Finite Group Databasebeta

← #10831707

Subgroups — #10831707

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×D4×Dic6C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic, complemented
4#74190C211yesmaximal, characteristic, complemented
5C3×#1804C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70172C212yesmaximal
8#70170C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68898C212yesmaximal, complemented
11#68827C212yesmaximal, complemented
12#24338—31nomaximal, Sylow
13C23×Dic6—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×D4×Dic3C2212yes—
16C22×#213C2211yescharacteristic
17C2×#924—21no—
18D4×Dic6—24no—
19D4×Dic6—24no—
20C2×#888C2212yes—
21C2×#878—22no—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2212yes—
27C3×#382C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#821C2212yes—
30C2×#817C2212yes—
31#5014—24no—
32#4988C2212yes—
33#4984C2212yes—
34C2×#809C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#803C2211yescharacteristic
37#4842—24no—
38#4677—21no—
39#4673—21no—
40#4668—22no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×D4×Q8—31no—
47C2×#476—31no—
48C2×#442—31no—
49#2580—31no—
50#2058—32no—
51#2023—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1864—32no—
56#1804D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×Dic3—22no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6—28no—
63C22×Dic6—21no—
64C22×Dic6—22no—
65C22×Dic6—48no—
66C22×Dic6D411yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4×D4—22no—
69C3×C4×D4D414yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75D4×Dic3—24no—
76D4×Dic3—24no—
77C2×#213—22no—
78C2×#213—22no—
79C2×#213Q814yescomplemented
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88#924—28no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4D412yes—
91C2×C4×Dic3C2312yes—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94#888—24no—
95#888—24no—
96#878—216no—
97C4×D12—22no—
98C4×D12—22no—
99#869—22no—
100#869D414yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×C4⋊C8—22no—
104C3×#102—24no—
105C3×#95—21no—
106C3×#95—21no—
107#832—22no—
108C3⋊#95—22no—
109#821—24no—
110#817—24no—
111#809—22no—
112#808—22no—
113#803—22no—
114C23×Q8—61no—
115C22×D8—31no—
116D4×Q8—64no—
117D4×Q8—64no—
118C2×C4⋊Q8—32no—
119C2×C4⋊Q8—31no—
120C2×C22⋊Q8—62no—
121C2×C22⋊Q8—61no—
122C2×C4×Q8—31no—
123C2×C4×D4—32no—
124C2×C4×D4—31no—
125C2×C4×D4C2×D311yescharacteristic
126#476—64no—
127#442—64no—
128C2×#106—32no—
129C2×#106—31no—
130C2×C4⋊C8C2×D311yescharacteristic
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#102C2×D312yes—
135C2×#97—31no—
136#399—61no—
137#393—61no—
138#390—32no—
139#389—32no—
140#385—62no—
141#382C2×D311yescharacteristic
142#340—32no—
143#340—32no—
144#340C2×D312yes—
145#336—32no—
146#336—32no—
147C24×C3—21no—
148C23×D3—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4C2×D412yes—
153C2×C3×D4C2×Q812yes—
154C2×C3×D4#4112yes—
155C22×C3×C4—21no—
156C22×C3×C4—24no—
157C22×C3×C4—22no—
158C22×C3×C4—44no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—28no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—44no—
166C22×Dic3—44no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12C2×Q812yescomplemented
175C2×C4×D3—42no—
176C2×C4×D3—82no—
177C2×Dic6—22no—
178C2×Dic6—24no—
179C2×Dic6—28no—
180C2×Dic6—24no—
181C2×Dic6—432no—
182C2×Dic6—44no—
183C2×Dic6—44no—
184C2×Dic6—44no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—24no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195#217—28no—
196#213—28no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—82no—
200C4⋊Dic3—22no—
201C4⋊Dic3—24no—
202C4⋊Dic3—22no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—82no—
206C4⋊Dic3C2×D414yes—
207Dic3⋊C4—28no—
208Dic3⋊C4—24no—
209C4×Dic3—22no—
210C4×Dic3—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8C2×Q812yescomplemented
217C22×Q8—31no—
218C22×Q8—62no—
219C22×Q8—68no—
220C22×Q8—61no—
221C22×Q8—128no—
222C22×D4—31no—
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C23×C4—62no—
226C23×C4—61no—
227C2×D8—34no—
228C2×D8—62no—
229C2×D8—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8C22×D312yes—
233C4⋊Q8—34no—
234C4⋊Q8—64no—
235C4⋊Q8—64no—
236C4⋊Q8—62no—
237C22⋊Q8—616no—
238C22⋊Q8—68no—
239C4×Q8—62no—
240C4×Q8—62no—
241C4×D4—22no—
242C4×D4—64no—
243C4×D4—64no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4D1214yes—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4C22×D312yes—
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—62no—
261C2×C42—32no—
262C2×C42C22×D311yescharacteristic
263#106—64no—
264#106—62no—
265C4⋊C8—22no—
266#102—24no—
267#102—64no—
268#102—62no—
269#102—62no—
270#97—62no—
271#95—21no—
272#95—21no—
273#95—62no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—41no—
282C22×D3—41no—
283C22×D3—42no—
284C22×D3—42no—
285C3×D4—22no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4C22⋊Q814yes—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—48no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×Q811yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C42−1+411yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—416no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3AGL(1,ℤ/8)14yes—
317D12—22no—
318D12—22no—
319D12—41no—
320D12—41no—
321C4×D3—84no—
322Dic6—28no—
323Dic6—416no—
324Dic6—48no—
325C3×C8—42no—
326C3⋊C8—22no—
327C3⋊C8—41no—
328C24—21no—
329C24—61no—
330C2×Q8—64no—
331C2×Q8—68no—
332C2×Q8—62no—
333C2×Q8—64no—
334C2×Q8—124no—
335C2×Q8—124no—
336C2×Q8—124no—
337C2×Q8—1232no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—32no—
342C2×D4—61no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4C2×D1212yes—
348C2×D4#23712yes—
349C2×D4D3×Q812yes—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—44no—
356C22×C4—68no—
357C22×C4—62no—
358C22×C4—61no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—122no—
362C22×C4—242no—
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C23×D311yescharacteristic
366D8—68no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—32no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—61no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—34no—
380C4⋊C4—68no—
381C4⋊C4—64no—
382C4⋊C4—62no—
383C4⋊C4—64no—
384C4⋊C4—62no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—242no—
388C4⋊C4C2×D1212yes—
389C22⋊C4—24no—
390C22⋊C4—68no—
391C22⋊C4—124no—
392C22⋊C4—122no—
393C22⋊C4—242no—
394C42—62no—
395C42—62no—
396C42C2×D1212yes—
397C42D3×D412yes—
398C22×C3—24no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3D4×Q811yescharacteristic
406C22×C3C2×AGL(1,ℤ/8)12yes—
407C22×C3#57712yes—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—82no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—82no—
417C3×C4C2×C22⋊Q811yescharacteristic
418C3×C4#53611yescharacteristic
419C3×C4#54811yescharacteristic
420C3×C4D4×Q811yescharacteristic
421Dic3—24no—
422Dic3—44no—
423Dic3—42no—
424Dic3—81no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—62no—
431C23—121no—
432C23—122no—
433C23—122no—
434C23—121no—
435C23C2×D3×D411yescharacteristic
436Q8—68no—
437Q8—128no—
438Q8—1216no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—62no—
443D4—62no—
444D4—121no—
445D4—121no—
446D4#89714yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—34no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—1216no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—244no—
468C2×C4#88212yes—
469C2×C4#88212yes—
470C2×C4C22×D1211yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×#23711yescharacteristic
473C2×C4C2×D3×Q811yescharacteristic
474C2×C4#101011yescharacteristic
475C2×C4#101011yescharacteristic
476C8—42no—
477C8—62no—
478C8—121no—
479C2×C3—44no—
480C2×C3#240911yescharacteristic
481C2×C3#300311yescharacteristic
482C2×C3#301011yescharacteristic
483C2×C3#337312yes—
484C2×C3#338412yes—
485D3—82no—
486C22—24no—
487C22—44no—
488C22—42no—
489C22—44no—
490C22—42no—
491C22—124no—
492C22—122no—
493C22—241no—
494C22—242no—
495C22—241no—
496C22C2×#88211yescharacteristic
497C22#559811yescharacteristic
498C22#563211yescharacteristic
499C22#577012yes—
500C22D3×AGL(1,ℤ/8)12yes—
501C4—22no—
502C4—22no—
503C4—42no—
504C4—64no—
505C4—82no—
506C4—124no—
507C4—122no—
508C4—241no—
509C4C2×#89711yescharacteristic
510C4#552611yescharacteristic
511C4#559811yescharacteristic
512C4#563211yescharacteristic
513C3#2433811yescharacteristic, complemented, Sylow
514C2—44no—
515C2—242no—
516C2#7401711yescharacteristic
517C2#7883611yescharacteristic
518C2#7892611yescharacteristic
519C2#8052112yes—
520C2#8115812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1804D3
hypocenterC3#24338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic6C2semidirect
3. C2×#4842C2semidirect
4. #74190C2semidirect
5. C3×#1804C2semidirect
6. #70473C2semidirect
8. #70170C2semidirect
9. #69555C2semidirect
10. #68898C2semidirect
11. #68827C2semidirect
56. #1804D3semidirect
79. C2×#213Q8semidirect
174. C2×D12C2×Q8semidirect
216. C2×C3⋊C8C2×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×D4×Dic62
C2×#50142
C2×#48422
#741902
C3×#18042
#704732
#701722
#701702
#695552
#688982
#688272
#243383