Finite Group Databasebeta

← #10823449

Subgroups — #10823449

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#5026C212yesmaximal, complemented
4#71699C212yesmaximal, complemented
5#71529C211yesmaximal, characteristic, complemented
6#69688C211yesmaximal, characteristic, complemented
7#69668C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69604C211yesmaximal, characteristic, complemented
10#69523C212yesmaximal, complemented
11#69474C212yesmaximal, complemented
12#23576—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#215C2212yes—
18C2×#930—21no—
19C2×#930C2212yes—
20#5477—28no—
21C2×#875—21no—
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×D12C2212yes—
25C2×#869C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#387—21no—
29C3×#385—21no—
30C3×#380C2211yescharacteristic
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C2×#835C2212yes—
35C2×#830C2212yes—
36#5026—28no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807C2212yes—
41C2×#803C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#440—32no—
48C4×C22⋊Q8D311yescharacteristic, complemented
49#2057—31no—
50#2041—31no—
51#2033—32no—
52#2024—31no—
53#2014—31no—
54#2000—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67—22no—
62C2×#235—28no—
63C2×#235—44no—
64C2×#235—44no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×Dic6—21no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C4×Q8—24no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—24no—
82C2×C3×C42—21no—
83C2×C3×C42—21no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#215—24no—
89C2×#215—24no—
90C2×#215—24no—
91#930—28no—
92#930—24no—
93#930—24no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2312yes—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2312yes—
100#875—28no—
101#874—24no—
102C4×D12—28no—
103C4×D12—24no—
104C4×D12D418yes—
105#869—24no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111#835—24no—
112#830—24no—
113#809—24no—
114#809—24no—
115#809—24no—
116#807—24no—
117#803—24no—
118C22×#41—61no—
119C22×SD16—32no—
120#532—68no—
121C2×C4⋊Q8—31no—
122C2×#127—31no—
123C2×#124—61no—
124C2×C22⋊Q8C2×D311yescharacteristic
125C2×#121—32no—
126C2×#121—61no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130C22×C42C2×D311yescharacteristic
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×C22⋊C8—32no—
138C2×#97—32no—
139#391C2×D311yescharacteristic
140#389C2×D311yescharacteristic
141#387—21no—
142#385—21no—
143#380C2×D311yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×D311yescharacteristic
146C4×C22⋊C4—21no—
147#340—32no—
148#340—32no—
149#336—31no—
150#336—31no—
151C24×C3C2×D411yescharacteristic
152C23×D3—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8#4112yescomplemented
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—28no—
178C2×#67—44no—
179C2×#67—44no—
180C22×Dic3—22no—
181#235—432no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—24no—
187C2×D12—24no—
188C2×D12—22no—
189C2×D12—42no—
190C2×D12—42no—
191C2×D12—42no—
192C2×D12—42no—
193C2×D12C2×D414yes—
194C2×C4×D3—28no—
195C2×C4×D3—44no—
196C2×C4×D3—44no—
197C2×Dic6—24no—
198C2×Dic6—42no—
199C2×Dic6—42no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4#4112yescomplemented
207C3×C22⋊C4—24no—
208C3×C22⋊C4—42no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—42no—
213C3×C42—42no—
214C3×C42C2×D414yes—
215#215—216no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—28no—
218C4⋊Dic3—24no—
219C4⋊Dic3—22no—
220C4⋊Dic3C2×D414yes—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8#4114yescomplemented
227C2×#41—68no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—61no—
231C22×Q8C22×D311yescharacteristic
232C22×D4—32no—
233C22×D4—62no—
234C22×D4—61no—
235C22×D4—61no—
236C23×C4—62no—
237C23×C4#6711yescharacteristic
238C23×C4#6711yescharacteristic
239C23×C4C22×D311yescharacteristic
240C2×SD16—64no—
241C2×SD16—64no—
242C2×SD16—64no—
243C22×C8—32no—
244C22×C8—32no—
245C4⋊Q8—64no—
246#127—64no—
247#124—68no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250#121—64no—
251#121—64no—
252#121—68no—
253C4×Q8—24no—
254C4×D4—38no—
255C4×D4—68no—
256C4×D4—64no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4C22×D311yescharacteristic
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—62no—
269C2×C42—24no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42#6711yescharacteristic
273C2×C42#6711yescharacteristic
274C2×C42C22×D311yescharacteristic
275C2×C42C22×D311yescharacteristic
276#107—64no—
277#103—64no—
278#102—64no—
279#102—64no—
280#102—64no—
281C22⋊C8—64no—
282#97—64no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3AGL(1,ℤ/8)12yes—
295C23×C3C22×D411yescharacteristic
296C22×D3—24no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C3×Q8—24no—
304C3×Q8—42no—
305C2×C3×C4—28no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—44no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4#12012yes—
333C2×C3×C4#12012yes—
334C2×C3×C4AGL(1,ℤ/8)12yes—
335C2×C3×C4#13712yes—
336C2×C3×C4C22×D412yes—
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4112yes—
340#67—416no—
341C2×Dic3—24no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344D12—24no—
345D12—24no—
346D12—48no—
347D12—48no—
348D12—42no—
349D12—42no—
350C4×D3—416no—
351Dic6—48no—
352C3⋊C8—24no—
353C3⋊C8—42no—
354C24—62no—
355C24C2×#6711yescharacteristic
356#41—1232no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—64no—
361C2×Q8—122no—
362C2×Q8—122no—
363C2×Q8#23512yescomplemented
364C2×D4—34no—
365C2×D4—64no—
366C2×D4—64no—
367C2×D4—68no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—122no—
374C2×D4—122no—
375C2×D4—124no—
376C2×D4—124no—
377C2×D4—122no—
378C2×D4—122no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—62no—
394C22×C4—68no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401C22×C4C2×#6711yescharacteristic
402C22×C4C23×D311yescharacteristic
403SD16—616no—
404C2×C8—34no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C2×C8—62no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—34no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—64no—
418C4⋊C4—62no—
419C4⋊C4#23512yescomplemented
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—68no—
423C22⋊C4—68no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—42no—
428C42—42no—
429C42D3×D414yes—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#52511yescharacteristic
439C22×C3#53211yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3C2×#13711yescharacteristic
443C22×C3#57511yescharacteristic
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—42no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4#52512yes—
461C3×C4#53212yes—
462Dic3—44no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—64no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23#94912yes—
477C23C22×#6711yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—128no—
481D4—64no—
482D4—64no—
483D4—128no—
484D4—128no—
485D4—1216no—
486D4—122no—
487D4—122no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—64no—
516C2×C4—1216no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4#93712yes—
520C2×C4#93712yes—
521C2×C4#94212yes—
522C2×C4#94912yes—
523C2×C4C2×#23511yescharacteristic
524C2×C4C2×#23511yescharacteristic
525C2×C4C2×D3×D412yes—
526C2×C4D3×#4112yes—
527C8—64no—
528C8—122no—
529C2×C3—42no—
530C2×C3#238811yescharacteristic
531C2×C3#300311yescharacteristic
532C2×C3#300711yescharacteristic
533C2×C3#311211yescharacteristic
534C2×C3#311511yescharacteristic
535C2×C3#318011yescharacteristic
536C2×C3#318211yescharacteristic
537D3—84no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—242no—
549C22—242no—
550C22—242no—
551C22—242no—
552C22#579111yescharacteristic
553C22C2×#93711yescharacteristic
554C22C2×#94211yescharacteristic
555C22C2×#94911yescharacteristic
556C22C2×#94911yescharacteristic
557C22#582211yescharacteristic
558C22#583211yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—124no—
569C4#554712yes—
570C4#554812yes—
571C3#2357611yescharacteristic, complemented, Sylow
572C2—42no—
573C2—244no—
574C2#8240011yescharacteristic
575C2#8244411yescharacteristic
576C2#8253111yescharacteristic
577C2#8273511yescharacteristic
578C2#8292711yescharacteristic
579C2#8313711yescharacteristic
580C2#8334711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenterC4×C22⋊Q8D3
hypocenterC3#23576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C4×C22⋊Q8C2semidirect
3. C2×#5026C2semidirect
4. #71699C2semidirect
5. #71529C2semidirect
6. #69688C2semidirect
7. #69668C2semidirect
8. #69627C2semidirect
9. #69604C2semidirect
10. #69523C2semidirect
11. #69474C2semidirect
48. C4×C22⋊Q8D3semidirect
156. C2×C3×Q8#41semidirect
206. C3×C4⋊C4#41semidirect
226. C2×C3⋊C8#41semidirect
363. C2×Q8#235semidirect
419. C4⋊C4#235semidirect
571. C3#23576semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C3×C4×C22⋊Q82
C2×#50262
#716992
#715292
#696882
#696682
#696272
#696042
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