Finite Group Databasebeta

← #10769430

Subgroups — #10769430

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic, complemented
2#83306C212yesmaximal, complemented
3#82260C211yesmaximal, characteristic, complemented
4#71880C212yesmaximal, complemented
5#71878C212yesmaximal, complemented
6#71846C212yesmaximal, complemented
7#71845C212yesmaximal, complemented
8#71812C212yesmaximal, complemented
9#71572C211yesmaximal, characteristic, complemented
10#18332—31nomaximal, Sylow
11C22×C3×#41—21no—
12C3×#537—21no—
13C3×#537C2212yes—
14C3×#536—22no—
15C3×#535—22no—
16C3×#534—21no—
17C3×#534C2212yes—
18C3×#532—21no—
19C3×#532C2212yes—
20C3×#525—24no—
21C2×C3×#117C2211yescharacteristic
22C2×#875C2211yescharacteristic, complemented
23#5213—21no—
24#5089—22no—
25#5074—22no—
26#5040C2214yescomplemented
27#5036C2214yescomplemented
28#5016—24no—
29#4991C2211yescharacteristic, complemented
30#4990—21no—
31#4979C2211yescharacteristic, complemented
32#4960C2211yescharacteristic
33#4957—21no—
34#4947C2211yescharacteristic
35#4940—21no—
36#4932C2211yescharacteristic, complemented
37#4926C2212yescomplemented
38#4537C2214yescomplemented
39#4537C2214yescomplemented
40#4528C2212yescomplemented
41#4525C2212yes—
42#3449D311yescharacteristic, complemented
43#3213—32no—
44#2610—31no—
45#1604—32no—
46#1601—32no—
47#1598—32no—
48#1595—32no—
49#1505—32no—
50#1464—31no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—44no—
56C2×C3×#41—44no—
57C22×C3×Q8—21no—
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C23×C3×C4—22no—
61C23×C3×C4D411yescharacteristic
62C22×#67—21no—
63C22×D12C2311yescharacteristic
64C22×Dic6C2311yescharacteristic
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8D412yescomplemented
67C3×#127—21no—
68C3×#127D412yescomplemented
69C3×#124—22no—
70C3×#124—22no—
71C3×#123—24no—
72C3×#123—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—24no—
75C3×#121—24no—
76C3×#121—24no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121C2314yes—
81C3×#120—24no—
82C3×#120—24no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4D414yescomplemented
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C22⋊C4C2311yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#214—24no—
97C2×#213—24no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105#875—22no—
106#875—22no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95C2311yescharacteristic
111C3⋊#95C2311yescharacteristic
112#830C2314yes—
113#810—24no—
114#809—24no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×#41—21no—
118#537—21no—
119#537C2×D312yes—
120#536—22no—
121#535—22no—
122#534—21no—
123#534C2×D312yes—
124#532—21no—
125#532C2×D312yes—
126C2×#124—31no—
127#525—24no—
128C2×#117C2×D311yescharacteristic
129#496—62no—
130#489—62no—
131#463—64no—
132#448—34no—
133#447—34no—
134#432—32no—
135#395—31no—
136#394—31no—
137#393—31no—
138#392—31no—
139#390—61no—
140#388—61no—
141#388—61no—
142#386—61no—
143#376—31no—
144#327—34no—
145#327—34no—
146#324—32no—
147#323—32no—
148C24×C3—22no—
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C3×#41—48no—
152C3×#41—48no—
153C3×#41—44no—
154C3×#41—44no—
155C3×#41—44no—
156C3×#41—44no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—44no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—24no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—44no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4C2×D414yes—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—48no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—81no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C2×D12—22no—
192C2×D12—81no—
193C2×Dic6—22no—
194C2×Dic6—81no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4C2×D414yes—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211#217—81no—
212#214—84no—
213#213—84no—
214(C2×D3)⋊C4—42no—
215(C2×D3)⋊C4—41no—
216(C2×D3)⋊C4—41no—
217(C2×D3)⋊C4—42no—
218(C2×D3)⋊C4—42no—
219(C2×D3)⋊C4—81no—
220Dic3⋊C4—81no—
221C4×Dic3—42no—
222C2×C3⋊C8—24no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—44no—
228C2×#41—44no—
229C22×Q8—21no—
230C22×Q8—31no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—31no—
234C22×D4—61no—
235C23×C4—22no—
236C23×C4#6711yescharacteristic
237C2×SD16—64no—
238C2×D8—64no—
239C4⋊Q8—21no—
240C4⋊Q8#6712yescomplemented
241#127—21no—
242#127#6712yescomplemented
243#124—22no—
244#124—22no—
245#124—62no—
246#124—62no—
247#123—24no—
248#123—24no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251#121—24no—
252#121—24no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121C22×D314yes—
257#120—24no—
258#120—24no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4#6714yescomplemented
263#117—22no—
264#117—22no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4#6711yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—61no—
269C2×C22⋊C4—61no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4#6711yescharacteristic
275C2×C22⋊C4C22×D311yescharacteristic
276C2×C42—61no—
277C2×C42#6711yescharacteristic
278C2×C42C22×D311yescharacteristic
279C4⋊C8—62no—
280#103—64no—
281#102—64no—
282C22⋊C8—34no—
283#97—62no—
284#95—31no—
285#95—31no—
286#95—61no—
287#95—61no—
288#95—61no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—24no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—44no—
295C23×C3—82no—
296C23×C3—82no—
297C23×C3AGL(1,ℤ/8)12yes—
298C23×C3C22×D411yescharacteristic
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C3×Q8—44no—
307C3×Q8—44no—
308C3×D4—44no—
309C3×D4—44no—
310C3×D4—44no—
311C3×D4—48no—
312C3×D4—44no—
313C3×D4—44no—
314C3×D4—82no—
315C3×D4—82no—
316C3×D4—82no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—48no—
329C2×C3×C4—48no—
330C2×C3×C4—44no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12012yes—
336C2×C3×C4AGL(1,ℤ/8)12yes—
337C2×C3×C4C22×D412yes—
338#67—84no—
339C2×Dic3—22no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348D12—44no—
349Dic6—44no—
350C3⋊C8—44no—
351C24—22no—
352C24—61no—
353C24C2×#6711yescharacteristic
354#41—48no—
355#41—48no—
356#41—44no—
357#41—44no—
358#41—44no—
359#41—44no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—44no—
363C2×Q8—62no—
364C2×Q8—241no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—62no—
376C2×D4—122no—
377C2×D4—122no—
378C2×D4—241no—
379C2×D4—241no—
380C2×D4C2×#6714yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—48no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398SD16—244no—
399D8—244no—
400C2×C8—64no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—241no—
407C4⋊C4C2×#6714yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—122no—
415C22⋊C4—122no—
416C22⋊C4—121no—
417C22⋊C4—121no—
418C22⋊C4—122no—
419C22⋊C4—241no—
420C22⋊C4—241no—
421C42—21no—
422C42—21no—
423C42—122no—
424C42C2×#6712yes—
425C42C2×#6712yes—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—81no—
436C22×C3—88no—
437C22×C3C2×#12011yescharacteristic
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C2×D3—81no—
445C2×D3—81no—
446C2×D3—81no—
447C3×C4—21no—
448C3×C4—22no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—44no—
453C3×C4—41no—
454C3×C4—41no—
455Dic3—42no—
456Dic3—81no—
457Dic3—81no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—41no—
462C23—41no—
463C23—44no—
464C23—82no—
465C23—82no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23—121no—
471C23—121no—
472C23—121no—
473C23#93212yes—
474C23C22×#6711yescharacteristic
475Q8—44no—
476Q8—44no—
477Q8—124no—
478D4—44no—
479D4—44no—
480D4—44no—
481D4—48no—
482D4—44no—
483D4—44no—
484D4—82no—
485D4—82no—
486D4—82no—
487D4—124no—
488D4—244no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—62no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4#93212yes—
515C2×C4#95311yescharacteristic
516C2×C4#95311yescharacteristic
517C2×C4#95312yes—
518C2×C4C22×#6712yes—
519C8—124no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—42no—
523C2×C3—84no—
524C2×C3#300311yescharacteristic
525C2×C3#301811yescharacteristic
526C2×C3#302711yescharacteristic
527D3—161no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—42no—
532C22—41no—
533C22—41no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—81no—
538C22—88no—
539C22—241no—
540C22—241no—
541C22—241no—
542C22—241no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22C2×#93211yescharacteristic
547C22C2×#93211yescharacteristic
548C22C2×#95311yescharacteristic
549C4—21no—
550C4—22no—
551C4—21no—
552C4—22no—
553C4—22no—
554C4—44no—
555C4—41no—
556C4—41no—
557C4—122no—
558C4—241no—
559C4—241no—
560C3#1833211yescharacteristic, complemented, Sylow
561C2—21no—
562C2—21no—
563C2—42no—
564C2—84no—
565C2—481no—
566C2#8403811yescharacteristic
567C2#8412111yescharacteristic
568C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3449C2
socleC22×C3C2×#120
hypercenter#3449D3
hypocenterC3#18332
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. C3×#3449C2semidirect
2. #83306C2semidirect
3. #82260C2semidirect
4. #71880C2semidirect
5. #71878C2semidirect
6. #71846C2semidirect
7. #71845C2semidirect
8. #71812C2semidirect
9. #71572C2semidirect
22. C2×#875C22semidirect
26. #5040C22semidirect
27. #5036C22semidirect
29. #4991C22semidirect
31. #4979C22semidirect
36. #4932C22semidirect
37. #4926C22semidirect
38. #4537C22semidirect
39. #4537C22semidirect
40. #4528C22semidirect
42. #3449D3semidirect
66. C3×C4⋊Q8D4semidirect
68. C3×#127D4semidirect
86. C3×C4×D4D4semidirect
240. C4⋊Q8#67semidirect
242. #127#67semidirect
262. C4×D4#67semidirect
560. C3#18332semidirect

Maximal subgroups

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

subgroupindex
C3×#34492
#833062
#822602
#718802
#718782
#718462
#718452
#718122
#715722
#183323