Finite Group Databasebeta

← #10822432

Subgroups — #10822432

Order 768.

2258 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5791C212yesmaximal, complemented
2#85735C211yesmaximal, characteristic, complemented
3C3×#2280C211yesmaximal, characteristic, complemented
4C2×#5006C212yesmaximal, complemented
5#71592C211yesmaximal, characteristic, complemented
6#69539C214yesmaximal, complemented
7#69500C214yesmaximal, complemented
8#23443—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C22×#235C2212yes—
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#953C2214yes—
14#5791—28no—
15#5791—28no—
16C2×#930—22no—
17C2×#929C2214yes—
18C2×C4×#67—24no—
19C2×#926C2212yes—
20C2×#924—22no—
21#5526—22no—
22#5525—22no—
23#5524—24no—
24#5510—22no—
25#5493C2214yes—
26#5477—22no—
27C3×#381C2212yes—
28C3×#380—22no—
29C3×#379C2212yes—
30C3×C4×C22⋊C4—22no—
31#5006—28no—
32C2×#806C2218yes—
33#4526C2214yes—
34#3449—31no—
35C2×#525—32no—
36C2×#410—32no—
37#2280D311yescharacteristic, complemented
38#2022—34no—
39#2021—34no—
40#1740—31no—
41C25×C3C2311yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C23×C3×C4—22no—
45C23×C3×C4—22no—
46C23×C3×C4—24no—
47C23×C3×C4—22no—
48C23×C3×C4D414yescomplemented
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×#67—24no—
52C22×#67C2314yes—
53C2×#235—22no—
54C2×#235—22no—
55C2×#235—24no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—24no—
59C2×#235—24no—
60C2×#235—24no—
61C22×D12D412yescomplemented
62C22×C4×D3D414yescomplemented
63C22×Dic6D412yescomplemented
64C3×#117—22no—
65C3×#117—22no—
66C3×#117—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4D412yescomplemented
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4C2312yes—
71C2×C3×C22⋊C4C2312yes—
72C2×C3×C42—21no—
73C2×C3×C42D412yescomplemented
74#953—28no—
75#953—28no—
76C2×#217—24no—
77C2×#217C2314yes—
78#930—216no—
79#930—28no—
80#929—28no—
81#929—28no—
82C4×#67—232no—
83C2×(C2×D3)⋊C4—24no—
84#926—24no—
85#926D418yescomplemented
86C2×C4⋊Dic3—22no—
87#924—216no—
88C2×Dic3⋊C4—24no—
89C2×C4×Dic3—22no—
90#897—28no—
91#895—28no—
92#886—28no—
93#882—28no—
94#875—24no—
95#874—22no—
96C4×D12—28no—
97#869—22no—
98C3×#95—24no—
99#806—216no—
100C22×#41—32no—
101C24×C4C2×D311yescharacteristic
102#537—62no—
103#536—62no—
104#535—62no—
105#534—34no—
106#532—62no—
107#525—64no—
108#525—68no—
109#525—68no—
110C2×#123—34no—
111C2×C22⋊Q8—62no—
112C2×#121—62no—
113C2×#120—34no—
114C2×C4×D4—64no—
115C2×#117—32no—
116C2×#117C2×D311yescharacteristic
117C2×#117C2×D311yescharacteristic
118#410—68no—
119C2×#99—38no—
120#381C2×D312yes—
121#380—22no—
122#379C2×D312yes—
123C4×C22⋊C4—22no—
124#342—34no—
125C24×C3—22no—
126C24×C3—21no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—44no—
130C24×C3C2×D412yes—
131C24×C3C2411yescharacteristic
132C23×D3C2×D414yes—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—28no—
138C22×C3×C4—28no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—41no—
143C22×C3×C4—42no—
144C22×C3×C4—41no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D414yes—
154C22×C3×C4C2×D414yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4#4112yes—
158C2×#67—216no—
159C2×#67—24no—
160C2×#67—24no—
161C2×#67—24no—
162C2×#67—24no—
163C2×#67—24no—
164C2×#67—24no—
165C2×#67—416no—
166C22×Dic3—24no—
167C22×Dic3C2×D414yes—
168#235—44no—
169#235—44no—
170#235—44no—
171#235—44no—
172#235—44no—
173#235—44no—
174#235—44no—
175#235—44no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—24no—
179C2×D12—24no—
180C2×D12—28no—
181C2×C4×D3—24no—
182C2×C4×D3—24no—
183C2×C4×D3—28no—
184C2×C4×D3—28no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—24no—
188C2×Dic6—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—42no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—48no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—42no—
197#217—216no—
198#217—24no—
199#217C2×D418yes—
200(C2×D3)⋊C4—216no—
201(C2×D3)⋊C4—216no—
202C4⋊Dic3—28no—
203C4⋊Dic3—28no—
204Dic3⋊C4—216no—
205C4×Dic3—28no—
206C25C22×D311yescharacteristic
207C2×#41—64no—
208C2×#41—64no—
209C2×#41—64no—
210C2×#41—62no—
211C2×#41—62no—
212C2×#41—64no—
213C2×#41—62no—
214C2×#41—62no—
215C22×Q8—32no—
216C22×D4—34no—
217C22×D4—32no—
218C22×D4—64no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—34no—
226C23×C4D1214yescomplemented
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D311yescharacteristic
229C4⋊Q8—62no—
230#127—62no—
231#124—64no—
232#123—68no—
233#123—68no—
234#123—68no—
235C22⋊Q8—68no—
236C22⋊Q8—616no—
237#121—68no—
238#121—616no—
239#121—68no—
240#120—68no—
241#120—68no—
242#120—68no—
243C4×D4—68no—
244C4×D4—632no—
245#117—22no—
246#117—22no—
247#117—24no—
248#117—38no—
249#117—64no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4#6712yescomplemented
254C2×C22⋊C4—24no—
255C2×C22⋊C4—34no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4C22×D312yes—
259C2×C22⋊C4C22×D312yes—
260C2×C42—21no—
261C2×C42—62no—
262C2×C42#6712yescomplemented
263#99—616no—
264#95—24no—
265C23×C3—24no—
266C23×C3—28no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—44no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—42no—
273C23×C3—48no—
274C23×C3—44no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3—41no—
278C23×C3C22×D411yescharacteristic
279C23×C3C22×D412yes—
280C23×C3C22×D412yes—
281C23×C3C22×D411yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C22×D3—24no—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—48no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—44no—
294C2×C3×C4—48no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—42no—
299C2×C3×C4#12014yes—
300C2×C3×C4#12014yes—
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4#12012yes—
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4#12114yes—
305C2×C3×C4#12114yes—
306C2×C3×C4#12112yes—
307C2×C3×C4#12112yes—
308C2×C3×C4#12712yes—
309C2×C3×C4#12712yes—
310#67—432no—
311#67—44no—
312#67—44no—
313#67—44no—
314#67—44no—
315C2×Dic3—28no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—48no—
320D12—44no—
321D12—44no—
322D12—88no—
323C4×D3—48no—
324C4×D3—48no—
325Dic6—44no—
326Dic6—44no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—34no—
332C24—44no—
333C24C2×D1212yes—
334C24C23×D311yescharacteristic
335#41—124no—
336#41—124no—
337#41—124no—
338#41—124no—
339#41—124no—
340#41—124no—
341#41—124no—
342#41—124no—
343C2×Q8—64no—
344C2×Q8—64no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×D4—68no—
348C2×D4—616no—
349C2×D4—64no—
350C2×D4—64no—
351C2×D4—64no—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—1216no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—34no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—64no—
378C22×C4—68no—
379C22×C4—68no—
380C22×C4—64no—
381C22×C4—64no—
382C22×C4C2×D1212yes—
383C22×C4C2×D1212yes—
384C22×C4#23512yes—
385C22×C4D3×D412yes—
386C22×C4D3×D412yes—
387C22×C4D3×D414yes—
388C22×C4C2×#6714yes—
389C22×C4C2×#6711yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C4⋊C4—24no—
392C4⋊C4—42no—
393C4⋊C4—616no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—38no—
399C22⋊C4—48no—
400C22⋊C4—616no—
401C22⋊C4—616no—
402C22⋊C4—616no—
403C22⋊C4—64no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—68no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—28no—
411C22×C3—24no—
412C22×C3—41no—
413C22×C3—42no—
414C22×C3—41no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—48no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—44no—
421C22×C3C2×#12012yes—
422C22×C3C2×#12011yescharacteristic
423C22×C3C2×#12112yes—
424C22×C3C2×#12111yescharacteristic
425C22×C3C2×#12711yescharacteristic
426C2×D3—48no—
427C2×D3—44no—
428C2×D3—48no—
429C2×D3—84no—
430C3×C4—42no—
431C3×C4—44no—
432C3×C4—82no—
433C3×C4#39212yes—
434C3×C4#39214yes—
435C3×C4#39212yes—
436Dic3—48no—
437Dic3—44no—
438C23—24no—
439C23—28no—
440C23—24no—
441C23—22no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—48no—
447C23—44no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—64no—
452C23—64no—
453C23—64no—
454C23—128no—
455C23C22×D1211yescharacteristic
456C23C2×#23511yescharacteristic
457C23C2×D3×D412yes—
458C23C2×D3×D412yes—
459C23C22×#6711yescharacteristic
460Q8—124no—
461Q8—124no—
462D4—1232no—
463D4—124no—
464D4—124no—
465D4—124no—
466D4—124no—
467D4—124no—
468D4—124no—
469D4—248no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—68no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—128no—
487C2×C4—128no—
488C2×C4—128no—
489C2×C4#88214yes—
490C2×C4#88214yes—
491C2×C4#88414yes—
492C2×C4#88414yes—
493C2×C4#93012yes—
494C2×C4#93012yes—
495C2×C4#93711yescharacteristic
496C2×C4#93712yes—
497C2×C4#93711yescharacteristic
498C2×C4#94012yes—
499C2×C4#94012yes—
500C2×C3—22no—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—21no—
504C2×C3—44no—
505C2×C3C2×#39211yescharacteristic
506C2×C3#302512yes—
507D3—84no—
508C22—22no—
509C22—24no—
510C22—28no—
511C22—24no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—44no—
516C22—44no—
517C22—42no—
518C22—48no—
519C22—42no—
520C22—44no—
521C22—128no—
522C22—124no—
523C22—128no—
524C22—244no—
525C22C2×#88212yes—
526C22C2×#88412yes—
527C22C2×#93011yescharacteristic
528C22C2×#93711yescharacteristic
529C22C2×#94011yescharacteristic
530C4—42no—
531C4—44no—
532C4—82no—
533C4—128no—
534C4—124no—
535C4#496012yes—
536C4#496014yes—
537C4#496012yes—
538C3#2344311yescharacteristic, complemented, Sylow
539C2—21no—
540C2—22no—
541C2—21no—
542C2—22no—
543C2—44no—
544C2—244no—
545C2C2×#496011yescharacteristic
546C2#8293612yes—
subgroupG/H
centerC2×C4#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2280C2
socleC22×C3C2×#120
hypercenter#2280D3
hypocenterC3#23443
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
1. C2×#5791C2semidirect
2. #85735C2semidirect
3. C3×#2280C2semidirect
4. C2×#5006C2semidirect
5. #71592C2semidirect
6. #69539C2semidirect
7. #69500C2semidirect
37. #2280D3semidirect
48. C23×C3×C4D4semidirect
61. C22×D12D4semidirect
62. C22×C4×D3D4semidirect
63. C22×Dic6D4semidirect
68. C2×C3×C4⋊C4D4semidirect
73. C2×C3×C42D4semidirect
85. #926D4semidirect
226. C23×C4D12semidirect
253. C2×C4⋊C4#67semidirect
262. C2×C42#67semidirect
538. C3#23443semidirect

Maximal subgroups

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

subgroupindex
C2×#57912
#857352
C3×#22802
C2×#50062
#715922
#695392
#695002
#234433