Finite Group Databasebeta

← #10847395

Subgroups — #10847395

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×#5470C211yesmaximal, characteristic, complemented
2C2×#5114C212yesmaximal
3#82080C211yesmaximal, characteristic
4C2×C3×#336C211yesmaximal, characteristic, complemented
5#71972C212yesmaximal, complemented
6#68687C211yesmaximal, characteristic, complemented
7#68686C212yesmaximal, complemented
8#68684C212yesmaximal, complemented
9#68678C211yesmaximal, characteristic, complemented
10#68633C211yesmaximal, characteristic, complemented
11#68631C211yesmaximal, characteristic, complemented
12#25497—31nomaximal, Sylow
13C23×C3×C8C2212yes—
14C22×C3×C42C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×#929—21no—
17C2×#926—21no—
18C2×#924—21no—
19C2×#924C2212yes—
20C22×#204C2212yes—
21#5470—28no—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2212yes—
27#5213—21no—
28#5212C2211yescharacteristic
29#5114—28no—
30C2×#821C2212yes—
31C2×#817C2212yes—
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35#4940—21no—
36#4940—21no—
37#4939C2211yescharacteristic
38#4939C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#533—31no—
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×#336D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×D12C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6C2312yes—
64C22×C3×C8—24no—
65C22×C3×C8—24no—
66C22×C3×C8—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C2×C3×C42—24no—
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—21no—
76#930—24no—
77#929—44no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83#926—28no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2312yes—
91#924—28no—
92#924—24no—
93#924D418yes—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—22no—
99C2×#204—24no—
100C2×#204—24no—
101C2×#204—42no—
102#870—28no—
103#869—24no—
104#869—24no—
105C4×Dic6—28no—
106C4×Dic6—24no—
107C4×Dic6—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111#821—24no—
112#817—24no—
113#816—24no—
114#816—24no—
115#816—42no—
116C22×SD16—32no—
117C23×C8C2×D312yes—
118#533—68no—
119C2×C4⋊Q8—31no—
120C2×C4⋊Q8—31no—
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×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#336C2×D311yescharacteristic
147#336C2×D312yes—
148#336C2×D311yescharacteristic
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4SD1614yescomplemented
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2411yescharacteristic
167C2×#67—42no—
168C2×#67—42no—
169C2×#67—42no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—24no—
184C2×Dic6—22no—
185C2×Dic6—42no—
186C2×Dic6—42no—
187C2×Dic6C2×D414yes—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—48no—
194C2×C3×C8#4114yes—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199#217—28no—
200#217—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—24no—
208C4⋊Dic3—24no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C2×D414yes—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217Dic3⋊C4—42no—
218Dic3⋊C4—42no—
219C4×Dic3—28no—
220#204—48no—
221C22×Q8—32no—
222C22×Q8—61no—
223C22×D4—31no—
224C22×D4—61no—
225C23×C4D1211yescharacteristic
226C23×C4D1211yescharacteristic
227C23×C4C22×D311yescharacteristic
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—42no—
234C22×C8C22×D312yes—
235C22×C8C22×D312yes—
236C4⋊Q8—64no—
237C4⋊Q8—64no—
238#125—68no—
239#123—124no—
240C22⋊Q8—38no—
241C22⋊Q8—68no—
242C22⋊Q8—64no—
243#121—64no—
244C4×Q8—64no—
245C4×Q8—64no—
246C4×Q8—68no—
247#117—68no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
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—24no—
267C2×C42—62no—
268C2×C42D1211yescharacteristic
269C2×C42D1211yescharacteristic
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—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—28no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—44no—
310C2×C3×C4#12012yes—
311C2×C3×C4C2×SD1612yes—
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
318#67—84no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334D12—42no—
335D12—42no—
336Dic6—24no—
337Dic6—24no—
338Dic6—48no—
339Dic6—42no—
340Dic6—42no—
341C3×C8—24no—
342C3×C8—42no—
343C24—61no—
344C24C2×D1211yescharacteristic
345C2×Q8—34no—
346C2×Q8—64no—
347C2×Q8—62no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—122no—
353C2×Q8—122no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—61no—
357C2×D4—61no—
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—41no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4#20414yescomplemented
375C22×C4C2×D1211yescharacteristic
376C22×C4C2×D1211yescharacteristic
377C22×C4C2×D1211yescharacteristic
378C22×C4C2×D1211yescharacteristic
379C22×C4C2×D1211yescharacteristic
380C22×C4D3×D414yes—
381C22×C4C23×D311yescharacteristic
382SD16—128no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—48no—
389C2×C8#23514yes—
390C4⋊C4—34no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—68no—
394C4⋊C4—64no—
395C4⋊C4—68no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—122no—
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—22no—
412C42—22no—
413C42—22no—
414C42—68no—
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—41no—
441C3×C4#54612yes—
442C3×C4#54712yes—
443Dic3—44no—
444Dic3—44no—
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—
459C23#88212yes—
460C23C2×#20412yes—
461C23#90912yes—
462C23C22×D1211yescharacteristic
463Q8—64no—
464Q8—64no—
465Q8—128no—
466Q8—122no—
467Q8—122no—
468D4—122no—
469D4—122no—
470D4—244no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—28no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—44no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
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×C4#88212yes—
502C2×C4C2×#20412yes—
503C2×C4C2×#20412yes—
504C2×C4#90412yes—
505C2×C4C2×#23512yes—
506C2×C4C2×D3×D412yes—
507C2×C4#100411yescharacteristic
508C2×C4#100711yescharacteristic
509C8—24no—
510C8—42no—
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#473814yes—
534C22#547711yescharacteristic
535C22C2×#88211yescharacteristic
536C22#551011yescharacteristic
537C22C22×#20411yescharacteristic
538C22C2×#90411yescharacteristic
539C22C2×#90911yescharacteristic
540C22#574811yescharacteristic
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—41no—
546C4—124no—
547C4—124no—
548C4—241no—
549C4—241no—
550C4—241no—
551C4#549012yes—
552C4#550712yes—
553C3#2549711yescharacteristic, complemented, Sylow
554C2—24no—
555C2—481no—
556C2#7337611yescharacteristic
557C2#7846811yescharacteristic
558C2#7850411yescharacteristic
559C2#7861011yescharacteristic
560C2C2×#473811yescharacteristic
561C2#7902711yescharacteristic
562C2#7902911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#336C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#336D3
hypocenterC3#25497
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
4. C2×C3×#336C2semidirect
5. #71972C2semidirect
6. #68687C2semidirect
7. #68686C2semidirect
8. #68684C2semidirect
9. #68678C2semidirect
10. #68633C2semidirect
11. #68631C2semidirect
56. C2×#336D3semidirect
159. C22×C3×C4SD16semidirect
374. C22×C4#204semidirect
553. C3#25497semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C2×#51142
#820802
C2×C3×#3362
#719722
#686872
#686862
#686842
#686782
#686332
#686312
#254973