Finite Group Databasebeta

← #10847399

Subgroups — #10847399

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#5188C212yesmaximal
3#76663C211yesmaximal, characteristic
4C3×#2016C211yesmaximal, characteristic, complemented
5#72065C212yesmaximal, complemented
6#71781C211yesmaximal, characteristic, complemented
7#70270C212yesmaximal
8#70236C212yesmaximal, complemented
9#69668C211yesmaximal, characteristic, complemented
10D3×#336C211yesmaximal, characteristic
11#68632C211yesmaximal, characteristic, complemented
12#25498—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#946C2211yescharacteristic
16C2×#946C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×D3×C8C2212yes—
19#5728—28no—
20C2×#897—21no—
21C2×#892—21no—
22C2×#890—21no—
23C2×#887—21no—
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#103C2212yes—
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5188—28no—
33C2×#818C2212yes—
34#4987C2211yescharacteristic
35#4984C2211yescharacteristic
36#4928C2211yescharacteristic
37#4668—21no—
38#4664—21no—
39#4654—21no—
40#4652—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#467—32no—
48#2855—31no—
49#2057—31no—
50#2043—31no—
51#2024—31no—
52#2016D311yescharacteristic, complemented
53#1892—32no—
54#1878—32no—
55#1863—32no—
56C2×#336—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×Dic6—21no—
63C2×C3×Q16—24no—
64C2×C3×Q16—24no—
65C2×C3×Q16—42no—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—24no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72#946—24no—
73#946—24no—
74#946D418yes—
75Q8×Dic3—24no—
76Q8×Dic3—24no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
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×Dic3⋊C4—21no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C2×D3×C8—24no—
95C2×D3×C8—24no—
96C2×D3×C8—42no—
97#897—28no—
98#892—28no—
99#890—28no—
100#887—28no—
101C42×D3—24no—
102#869—24no—
103C3×#103—24no—
104C3×#95—21no—
105C3×#95—21no—
106#835—24no—
107#835—24no—
108#835—42no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#818—24no—
115C22×Q16C2×D312yes—
116C23×C8—32no—
117#533—68no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×#125—61no—
121C2×C22⋊Q8—32no—
122C2×C22⋊Q8—31no—
123C2×C22⋊Q8—61no—
124C2×C4×Q8—32no—
125C2×C4×Q8—61no—
126C2×#117—61no—
127C22×C42—31no—
128#467—68no—
129C2×#107—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103C2×D312yes—
134#391—31no—
135#391C2×D311yescharacteristic
136#389—31no—
137#389—31no—
138#387—61no—
139#387—61no—
140#385—61no—
141#381—61no—
142#380—31no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—31no—
147#336C2×D311yescharacteristic
148C23×D3C2×D411yescharacteristic
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8C2×D414yes—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3C2×D411yescharacteristic
169C22×Dic3C2×D411yescharacteristic
170C22×Dic3C2×D411yescharacteristic
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—41no—
179C2×C4×D3Q1614yescomplemented
180C2×C4×D3C2×D414yes—
181C2×Dic6—24no—
182C2×Dic6—42no—
183C2×Dic6—42no—
184C3×Q16—48no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8#4114yes—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C42—22no—
192(C2×D3)⋊C4—28no—
193(C2×D3)⋊C4—42no—
194(C2×D3)⋊C4—42no—
195C4⋊Dic3—24no—
196C4⋊Dic3—28no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—42no—
202C4⋊Dic3C2×D414yes—
203Dic3⋊C4—28no—
204Dic3⋊C4—42no—
205Dic3⋊C4—42no—
206C4×Dic3—28no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213D3×C8—48no—
214C22×Q8—61no—
215C22×Q8C22×D311yescharacteristic
216C22×Q8C22×D312yes—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—31no—
220C2×Q16—24no—
221C2×Q16—24no—
222C2×Q16—42no—
223C22×C8—32no—
224C22×C8—64no—
225C22×C8—64no—
226C22×C8—122no—
227C22×C8C22×D312yes—
228C4⋊Q8—24no—
229C4⋊Q8—64no—
230#125—68no—
231C22⋊Q8—38no—
232C22⋊Q8—68no—
233C22⋊Q8—64no—
234C22⋊Q8—64no—
235C4×Q8—68no—
236C4×Q8—64no—
237C4×Q8—64no—
238#117—68no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C22⋊C4—62no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C42—31no—
255C2×C42—31no—
256C2×C42—31no—
257C2×C42—64no—
258C2×C42—62no—
259C2×C42C22×D311yescharacteristic
260#107—64no—
261#103—24no—
262#103—64no—
263#103—64no—
264#103—122no—
265#95—21no—
266#95—21no—
267#95—61no—
268#95—61no—
269#95—61no—
270#95—61no—
271#95—61no—
272C23×C3C22×D411yescharacteristic
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3#12012yes—
278C22×D3C2×Q1612yes—
279C22×D3#13712yes—
280C3×Q8—24no—
281C3×Q8—24no—
282C3×Q8—42no—
283C3×Q8—42no—
284C3×Q8—42no—
285C3×Q8—42no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4C2×Q1612yes—
300C2×C3×C4#13512yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4C2×#4112yes—
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—24no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3#12012yes—
322C2×Dic3C2×Q1612yes—
323C4×D3—28no—
324C4×D3—24no—
325C4×D3—24no—
326C4×D3—44no—
327Dic6—48no—
328C3×C8—24no—
329C3⋊C8—42no—
330C24—31no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—64no—
341C2×Q8—122no—
342C2×Q8—122no—
343C2×Q8D3×D414yes—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—34no—
350C22×C4—34no—
351C22×C4—31no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—121no—
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C23×D311yescharacteristic
366Q16—48no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—128no—
373C2×C8#23714yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—34no—
378C4⋊C4—68no—
379C4⋊C4—64no—
380C4⋊C4—68no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—122no—
386C4⋊C4—122no—
387C4⋊C4—122no—
388C22⋊C4—68no—
389C22⋊C4—122no—
390C22⋊C4—122no—
391C42—22no—
392C42—62no—
393C42—62no—
394C42—68no—
395C42—62no—
396C22×C3C2×#12011yescharacteristic
397C22×C3#53211yescharacteristic
398C22×C3#53711yescharacteristic
399C22×C3C22×Q1611yescharacteristic
400C22×C3C2×#13511yescharacteristic
401C22×C3C2×#13711yescharacteristic
402C22×C3#57511yescharacteristic
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—22no—
409C2×D3#45114yes—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—44no—
413C3×C4—81no—
414C3×C4—81no—
415C3×C4#54612yes—
416C3×C4#54712yes—
417Dic3—22no—
418Dic3—22no—
419Dic3—44no—
420Dic3—41no—
421Dic3—81no—
422Dic3—81no—
423C23—32no—
424C23—32no—
425C23—32no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23C2×D3×D411yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—42no—
437Q8—128no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—68no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—64no—
464C2×C4—124no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4D3×Q1612yes—
474C2×C4#91912yes—
475C2×C4C2×D3×D412yes—
476C2×C4C2×#23712yes—
477C2×C4#100411yescharacteristic
478C2×C4#101011yescharacteristic
479C8—24no—
480C8—122no—
481C2×C3#243011yescharacteristic
482C2×C3C2×#45111yescharacteristic
483C2×C3#300211yescharacteristic
484C2×C3#317011yescharacteristic
485C2×C3#318111yescharacteristic
486C2×C3#324811yescharacteristic
487C2×C3#324911yescharacteristic
488D3—24no—
489C22—34no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22D3×#12011yescharacteristic
497C22#560811yescharacteristic
498C22#571611yescharacteristic
499C22C2×D3×Q1611yescharacteristic
500C22C2×#91911yescharacteristic
501C22D3×#13711yescharacteristic
502C22#577711yescharacteristic
503C4—22no—
504C4—22no—
505C4—44no—
506C4—62no—
507C4—62no—
508C4—81no—
509C4—81no—
510C4—121no—
511C4—124no—
512C4—241no—
513C4—241no—
514C4#559112yes—
515C4#561112yes—
516C3#2549811yescharacteristic, complemented, Sylow
517C2—64no—
518C2#7675711yescharacteristic
519C2D3×#45111yescharacteristic
520C2#8018211yescharacteristic
521C2#8048611yescharacteristic
522C2#8049711yescharacteristic
523C2#8179211yescharacteristic
524C2#8186311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2016C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2016D3
hypocenterC3#25498
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
4. C3×#2016C2semidirect
5. #72065C2semidirect
6. #71781C2semidirect
8. #70236C2semidirect
9. #69668C2semidirect
11. #68632C2semidirect
52. #2016D3semidirect
179. C2×C4×D3Q16semidirect
516. C3#25498semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#51882
#766632
C3×#20162
#720652
#717812
#702702
#702362
#696682
D3×#3362
#686322
#254983