Finite Group Databasebeta

← #10829272

Subgroups — #10829272

Order 768.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5551C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic, complemented
3#82499C211yesmaximal, characteristic, complemented
4C2×#5021C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#71605C211yesmaximal, characteristic, complemented
7#70632C212yesmaximal, complemented
8#70268C212yesmaximal, complemented
9#70206C212yesmaximal, complemented
10#69837C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24132—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#946C2212yes—
18C2×#938C2212yes—
19C2×#924—21no—
20#5551—24no—
21#5551—24no—
22C2×D3×C4⋊C4C2212yes—
23C2×#883—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5021—24no—
36#4991—21no—
37#4987C2212yes—
38#4981—21no—
39C2×#807C2212yes—
40#4940—22no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#477—31no—
48C2×#440D311yescharacteristic, complemented
49#2536—31no—
50#2033—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1872—32no—
55#1852—32no—
56#1804—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C2×#235—28no—
62C2×#235—48no—
63C22×D12D411yescharacteristic
64C22×C4×D3—22no—
65C22×C4×D3C2312yes—
66C22×Dic6—21no—
67C2×C3×SD16—22no—
68C2×C3×SD16—22no—
69C2×C3×SD16—22no—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83#946—24no—
84#946—42no—
85#938—24no—
86#938—24no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4D412yes—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94#924—28no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8C2312yes—
98D3×C4⋊C4—24no—
99D3×C4⋊C4—24no—
100#883—216no—
101C4×D12—22no—
102C4×D12—22no—
103#869—22no—
104#869D414yes—
105C3×#107—22no—
106C3×#103—22no—
107C3×#102—22no—
108C3×#97—22no—
109#835—24no—
110#832—42no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—22no—
114(C2×D3)⋊C8—24no—
115(C2×D3)⋊C8—42no—
116#807—24no—
117C4⋊(C3⋊C8)—22no—
118C22×#41—61no—
119C22×SD16C2×D311yescharacteristic
120#547—64no—
121#547—64no—
122C2×C4⋊Q8—31no—
123C2×#123—62no—
124C2×C22⋊Q8—32no—
125C2×C22⋊Q8—61no—
126C2×#121—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#477—64no—
132#440—24no—
133C2×#107—32no—
134C2×#107C2×D311yescharacteristic
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103C2×D311yescharacteristic
138C2×#102—32no—
139C2×#102C2×D311yescharacteristic
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97C2×D311yescharacteristic
143#398—61no—
144#394—61no—
145#388—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C23×D3C2×D412yes—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8#4112yescomplemented
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—44no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C2×#67—28no—
174C2×#67—44no—
175C2×#67—44no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3C2×D412yes—
179#235—432no—
180C2×D12—24no—
181C2×D12—22no—
182C2×C4×D3—28no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×Dic6—24no—
192C2×Dic6—44no—
193C3×SD16—28no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8#4112yescomplemented
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4C2×D412yes—
205C3×C22⋊C4—24no—
206C3×C42—42no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209#217—28no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—44no—
212(C2×D3)⋊C4—44no—
213C4⋊Dic3—24no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3C2×D414yes—
219Dic3⋊C4—28no—
220Dic3⋊C4—44no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×#41—68no—
225C2×#41—128no—
226C22×Q8—61no—
227C22×Q8C22×D311yescharacteristic
228C22×D4—31no—
229C22×D4—62no—
230C22×D4C22×D311yescharacteristic
231C23×C4—21no—
232C23×C4—32no—
233C23×C4—62no—
234C2×SD16—22no—
235C2×SD16—22no—
236C2×SD16—22no—
237C22×C8—32no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242#123—616no—
243C22⋊Q8—68no—
244C22⋊Q8—64no—
245C22⋊Q8—122no—
246#121—64no—
247#121—64no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4#6714yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—32no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—21no—
271C2×C42C22×D311yescharacteristic
272#107—22no—
273#107—64no—
274C4⋊C8—62no—
275#103—22no—
276#103—64no—
277#102—22no—
278#102—122no—
279C22⋊C8—64no—
280C22⋊C8—122no—
281#97—22no—
282#95—62no—
283#95—61no—
284#95—61no—
285C23×C3—22no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3AGL(1,ℤ/8)14yes—
298C3×Q8—22no—
299C3×Q8—22no—
300C3×Q8—41no—
301C3×Q8—41no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—41no—
305C3×D4—41no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—48no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324#67—416no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334D12—28no—
335C4×D3—416no—
336C4×D3—42no—
337C4×D3—42no—
338C4×D3—42no—
339C4×D3—42no—
340Dic6—48no—
341C3×C8—22no—
342C3×C8—41no—
343C3⋊C8—42no—
344C24—21no—
345C24—32no—
346#41—1232no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—64no—
353C2×Q8—124no—
354C2×Q8#23512yescomplemented
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—64no—
361C2×D4—68no—
362C2×D4—62no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4C2×#6712yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—32no—
371C22×C4—44no—
372C22×C4—68no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386SD16—28no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8#23512yescomplemented
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—34no—
400C4⋊C4—42no—
401C4⋊C4—68no—
402C4⋊C4—62no—
403C4⋊C4—64no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—124no—
408C4⋊C4C2×#6712yes—
409C22⋊C4—24no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—124no—
413C22⋊C4—124no—
414C42—42no—
415C42D3×D412yes—
416C42C2×#6712yes—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3C2×#12011yescharacteristic
423C22×C3#54611yescharacteristic
424C22×C3#54711yescharacteristic
425C22×C3C2×AGL(1,ℤ/8)12yes—
426C22×C3#57812yes—
427C2×D3—24no—
428C2×D3—24no—
429C2×D3—24no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—81no—
439C3×C4#52511yescharacteristic
440C3×C4#53711yescharacteristic
441C3×C4#54811yescharacteristic
442C3×C4#55011yescharacteristic
443Dic3—42no—
444Dic3—44no—
445Dic3—82no—
446C23—22no—
447C23—34no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23C2×D3×D411yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—128no—
464D4—22no—
465D4—22no—
466D4—41no—
467D4—41no—
468D4—68no—
469D4—1216no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—48no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—1216no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4#93712yes—
495C2×C4#93712yes—
496C2×C4C2×#23511yescharacteristic
497C2×C4C2×#23511yescharacteristic
498C2×C4C2×D3×D411yescharacteristic
499C2×C4#100911yescharacteristic
500C2×C4#101011yescharacteristic
501C2×C4C22×#6711yescharacteristic
502C8—22no—
503C8—41no—
504C8—122no—
505C2×C3—82no—
506C2×C3#240711yescharacteristic
507C2×C3#300311yescharacteristic
508C2×C3#301111yescharacteristic
509C2×C3#330812yes—
510C2×C3#334312yes—
511D3—44no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—64no—
516C22—64no—
517C22—64no—
518C22—82no—
519C22—82no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—122no—
524C22#555011yescharacteristic
525C22#555111yescharacteristic
526C22D3×AGL(1,ℤ/8)12yes—
527C22#577912yes—
528C22C2×#93711yescharacteristic
529C4—22no—
530C4—22no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—122no—
535C4—124no—
536C4—242no—
537C4#555211yescharacteristic
538C4#555311yescharacteristic
539C4#579111yescharacteristic
540C4#583311yescharacteristic
541C3#2413211yescharacteristic, complemented, Sylow
542C2—82no—
543C2—124no—
544C2#7979812yes—
545C2#7980612yes—
546C2#8240211yescharacteristic
547C2#8394411yescharacteristic
548C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC2×#5551C2
hypercenterC2×#440D3
hypocenterC3#24132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C2×C3×#440C2semidirect
3. #82499C2semidirect
4. C2×#5021C2semidirect
5. #72054C2semidirect
6. #71605C2semidirect
7. #70632C2semidirect
8. #70268C2semidirect
9. #70206C2semidirect
10. #69837C2semidirect
11. #69828C2semidirect
48. C2×#440D3semidirect
158. C2×C3×Q8#41semidirect
199. C2×C3×C8#41semidirect
354. C2×Q8#235semidirect
395. C2×C8#235semidirect
541. C3#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#55512
C2×C3×#4402
#824992
C2×#50212
#720542
#716052
#706322
#702682
#702062
#698372
#698282
#241323