Finite Group Databasebeta

← #10847375

Subgroups — #10847375

Order 768.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2C3×#2855C211yesmaximal, characteristic, complemented
3C2×#5048C212yesmaximal, complemented
4C2×#4524C211yesmaximal, characteristic, complemented
5#69688C211yesmaximal, characteristic, complemented
6#69687C212yesmaximal, complemented
7#69668C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69626C212yesmaximal, complemented
10#69605C211yesmaximal, characteristic, complemented
11#69515C212yesmaximal, complemented
12#25495—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×C22⋊Q8C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×#215C2212yes—
17C2×#930—21no—
18C2×#930C2212yes—
19C23×C3⋊C8C2212yes—
20#5477—28no—
21C2×#875—21no—
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×D12C2212yes—
25C2×#869C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#387—21no—
31C3×#387—21no—
32C3×#385—21no—
33C3×#381—21no—
34C3×#380C2211yescharacteristic
35C2×#835C2212yes—
36#5048—28no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#465—32no—
48#2855D311yescharacteristic, complemented
49#2058—31no—
50#2041—31no—
51#2024—31no—
52#2014—31no—
53#1893—32no—
54#1877—32no—
55#1864—32no—
56C2×#336—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—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—24no—
80C2×C3×C42D411yescharacteristic
81C2×C3×C42D411yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#215—24no—
85C2×#215—24no—
86C2×#215—42no—
87#930—28no—
88#930—24no—
89#930D418yes—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C22×C3⋊C8—24no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8—42no—
97C22×C3⋊C8C2312yes—
98C22×C3⋊C8C2312yes—
99#875—28no—
100#874—24no—
101C4×D12—28no—
102C4×D12—24no—
103C4×D12—24no—
104#869—24no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112#835—24no—
113#809—24no—
114#809—24no—
115#809—42no—
116#808—24no—
117C22×#41—61no—
118C22×SD16—32no—
119C23×C8—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×D4—32no—
128C2×C4×D4—61no—
129C22×C42C2×D311yescharacteristic
130#465—68no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136#391C2×D311yescharacteristic
137#391C2×D311yescharacteristic
138#389C2×D311yescharacteristic
139#389C2×D311yescharacteristic
140#387—21no—
141#387—21no—
142#385—21no—
143#381—21no—
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—31no—
148#336—32no—
149#336—31no—
150C24×C3C2×D411yescharacteristic
151C23×D3—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—41no—
168C22×C3×C4SD1614yescomplemented
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—28no—
177C2×#67—44no—
178C2×#67—44no—
179C22×Dic3—22no—
180#235—432no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—24no—
186C2×D12—24no—
187C2×D12—22no—
188C2×D12—42no—
189C2×D12—42no—
190C2×D12—42no—
191C2×D12—42no—
192C2×D12C2×D414yes—
193C2×C4×D3—28no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×Dic6—24no—
197C2×Dic6—42no—
198C2×Dic6—42no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—42no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—22no—
212#215—48no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215C4⋊Dic3—24no—
216C4⋊Dic3—22no—
217C4⋊Dic3C2×D414yes—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—48no—
224C2×C3⋊C8#4114yescomplemented
225C2×#41—68no—
226C2×#41—124no—
227C2×#41—124no—
228C22×Q8—61no—
229C22×Q8C22×D311yescharacteristic
230C22×D4—32no—
231C22×D4—62no—
232C22×D4—61no—
233C22×D4—61no—
234C23×C4—62no—
235C23×C4#6711yescharacteristic
236C23×C4#6711yescharacteristic
237C23×C4C22×D311yescharacteristic
238C2×SD16—64no—
239C2×SD16—64no—
240C2×SD16—122no—
241C22×C8—32no—
242C22×C8—32no—
243C22×C8—64no—
244C22×C8—64no—
245C22×C8—122no—
246C4⋊Q8—64no—
247#127—64no—
248#124—68no—
249C22⋊Q8—24no—
250#121—38no—
251#121—68no—
252#121—64no—
253C4×D4—64no—
254C4×D4—64no—
255C4×D4—68no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C4⋊C4C22×D311yescharacteristic
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—24no—
269C2×C42#6711yescharacteristic
270C2×C42#6711yescharacteristic
271C2×C42C22×D311yescharacteristic
272C2×C42C22×D311yescharacteristic
273#106—64no—
274#103—64no—
275#102—64no—
276#102—64no—
277#102—122no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3#12012yes—
290C23×C3C2×SD1612yes—
291C23×C3AGL(1,ℤ/8)12yes—
292C23×C3C22×D411yescharacteristic
293C22×D3—24no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C3×Q8—42no—
301C3×Q8—42no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—28no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—44no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4#12012yes—
330C2×C3×C4C2×SD1612yes—
331C2×C3×C4C2×SD1612yes—
332C2×C3×C4#13512yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4C2×#4112yes—
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336C2×C3×C42−1+411yescharacteristic
337#67—416no—
338C2×Dic3—24no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341D12—24no—
342D12—24no—
343D12—48no—
344D12—48no—
345D12—42no—
346D12—42no—
347C4×D3—416no—
348Dic6—48no—
349C3⋊C8—24no—
350C3⋊C8—42no—
351C24—62no—
352C24C2×#6711yescharacteristic
353#41—1232no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—64no—
359C2×Q8—122no—
360C2×Q8—122no—
361C2×D4—34no—
362C2×D4—64no—
363C2×D4—64no—
364C2×D4—68no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—122no—
371C2×D4—122no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—122no—
375C2×D4—122no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—68no—
389C22×C4—62no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4#21514yescomplemented
393C22×C4D3×D414yes—
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C23×D311yescharacteristic
400SD16—128no—
401C2×C8—34no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—128no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—34no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—64no—
417C4⋊C4—62no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—68no—
421C22⋊C4—68no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3#45014yes—
433C22×C3C2×#12011yescharacteristic
434C22×C3#53211yescharacteristic
435C22×C3#53711yescharacteristic
436C22×C3C22×SD1611yescharacteristic
437C22×C3C2×#13511yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3#57511yescharacteristic
440C2×D3—44no—
441C2×D3—44no—
442C2×D3—44no—
443C2×D3—82no—
444C2×D3—82no—
445C2×D3—82no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—81no—
453C3×C4—81no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4#54612yes—
457C3×C4#54712yes—
458Dic3—44no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—64no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23#93712yes—
471C23C2×#21512yes—
472C23#94912yes—
473C23C22×#6711yescharacteristic
474Q8—42no—
475Q8—42no—
476Q8—128no—
477D4—64no—
478D4—64no—
479D4—128no—
480D4—128no—
481D4—1216no—
482D4—122no—
483D4—122no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—64no—
512C2×C4—1216no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4#93712yes—
516C2×C4C2×#21512yes—
517C2×C4C2×#21512yes—
518C2×C4#95012yes—
519C2×C4C2×D3×D412yes—
520C2×C4D3×#4112yes—
521C2×C4#100911yescharacteristic
522C2×C4#101011yescharacteristic
523C8—64no—
524C8—122no—
525C2×C3—24no—
526C2×C3#243011yescharacteristic
527C2×C3C2×#45011yescharacteristic
528C2×C3#300211yescharacteristic
529C2×C3#316711yescharacteristic
530C2×C3#318011yescharacteristic
531C2×C3#324611yescharacteristic
532C2×C3#324711yescharacteristic
533D3—84no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—124no—
541C22—124no—
542C22—124no—
543C22—242no—
544C22—242no—
545C22—242no—
546C22—242no—
547C22#504614yes—
548C22C2×#93711yescharacteristic
549C22C22×#21511yescharacteristic
550C22C2×#94911yescharacteristic
551C22C2×#95011yescharacteristic
552C22#582211yescharacteristic
553C22#583211yescharacteristic
554C22#583311yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—124no—
565C4#565412yes—
566C4#565512yes—
567C3#2549511yescharacteristic, complemented, Sylow
568C2—24no—
569C2—244no—
570C2C2×#504611yescharacteristic
571C2#8284811yescharacteristic
572C2#8292611yescharacteristic
573C2#8313711yescharacteristic
574C2#8314811yescharacteristic
575C2#8334311yescharacteristic
576C2#8336311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2855C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenter#2855D3
hypocenterC3#25495
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
2. C3×#2855C2semidirect
3. C2×#5048C2semidirect
4. C2×#4524C2semidirect
5. #69688C2semidirect
6. #69687C2semidirect
7. #69668C2semidirect
8. #69627C2semidirect
9. #69626C2semidirect
10. #69605C2semidirect
11. #69515C2semidirect
48. #2855D3semidirect
168. C22×C3×C4SD16semidirect
224. C2×C3⋊C8#41semidirect
392. C22×C4#215semidirect
567. C3#25495semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C3×#28552
C2×#50482
C2×#45242
#696882
#696872
#696682
#696272
#696262
#696052
#695152
#254953