Finite Group Databasebeta

← #10830074

Subgroups — #10830074

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×#5577C211yesmaximal, characteristic, complemented
2C3×#2580C211yesmaximal, characteristic, complemented
3C2×#5015C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69658C212yesmaximal, complemented
7#69657C212yesmaximal, complemented
8#69627C212yesmaximal, complemented
9#69626C212yesmaximal, complemented
10#69556C211yesmaximal, characteristic, complemented
11#69553C211yesmaximal, characteristic, complemented
12#24196—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#213C2211yescharacteristic
17C22×#213C2212yes—
18#5577—24no—
19#5577—24no—
20C2×#895—22no—
21C2×#895—21no—
22C2×#874C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×D12C2211yescharacteristic
26C3×#399—21no—
27C3×#393—21no—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#385—22no—
31C3×#382C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5015—24no—
34#5014—24no—
35C2×#809C2212yes—
36C2×#809C2211yescharacteristic
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550—31no—
47C2×#459—31no—
48C2×#442—31no—
49#2580D311yescharacteristic, complemented
50#2041—32no—
51#2011—32no—
52#1985—31no—
53#1961—31no—
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×#239—28no—
61C2×#239—48no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12—22no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×D12C2311yescharacteristic
68C22×C4×D3—22no—
69C22×C4×D3—21no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4D412yes—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42C2312yes—
83C2×C3×C42C2311yescharacteristic
84C2×#213—22no—
85C2×#213—22no—
86C2×#213—24no—
87C2×#213—24no—
88C2×#213—22no—
89C2×#213—42no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2311yescharacteristic
96#895—28no—
97#895—216no—
98#874—24no—
99#874—24no—
100#874—22no—
101#874D414yes—
102C4×D12—24no—
103C4×D12—22no—
104C4×D12—24no—
105C4×D12D414yes—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—22no—
109#832—22no—
110#809—22no—
111#809—24no—
112#809—24no—
113#809—42no—
114#809—42no—
115#808—22no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—61no—
119C22×D8—31no—
120C22×D8—32no—
121#550—64no—
122#550—64no—
123C2×#127—31no—
124C2×#127—32no—
125C2×#121—62no—
126C2×#121—61no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×C4×D4—32no—
130C2×C4×D4C2×D311yescharacteristic
131#459—64no—
132#442—64no—
133C2×#106—31no—
134C2×C4⋊C8—31no—
135C2×#102—31no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#97—31no—
142#399—21no—
143#393—21no—
144#390C2×D312yes—
145#389C2×D312yes—
146#385—22no—
147#382C2×D311yescharacteristic
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—22no—
153C23×D3—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—24no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4#4112yescomplemented
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—42no—
165C22×C3×C4—82no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—21no—
172#239—432no—
173C2×D12—22no—
174C2×D12—21no—
175C2×D12—22no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—22no—
179C2×D12—21no—
180C2×D12—22no—
181C2×D12—28no—
182C2×D12—24no—
183C2×D12—22no—
184C2×D12—21no—
185C2×D12—28no—
186C2×D12—24no—
187C2×D12—44no—
188C2×D12—44no—
189C2×D12—44no—
190C2×D12—82no—
191C2×D12—82no—
192C2×D12—82no—
193C2×D12C2×D414yes—
194C2×D12C2×D412yes—
195C2×C4×D3—28no—
196C2×C4×D3—24no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×C4×D3—44no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—82no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—44no—
209C3×C22⋊C4—82no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42C2×D412yes—
213C3×C42C2×D412yes—
214#213—28no—
215#213—48no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3C2×D412yes—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8#4112yescomplemented
229C2×#41—68no—
230C2×#41—128no—
231C22×Q8#6711yescharacteristic
232C22×D4—31no—
233C22×D4—32no—
234C22×D4—61no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4—61no—
238C22×D4C22×D311yescharacteristic
239C23×C4—21no—
240C23×C4—62no—
241C23×C4—61no—
242C2×D8—62no—
243C2×D8—62no—
244C2×D8—62no—
245C2×D8—64no—
246C2×D8—64no—
247C2×D8—122no—
248C22×C8—31no—
249C22×C8—32no—
250C22×C8—31no—
251#127—34no—
252#127—64no—
253#127—64no—
254#127—62no—
255#121—68no—
256#121—616no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—34no—
262C4×D4—64no—
263C4×D4—64no—
264C4×D4—62no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4#6712yes—
268C2×C4⋊C4#6711yescharacteristic
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C42C22×D312yes—
277C2×C42C22×D311yescharacteristic
278#106—62no—
279C4⋊C8—62no—
280#102—64no—
281#102—62no—
282#102—64no—
283#102—62no—
284#102—122no—
285#102—122no—
286#97—62no—
287#95—21no—
288#95—21no—
289#95—22no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3C22×D411yescharacteristic
296C22×D3—24no—
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C3×Q8—28no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—41no—
312C3×D4—41no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—84no—
326C2×C3×C4#12012yes—
327C2×C3×C4#12012yes—
328C2×C3×C4AGL(1,ℤ/8)14yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335C2×Dic3—22no—
336C2×Dic3—42no—
337D12—22no—
338D12—22no—
339D12—24no—
340D12—24no—
341D12—48no—
342D12—416no—
343D12—42no—
344D12—42no—
345D12—41no—
346D12—41no—
347D12—84no—
348D12—88no—
349C4×D3—48no—
350C4×D3—416no—
351C3⋊C8—22no—
352C3⋊C8—41no—
353C3⋊C8—42no—
354C24—21no—
355C24—62no—
356C24—61no—
357#41—1232no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—32no—
366C2×D4—34no—
367C2×D4—68no—
368C2×D4—68no—
369C2×D4—64no—
370C2×D4—64no—
371C2×D4—62no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—62no—
375C2×D4—61no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—62no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—242no—
385C2×D4—242no—
386C2×D4—242no—
387C2×D4#23912yescomplemented
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—42no—
391C22×C4—68no—
392C22×C4—61no—
393C22×C4—64no—
394C22×C4—82no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4—124no—
398C22×C4C2×#6712yes—
399C22×C4C2×#6711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401C22×C4C2×#6712yes—
402C22×C4C23×D311yescharacteristic
403D8—68no—
404D8—128no—
405C2×C8—32no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—62no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—32no—
421C4⋊C4—61no—
422C4⋊C4—82no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—68no—
426C22⋊C4—64no—
427C22⋊C4—82no—
428C42—22no—
429C42—22no—
430C42D3×D412yes—
431C42D3×D412yes—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—82no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54811yescharacteristic
439C22×C3#55011yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)12yes—
441C22×C3#57612yes—
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—42no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—42no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—82no—
451C2×D3—82no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—24no—
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—82no—
459C3×C4—81no—
460C3×C4#52511yescharacteristic
461C3×C4#53711yescharacteristic
462C3×C4#54611yescharacteristic
463C3×C4#54711yescharacteristic
464Dic3—42no—
465C23—22no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23—64no—
471C23—62no—
472C23—121no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23—122no—
478C23—121no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23C22×#6711yescharacteristic
483Q8—28no—
484D4—22no—
485D4—22no—
486D4—41no—
487D4—41no—
488D4—62no—
489D4—62no—
490D4—64no—
491D4—64no—
492D4—128no—
493D4—1216no—
494D4—122no—
495D4—122no—
496D4—121no—
497D4—121no—
498D4—244no—
499D4—248no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—62no—
513C2×C4—84no—
514C2×C4—128no—
515C2×C4—1216no—
516C2×C4—122no—
517C2×C4#93712yes—
518C2×C4#93712yes—
519C2×C4#94914yes—
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×#23911yescharacteristic
523C2×C4#100711yescharacteristic
524C2×C4D3×#4111yescharacteristic
525C2×C4#100911yescharacteristic
526C8—62no—
527C8—121no—
528C8—122no—
529C2×C3—82no—
530C2×C3#240711yescharacteristic
531C2×C3#300311yescharacteristic
532C2×C3#300811yescharacteristic
533C2×C3#336612yes—
534C2×C3#339912yes—
535D3—82no—
536D3—84no—
537C22—42no—
538C22—44no—
539C22—81no—
540C22—81no—
541C22—82no—
542C22—122no—
543C22—124no—
544C22—124no—
545C22—122no—
546C22—122no—
547C22—124no—
548C22—242no—
549C22—242no—
550C22—242no—
551C22—242no—
552C22—242no—
553C22—242no—
554C22C2×#93711yescharacteristic
555C22#581711yescharacteristic
556C22C2×#94912yes—
557C22#583411yescharacteristic
558C22#583512yes—
559C4—22no—
560C4—22no—
561C4—24no—
562C4—42no—
563C4—81no—
564C4—82no—
565C4—122no—
566C4#562711yescharacteristic
567C4#562911yescharacteristic
568C4#563011yescharacteristic
569C4#563111yescharacteristic
570C3#2419611yescharacteristic, complemented, Sylow
571C2—82no—
572C2—242no—
573C2—244no—
574C2#8281211yescharacteristic
575C2#8292711yescharacteristic
576C2#8338311yescharacteristic
577C2#8363812yes—
578C2#8371112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2580C2
socleC23×C3C22×D4
hypercenter#2580D3
hypocenterC3#24196
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#5577C2semidirect
2. C3×#2580C2semidirect
3. C2×#5015C2semidirect
4. C2×#5014C2semidirect
5. #71635C2semidirect
6. #69658C2semidirect
7. #69657C2semidirect
8. #69627C2semidirect
9. #69626C2semidirect
10. #69556C2semidirect
11. #69553C2semidirect
49. #2580D3semidirect
161. C2×C3×D4#41semidirect
228. C2×C3⋊C8#41semidirect
387. C2×D4#239semidirect
570. C3#24196semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
C3×#25802
C2×#50152
C2×#50142
#716352
#696582
#696572
#696272
#696262
#695562
#695532
#241963