Finite Group Databasebeta

← #10849691

Subgroups — #10849691

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86259C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3#83259C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5#71841C211yesmaximal, characteristic, complemented
6#70680C214yesmaximal, complemented
7#70675C214yesmaximal, complemented
8C4×#823C211yesmaximal, characteristic, complemented
9#69867C211yesmaximal, characteristic, complemented
10#25694—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×#946—22no—
14#5734—21no—
15#5726—21no—
16#5695—22no—
17#5644—22no—
18#5627—22no—
19#5585—22no—
20#5584—21no—
21#5577—21no—
22#5525C2212yes—
23#5523C2212yes—
24#5477C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C3×#501—22no—
27C3×#500—21no—
28C3×#498—21no—
29C3×#428C2214yes—
30C3×C4×M16C2211yescharacteristic
31C2×#825C2214yes—
32C2×#823C2212yes—
33#5093—21no—
34#5089—22no—
35#5088—21no—
36#4993—22no—
37#4988C2212yes—
38#4975C2212yes—
39#4966C2214yes—
40C4×C4⋊Dic3C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4554C2212yes—
43#4532C2214yes—
44#3506—31no—
45#3165D311yescharacteristic, complemented
46#3141—31no—
47#2542—31no—
48#2037—34no—
49#1986—31no—
50#1980—31no—
51#1859—34no—
52C4×#100—31no—
53C22×C3×Q8—21no—
54C2×#239—22no—
55C2×#235C2312yes—
56C22×D12C2311yescharacteristic
57C22×C4×D3D412yes—
58C2×C3×M16C2312yes—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8D411yescharacteristic
65C3×#125D412yes—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×#117C2312yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74#947—24no—
75#946—22no—
76#946—48no—
77#930—21no—
78#930—21no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4D412yes—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—24no—
84C2×C4×Dic3C2312yes—
85C2×#208C2312yes—
86#898—24no—
87#895—24no—
88#895—22no—
89#895—22no—
90#895—22no—
91#894—24no—
92#894—24no—
93#893—24no—
94#892—22no—
95#892—22no—
96D3×C4⋊C4—22no—
97#882—24no—
98#67⋊C4—24no—
99#875—22no—
100#875D412yes—
101#874—22no—
102#874D411yescharacteristic
103C4×D12—22no—
104C4×D12—21no—
105C4×D12—21no—
106C4×D12—21no—
107C4×D12—21no—
108C4×D12—21no—
109C4×D12—21no—
110#869D411yescharacteristic
111C4×Dic6—22no—
112C3×#107—24no—
113C3×#106—24no—
114C3×#97—21no—
115C3×C4×C8—21no—
116C3⋊#95—21no—
117#825—28no—
118#823—24no—
119#810—24no—
120#809—24no—
121#803—21no—
122C4×C3⋊C8—21no—
123#559—62no—
124#556—61no—
125#555—61no—
126#553—62no—
127#550—61no—
128#547—61no—
129#538—62no—
130#535—32no—
131#533C2×D311yescharacteristic
132#532—31no—
133C2×C4⋊Q8C2×D311yescharacteristic
134#525—62no—
135C2×C22⋊Q8—62no—
136#520—32no—
137C2×C4×D4—31no—
138#501—22no—
139#500—21no—
140#498—21no—
141#489—62no—
142#487—61no—
143#486—61no—
144#428C2×D314yes—
145C2×#104—34no—
146#417—34no—
147C2×#100—32no—
148C4×M16—31no—
149C4×M16C2×D311yescharacteristic
150#399—62no—
151#390—32no—
152#384—32no—
153C4×C4⋊C4—31no—
154#343—32no—
155#339—34no—
156C23×D3C2×D412yes—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—44no—
160C2×C3×Q8—41no—
161C2×C3×Q8—41no—
162C22×C3×C4—22no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—22no—
167C22×Dic3—22no—
168C22×Dic3C2×D412yes—
169#239—84no—
170#235—24no—
171#235—24no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—24no—
178C2×D12—24no—
179C2×D12—21no—
180C2×D12C2×D412yes—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—24no—
184C2×C4×D3—22no—
185C2×C4×D3—44no—
186C2×Dic6C2×D412yes—
187C3×M16—22no—
188C3×M16#4114yescomplemented
189C2×C3×C8—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4C2×D412yes—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—22no—
201C3×C42—22no—
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—22no—
210(C2×D3)⋊C4—22no—
211(C2×D3)⋊C4—22no—
212(C2×D3)⋊C4—22no—
213(C2×D3)⋊C4—48no—
214C4⋊Dic3—24no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219Dic3⋊C4—24no—
220Dic3⋊C4—48no—
221C4×Dic3—24no—
222C4×Dic3—24no—
223#208—22no—
224#208—22no—
225C2×C3⋊C8—22no—
226C2×#41—32no—
227C2×#41—62no—
228C22×Q8—21no—
229C22×D4—31no—
230C23×C4—32no—
231C2×M16—32no—
232C2×M16C22×D312yes—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—31no—
238C4⋊Q8#6711yescharacteristic
239C4⋊Q8#6711yescharacteristic
240#127—31no—
241#127—62no—
242#126—64no—
243#125#6712yes—
244#124—32no—
245#124—62no—
246#124—64no—
247#123—64no—
248#123—64no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—62no—
252C22⋊Q8—128no—
253#121—62no—
254#121—64no—
255#121—61no—
256#121—61no—
257#121—62no—
258#121—62no—
259#120—64no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—62no—
263C4×D4—62no—
264C4×D4—64no—
265C4×D4—64no—
266C4×D4—61no—
267C4×D4—61no—
268C4×D4—61no—
269C4×D4—61no—
270C4×D4—61no—
271C4×D4—61no—
272#117—62no—
273#117—62no—
274#117C22×D312yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4C22×D312yes—
281C2×C22⋊C4—32no—
282C2×C22⋊C4—64no—
283C2×C42—32no—
284C2×C42C22×D311yescharacteristic
285#107—24no—
286#106—24no—
287#104—68no—
288#103—64no—
289#102—64no—
290#100—64no—
291#97—21no—
292#97—61no—
293C4×C8—21no—
294C4×C8—61no—
295#95—61no—
296C23×C3C22×D411yescharacteristic
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—42no—
302C22×D3—42no—
303C3×Q8—42no—
304C3×Q8—81no—
305C3×Q8—82no—
306C3×Q8—82no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—41no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4#12011yescharacteristic
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×C3×C42−1+412yes—
321#67—42no—
322#67—42no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329D12—24no—
330D12—42no—
331D12—42no—
332D12—42no—
333D12—81no—
334D12—84no—
335D12—84no—
336C4×D3—44no—
337C4×D3—44no—
338C4×D3—82no—
339C4×D3—84no—
340Dic6—24no—
341C3×C8—24no—
342C3⋊C8—42no—
343C24—32no—
344#41—64no—
345#41—64no—
346#41—244no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—32no—
350C2×Q8—44no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×D4—32no—
354C2×D4—64no—
355C2×D4—64no—
356C2×D4—62no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—61no—
363C22×C4—22no—
364C22×C4—32no—
365C22×C4—64no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—124no—
371C22×C4D3×D411yescharacteristic
372C22×C4D3×D411yescharacteristic
373C22×C4C23×D311yescharacteristic
374M16—22no—
375M16—62no—
376M16—62no—
377M16#23914yescomplemented
378C2×C8—22no—
379C2×C8—62no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—128no—
395C4⋊C4C2×#6712yes—
396C4⋊C4C2×#6712yes—
397C22⋊C4—22no—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—62no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—62no—
405C22⋊C4—128no—
406C42—22no—
407C42—64no—
408C42—64no—
409C42D3×D411yescharacteristic
410C42D3×D411yescharacteristic
411C42C2×#6711yescharacteristic
412C42C2×#6711yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#53211yescharacteristic
417C22×C3#53611yescharacteristic
418C2×D3—44no—
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—82no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4#45312yes—
433C3×C4#54712yes—
434C3×C4#54812yes—
435Dic3—44no—
436Dic3—42no—
437Dic3—42no—
438C23—62no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23—122no—
443C23—122no—
444C23C2×D3×D411yescharacteristic
445Q8—42no—
446Q8—64no—
447Q8—81no—
448Q8—82no—
449Q8—82no—
450D4—64no—
451D4—122no—
452D4—122no—
453D4—122no—
454D4—122no—
455D4—122no—
456D4—241no—
457D4—244no—
458D4—244no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—124no—
471C2×C4—124no—
472C2×C4—122no—
473C2×C4—242no—
474C2×C4—244no—
475C2×C4#93711yescharacteristic
476C2×C4#93711yescharacteristic
477C2×C4#93711yescharacteristic
478C2×C4#93711yescharacteristic
479C2×C4C2×D3×D411yescharacteristic
480C2×C4C2×#23912yes—
481C2×C4#100712yes—
482C2×C4C22×#6711yescharacteristic
483C8—24no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3#243211yescharacteristic
488C2×C3C2×#45311yescharacteristic
489C2×C3#302211yescharacteristic
490D3—82no—
491D3—82no—
492C22—21no—
493C22—21no—
494C22—124no—
495C22—122no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22—122no—
501C22—242no—
502C22#572411yescharacteristic
503C22#572511yescharacteristic
504C22C2×#93711yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—82no—
511C4—124no—
512C4—122no—
513C4—122no—
514C4#519812yes—
515C4#572612yes—
516C4#581712yes—
517C3#2569411yescharacteristic, complemented, Sylow
518C2—21no—
519C2—21no—
520C2—242no—
521C2—242no—
522C2#8325511yescharacteristic
523C2C2×#519811yescharacteristic
524C2#8399311yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3165C2
socleC22×C3C2×#120
prime radical#86259C2
hypercenter#3165D3
hypocenterC3#25694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
2. C3×#3165C2semidirect
3. #83259C2semidirect
4. #83199C2semidirect
5. #71841C2semidirect
6. #70680C2semidirect
7. #70675C2semidirect
8. C4×#823C2semidirect
9. #69867C2semidirect
45. #3165D3semidirect
188. C3×M16#41semidirect
377. M16#239semidirect
517. C3#25694semidirect

Maximal subgroups

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

subgroupindex
#862592
C3×#31652
#832592
#831992
#718412
#706802
#706752
C4×#8232
#698672
#256943