Finite Group Databasebeta

← #10691609

Subgroups — #10691609

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5046C211yesmaximal, characteristic
2C22⋊C4×D12C211yesmaximal, characteristic
3C3×#1996C211yesmaximal, characteristic, complemented
4#71690C211yesmaximal, characteristic, complemented
5#69521C211yesmaximal, characteristic
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9255—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#215C412yescomplemented
12C2×D3×C22⋊C4—21no—
13C2×C4×D12—21no—
14C3×#380C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5046—21no—
20#5046—22no—
21#5046—21no—
22C2×#809—22no—
23C2×#809C412yescomplemented
24#4955—21no—
25#4939—21no—
26#4665—22no—
27C3×#342—21no—
28C3×#342—21no—
29#4538—22no—
30C2×#450—31no—
31D4×C22⋊C4—31no—
32#1996D311yescharacteristic, complemented
33#1786—31no—
34#1520—31no—
35#1289—31no—
36#1282—31no—
37C24×D3—21no—
38C22×C3×Q8C2×C411yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—44no—
44C22×D12C2×C412yes—
45C22×D12D411yescharacteristic
46C22×D12D412yes—
47C22×D12D411yescharacteristic
48C22×C4×D3—22no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4D411yescharacteristic
59C2×C3×C22⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×#217—21no—
62C2×#215—22no—
63C2×#215—22no—
64C2×#215—44no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8C2×C412yescomplemented
71D3×C22⋊C4—42no—
72D3×C22⋊C4—42no—
73C4×D12—44no—
74C3×#95—21no—
75C3×#95—21no—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#809—22no—
82#809—22no—
83#809—44no—
84C23×D4—31no—
85C22×SD16—32no—
86C2×C22⋊Q8C2×D311yescharacteristic
87C2×C4×D4—61no—
88C22×C22⋊C4—61no—
89#450—61no—
90#450—62no—
91#450—61no—
92C2×#102—32no—
93C2×#102—62no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8—31no—
96#390—61no—
97#386—62no—
98#380—31no—
99#380C2×D311yescharacteristic
100#379—61no—
101C4×C22⋊C4C2×D311yescharacteristic
102#342—21no—
103#342—21no—
104#328—62no—
105C24×C3C2×D411yescharacteristic
106C23×D3—21no—
107C23×D3—21no—
108C23×D3—22no—
109C23×D3—22no—
110C23×D3—44no—
111C23×D3—41no—
112C23×D3—41no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3—82no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—42no—
124C22×C3×C4—41no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×Dic3—22no—
132C2×D12—21no—
133C2×D12—24no—
134C2×D12—24no—
135C2×D12—24no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—21no—
139C2×D12—42no—
140C2×D12—42no—
141C2×D12—42no—
142C2×D12—44no—
143C2×D12—44no—
144C2×D12—44no—
145C2×D12C22⋊C412yes—
146C2×D12C22⋊C412yes—
147C2×C4×D3—44no—
148C2×C4×D3—44no—
149C2×C4×D3—82no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4SD1612yescomplemented
163C3×C42—42no—
164#217—41no—
165#217—41no—
166#215—44no—
167#215—44no—
168(C2×D3)⋊C4—44no—
169(C2×D3)⋊C4—44no—
170(C2×D3)⋊C4—82no—
171C4⋊Dic3—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C25—61no—
177C22×Q8C4×D311yescharacteristic, complemented
178C22×D4—32no—
179C22×D4—31no—
180C22×D4—32no—
181C22×D4—31no—
182C22×D4—61no—
183C22×D4—61no—
184C22×D4—61no—
185C22×D4—124no—
186C23×C4—62no—
187C23×C4C22×D311yescharacteristic
188C2×SD16—62no—
189C2×SD16—62no—
190C2×SD16—124no—
191C22×C8—32no—
192C22×C8—61no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C4×D4—124no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4C4×D311yescharacteristic, complemented
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—122no—
209C2×C22⋊C4—122no—
210C2×C22⋊C4#6711yescharacteristic
211C2×C22⋊C4#6711yescharacteristic
212C2×C42—21no—
213#102—62no—
214#102—62no—
215#102—124no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220#95—21no—
221#95—21no—
222#95—32no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3C4×D411yescharacteristic
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3C2×SD1611yescharacteristic
229C23×C3#13511yescharacteristic
230C23×C3AGL(1,ℤ/8)11yescharacteristic
231C23×C3#13711yescharacteristic
232C22×D3—24no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—44no—
236C22×D3—44no—
237C22×D3—42no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—82no—
249C22×D3—82no—
250C22×D3—84no—
251C22×D3—84no—
252C22×D3—84no—
253C22×D3—84no—
254C3×Q8—42no—
255C3×Q8—81no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—81no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4C2×C22⋊C411yescharacteristic
276C2×C3×C4C4×D411yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12311yescharacteristic
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282D12—24no—
283D12—22no—
284D12—22no—
285D12—22no—
286D12—44no—
287D12—44no—
288D12—42no—
289D12#9914yes—
290C4×D3—88no—
291C3⋊C8—42no—
292C3⋊C8—81no—
293C24—61no—
294C24—61no—
295C24—62no—
296C24—62no—
297C24—124no—
298C24—121no—
299C24—121no—
300C24—121no—
301C24—121no—
302C24—242no—
303C24C2×#6711yescharacteristic
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—42no—
307C2×D4—32no—
308C2×D4—32no—
309C2×D4—64no—
310C2×D4—64no—
311C2×D4—64no—
312C2×D4—62no—
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—62no—
316C2×D4—122no—
317C2×D4—122no—
318C2×D4—122no—
319C2×D4—124no—
320C2×D4—124no—
321C2×D4—124no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—62no—
329C22×C4—124no—
330C22×C4—124no—
331C22×C4—242no—
332C22×C4C2×C4×D311yescharacteristic
333C22×C4#23511yescharacteristic
334C22×C4D3×D411yescharacteristic
335C22×C4D3×D411yescharacteristic
336C22×C4D3×D411yescharacteristic
337C22×C4#23711yescharacteristic
338SD16—124no—
339SD16—124no—
340C2×C8—62no—
341C2×C8—122no—
342C2×C8—122no—
343C2×C8—121no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—122no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—124no—
358C22⋊C4—124no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C22⋊C4—242no—
362C22⋊C4#21512yescomplemented
363C42—42no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3#38611yescharacteristic
375C22×C3C4×SD1611yescharacteristic
376C22×C3#44011yescharacteristic
377C22×C3#45011yescharacteristic
378C22×C3#45211yescharacteristic
379C22×C3#46111yescharacteristic
380C22×C3#46311yescharacteristic
381C2×D3—24no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—84no—
390C2×D3—84no—
391C2×D3—84no—
392C2×D3—84no—
393C2×D3—82no—
394C2×D3—82no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—42no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4—81no—
402C3×C4—161no—
403C3×C4C2×#9912yes—
404Dic3—82no—
405C23—41no—
406C23—41no—
407C23—64no—
408C23—62no—
409C23—62no—
410C23—124no—
411C23—124no—
412C23—122no—
413C23—122no—
414C23—122no—
415C23—121no—
416C23—121no—
417C23—122no—
418C23—122no—
419C23—121no—
420C23—121no—
421C23—122no—
422C23—122no—
423C23—242no—
424C23—242no—
425C23—244no—
426C23—244no—
427C23—244no—
428C23—244no—
429C23C4×#6711yescharacteristic
430C23#93711yescharacteristic
431C23#93811yescharacteristic
432C23C2×#21511yescharacteristic
433C23#94211yescharacteristic
434C23#94911yescharacteristic
435C23#95011yescharacteristic
436Q8—42no—
437Q8—81no—
438D4—34no—
439D4—64no—
440D4—62no—
441D4—62no—
442D4—62no—
443D4—124no—
444D4—124no—
445D4—122no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—248no—
469C2×C4D3×C22⋊C411yescharacteristic
470C2×C4#67⋊C411yescharacteristic
471C2×C4#88311yescharacteristic
472C2×C4#88411yescharacteristic
473C8—122no—
474C8—241no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3#187711yescharacteristic
482C2×C3#189811yescharacteristic
483C2×C3#190411yescharacteristic
484D3—44no—
485D3—82no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—64no—
497C22—124no—
498C22—124no—
499C22—122no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—244no—
505C22—244no—
506C22—244no—
507C22—244no—
508C22—242no—
509C22—242no—
510C22#495611yescharacteristic
511C22C4×#21511yescharacteristic
512C22#502611yescharacteristic
513C22#504611yescharacteristic
514C22#504711yescharacteristic
515C22#505711yescharacteristic
516C22#508311yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4—161no—
525C4—242no—
526C4D3×#9912yes—
527C3#925511yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—124no—
535C2—242no—
536C2#6968711yescharacteristic
537C2#7003611yescharacteristic
538C2#7004911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1996C2
socleC22×C3#386
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#1996D3
hypocenterC3#9255
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
3. C3×#1996C2semidirect
4. #71690C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
11. C22×#215C4semidirect
23. C2×#809C4semidirect
32. #1996D3semidirect
38. C22×C3×Q8C2×C4semidirect
55. C2×C3×C4⋊C4C2×C4semidirect
70. C22×C3⋊C8C2×C4semidirect
162. C3×C22⋊C4SD16semidirect
177. C22×Q8C4×D3semidirect
201. C2×C4⋊C4C4×D3semidirect
362. C22⋊C4#215semidirect
527. C3#9255semidirect

Maximal subgroups

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

subgroupindex
C2×#50462
C22⋊C4×D122
C3×#19962
#716902
#695212
#676252
#676172
#92553