Finite Group Databasebeta

← #10883497

Subgroups — #10883497

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79067C211yesmaximal, characteristic, complemented
2#79055C211yesmaximal, characteristic
3C2×C4×#204C211yesmaximal, characteristic, complemented
4#78458C211yesmaximal, characteristic, complemented
5C3×#1837C211yesmaximal, characteristic, complemented
6#70528C211yesmaximal, characteristic, complemented
7#70527C211yesmaximal, characteristic, complemented
8#68949C212yesmaximal, complemented
9#68948C212yesmaximal, complemented
10#68943C212yesmaximal, complemented
11#68922C212yesmaximal, complemented
12#28175—31nomaximal, Sylow
13C2×#909C2212yes—
14C2×#908C2212yes—
15C22×#204C2211yescharacteristic
16C2×#901C2212yes—
17#5477C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C2×#824C2212yes—
24#5133C2212yes—
25#5133C2212yes—
26C2×#823C2212yes—
27C2×#821C2211yescharacteristic
28#5110C2212yes—
29C2×#817C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4924C2211yescharacteristic
32#4849—21no—
33#4848—21no—
34#4845—22no—
35#4844—22no—
36#4843—21no—
37#4841—21no—
38C4×#204—21no—
39C4×#204—21no—
40C4×#204—21no—
41C4×#204—21no—
42#4688—21no—
43#4688—21no—
44C8×D12—21no—
45C8×D12—21no—
46C3×#341C2212yes—
47C3×#341C2212yes—
48#4554C2212yes—
49#4554C2212yes—
50#3037—31no—
51#3036—31no—
52C2×C4×SD16—31no—
53#2931—31no—
54#2079—32no—
55#2019—32no—
56#1908—32no—
57#1907—32no—
58#1837D311yescharacteristic, complemented
59#1801—31no—
60#1800—31no—
61C2×#235C2312yes—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×Dic6C2311yescharacteristic
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70#930—21no—
71#930—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×Dic3⋊C4—21no—
76C2×C4×Dic3—21no—
77C2×#208C2312yes—
78#909—24no—
79#909—42no—
80#908—24no—
81#908—42no—
82C2×Dic12—22no—
83C2×D24—22no—
84C2×#204—22no—
85C2×#204—22no—
86C2×#204—21no—
87C2×#204—21no—
88C2×#204—22no—
89C2×#204—22no—
90#901—42no—
91#901—42no—
92C2×ΓL(1,𝔽25)—22no—
93C2×D3×C8—22no—
94#875D412yes—
95#874D411yescharacteristic
96C4×D12—21no—
97C4×D12—21no—
98C4×D12—21no—
99C4×D12—21no—
100C4×D12—21no—
101C4×D12—21no—
102#869D411yescharacteristic
103C4×Dic6—21no—
104C4×Dic6—21no—
105C4×Dic6—21no—
106C4×Dic6—21no—
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×C4⋊C8—21no—
110C3×C4⋊C8—21no—
111C3×C4×C8D411yescharacteristic
112C3×C4×C8D411yescharacteristic
113C3×C4×C8D411yescharacteristic
114C3×C4×C8D411yescharacteristic
115#824—42no—
116#823—42no—
117#821—22no—
118#821—22no—
119#821—21no—
120#821—21no—
121(C2×D3)⋊C8—22no—
122(C2×D3)⋊C8—22no—
123#817—21no—
124#817—21no—
125#816—22no—
126#816—22no—
127#816—21no—
128#816—21no—
129C4⋊(C3⋊C8)—21no—
130C4⋊(C3⋊C8)—21no—
131C2×#137—32no—
132C2×AGL(1,ℤ/8)—32no—
133C22×SD16—31no—
134C2×#131—32no—
135#532—31no—
136C2×C4×Q8—31no—
137C2×C4×D4—31no—
138#464—62no—
139#463—62no—
140#462—61no—
141#461—61no—
142#460—61no—
143#459—61no—
144C4×SD16—61no—
145C4×SD16—61no—
146C4×SD16—61no—
147C4×SD16—61no—
148#436—61no—
149#436—61no—
150C8×D4—61no—
151C8×D4—61no—
152C2×#106—31no—
153#423—31no—
154#423C2×D311yescharacteristic
155#423C2×D311yescharacteristic
156#417—32no—
157#417—32no—
158C2×#103—31no—
159C2×#102—31no—
160C2×#101—32no—
161C2×#100—32no—
162#408—32no—
163C2×C4×C8C2×D311yescharacteristic
164#343—32no—
165#343—32no—
166#341C2×D312yes—
167#341C2×D312yes—
168C23×D3—21no—
169C22×C3×C4#4111yescharacteristic
170C22×C3×C4#4111yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—22no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176#235—22no—
177#235—22no—
178#235—22no—
179#235—22no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12C2×D412yes—
188C2×D12#4112yes—
189C2×C4×D3—22no—
190C2×C4×D3—41no—
191C2×C4×D3—41no—
192C2×C4×D3—82no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6C2×D412yes—
199C2×Dic6#4112yes—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16—22no—
203C3×M16—22no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8C2×D411yescharacteristic
211C2×C3×C8C2×D411yescharacteristic
212C2×C3×C8C2×D411yescharacteristic
213C2×C3×C8C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216C3×C42C2×D411yescharacteristic
217C3×C42C2×D411yescharacteristic
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—41no—
221(C2×D3)⋊C4—41no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228Dic3⋊C4—41no—
229Dic3⋊C4—41no—
230C4×Dic3—41no—
231C4×Dic3—41no—
232#208—22no—
233#208—22no—
234C2×C3⋊C8—22no—
235Dic12—44no—
236D24—44no—
237#204—24no—
238#204—24no—
239#204—44no—
240#204—44no—
241ΓL(1,𝔽25)—44no—
242D3×C8—44no—
243C2×#41—32no—
244C22×Q8—31no—
245C22×D4—31no—
246C23×C4—61no—
247#137—64no—
248#137—122no—
249AGL(1,ℤ/8)—64no—
250AGL(1,ℤ/8)—122no—
251C2×Q16—62no—
252C2×SD16—62no—
253C2×SD16—62no—
254C2×SD16—62no—
255C2×SD16—62no—
256C2×SD16—61no—
257C2×SD16—61no—
258C2×D8—62no—
259#131—122no—
260#131—122no—
261C2×M16—32no—
262C2×M16—62no—
263C2×M16C22×D312yes—
264C2×M16C22×D312yes—
265C22×C8—62no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C4⋊Q8—31no—
269#127—31no—
270#124—32no—
271#121—61no—
272#121—61no—
273C4×Q8—61no—
274C4×Q8—61no—
275C4×Q8—61no—
276C4×Q8—61no—
277C4×D4—61no—
278C4×D4—61no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#106—61no—
290#106—61no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—61no—
296C4⋊C8—61no—
297#103—62no—
298#103—62no—
299#103—61no—
300#103—61no—
301#102—62no—
302#102—62no—
303#102—61no—
304#102—61no—
305#101—122no—
306#100—122no—
307C22⋊C8—62no—
308C22⋊C8—62no—
309C4×C8D1211yescharacteristic
310C4×C8D1211yescharacteristic
311C4×C8D1211yescharacteristic
312C4×C8D1211yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—81no—
319C22×D3—81no—
320C2×C3×C4—22no—
321C2×C3×C4—41no—
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4H(5,𝔽2)11yescharacteristic
331C2×C3×C4H(5,𝔽2)11yescharacteristic
332#67—42no—
333#67—42no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—81no—
341C2×Dic3—81no—
342D12—22no—
343D12—22no—
344D12—22no—
345D12—22no—
346D12—42no—
347D12—41no—
348D12—41no—
349D12—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—82no—
353C4×D3—82no—
354Dic6—22no—
355Dic6—22no—
356Dic6—22no—
357Dic6—22no—
358Dic6—41no—
359Dic6—41no—
360C3×C8—21no—
361C3×C8—21no—
362C3×C8—21no—
363C3×C8—21no—
364C3×C8—42no—
365C3×C8—42no—
366C3⋊C8—42no—
367C24—61no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—62no—
372C2×Q8—32no—
373C2×Q8—32no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×D4—32no—
380C2×D4—32no—
381C2×D4—62no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—121no—
394C22×C4—121no—
395C22×C4—242no—
396C22×C4#23911yescharacteristic
397C22×C4#23911yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—124no—
400SD16—64no—
401SD16—64no—
402SD16—124no—
403SD16—124no—
404D8—124no—
405M16—22no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—62no—
410M16—62no—
411M16—124no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—62no—
419C2×C8—124no—
420C2×C8C2×D1211yescharacteristic
421C2×C8C2×D1211yescharacteristic
422C2×C8C2×D1211yescharacteristic
423C2×C8C2×D1211yescharacteristic
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—121no—
431C4⋊C4—121no—
432C22⋊C4—62no—
433C22⋊C4—62no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C42—121no—
437C42—121no—
438C42C2×D1211yescharacteristic
439C42C2×D1211yescharacteristic
440C42D3×D411yescharacteristic
441C42D3×D411yescharacteristic
442C22×C3—21no—
443C22×C3—21no—
444C22×C3C2×#12111yescharacteristic
445C22×C3#55311yescharacteristic
446C22×C3#55311yescharacteristic
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—82no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4#47112yes—
459C3×C4#52511yescharacteristic
460C3×C4#53511yescharacteristic
461C3×C4#54611yescharacteristic
462C3×C4#54611yescharacteristic
463Dic3—42no—
464Dic3—42no—
465Dic3—81no—
466Dic3—81no—
467C23—62no—
468C23—62no—
469C23—121no—
470C23—121no—
471C23—241no—
472C23—241no—
473C23C2×#23911yescharacteristic
474Q8—62no—
475Q8—62no—
476Q8—62no—
477Q8—62no—
478Q8—121no—
479Q8—121no—
480D4—62no—
481D4—62no—
482D4—62no—
483D4—62no—
484D4—122no—
485D4—122no—
486D4—122no—
487D4—122no—
488D4—121no—
489D4—121no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—242no—
501C2×C4—242no—
502C2×C4—241no—
503C2×C4—241no—
504C2×C4#89511yescharacteristic
505C2×C4#89511yescharacteristic
506C2×C4#89511yescharacteristic
507C2×C4#89511yescharacteristic
508C2×C4C22×D1211yescharacteristic
509C2×C4C2×D3×D411yescharacteristic
510C2×C4D3×#4111yescharacteristic
511C2×C4D3×#4111yescharacteristic
512C2×C4#100911yescharacteristic
513C2×C4#100911yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—42no—
519C8—42no—
520C8—122no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3#253511yescharacteristic
524C2×C3C2×#47111yescharacteristic
525C2×C3#306711yescharacteristic
526D3—82no—
527D3—82no—
528C22—21no—
529C22—21no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22—242no—
535C22—242no—
536C22—241no—
537C22—241no—
538C22C2×#89511yescharacteristic
539C22#569511yescharacteristic
540C22#569511yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—122no—
545C4—122no—
546C4—241no—
547C4—241no—
548C4#489812yes—
549C4#552411yescharacteristic
550C4#552511yescharacteristic
551C4#567611yescharacteristic
552C4#567611yescharacteristic
553C3#2817511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—242no—
557C2—242no—
558C2#7423911yescharacteristic
559C2C2×#489811yescharacteristic
560C2#7925611yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1837C2
socleC22×C3C2×#121
prime radical#79055C2
hypercenter#1837D3
hypocenterC3#28175
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1837

Complements

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

NH ≅ G/Ntype
1. #79067C2semidirect
3. C2×C4×#204C2semidirect
4. #78458C2semidirect
5. C3×#1837C2semidirect
6. #70528C2semidirect
7. #70527C2semidirect
8. #68949C2semidirect
9. #68948C2semidirect
10. #68943C2semidirect
11. #68922C2semidirect
58. #1837D3semidirect
553. C3#28175semidirect

Maximal subgroups

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

subgroupindex
#790672
#790552
C2×C4×#2042
#784582
C3×#18372
#705282
#705272
#689492
#689482
#689432
#689222
#281753