Finite Group Databasebeta

← #10831709

Subgroups — #10831709

Order 768.

1296 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D12C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic, complemented
3C2×#4842C211yesmaximal, characteristic, complemented
4C2×#4711C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69828C211yesmaximal, characteristic, complemented
8#68678C212yesmaximal, complemented
9#68673C212yesmaximal
10#68671C212yesmaximal, complemented
11#68633C212yesmaximal, complemented
12#24338—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C22×D24C2211yescharacteristic
16Q8×D12—24no—
17Q8×D12—24no—
18C2×#897—22no—
19C2×#897—21no—
20C2×C4×D12C2212yes—
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#869C2212yes—
24C2×#869C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27#5231—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2212yes—
34#4993—21no—
35#4989—22no—
36#4939C2212yes—
37#4936C2211yescharacteristic
38#4928C2212yes—
39#4842—24no—
40#4711—24no—
41C3×#336C2212yes—
42C3×#336C2212yes—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×D4×Q8—31no—
47C2×#476—31no—
48C2×#442—31no—
49#2580—31no—
50#2058—32no—
51#2023—32no—
52#1993—31no—
53#1985D311yescharacteristic, complemented
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C2×D3×Q8—28no—
59C2×D3×Q8—48no—
60C22×D12C2311yescharacteristic
61C22×D12C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—21no—
64C22×C4×D3—21no—
65C22×Dic6—22no—
66C22×Dic6—21no—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C2×D24—22no—
89C2×D24—22no—
90C2×D24Q814yescomplemented
91#897—28no—
92#897—216no—
93C4×D12—24no—
94C4×D12—22no—
95C4×D12—22no—
96C4×D12—22no—
97C4×D12—24no—
98C4×D12D414yes—
99#869—22no—
100#869—24no—
101#869—24no—
102#869D414yes—
103C3×C4⋊C8—22no—
104C3×#97—22no—
105C3⋊#95—22no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108#821—22no—
109#821—24no—
110#821—22no—
111#821—24no—
112#817—22no—
113#817—24no—
114C23×Q8—61no—
115C22×D8—31no—
116D4×Q8—64no—
117D4×Q8—64no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8—32no—
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—61no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×D4—31no—
124C2×C4×D4—31no—
125C2×C4×D4—32no—
126#476—64no—
127#442—64no—
128C2×#106—31no—
129C2×#106—32no—
130C2×C4⋊C8C2×D311yescharacteristic
131C2×#102—32no—
132C2×#102—31no—
133C2×#102—32no—
134C2×#102—31no—
135C2×#97C2×D311yescharacteristic
136#399—61no—
137#393—61no—
138#390—32no—
139#389—32no—
140#385—62no—
141#382—31no—
142#340—32no—
143#340—32no—
144#340—32no—
145#336C2×D312yes—
146#336C2×D312yes—
147C23×D3—21no—
148C23×D3—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160D3×Q8—432no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12C2×D412yes—
170C2×D12C2×Q812yes—
171C2×D12C2×Q812yescomplemented
172C2×D12#4112yes—
173C2×C4×D3—28no—
174C2×C4×D3—24no—
175C2×C4×D3—44no—
176C2×C4×D3—44no—
177C2×C4×D3—44no—
178C2×C4×D3—42no—
179C2×C4×D3—82no—
180C2×Dic6—28no—
181C2×Dic6—24no—
182C2×Dic6—44no—
183C2×Dic6—44no—
184C2×Dic6—44no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8C2×Q812yescomplemented
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4—42no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—82no—
205C4⋊Dic3—28no—
206C4⋊Dic3—24no—
207C4⋊Dic3—22no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3C2×D412yes—
214C4⋊Dic3C2×D414yes—
215Dic3⋊C4—82no—
216D24—28no—
217C22×Q8—68no—
218C22×Q8—61no—
219C22×Q8—62no—
220C22×Q8—128no—
221C22×Q8D1211yescharacteristic
222C22×D4—31no—
223C22×D4—31no—
224C23×C4—62no—
225C23×C4—61no—
226C23×C4—61no—
227C2×D8—34no—
228C2×D8—62no—
229C2×D8—62no—
230C22×C8C22×D311yescharacteristic
231C22×C8C22×D312yes—
232C22×C8C22×D311yescharacteristic
233C4⋊Q8—34no—
234C4⋊Q8—64no—
235C4⋊Q8—64no—
236C4⋊Q8—62no—
237C22⋊Q8—68no—
238C22⋊Q8—616no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×D4—34no—
242C4×D4—64no—
243C4×D4—64no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4D1212yes—
255C2×C4⋊C4D1211yescharacteristic
256C2×C22⋊C4—62no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42C22×D312yes—
262C2×C42C22×D311yescharacteristic
263#106—62no—
264#106—64no—
265C4⋊C8—22no—
266#102—62no—
267#102—64no—
268#102—64no—
269#102—62no—
270#97—22no—
271#95—61no—
272#95—61no—
273#95—62no—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—41no—
282C22×D3—41no—
283C22×D3—42no—
284C22×D3—42no—
285C3×Q8—28no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4#12012yes—
294C2×C3×C4#12012yes—
295C2×C3×C4AGL(1,ℤ/8)14yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×Q811yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C2×C3×C42−1+411yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—24no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312D12—22no—
313D12—22no—
314D12—22no—
315D12—22no—
316D12—22no—
317D12—41no—
318D12—41no—
319D12C22⋊Q814yes—
320C4×D3—48no—
321C4×D3—416no—
322C4×D3—84no—
323Dic6—48no—
324Dic6—416no—
325C3×C8—22no—
326C3×C8—41no—
327C3×C8—42no—
328C24—61no—
329C24—61no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—68no—
333C2×Q8—64no—
334C2×Q8—124no—
335C2×Q8—124no—
336C2×Q8—1232no—
337C2×Q8—124no—
338C2×D4—32no—
339C2×D4—32no—
340C2×D4—32no—
341C2×D4—32no—
342C2×D4—61no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C22×C4—68no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—64no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—124no—
357C22×C4—124no—
358C22×C4—124no—
359C22×C4—122no—
360C22×C4—242no—
361C22×C4C2×D1212yes—
362C22×C4C2×D1211yescharacteristic
363C22×C4C2×D1211yescharacteristic
364C22×C4C2×D1212yes—
365C22×C4C23×D311yescharacteristic
366D8—68no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8D3×Q812yescomplemented
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—32no—
379C4⋊C4—34no—
380C4⋊C4—68no—
381C4⋊C4—64no—
382C4⋊C4—61no—
383C4⋊C4—62no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—242no—
389C22⋊C4—68no—
390C22⋊C4—64no—
391C22⋊C4—122no—
392C22⋊C4—124no—
393C22⋊C4—242no—
394C42—22no—
395C42—22no—
396C42D3×D412yes—
397C42D3×D412yes—
398C22×C3C2×#12011yescharacteristic
399C22×C3#54811yescharacteristic
400C22×C3D4×Q811yescharacteristic
401C22×C3C2×AGL(1,ℤ/8)12yes—
402C22×C3#57712yes—
403C2×D3—24no—
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—42no—
407C2×D3—42no—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—82no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4C2×C22⋊Q811yescharacteristic
418C3×C4#53611yescharacteristic
419C3×C4#54811yescharacteristic
420C3×C4D4×Q811yescharacteristic
421Dic3—44no—
422Dic3—42no—
423Dic3—82no—
424Dic3—81no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23—122no—
430C23—122no—
431C23—121no—
432C23—122no—
433C23—122no—
434C23—121no—
435C23C22×D1211yescharacteristic
436Q8—28no—
437Q8—128no—
438Q8—1216no—
439D4—34no—
440D4—62no—
441D4—62no—
442D4—62no—
443D4—62no—
444D4—62no—
445D4—121no—
446D4—121no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—64no—
455C2×C4—62no—
456C2×C4—128no—
457C2×C4—1216no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—244no—
467C2×C4#88212yes—
468C2×C4#88212yes—
469C2×C4#90814yes—
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×Q811yescharacteristic
473C2×C4#100711yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4#101011yescharacteristic
476C8—22no—
477C8—42no—
478C8—41no—
479C2×C3#240911yescharacteristic
480C2×C3#300311yescharacteristic
481C2×C3#301011yescharacteristic
482C2×C3#337312yes—
483C2×C3#338412yes—
484D3—44no—
485D3—82no—
486C22—64no—
487C22—122no—
488C22—122no—
489C22—124no—
490C22—124no—
491C22—122no—
492C22—124no—
493C22—241no—
494C22—241no—
495C22—242no—
496C22C2×#88211yescharacteristic
497C22Q8×D1211yescharacteristic
498C22#557611yescharacteristic
499C22C2×#90812yes—
500C22#575112yes—
501C4—22no—
502C4—24no—
503C4—22no—
504C4—42no—
505C4—124no—
506C4—122no—
507C4—241no—
508C4—242no—
509C4D3×C22⋊Q811yescharacteristic
510C4#562811yescharacteristic
511C4#567711yescharacteristic
512C4#572711yescharacteristic
513C3#2433811yescharacteristic, complemented, Sylow
514C2—124no—
515C2—242no—
516C2#7661911yescharacteristic
517C2#7863711yescharacteristic
518C2#7867811yescharacteristic
519C2#7997312yes—
520C2#7997512yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1985D3
hypocenterC3#24338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D12C2semidirect
2. #82501C2semidirect
3. C2×#4842C2semidirect
5. C3×#1985C2semidirect
6. #70480C2semidirect
7. #69828C2semidirect
8. #68678C2semidirect
10. #68671C2semidirect
11. #68633C2semidirect
53. #1985D3semidirect
90. C2×D24Q8semidirect
171. C2×D12C2×Q8semidirect
193. C2×C3×C8C2×Q8semidirect
375. C2×C8D3×Q8semidirect
513. C3#24338semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D122
#825012
C2×#48422
C2×#47112
C3×#19852
#704802
#698282
#686782
#686732
#686712
#686332
#243383