Finite Group Databasebeta

← #10847389

Subgroups — #10847389

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×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5048C212yesmaximal
3#82080C211yesmaximal, characteristic
4#71792C211yesmaximal, characteristic, complemented
5#71791C211yesmaximal, characteristic, complemented
6#71783C211yesmaximal, characteristic, complemented
7#71781C211yesmaximal, characteristic, complemented
8#71696C212yesmaximal, complemented
9#71633C212yesmaximal, complemented
10#71629C212yesmaximal, complemented
11C2×#4524C211yesmaximal, characteristic, complemented
12#25497—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#215C2212yes—
24C2×#930C2211yescharacteristic
25C2×#929—21no—
26C23×C3⋊C8C2212yes—
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#835C2212yes—
30#5213—21no—
31#5212C2211yescharacteristic
32#5048—28no—
33C2×#809C2212yes—
34C2×#808C2212yes—
35#4940—21no—
36#4940—21no—
37#4939C2211yescharacteristic
38#4939C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#533D311yescharacteristic, complemented
47C2×#465—32no—
48#2854—31no—
49#2058—31no—
50#2043—31no—
51#2023—31no—
52#2013—31no—
53#1892—32no—
54#1880—32no—
55#1865—32no—
56C2×#336—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×D12C2311yescharacteristic
64C3×C4⋊Q8—24no—
65C3×C4⋊Q8—24no—
66C3×#125—28no—
67C3×C22⋊Q8—28no—
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8D418yes—
70C3×C4×Q8—28no—
71C3×C4×Q8—24no—
72C3×C4×Q8—24no—
73C3×#117—28no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—24no—
82C2×C3×C42—22no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#215—24no—
89C2×#215—24no—
90C2×#215—42no—
91#930—24no—
92#929—44no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8—24no—
105C22×C3⋊C8—42no—
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2312yes—
108#835—24no—
109#835—24no—
110#835—42no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114#809—24no—
115#808—24no—
116C22×SD16—32no—
117C23×C8—32no—
118#533—28no—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—21no—
122C2×#123—61no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C2×D312yes—
125C2×#121—31no—
126C2×C4×Q8—21no—
127C2×C4×Q8C2×D312yes—
128C2×#117—21no—
129C22×C42C2×D311yescharacteristic
130#465—68no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#102—32no—
136#390—31no—
137#390—31no—
138#389—31no—
139#389—31no—
140#388—61no—
141#388—61no—
142#386—61no—
143#380—31no—
144#340—32no—
145#340—32no—
146#336—31no—
147#336—32no—
148#336—31no—
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—42no—
158C2×C3×Q8—42no—
159C2×C3×Q8C2×D414yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—41no—
170C22×C3×C4SD1614yescomplemented
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—42no—
179C2×#67—42no—
180C2×#67—42no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—21no—
187C2×D12—21no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C42—28no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201#217—42no—
202#215—48no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—42no—
205(C2×D3)⋊C4—42no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—42no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212Dic3⋊C4—42no—
213Dic3⋊C4—42no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—48no—
220C2×C3⋊C8#4114yes—
221C22×Q8—21no—
222C22×Q8C22×D312yes—
223C22×D4—31no—
224C22×D4—61no—
225C23×C4#6711yescharacteristic
226C23×C4#6711yescharacteristic
227C23×C4C22×D311yescharacteristic
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8—32no—
232C22×C8—32no—
233C22×C8—64no—
234C22×C8—64no—
235C22×C8—122no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238#125—28no—
239#123—124no—
240C22⋊Q8—28no—
241C22⋊Q8—24no—
242C22⋊Q8#6718yes—
243#121—64no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247#117—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—22no—
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—22no—
267C2×C42—24no—
268C2×C42#6711yescharacteristic
269C2×C42#6711yescharacteristic
270C2×C42C22×D311yescharacteristic
271C2×C42C22×D311yescharacteristic
272#106—64no—
273#103—64no—
274#103—64no—
275#103—122no—
276#102—64no—
277#95—61no—
278#95—61no—
279#95—61no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3#12012yes—
285C23×C3C2×SD1612yes—
286C23×C3#13712yes—
287C23×C3C22×D411yescharacteristic
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C3×Q8—24no—
296C3×Q8—24no—
297C3×Q8—48no—
298C3×Q8—42no—
299C3×Q8—42no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—28no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—44no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4#12012yes—
322C2×C3×C4C2×SD1612yes—
323C2×C3×C4C2×SD1612yes—
324C2×C3×C4#13512yes—
325C2×C3×C4C22×D412yes—
326C2×C3×C4C2×#4112yes—
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C42−1+411yescharacteristic
329#67—84no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339D12—42no—
340D12—42no—
341C3⋊C8—24no—
342C3⋊C8—42no—
343C24—61no—
344C24C2×#6711yescharacteristic
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8C2×#6714yes—
354C2×D4—62no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—62no—
358C2×D4—122no—
359C2×D4—122no—
360C2×D4—122no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4#21514yescomplemented
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377C22×C4C2×#6711yescharacteristic
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6714yes—
381C22×C4C23×D311yescharacteristic
382SD16—128no—
383C2×C8—34no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—128no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—62no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—122no—
401C4⋊C4—122no—
402C4⋊C4C2×#6714yes—
403C22⋊C4—28no—
404C22⋊C4—122no—
405C22⋊C4—122no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C22⋊C4—122no—
409C22⋊C4—122no—
410C42—28no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3#44814yes—
422C22×C3C2×#12011yescharacteristic
423C22×C3#53211yescharacteristic
424C22×C3#53711yescharacteristic
425C22×C3C22×SD1611yescharacteristic
426C22×C3C2×#13511yescharacteristic
427C22×C3C2×#13711yescharacteristic
428C22×C3#57511yescharacteristic
429C2×D3—81no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C2×D3—81no—
435C2×D3—81no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—44no—
441C3×C4—41no—
442C3×C4—44no—
443C3×C4#54612yes—
444C3×C4#54712yes—
445Dic3—81no—
446Dic3—81no—
447Dic3—81no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—121no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23C2×#21512yes—
460C23#94212yes—
461C23#95312yes—
462C23C22×#6711yescharacteristic
463Q8—24no—
464Q8—24no—
465Q8—48no—
466Q8—42no—
467Q8—42no—
468D4—122no—
469D4—122no—
470D4—244no—
471C2×C4—28no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4C2×#21512yes—
502C2×C4C2×#21512yes—
503C2×C4#95012yes—
504C2×C4#95312yes—
505C2×C4D3×#4112yes—
506C2×C4#100911yescharacteristic
507C2×C4#101011yescharacteristic
508C2×C4C22×#6712yes—
509C8—64no—
510C8—122no—
511C2×C3—24no—
512C2×C3#243011yescharacteristic
513C2×C3C2×#44811yescharacteristic
514C2×C3#300211yescharacteristic
515C2×C3#316711yescharacteristic
516C2×C3#318111yescharacteristic
517C2×C3#324611yescharacteristic
518C2×C3#324811yescharacteristic
519D3—161no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—241no—
527C22—241no—
528C22—241no—
529C22—241no—
530C22—241no—
531C22—241no—
532C22—241no—
533C22#522714yes—
534C22C22×#21511yescharacteristic
535C22C2×#94211yescharacteristic
536C22C2×#95011yescharacteristic
537C22#582211yescharacteristic
538C22#583211yescharacteristic
539C22#583311yescharacteristic
540C22C2×#95311yescharacteristic
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—41no—
546C4—44no—
547C4—44no—
548C4—241no—
549C4—241no—
550C4—241no—
551C4#582412yes—
552C4#582912yes—
553C3#2549711yescharacteristic, complemented, Sylow
554C2—24no—
555C2—481no—
556C2#8334311yescharacteristic
557C2#8334611yescharacteristic
558C2#8336311yescharacteristic
559C2#8336411yescharacteristic
560C2C2×#522711yescharacteristic
561C2#8408011yescharacteristic
562C2#8409111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#533D3
hypocenterC3#25497
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
4. #71792C2semidirect
5. #71791C2semidirect
6. #71783C2semidirect
7. #71781C2semidirect
8. #71696C2semidirect
9. #71633C2semidirect
10. #71629C2semidirect
11. C2×#4524C2semidirect
46. C2×#533D3semidirect
170. C22×C3×C4SD16semidirect
374. C22×C4#215semidirect
553. C3#25497semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50482
#820802
#717922
#717912
#717832
#717812
#716962
#716332
#716292
C2×#45242
#254973