Finite Group Databasebeta

← #10830072

Subgroups — #10830072

Order 768.

1428 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5553C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic, complemented
5#72044C212yesmaximal, complemented
6#71605C211yesmaximal, characteristic, complemented
7#70584C212yesmaximal, complemented
8#70211C212yesmaximal, complemented
9#70208C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69823C211yesmaximal, characteristic, complemented
12#24196—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#940C2212yes—
18C22×#213C2212yes—
19C2×#930—21no—
20#5553—24no—
21#5553—24no—
22C2×#893C2212yes—
23C2×#884—22no—
24C2×#874C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#442—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5221C2212yes—
32C2×#832C2212yes—
33#5210—22no—
34C2×#821C2212yes—
35C2×#821C2212yes—
36#5015—24no—
37#4972C2212yes—
38C2×#809C2212yes—
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#550—31no—
47C2×#459—31no—
48C2×#442D311yescharacteristic, complemented
49#2580—31no—
50#2041—32no—
51#2011—32no—
52#1985—31no—
53#1961—31no—
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C22×#67—22no—
62C22×#67—22no—
63C2×#235—28no—
64C2×#235—48no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×Dic6D411yescharacteristic
70C2×C3×D8—22no—
71C2×C3×D8—22no—
72C2×C3×D8—22no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4D414yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—22no—
85C2×#217—22no—
86#940—24no—
87#940—24no—
88C2×#213—24no—
89C2×#213—24no—
90C2×#213—42no—
91#930—28no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D412yes—
97C2×C4×Dic3C2312yes—
98C22×C3⋊C8C2312yes—
99#893—24no—
100#893—24no—
101#884—216no—
102#874—22no—
103#874D414yes—
104C4×Dic6—22no—
105C4×Dic6—22no—
106C3×#106—22no—
107C3×#102—22no—
108C3×#102—22no—
109C3×#97—22no—
110#832—42no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—22no—
114#821—24no—
115#821—42no—
116#809—24no—
117C4⋊(C3⋊C8)—22no—
118C22×#41—61no—
119C22×D8—32no—
120C22×D8C2×D311yescharacteristic
121#550—64no—
122#550—64no—
123C2×#127—32no—
124C2×#127—31no—
125C2×#121—62no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×D4—32no—
129C2×C4×D4C2×D311yescharacteristic
130C2×C4×D4C2×D311yescharacteristic
131#459—64no—
132#442—24no—
133C2×#106C2×D311yescharacteristic
134C2×C4⋊C8—31no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102C2×D311yescharacteristic
140C2×#102C2×D311yescharacteristic
141C2×#97C2×D311yescharacteristic
142#399—61no—
143#393—61no—
144#390—32no—
145#389—32no—
146#385—62no—
147#382—31no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C24×C3—21no—
153C23×D3—22no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4C2×D412yes—
165C2×C3×D4#4112yescomplemented
166C22×C3×C4—21no—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—44no—
170C22×C3×C4—42no—
171C22×C3×C4—82no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—28no—
176C2×#67—28no—
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—82no—
180C2×#67—82no—
181C22×Dic3—22no—
182C22×Dic3C2×D412yes—
183C22×Dic3C2×D412yes—
184#235—432no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—24no—
190C2×D12—22no—
191C2×D12—24no—
192C2×D12—44no—
193C2×D12—82no—
194C2×D12C2×D414yes—
195C2×C4×D3—28no—
196C2×C4×D3—44no—
197C2×C4×D3—44no—
198C2×Dic6—22no—
199C2×Dic6—24no—
200C3×D8—28no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8#4112yescomplemented
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—42no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215#217—44no—
216#217—82no—
217#213—48no—
218(C2×D3)⋊C4—28no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222Dic3⋊C4—24no—
223Dic3⋊C4—82no—
224C4×Dic3—22no—
225C4×Dic3—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×#41—68no—
230C2×#41—128no—
231C22×Q8—31no—
232C22×D4—32no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4—61no—
236C22×D4—61no—
237C22×D4C22×D311yescharacteristic
238C22×D4C22×D311yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—62no—
242C2×D8—22no—
243C2×D8—22no—
244C2×D8—22no—
245C2×D8—64no—
246C2×D8—64no—
247C2×D8—122no—
248C22×C8—32no—
249C22×C8C22×D311yescharacteristic
250C22×C8C22×D311yescharacteristic
251#127—34no—
252#127—64no—
253#127—64no—
254#127—62no—
255#121—616no—
256#121—68no—
257C4×Q8—62no—
258C4×Q8—62no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—64no—
263C4×D4—64no—
264C4×D4#6714yes—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C42—32no—
277C2×C42C22×D311yescharacteristic
278#106—22no—
279C4⋊C8—62no—
280#102—22no—
281#102—22no—
282#102—64no—
283#102—64no—
284#102—122no—
285#102—122no—
286#97—22no—
287#95—62no—
288#95—61no—
289#95—61no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3C22×D411yescharacteristic
301C22×D3—24no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—41no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—48no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—84no—
324C2×C3×C4#12012yes—
325C2×C3×C4#12012yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4H(5,𝔽2)11yescharacteristic
331C2×C3×C42−1+411yescharacteristic
332#67—416no—
333#67—88no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3AGL(1,ℤ/8)14yes—
343D12—24no—
344D12—24no—
345D12—48no—
346D12—42no—
347D12—42no—
348D12—84no—
349C4×D3—416no—
350Dic6—28no—
351C3×C8—22no—
352C3×C8—41no—
353C3⋊C8—42no—
354C24—21no—
355C24—21no—
356C24—62no—
357#41—1232no—
358C2×Q8—64no—
359C2×Q8—62no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—34no—
371C2×D4—68no—
372C2×D4—68no—
373C2×D4—64no—
374C2×D4—64no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—124no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—242no—
384C2×D4—242no—
385C2×D4—242no—
386C2×D4#23512yescomplemented
387C2×D4C2×#6712yes—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—32no—
392C22×C4—32no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—68no—
396C22×C4—62no—
397C22×C4—82no—
398C22×C4—124no—
399C22×C4—124no—
400C22×C4D3×D411yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4C23×D311yescharacteristic
403D8—28no—
404D8—128no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8#23512yescomplemented
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—64no—
418C4⋊C4—62no—
419C4⋊C4—62no—
420C4⋊C4—62no—
421C4⋊C4—242no—
422C4⋊C4C2×#6712yes—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—68no—
426C22⋊C4—124no—
427C22⋊C4—242no—
428C42—62no—
429C42—62no—
430C42D3×D412yes—
431C42C2×#6712yes—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3C2×#12011yescharacteristic
443C22×C3#54811yescharacteristic
444C22×C3#55011yescharacteristic
445C22×C3C2×AGL(1,ℤ/8)12yes—
446C22×C3#57612yes—
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—82no—
451C2×D3—82no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—81no—
458C3×C4#52511yescharacteristic
459C3×C4#53711yescharacteristic
460C3×C4#54611yescharacteristic
461C3×C4#54711yescharacteristic
462Dic3—24no—
463Dic3—42no—
464Dic3—82no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—64no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23C2×D3×D411yescharacteristic
483Q8—68no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—64no—
493D4—64no—
494D4—128no—
495D4—1216no—
496D4—122no—
497D4—122no—
498D4—244no—
499D4—248no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—34no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—84no—
514C2×C4—1216no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4#93712yes—
519C2×C4#93712yes—
520C2×C4C2×#23511yescharacteristic
521C2×C4C2×#23511yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4#100911yescharacteristic
524C2×C4#101011yescharacteristic
525C2×C4C22×#6711yescharacteristic
526C8—22no—
527C8—41no—
528C8—122no—
529C2×C3—82no—
530C2×C3—82no—
531C2×C3#240711yescharacteristic
532C2×C3#300311yescharacteristic
533C2×C3#300811yescharacteristic
534C2×C3#336612yes—
535C2×C3#339912yes—
536D3—84no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—44no—
541C22—42no—
542C22—81no—
543C22—82no—
544C22—81no—
545C22—82no—
546C22—82no—
547C22—124no—
548C22—124no—
549C22—124no—
550C22—242no—
551C22—242no—
552C22—242no—
553C22—242no—
554C22#555211yescharacteristic
555C22#555311yescharacteristic
556C22D3×AGL(1,ℤ/8)12yes—
557C22#577412yes—
558C22C2×#93711yescharacteristic
559C4—22no—
560C4—22no—
561C4—42no—
562C4—64no—
563C4—81no—
564C4—122no—
565C4—242no—
566C4#555011yescharacteristic
567C4#555111yescharacteristic
568C4#579111yescharacteristic
569C4#583311yescharacteristic
570C3#2419611yescharacteristic, complemented, Sylow
571C2—82no—
572C2—82no—
573C2—244no—
574C2#7985112yes—
575C2#7985712yes—
576C2#8240211yescharacteristic
577C2#8394411yescharacteristic
578C2#8394711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
hypercenterC2×#442D3
hypocenterC3#24196
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#5553C2semidirect
2. C2×C3×#442C2semidirect
3. #82388C2semidirect
4. C2×#5015C2semidirect
5. #72044C2semidirect
6. #71605C2semidirect
7. #70584C2semidirect
8. #70211C2semidirect
9. #70208C2semidirect
10. #69838C2semidirect
11. #69823C2semidirect
48. C2×#442D3semidirect
165. C2×C3×D4#41semidirect
206. C2×C3×C8#41semidirect
386. C2×D4#235semidirect
413. C2×C8#235semidirect
570. C3#24196semidirect

Maximal subgroups

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

subgroupindex
C2×#55532
C2×C3×#4422
#823882
C2×#50152
#720442
#716052
#705842
#702112
#702082
#698382
#698232
#241963