Finite Group Databasebeta

← #10852215

Subgroups — #10852215

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic, complemented
2#82761C212yesmaximal, complemented
3#82403C211yesmaximal, characteristic
4#71897C214yesmaximal, complemented
5#71701C212yesmaximal, complemented
6#71654C214yesmaximal, complemented
7C4×#834C211yesmaximal, characteristic, complemented
8#25947—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C3×#554—22no—
11C3×#554—24no—
12C3×D4×Q8—22no—
13C3×#544—21no—
14C3×#543—21no—
15C3×#543C2212yes—
16C3×#533—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C3×#518C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#947—22no—
22C2×#937C2212yes—
23C2×#930—21no—
24C2×#834C2212yes—
25#5212C2211yescharacteristic
26C4×#217C2212yes—
27#4991—21no—
28#4991—24no—
29#4988—22no—
30#4960—21no—
31#4955C2218yes—
32#4955C2212yes—
33#4939—21no—
34#4526C2218yes—
35#4526C2212yes—
36#3501D311yescharacteristic, complemented
37#2435—32no—
38#2408—31no—
39#2029—34no—
40#1996—32no—
41#1889—34no—
42C4×#99—31no—
43C24×D3C2312yes—
44C22×C3×Q8—21no—
45C22×C3×Q8D412yescomplemented
46C22×C3×D4C2311yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50C22×D12—21no—
51C22×D12—22no—
52C22×D12—21no—
53C22×D12—22no—
54C22×D12—44no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—24no—
58C3×#125—22no—
59C3×#125—24no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—24no—
71C3×C22⋊Q8—24no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C3×#117—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D412yescomplemented
83C2×C3×C4⋊C4D414yescomplemented
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C22⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4C2314yes—
88C2×C3×C42D411yescharacteristic
89C2×#217C2314yes—
90C2×#217C2312yes—
91#947—48no—
92#937—28no—
93#930—44no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—28no—
99C2×C4⋊Dic3—21no—
100C2×C4×Dic3—22no—
101#834—24no—
102C3⋊#95—22no—
103C3⋊#95—21no—
104C23×D4—31no—
105#554—22no—
106#554—24no—
107D4×Q8—22no—
108#544—21no—
109#543—21no—
110#543C2×D312yes—
111#533—21no—
112C2×#124—62no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2×D312yes—
115C2×#121—61no—
116C2×#120—32no—
117#518C2×D312yes—
118C2×C4×D4C2×D311yescharacteristic
119C2×#99—32no—
120#394—61no—
121#394—64no—
122#392—61no—
123#390—62no—
124#390—61no—
125#380—31no—
126#379—38no—
127#379—32no—
128C4×C22⋊C4—32no—
129#342—38no—
130#342—32no—
131C24×C3C2×D411yescharacteristic
132C24×C3C2411yescharacteristic
133C23×D3—24no—
134C23×D3—22no—
135C23×D3—24no—
136C23×D3—24no—
137C23×D3—44no—
138C23×D3—82no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—44no—
144C2×C3×Q8—82no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4#4114yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C2×#67—44no—
164C2×#67—44no—
165C2×#67—82no—
166C22×Dic3—24no—
167C22×Dic3—22no—
168C2×D12—24no—
169C2×D12—28no—
170C2×D12—44no—
171C2×D12—42no—
172C2×D12—48no—
173C2×D12—44no—
174C2×D12—41no—
175C2×D12—42no—
176C2×D12—41no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—22no—
193C3×C42—24no—
194C3×C42—21no—
195C3×C42—21no—
196#217—24no—
197#217—24no—
198#217—24no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—416no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—88no—
203(C2×D3)⋊C4—82no—
204C4⋊Dic3—42no—
205C4×Dic3—44no—
206C25—32no—
207C22×Q8—21no—
208C22×Q8#6712yescomplemented
209C22×D4—62no—
210C22×D4—61no—
211C22×D4—62no—
212C22×D4—61no—
213C22×D4—62no—
214C22×D4—124no—
215C22×D4C22×D311yescharacteristic
216C23×C4#6711yescharacteristic
217C23×C4C22×D311yescharacteristic
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—24no—
221#125—22no—
222#125—24no—
223#124—128no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236#121—124no—
237#120—68no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—24no—
243C4×D4—24no—
244#117—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4#6711yescharacteristic
249C2×C4⋊C4#6712yescomplemented
250C2×C4⋊C4#6714yescomplemented
251C2×C22⋊C4—22no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—68no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4#6711yescharacteristic
260C2×C22⋊C4C22×D311yescharacteristic
261C2×C22⋊C4C22×D314yes—
262C2×C42—62no—
263C2×C42#6711yescharacteristic
264#99—64no—
265#95—62no—
266#95—61no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3#12412yescomplemented
274C23×C3C22×D411yescharacteristic
275C23×C3C22×D412yes—
276C23×C3C2×#4112yes—
277C23×C32−1+412yes—
278C22×D3—28no—
279C22×D3—24no—
280C22×D3—48no—
281C22×D3—42no—
282C22×D3—44no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—48no—
289C22×D3—88no—
290C22×D3—88no—
291C3×Q8—44no—
292C3×Q8—82no—
293C3×Q8—82no—
294C3×Q8—82no—
295C3×D4—28no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—21no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—41no—
311C2×C3×C4—81no—
312C2×C3×C4—82no—
313C2×C3×C4—82no—
314C2×C3×C4#12011yescharacteristic
315C2×C3×C4#12012yes—
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12412yescomplemented
318#67—88no—
319C2×Dic3—44no—
320C2×Dic3—42no—
321C2×Dic3—44no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—44no—
325D12—44no—
326D12—48no—
327D12—44no—
328C24—64no—
329C24—62no—
330C24—64no—
331C24—64no—
332C24—124no—
333C24—242no—
334C24C2×#6711yescharacteristic
335C24C23×D311yescharacteristic
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—44no—
341C2×Q8—82no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—64no—
345C2×D4—68no—
346C2×D4—124no—
347C2×D4—122no—
348C2×D4—128no—
349C2×D4—124no—
350C2×D4—124no—
351C2×D4—124no—
352C2×D4—121no—
353C2×D4—122no—
354C2×D4—121no—
355C2×D4—242no—
356C2×D4#23914yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—64no—
364C22×C4—62no—
365C22×C4C2×#6711yescharacteristic
366C22×C4C2×#6712yes—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6714yes—
371C22×C4C2×#6714yes—
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6712yes—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—122no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—22no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—64no—
392C22⋊C4—64no—
393C22⋊C4—64no—
394C22⋊C4—124no—
395C22⋊C4—1216no—
396C22⋊C4—124no—
397C22⋊C4—248no—
398C22⋊C4—242no—
399C42—24no—
400C42—21no—
401C42—21no—
402C42—124no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—81no—
414C22×C3C2×#12011yescharacteristic
415C22×C3C2×#12411yescharacteristic
416C22×C3#53611yescharacteristic
417C22×C3#54814yes—
418C2×D3—44no—
419C2×D3—416no—
420C2×D3—44no—
421C2×D3—44no—
422C2×D3—82no—
423C2×D3—88no—
424C2×D3—82no—
425C2×D3—82no—
426C2×D3—82no—
427C3×C4—22no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—41no—
431C3×C4—44no—
432C3×C4—44no—
433C3×C4—41no—
434C3×C4B(4,𝔽2)12yes—
435Dic3—84no—
436Dic3—82no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—68no—
444C23—64no—
445C23—128no—
446C23—122no—
447C23—124no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—128no—
453C23—122no—
454C23—248no—
455C23—248no—
456C23#94712yescomplemented
457C23C2×#23912yes—
458C23#100712yes—
459C23C22×#6711yescharacteristic
460C23C22×#6712yes—
461Q8—44no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465D4—28no—
466D4—124no—
467D4—128no—
468D4—124no—
469D4—248no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—124no—
489C2×C4—122no—
490C2×C4—124no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—124no—
494C2×C4#94712yescomplemented
495C2×C4#95311yescharacteristic
496C2×C4#95312yes—
497C2×C4#95311yescharacteristic
498C2×C3—21no—
499C2×C3—24no—
500C2×C3—21no—
501C2×C3—42no—
502C2×C3#243611yescharacteristic
503C2×C3C2×B(4,𝔽2)11yescharacteristic
504C2×C3#302711yescharacteristic
505D3—84no—
506C22—22no—
507C22—21no—
508C22—22no—
509C22—21no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—124no—
518C22—1216no—
519C22—124no—
520C22—124no—
521C22—242no—
522C22—248no—
523C22—242no—
524C22—242no—
525C22—242no—
526C22C2×#94711yescharacteristic
527C22#581611yescharacteristic
528C22#581714yes—
529C22C2×#95311yescharacteristic
530C4—22no—
531C4—21no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—44no—
536C4—41no—
537C4—242no—
538C4—244no—
539C4#524412yes—
540C3#2594711yescharacteristic, complemented, Sylow
541C2—21no—
542C2—24no—
543C2—21no—
544C2—42no—
545C2—244no—
546C2#8406611yescharacteristic
547C2C2×#524411yescharacteristic
548C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3501C2
socleC22×C3C2×#120
prime radical#82403C2
hypercenter#3501D3
hypocenterC3#25947
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. C3×#3501C2semidirect
2. #82761C2semidirect
4. #71897C2semidirect
5. #71701C2semidirect
6. #71654C2semidirect
7. C4×#834C2semidirect
36. #3501D3semidirect
45. C22×C3×Q8D4semidirect
82. C2×C3×C4⋊C4D4semidirect
83. C2×C3×C4⋊C4D4semidirect
208. C22×Q8#67semidirect
249. C2×C4⋊C4#67semidirect
250. C2×C4⋊C4#67semidirect
273. C23×C3#124semidirect
317. C2×C3×C4#124semidirect
456. C23#947semidirect
494. C2×C4#947semidirect
540. C3#25947semidirect

Maximal subgroups

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

subgroupindex
C3×#35012
#827612
#824032
#718972
#717012
#716542
C4×#8342
#259473