Finite Group Databasebeta

← #10847390

Subgroups — #10847390

Order 768.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic, complemented
2C2×#4807C212yesmaximal
3#76286C211yesmaximal, characteristic
4C3×#2043C211yesmaximal, characteristic, complemented
5#71791C211yesmaximal, characteristic, complemented
6#70270C212yesmaximal, complemented
7#70240C212yesmaximal, complemented
8#70237C212yesmaximal, complemented
9#69605C211yesmaximal, characteristic, complemented
10#68678C211yesmaximal, characteristic, complemented
11D3×#336C211yesmaximal, characteristic, complemented
12#25497—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×#215C2212yes—
17C22×D3×C8C2212yes—
18#5728—28no—
19C2×#897—21no—
20C2×#895C2211yescharacteristic
21C2×#894—21no—
22C2×#892—21no—
23C2×#890—21no—
24C2×#887—21no—
25C2×C42×D3C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C3×#389C2211yescharacteristic
30C2×#835C2212yes—
31C2×#821C2212yes—
32#4990—21no—
33#4988C2211yescharacteristic
34#4987C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#808C2212yes—
37#4939C2211yescharacteristic
38#4807—28no—
39#4669—21no—
40#4665—21no—
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#465—32no—
48#2854—31no—
49#2058—31no—
50#2043D311yescharacteristic, complemented
51#2023—31no—
52#2013—31no—
53#1892—32no—
54#1880—32no—
55#1865—32no—
56C2×#336—31no—
57C22×C3×Q8C2312yes—
58C22×D12—21no—
59C22×D12C2311yescharacteristic
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⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72#946—24no—
73#946D418yes—
74Q8×Dic3—24no—
75Q8×Dic3—24no—
76C2×#215—24no—
77C2×#215—24no—
78C2×#215—42no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—21no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3D411yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C2×D3×C8—24no—
98C2×D3×C8—24no—
99C2×D3×C8—42no—
100#897—28no—
101#895—24no—
102#894—44no—
103#892—28no—
104#890—28no—
105#887—28no—
106C42×D3—24no—
107#869—24no—
108C3×#103—24no—
109C3×#103—42no—
110C3×#95—21no—
111#835—24no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#821—24no—
115#808—24no—
116C22×SD16—32no—
117C23×C8—32no—
118#533—68no—
119C2×C4⋊Q8—31no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—61no—
122C2×#123—61no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—61no—
125C2×#121—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—61no—
128C2×#117—61no—
129C22×C42—31no—
130#465—68no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#103C2×D312yes—
134C2×#103C2×D312yes—
135C2×#102—32no—
136#390—31no—
137#390—31no—
138#389—31no—
139#389C2×D311yescharacteristic
140#388—61no—
141#388—61no—
142#386—61no—
143#380—31no—
144#340—32no—
145#340—32no—
146#336—32no—
147#336—31no—
148#336C2×D311yescharacteristic
149C23×D3—21no—
150C23×D3C2×D411yescharacteristic
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D414yes—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—42no—
173C2×D12—42no—
174C2×D12—42no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—41no—
183C2×C4×D3SD1614yescomplemented
184C2×C4×D3C2×D414yes—
185C2×Dic6—24no—
186C2×Dic6—42no—
187C2×Dic6—42no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—42no—
197C3×C42—22no—
198#215—48no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—42no—
201(C2×D3)⋊C4—42no—
202(C2×D3)⋊C4—42no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—42no—
205(C2×D3)⋊C4—42no—
206C4⋊Dic3—24no—
207C4⋊Dic3—28no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D414yes—
211Dic3⋊C4—28no—
212Dic3⋊C4—42no—
213C4×Dic3—28no—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8#4114yes—
220D3×C8—48no—
221C22×Q8—61no—
222C22×Q8C22×D312yes—
223C22×D4—31no—
224C22×D4—61no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—31no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8—32no—
232C22×C8—64no—
233C22×C8—64no—
234C22×C8—122no—
235C22×C8C22×D312yes—
236C4⋊Q8—24no—
237C4⋊Q8—64no—
238#125—68no—
239#123—124no—
240C22⋊Q8—38no—
241C22⋊Q8—68no—
242C22⋊Q8—64no—
243#121—64no—
244C4×Q8—68no—
245C4×Q8—64no—
246C4×Q8—64no—
247#117—68no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—62no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—61no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42—64no—
270C2×C42—62no—
271C2×C42C22×D311yescharacteristic
272#106—64no—
273#103—24no—
274#103—42no—
275#103—64no—
276#102—64no—
277#95—21no—
278#95—61no—
279#95—61no—
280C23×C3C22×D411yescharacteristic
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—21no—
284C22×D3—21no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3#12012yes—
293C22×D3C2×SD1612yes—
294C22×D3#13712yes—
295C3×Q8—24no—
296C3×Q8—24no—
297C3×Q8—42no—
298C3×Q8—42no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—24no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4C2×SD1612yes—
313C2×C3×C4#13512yes—
314C2×C3×C4C22×D412yes—
315C2×C3×C4C2×#4112yes—
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—24no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3#12012yes—
332C2×Dic3C2×SD1612yes—
333D12—42no—
334D12—42no—
335D12—84no—
336C4×D3—28no—
337C4×D3—24no—
338C4×D3—24no—
339C4×D3—44no—
340Dic6—48no—
341C3×C8—42no—
342C3⋊C8—24no—
343C24—31no—
344C24—61no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—64no—
351C2×Q8—122no—
352C2×Q8—122no—
353C2×Q8D3×D414yes—
354C2×D4—62no—
355C2×D4—61no—
356C2×D4—62no—
357C2×D4—61no—
358C2×D4—122no—
359C2×D4—122no—
360C2×D4—122no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—31no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—121no—
379C22×C4D3×D411yescharacteristic
380C22×C4D3×D411yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—128no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—34no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—128no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—34no—
396C4⋊C4—42no—
397C4⋊C4—68no—
398C4⋊C4—64no—
399C4⋊C4—68no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—122no—
403C22⋊C4—68no—
404C22⋊C4—122no—
405C22⋊C4—122no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C22⋊C4—122no—
409C22⋊C4—122no—
410C42—22no—
411C42—62no—
412C42—62no—
413C42—68no—
414C42—62no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#53211yescharacteristic
417C22×C3#53711yescharacteristic
418C22×C3C22×SD1611yescharacteristic
419C22×C3C2×#13511yescharacteristic
420C22×C3C2×#13711yescharacteristic
421C22×C3#57511yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—81no—
429C2×D3—81no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C2×D3—81no—
435C2×D3#44814yes—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—44no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4#54612yes—
442C3×C4#54712yes—
443Dic3—22no—
444Dic3—22no—
445Dic3—44no—
446Dic3—41no—
447Dic3—81no—
448C23—32no—
449C23—32no—
450C23—32no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23—61no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23C2×D3×D411yescharacteristic
463Q8—24no—
464Q8—24no—
465Q8—42no—
466Q8—42no—
467Q8—128no—
468D4—122no—
469D4—122no—
470D4—244no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—68no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—64no—
497C2×C4—124no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4D3×SD1612yes—
504C2×C4#91912yes—
505C2×C4C2×D3×D412yes—
506C2×C4#100411yescharacteristic
507C2×C4D3×#4112yes—
508C2×C4#101011yescharacteristic
509C8—42no—
510C8—64no—
511C2×C3#243011yescharacteristic
512C2×C3C2×#44811yescharacteristic
513C2×C3#300211yescharacteristic
514C2×C3#316711yescharacteristic
515C2×C3#318111yescharacteristic
516C2×C3#324611yescharacteristic
517C2×C3#324811yescharacteristic
518D3—24no—
519D3—161no—
520C22—34no—
521C22—62no—
522C22—62no—
523C22—62no—
524C22—62no—
525C22—62no—
526C22—62no—
527C22—241no—
528C22—241no—
529C22—241no—
530C22—241no—
531C22—241no—
532C22—241no—
533C22—241no—
534C22D3×#12011yescharacteristic
535C22#560811yescharacteristic
536C22#566811yescharacteristic
537C22C2×D3×SD1611yescharacteristic
538C22C2×#91911yescharacteristic
539C22D3×#13711yescharacteristic
540C22#577711yescharacteristic
541C4—22no—
542C4—22no—
543C4—44no—
544C4—62no—
545C4—62no—
546C4—81no—
547C4—81no—
548C4—121no—
549C4—124no—
550C4—241no—
551C4#561012yes—
552C4#565512yes—
553C3#2549711yescharacteristic, complemented, Sylow
554C2—64no—
555C2—481no—
556C2#7711611yescharacteristic
557C2D3×#44811yescharacteristic
558C2#8018211yescharacteristic
559C2#8043811yescharacteristic
560C2#8049711yescharacteristic
561C2#8159211yescharacteristic
562C2#8173111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2043C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2043D3
hypocenterC3#25497
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
1. C2×#5728C2semidirect
4. C3×#2043C2semidirect
5. #71791C2semidirect
6. #70270C2semidirect
7. #70240C2semidirect
8. #70237C2semidirect
9. #69605C2semidirect
10. #68678C2semidirect
11. D3×#336C2semidirect
50. #2043D3semidirect
183. C2×C4×D3SD16semidirect
553. C3#25497semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#48072
#762862
C3×#20432
#717912
#702702
#702402
#702372
#696052
#686782
D3×#3362
#254973