Finite Group Databasebeta

← #10849682

Subgroups — #10849682

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86259C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3#83260C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5#71841C211yesmaximal, characteristic, complemented
6#70681C214yesmaximal, complemented
7#70675C214yesmaximal, complemented
8C4×#823C211yesmaximal, characteristic, complemented
9#69867C211yesmaximal, characteristic, complemented
10#25693—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5734—21no—
14#5726—21no—
15#5695—22no—
16#5644—22no—
17#5627—22no—
18#5585—22no—
19#5584—21no—
20#5577—21no—
21#5525C2212yes—
22#5523C2212yes—
23#5477C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#501—22no—
26C3×#500—21no—
27C3×#498—21no—
28C3×#428C2214yes—
29C3×C4×M16C2211yescharacteristic
30#5231—22no—
31C2×#825C2214yes—
32C2×#823C2212yes—
33#5093—21no—
34#5089—22no—
35#5088—21no—
36#4988C2212yes—
37#4987—22no—
38#4977C2212yes—
39#4966C2214yes—
40C4×C4⋊Dic3C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4554C2212yes—
43#4532C2214yes—
44#3506—31no—
45#3165D311yescharacteristic, complemented
46#3141—31no—
47#2563—31no—
48#2037—34no—
49#1986—31no—
50#1980—31no—
51#1860—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#930—21no—
77#930—21no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4D412yes—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4×Dic3C2312yes—
83C2×#208C2312yes—
84#898—24no—
85#895—24no—
86#895—22no—
87#895—22no—
88#895—22no—
89#894—24no—
90#894—24no—
91#893—24no—
92#892—22no—
93#892—22no—
94D3×C4⋊C4—22no—
95#882—24no—
96#67⋊C4—24no—
97#875—22no—
98#875D412yes—
99#874—22no—
100#874D411yescharacteristic
101C4×D12—22no—
102C4×D12—21no—
103C4×D12—21no—
104C4×D12—21no—
105C4×D12—21no—
106C4×D12—21no—
107C4×D12—21no—
108#869D411yescharacteristic
109C4×Dic6—22no—
110C3×#107—24no—
111C3×#106—24no—
112C3×#97—21no—
113C3×C4×C8—21no—
114C3⋊#95—21no—
115C3⋊#95—24no—
116#825—28no—
117#823—24no—
118#810—24no—
119#809—24no—
120#803—21no—
121C4×C3⋊C8—21no—
122#559—62no—
123#556—61no—
124#555—61no—
125#553—62no—
126#550—61no—
127#547—61no—
128#538—62no—
129#535—32no—
130#533C2×D311yescharacteristic
131#532—31no—
132C2×C4⋊Q8C2×D311yescharacteristic
133#525—62no—
134#520—32no—
135C2×C4×D4—31no—
136#501—22no—
137#500—21no—
138#498—21no—
139#489—62no—
140#487—61no—
141#486—61no—
142#428C2×D314yes—
143C2×#104—34no—
144#417—34no—
145C2×#100—32no—
146C4×M16—31no—
147C4×M16C2×D311yescharacteristic
148#393—62no—
149#390—32no—
150#382—32no—
151#380—62no—
152C4×C4⋊C4—31no—
153#343—32no—
154#339—34no—
155C23×D3C2×D412yes—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—41no—
159C2×C3×Q8—41no—
160C2×C3×Q8—82no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—22no—
166C22×Dic3—22no—
167C22×Dic3C2×D412yes—
168#239—84no—
169#235—24no—
170#235—24no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—24no—
177C2×D12—24no—
178C2×D12—21no—
179C2×D12C2×D412yes—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—24no—
183C2×C4×D3—22no—
184C2×C4×D3—44no—
185C2×Dic6C2×D412yes—
186C3×M16—22no—
187C3×M16#4114yescomplemented
188C2×C3×C8—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—22no—
200C3×C42—22no—
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—22no—
209(C2×D3)⋊C4—22no—
210(C2×D3)⋊C4—22no—
211(C2×D3)⋊C4—22no—
212(C2×D3)⋊C4—84no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—42no—
218Dic3⋊C4—24no—
219C4×Dic3—24no—
220C4×Dic3—24no—
221#208—22no—
222#208—22no—
223C2×C3⋊C8—22no—
224C2×#41—32no—
225C2×#41—62no—
226C22×Q8—21no—
227C22×D4—31no—
228C23×C4—32no—
229C2×M16—32no—
230C2×M16C22×D312yes—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—31no—
236C4⋊Q8#6711yescharacteristic
237C4⋊Q8#6711yescharacteristic
238#127—31no—
239#127—62no—
240#126—64no—
241#125#6712yes—
242#124—32no—
243#124—62no—
244#124—64no—
245#123—64no—
246#123—64no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—62no—
250#121—62no—
251#121—64no—
252#121—61no—
253#121—61no—
254#121—62no—
255#121—62no—
256#120—64no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—62no—
260C4×D4—62no—
261C4×D4—64no—
262C4×D4—64no—
263C4×D4—61no—
264C4×D4—61no—
265C4×D4—61no—
266C4×D4—61no—
267C4×D4—61no—
268C4×D4—61no—
269#117—62no—
270#117—62no—
271#117C22×D312yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—64no—
279C2×C42—32no—
280C2×C42C22×D311yescharacteristic
281#107—24no—
282#106—24no—
283#104—68no—
284#103—64no—
285#102—64no—
286#100—64no—
287#97—21no—
288#97—61no—
289C4×C8—21no—
290C4×C8—61no—
291#95—64no—
292#95—61no—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—42no—
299C22×D3—42no—
300C3×Q8—42no—
301C3×Q8—81no—
302C3×Q8—82no—
303C3×Q8—82no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—44no—
309C2×C3×C4—42no—
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4112yes—
317C2×C3×C42−1+412yes—
318#67—42no—
319#67—42no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326D12—24no—
327D12—42no—
328D12—42no—
329D12—42no—
330D12—81no—
331D12—84no—
332D12—84no—
333C4×D3—44no—
334C4×D3—44no—
335C4×D3—82no—
336C4×D3—84no—
337Dic6—24no—
338C3×C8—24no—
339C3⋊C8—42no—
340C24—32no—
341#41—64no—
342#41—64no—
343#41—244no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—32no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—82no—
350C2×D4—32no—
351C2×D4—64no—
352C2×D4—64no—
353C2×D4—62no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—61no—
360C22×C4—22no—
361C22×C4—32no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—124no—
368C22×C4D3×D411yescharacteristic
369C22×C4D3×D411yescharacteristic
370C22×C4C23×D311yescharacteristic
371M16—22no—
372M16—62no—
373M16—62no—
374M16#23914yescomplemented
375C2×C8—22no—
376C2×C8—62no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—64no—
386C4⋊C4—61no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—122no—
391C4⋊C4C2×#6712yes—
392C4⋊C4C2×#6712yes—
393C22⋊C4—22no—
394C22⋊C4—64no—
395C22⋊C4—64no—
396C22⋊C4—64no—
397C22⋊C4—62no—
398C22⋊C4—62no—
399C22⋊C4—62no—
400C22⋊C4—62no—
401C22⋊C4—244no—
402C42—22no—
403C42—64no—
404C42—64no—
405C42D3×D411yescharacteristic
406C42D3×D411yescharacteristic
407C42C2×#6711yescharacteristic
408C42C2×#6711yescharacteristic
409C22×C3—21no—
410C22×C3—21no—
411C22×C3C2×#12011yescharacteristic
412C22×C3#53211yescharacteristic
413C22×C3#53611yescharacteristic
414C2×D3—44no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—82no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—82no—
428C3×C4#45512yes—
429C3×C4#54712yes—
430C3×C4#54812yes—
431Dic3—42no—
432Dic3—42no—
433Dic3—82no—
434C23—62no—
435C23—62no—
436C23—62no—
437C23—62no—
438C23—122no—
439C23—122no—
440C23C2×D3×D411yescharacteristic
441Q8—42no—
442Q8—64no—
443Q8—81no—
444Q8—82no—
445Q8—82no—
446D4—64no—
447D4—122no—
448D4—122no—
449D4—122no—
450D4—122no—
451D4—122no—
452D4—241no—
453D4—244no—
454D4—244no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—124no—
464C2×C4—124no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—242no—
470C2×C4—244no—
471C2×C4#93711yescharacteristic
472C2×C4#93711yescharacteristic
473C2×C4#93711yescharacteristic
474C2×C4#93711yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4C2×#23912yes—
477C2×C4#100712yes—
478C2×C4C22×#6711yescharacteristic
479C8—24no—
480C8—122no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3#243211yescharacteristic
484C2×C3C2×#45511yescharacteristic
485C2×C3#302111yescharacteristic
486D3—82no—
487D3—82no—
488C22—21no—
489C22—21no—
490C22—124no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—242no—
498C22#572411yescharacteristic
499C22#572511yescharacteristic
500C22C2×#93711yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—122no—
508C4—122no—
509C4—242no—
510C4#520212yes—
511C4#572612yes—
512C4#581712yes—
513C3#2569311yescharacteristic, complemented, Sylow
514C2—21no—
515C2—21no—
516C2—242no—
517C2—242no—
518C2#8325511yescharacteristic
519C2#8395711yescharacteristic
520C2C2×#520211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3165C2
socleC22×C3C2×#120
prime radical#86259C2
hypercenter#3165D3
hypocenterC3#25693
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. #83260C2semidirect
4. #83199C2semidirect
5. #71841C2semidirect
6. #70681C2semidirect
7. #70675C2semidirect
8. C4×#823C2semidirect
9. #69867C2semidirect
45. #3165D3semidirect
187. C3×M16#41semidirect
374. M16#239semidirect
513. C3#25693semidirect

Maximal subgroups

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

subgroupindex
#862592
C3×#31652
#832602
#831992
#718412
#706812
#706752
C4×#8232
#698672
#256933