Finite Group Databasebeta

← #10845926

Subgroups — #10845926

Order 768.

1018 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×C3×#462C211yesmaximal, characteristic, complemented
2C2×#5188C212yesmaximal, complemented
3#83259C211yesmaximal, characteristic
4#82500C211yesmaximal, characteristic
5C2×#4834C211yesmaximal, characteristic
6#72071C212yesmaximal, complemented
7#71785C211yesmaximal, characteristic, complemented
8C4×#835C211yesmaximal, characteristic, complemented
9#70453C211yesmaximal, characteristic, complemented
10#70270C212yesmaximal, complemented
11#70244C212yesmaximal, complemented
12#25386—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946C2212yes—
17C2×#946C2212yes—
18C2×#946C2212yes—
19C22×D3×C8C2212yes—
20C2×C4×D12C2211yescharacteristic
21C3×#462—24no—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×#835C2212yes—
25C2×#835C2211yescharacteristic
26C2×#835C2212yes—
27C2×#835C2211yescharacteristic
28#5188—28no—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#818C2212yes—
31#4994—22no—
32#4993—22no—
33#4988—21no—
34#4987C2212yes—
35#4975—21no—
36#4939—21no—
37#4935C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4928—21no—
40C2×C4×C3⋊C8C2211yescharacteristic
41#4834—24no—
42#4551C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#467—32no—
46C2×#462D311yescharacteristic, complemented
47C2×C8×D4—31no—
48#2542—31no—
49#2538—31no—
50#2036—32no—
51#2001—32no—
52#1956—31no—
53#1947—31no—
54#1892—32no—
55C4×#103—31no—
56C22×C3×Q8C2312yes—
57C22×C3×Q8C2311yescharacteristic
58C22×D12D411yescharacteristic
59C22×C4×D3C2312yes—
60C2×C3×Q16—24no—
61C2×C3×Q16—24no—
62C2×C3×Q16—42no—
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74#946—24no—
75#946—24no—
76#946—24no—
77#946—42no—
78#946—42no—
79#946—42no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4D412yes—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97C2×D3×C8—24no—
98C2×D3×C8—24no—
99C2×D3×C8—42no—
100C4×D12—22no—
101C4×D12—22no—
102C3×C4⋊C8D414yes—
103C3×#103—24no—
104#835—22no—
105#835—24no—
106#835—42no—
107#835D414yescomplemented
108C3⋊#95—21no—
109C3⋊#95—22no—
110(C2×D3)⋊C8—24no—
111#818—24no—
112C4×C3⋊C8—22no—
113C22×Q16C2×D312yes—
114C23×C8—32no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×C4×Q8C2×D311yescharacteristic
120C2×C4×D4—31no—
121#467—68no—
122#462—24no—
123C8×D4—64no—
124C2×#107—32no—
125C2×C4⋊C8C2×D311yescharacteristic
126C2×#103—31no—
127C2×#103—32no—
128C2×#103—31no—
129C2×#103—32no—
130C2×#103C2×D312yes—
131C2×C22⋊C8—32no—
132C2×C4×C8—31no—
133#399—62no—
134#396—62no—
135#390—61no—
136#390—61no—
137#389—61no—
138#384—31no—
139#384—61no—
140#380—32no—
141C4×C4⋊C4—31no—
142#340—32no—
143#340—31no—
144#340—31no—
145C23×D3C2×D412yes—
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D412yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3C2×D412yes—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12Q1614yescomplemented
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C3×Q16—48no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8C2×D414yes—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C42—42no—
187C3×C42C2×D412yes—
188C3×C42#4112yes—
189(C2×D3)⋊C4—24no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—82no—
194C4⋊Dic3—22no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—42no—
200C4⋊Dic3—81no—
201C4⋊Dic3C2×D412yes—
202C4⋊Dic3#4112yes—
203Dic3⋊C4—44no—
204Dic3⋊C4—44no—
205Dic3⋊C4—44no—
206C4×Dic3—42no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8C2×D412yes—
213D3×C8—48no—
214C22×Q8C22×D312yes—
215C22×Q8C22×D311yescharacteristic
216C22×D4—31no—
217C23×C4—32no—
218C2×Q16—24no—
219C2×Q16—24no—
220C2×Q16—42no—
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—64no—
224C22×C8—64no—
225C22×C8—122no—
226C22×C8C22×D312yes—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C22⋊Q8—64no—
230C22⋊Q8—64no—
231C22⋊Q8—64no—
232C22⋊Q8—122no—
233C22⋊Q8—122no—
234C22⋊Q8—122no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×D4—62no—
238C4×D4—62no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C22⋊C4—32no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C42—21no—
258C2×C42—61no—
259C2×C42C22×D311yescharacteristic
260#107—64no—
261C4⋊C8#6714yes—
262#103—24no—
263#103—34no—
264#103—62no—
265#103—64no—
266#103—122no—
267C22⋊C8—64no—
268C4×C8—62no—
269#95—62no—
270#95—61no—
271C23×C3C22×D411yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3C2×Q1614yes—
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—41no—
282C3×Q8—41no—
283C3×Q8—42no—
284C3×Q8—42no—
285C3×Q8—42no—
286C3×Q8—42no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4#12112yes—
299C2×C3×C4#12112yes—
300C2×C3×C4C2×Q1612yes—
301C2×C3×C4#13512yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307C2×C3×C4H(5,𝔽2)11yescharacteristic
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320D12—28no—
321C4×D3—42no—
322C4×D3—42no—
323C4×D3—42no—
324C4×D3—42no—
325C3×C8—24no—
326C3⋊C8—22no—
327C3⋊C8—41no—
328C24—32no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8D3×D412yes—
341C2×D4—34no—
342C2×D4—62no—
343C2×D4—62no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—32no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—48no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—32no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—128no—
370C2×C8C2×#6714yes—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—32no—
374C4⋊C4—32no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—62no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—124no—
384C4⋊C4—124no—
385C4⋊C4—124no—
386C4⋊C4—122no—
387C4⋊C4—241no—
388C22⋊C4—64no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C22⋊C4—242no—
393C42—42no—
394C42—122no—
395C42#23712yes—
396C42C2×#6712yes—
397C22×C3C2×#12111yescharacteristic
398C22×C3D4211yescharacteristic
399C22×C3#54611yescharacteristic
400C22×C3C22×Q1611yescharacteristic
401C22×C3C2×#13511yescharacteristic
402C22×C3#57711yescharacteristic
403C22×C3#57811yescharacteristic
404C2×D3—24no—
405C2×D3—24no—
406C2×D3—24no—
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—42no—
411C3×C4—22no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—81no—
415C3×C4#46714yescomplemented
416C3×C4#53211yescharacteristic
417C3×C4#54011yescharacteristic
418C3×C4D4211yescharacteristic
419C3×C4#55011yescharacteristic
420Dic3—42no—
421Dic3—42no—
422Dic3—42no—
423Dic3—82no—
424C23—34no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23—62no—
430C23—62no—
431C23C2×D3×D411yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—41no—
435Q8—41no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440D4—68no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4D3×Q1612yes—
469C2×C4#91912yes—
470C2×C4#93812yes—
471C2×C4#93812yes—
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4#100411yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#100911yescharacteristic
477C2×C4C22×#6711yescharacteristic
478C8—24no—
479C8—62no—
480C8—121no—
481C2×C3#242811yescharacteristic
482C2×C3C2×#46711yescharacteristic
483C2×C3#306311yescharacteristic
484C2×C3#328411yescharacteristic
485C2×C3D4×Q1611yescharacteristic
486C2×C3#329911yescharacteristic
487C2×C3#330211yescharacteristic
488D3—44no—
489C22—64no—
490C22—64no—
491C22—64no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22#560911yescharacteristic
497C22#561211yescharacteristic
498C22C2×D3×Q1611yescharacteristic
499C22C2×#91911yescharacteristic
500C22#577811yescharacteristic
501C22#577911yescharacteristic
502C22C2×#93811yescharacteristic
503C4—22no—
504C4—42no—
505C4—81no—
506C4—82no—
507C4—122no—
508C4—122no—
509C4—122no—
510C4—242no—
511C4#518814yescomplemented
512C4#559611yescharacteristic
513C4#561911yescharacteristic
514C4D4×#6711yescharacteristic
515C4#583411yescharacteristic
516C3#2538611yescharacteristic, complemented, Sylow
517C2—124no—
518C2#8049011yescharacteristic
519C2#8049111yescharacteristic
520C2#8049211yescharacteristic
521C2#8050211yescharacteristic
522C2#8264811yescharacteristic
523C2C2×#518811yescharacteristic
524C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#462C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenterC2×#462D3
hypocenterC3#25386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×C3×#462C2semidirect
2. C2×#5188C2semidirect
6. #72071C2semidirect
7. #71785C2semidirect
8. C4×#835C2semidirect
9. #70453C2semidirect
10. #70270C2semidirect
11. #70244C2semidirect
46. C2×#462D3semidirect
107. #835D4semidirect
170. C2×D12Q16semidirect
415. C3×C4#467semidirect
511. C4#5188semidirect
516. C3#25386semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4622
C2×#51882
#832592
#825002
C2×#48342
#720712
#717852
C4×#8352
#704532
#702702
#702442
#253863