64no—
255C2×C42—62no—
256C2×C42C22×D311yescharacteristic
257#106—64no—
258#106—64no—
259C4⋊C8—64no—
260#103—24no—
261#103—42no—
262#103—64no—
263#103—122no—
264C22⋊C8—122no—
265#95—21no—
266#95—61no—
267#95—61no—
268#95—61no—
269C23×C3C22×D411yescharacteristic
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—21no—
276C22×D3C2×SD1612yes—
277C3×Q8—24no—
278C3×Q8—24no—
279C3×Q8—42no—
280C3×Q8—42no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4C2×SD1612yes—
295C2×C3×C4#13712yes—
296C2×C3×C4C22×D412yes—
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299C2×C3×C42−1+412yes—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—24no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3#12012yes—
315C2×Dic3#12012yes—
316C2×Dic3C2×SD1612yes—
317C2×Dic3#13712yes—
318C4×D3—28no—
319C4×D3—44no—
320C4×D3—42no—
321C4×D3—42no—
322Dic6—48no—
323Dic6—84no—
324C3×C8—42no—
325C3⋊C8—42no—
326C24—31no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—64no—
333C2×Q8—122no—
334C2×Q8—122no—
335C2×Q8—122no—
336C2×Q8—122no—
337C2×Q8—122no—
338C2×Q8D3×D414yes—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—34no—
345C22×C4—31no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—121no—
358C22×C4D3×D411yescharacteristic
359C22×C4D3×D411yescharacteristic
360C22×C4C23×D311yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—62no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—34no—
373C4⋊C4—42no—
374C4⋊C4—68no—
375C4⋊C4—64no—
376C4⋊C4—68no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—122no—
382C4⋊C4—122no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C22⋊C4—68no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C42—22no—
389C42—34no—
390C42—62no—
391C42—62no—
392C42—68no—
393C22×C3C2×#12011yescharacteristic
394C22×C3#53611yescharacteristic
395C22×C3#53711yescharacteristic
396C22×C3C22×SD1611yescharacteristic
397C22×C3C2×#13711yescharacteristic
398C22×C3C2×#13711yescharacteristic
399C22×C3#57511yescharacteristic
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—41no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—44no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4#53612yes—
413C3×C4#53712yes—
414Dic3—22no—
415Dic3—41no—
416Dic3—44no—
417Dic3—81no—
418Dic3—81no—
419Dic3#44814yes—
420C23—32no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23—61no—
427C23C2×D3×D411yescharacteristic
428Q8—24no—
429Q8—24no—
430Q8—42no—
431Q8—42no—
432Q8—128no—
433Q8—244no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—68no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—64no—
457C2×C4—61no—
458C2×C4—124no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4D3×SD1612yes—
470C2×C4#91812yes—
471C2×C4C2×D3×D412yes—
472C2×C4#100411yescharacteristic
473C2×C4#101011yescharacteristic
474C2×C4#101012yes—
475C8—42no—
476C8—122no—
477C2×C3#261611yescharacteristic
478C2×C3C2×#44811yescharacteristic
479C2×C3#300711yescharacteristic
480C2×C3#322911yescharacteristic
481C2×C3#323711yescharacteristic
482C2×C3#324611yescharacteristic
483C2×C3#325511yescharacteristic
484D3—42no—
485C22—62no—
486C22—62no—
487C22—62no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22D3×#12011yescharacteristic
493C22#560811yescharacteristic
494C22#567511yescharacteristic
495C22C2×D3×SD1611yescharacteristic
496C22C2×#91811yescharacteristic
497C22D3×#13711yescharacteristic
498C22#577711yescharacteristic
499C4—22no—
500C4—22no—
501C4—34no—
502C4—44no—
503C4—62no—
504C4—81no—
505C4—81no—
506C4—124no—
507C4—121no—
508C4—241no—
509C4—241no—
510C4#559712yes—
511C4#564812yes—
512C3#2968811yescharacteristic, complemented, Sylow
513C2—122no—
514C2#7697011yescharacteristic
515C2D3×#44811yescharacteristic
516C2#8018411yescharacteristic
517C2#8043811yescharacteristic
518C2#8048711yescharacteristic
519C2#8160711yescharacteristic
520C2#8174611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2043C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2043D3
hypocenterC3#29688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
2. C2×#4791C2semidirect
4. C3×#2043C2semidirect
5. #71781C2semidirect
8. #70221C2semidirect
9. #69605C2semidirect
11. #68633C2semidirect
52. #2043D3semidirect
512. C3#29688semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#47912
#762882
C3×#20432
#717812
#702742
#702432
#702212
#696052
#686622
#686332
#296883