Finite Group Databasebeta

← #10948005

Subgroups — #10948005

Order 768.

1220 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86317C211yesmaximal, characteristic
2#78689C212yesmaximal, complemented
3C3×#2128C211yesmaximal, characteristic, complemented
4#71861C211yesmaximal, characteristic, complemented
5#70060C212yesmaximal, complemented
6#69152C214yesmaximal, complemented
7#69151C214yesmaximal, complemented
8#33362—31nomaximal, Sylow
9C2×#1010C2211yescharacteristic
10C2×#1004C411yescharacteristic, complemented
11C3×#535C2211yescharacteristic
12#5832C2211yescharacteristic
13C2×#936C411yescharacteristic, complemented
14#5716C2212yes—
15#5715C2212yes—
16#5675C412yescomplemented
17D3×#124C412yescomplemented
18#5612—22no—
19#5598—22no—
20#5591—24no—
21#5526C411yescharacteristic, complemented
22#5522C411yescharacteristic, complemented
23#5516—22no—
24C3×#410C2212yescomplemented
25#5240C2212yescomplemented
26#5006—22no—
27#4740—24no—
28#4740C2218yescomplemented
29C3×#353C2214yescomplemented
30#4545C2214yescomplemented
31#4474C2214yescomplemented
32#4473C2214yescomplemented
33#3500—31no—
34#2884—32no—
35#2128—34no—
36#2128—31no—
37#2128D311yescharacteristic, complemented
38#2126—32no—
39#2126—34no—
40C2×C3×#41C2311yescharacteristic
41C22×C3×D4C2×C411yescharacteristic
42C22×#67—22no—
43#1010—44no—
44C2×D3×Q8C2×C412yes—
45#1004—44no—
46C2×#237—21no—
47C2×#237C2×C412yes—
48C2×#237D414yes—
49C2×#237C2312yes—
50C2×D3×D4—21no—
51C2×D3×D4C2×C412yes—
52C2×#235C2×C411yescharacteristic
53C2×#235C2×C411yescharacteristic
54C2×#235D412yes—
55C22×Dic6C2×C411yescharacteristic
56C22×Dic6D412yes—
57C3×#127C2312yes—
58C3×#124C2×C412yes—
59C3×#121—22no—
60C3×#120—22no—
61C3×#117—22no—
62C3×#117C2×C411yescharacteristic
63C2×#217—22no—
64#947C2×C411yescharacteristic
65#944C2×C411yescharacteristic
66#940C2311yescharacteristic
67#939—22no—
68#939—24no—
69#938—24no—
70#937—24no—
71#936—22no—
72#936C2×C412yes—
73#936C2×C412yes—
74#936C2311yescharacteristic
75D4×Dic3—24no—
76C4×#67—22no—
77#926—22no—
78C2×C4×Dic3C2×C411yescharacteristic
79#897—22no—
80#893—21no—
81#887—21no—
82#887—44no—
83#886—22no—
84#885—24no—
85#883—24no—
86#67⋊C4—22no—
87D3×C22⋊C4—24no—
88#878—24no—
89#877—28no—
90#875C2×C412yes—
91#872C2×C412yes—
92C42×D3C2×C412yes—
93#869C2×C412yes—
94C4×Dic6—44no—
95C3×#99—22no—
96C3×#99C2314yes—
97#834—22no—
98#834C2314yes—
99#806—28no—
100C2×2−1+4—31no—
101C2×H(5,𝔽2)—31no—
102#548—62no—
103#546—64no—
104#546—62no—
105#536—32no—
106#536—31no—
107#535—32no—
108#535C2×D311yescharacteristic
109#532—31no—
110#532—32no—
111C2×#124—31no—
112C2×#124—32no—
113#519—62no—
114C4×#41—31no—
115#410—32no—
116#410—38no—
117#410—64no—
118#410—62no—
119#410C2×D312yescomplemented
120#354—34no—
121#353—34no—
122#353—34no—
123#353C2×D314yescomplemented
124C24×C3—21no—
125C23×D3—22no—
126C3×#41—42no—
127C2×C3×Q8C22×C411yescharacteristic
128C2×C3×D4—24no—
129C2×C3×D4—21no—
130C2×C3×D4—42no—
131C2×C3×D4C22×C412yes—
132C2×C3×D4C2×D412yes—
133C2×C3×D4C2411yescharacteristic
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D412yes—
136C2×#67—21no—
137C2×#67—22no—
138C2×#67—22no—
139C2×#67—22no—
140C2×#67—48no—
141C2×#67C22⋊C412yes—
142C2×#67C2×D414yes—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C22×Dic3C22⋊C412yes—
146C22×Dic3C2×D414yes—
147D3×Q8—44no—
148#237—48no—
149#237—44no—
150#237—44no—
151#237—44no—
152D3×D4—44no—
153D3×D4—44no—
154#235—24no—
155#235—42no—
156#235—42no—
157#235—42no—
158C2×D12C22⋊C411yescharacteristic
159C2×D12C22×C411yescharacteristic
160C2×C4×D3—21no—
161C2×C4×D3—22no—
162C2×C4×D3C22⋊C414yes—
163C2×C4×D3C22×C412yes—
164C2×Dic6—24no—
165C2×Dic6—24no—
166C2×Dic6—42no—
167C2×Dic6C22⋊C414yes—
168C2×Dic6C22⋊C412yes—
169C2×Dic6C22⋊C411yescharacteristic
170C2×Dic6C22×C411yescharacteristic
171C2×Dic6C22×C412yes—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—42no—
174C3×C22⋊C4—21no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C42—42no—
178C3×C42C22×C412yes—
179#217—22no—
180#217—24no—
181#217—24no—
182#217—24no—
183(C2×D3)⋊C4—22no—
184(C2×D3)⋊C4—22no—
185(C2×D3)⋊C4—24no—
186C4⋊Dic3—24no—
187C4⋊Dic3—42no—
188Dic3⋊C4—22no—
189Dic3⋊C4—22no—
190Dic3⋊C4—48no—
191C4×Dic3—21no—
192C4×Dic3—22no—
193C4×Dic3—48no—
194C4×Dic3—42no—
195C4×Dic3C22×C411yescharacteristic
196C4×Dic3C22×C411yescharacteristic
1972−1+4—124no—
198H(5,𝔽2)—124no—
199C2×#41—34no—
200C2×#41—32no—
201C2×#41—31no—
202C2×#41—32no—
203C2×#41—32no—
204C2×#41—31no—
205C2×#41—61no—
206C2×#41C22×D311yescharacteristic
207C22×Q8—32no—
208C22×Q8—31no—
209C22×Q8—32no—
210C22×D4—32no—
211C22×D4—61no—
212C22×D4—62no—
213C22×D4C4×D311yescharacteristic
214C4⋊Q8—31no—
215C4⋊Q8—32no—
216#127—31no—
217#127C22×D312yes—
218#124—31no—
219#124—32no—
220#124—32no—
221#124—31no—
222#124—32no—
223#124—64no—
224#124—62no—
225#124C4×D312yes—
226#123—64no—
227#123—62no—
228C22⋊Q8—64no—
229C22⋊Q8—62no—
230#121—22no—
231#121—64no—
232#121—64no—
233#121—62no—
234#120—22no—
235#120—64no—
236C4×Q8—61no—
237C4×Q8—124no—
238C4×Q8—124no—
239C4×D4—62no—
240C4×D4—61no—
241C4×D4—64no—
242C4×D4—62no—
243#117—22no—
244#117—32no—
245#117—68no—
246#117—62no—
247#117C4×D311yescharacteristic
248C2×C22⋊C4—64no—
249C2×C22⋊C4—62no—
250C2×C42—31no—
251C2×C42—32no—
252#99—22no—
253#99—34no—
254#99—68no—
255#99—62no—
256#99C22×D314yes—
257C23×C3—22no—
258C23×C3—42no—
259C23×C3—82no—
260C23×C3C2×C22⋊C411yescharacteristic
261C23×C3C22×D412yes—
262C22×D3—21no—
263C22×D3—44no—
264C22×D3—84no—
265C22×D3C2×C22⋊C412yes—
266C3×Q8—41no—
267C3×D4—42no—
268C3×D4—41no—
269C3×D4—42no—
270C3×D4—42no—
271C3×D4—84no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—44no—
277C2×C3×C4C2×C22⋊C412yes—
278C2×C3×C4C2×C22⋊C411yescharacteristic
279C2×C3×C4C23×C411yescharacteristic
280#67—24no—
281#67—44no—
282#67—44no—
283#67—42no—
284#67—44no—
285#67—82no—
286C2×Dic3—28no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—42no—
291C2×Dic3—44no—
292C2×Dic3—44no—
293C2×Dic3—81no—
294C2×Dic3C2×C22⋊C414yes—
295C2×Dic3C2×C22⋊C412yes—
296D12—22no—
297D12—41no—
298C4×D3—24no—
299C4×D3—44no—
300C4×D3—42no—
301C4×D3—82no—
302Dic6—22no—
303Dic6—22no—
304Dic6—28no—
305Dic6—42no—
306Dic6—41no—
307Dic6#9914yes—
308C24—21no—
309C24—62no—
310#41—42no—
311#41—64no—
312#41—122no—
313#41—122no—
314#41—122no—
315#41—124no—
316#41—128no—
317#41—124no—
318#41—124no—
319C2×Q8—34no—
320C2×Q8—31no—
321C2×Q8—32no—
322C2×Q8—32no—
323C2×Q8—31no—
324C2×Q8—64no—
325C2×Q8—64no—
326C2×Q8—122no—
327C2×Q8—124no—
328C2×Q8C2×C4×D311yescharacteristic
329C2×D4—24no—
330C2×D4—21no—
331C2×D4—32no—
332C2×D4—34no—
333C2×D4—31no—
334C2×D4—31no—
335C2×D4—42no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—61no—
339C2×D4—62no—
340C2×D4—124no—
341C2×D4—124no—
342C2×D4—128no—
343C2×D4C2×C4×D312yes—
344C2×D4D3×D412yes—
345C2×D4C23×D311yescharacteristic
346C22×C4—34no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—34no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4C2×C4×D311yescharacteristic
355C22×C4D3×D412yes—
356C4⋊C4—21no—
357C4⋊C4—42no—
358C4⋊C4—62no—
359C4⋊C4—64no—
360C4⋊C4—62no—
361C4⋊C4—122no—
362C4⋊C4—128no—
363C22⋊C4—21no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—64no—
367C22⋊C4—62no—
368C22⋊C4—64no—
369C22⋊C4—64no—
370C22⋊C4—62no—
371C22⋊C4—62no—
372C22⋊C4—64no—
373C42—31no—
374C42—31no—
375C42—42no—
376C42—62no—
377C42—61no—
378C42—122no—
379C42—128no—
380C42C2×C4×D312yes—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—41no—
384C22×C3—42no—
385C22×C3—44no—
386C22×C3—42no—
387C22×C3—162no—
388C22×C3C22×C22⋊C411yescharacteristic
389C2×D3—22no—
390C2×D3—42no—
391C2×D3—42no—
392C2×D3—41no—
393C2×D3—164no—
394C3×C4—22no—
395C3×C4—21no—
396C3×C4—42no—
397C3×C4—84no—
398C3×C4C2×#9912yes—
399Dic3—24no—
400Dic3—41no—
401Dic3—41no—
402Dic3—41no—
403Dic3—84no—
404Dic3C2×#9914yes—
405C23—22no—
406C23—32no—
407C23—42no—
408C23—61no—
409C23—82no—
410C23—124no—
411C23—244no—
412C23D3×C22⋊C411yescharacteristic
413C23C2×D3×D412yes—
414Q8—34no—
415Q8—41no—
416Q8—68no—
417Q8—62no—
418Q8—62no—
419Q8—122no—
420Q8—121no—
421D4—42no—
422D4—41no—
423D4—42no—
424D4—42no—
425D4—64no—
426D4—62no—
427D4—84no—
428D4—121no—
429D4—124no—
430D4—124no—
431D4—122no—
432D4—124no—
433D4—242no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—32no—
439C2×C4—34no—
440C2×C4—44no—
441C2×C4—64no—
442C2×C4—68no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—124no—
447C2×C4—124no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—124no—
451C2×C4—242no—
452C2×C4—241no—
453C2×C4D3×C22⋊C412yes—
454C2×C4D3×C22⋊C411yescharacteristic
455C2×C4C22×C4×D311yescharacteristic
456C2×C3—21no—
457C2×C3—42no—
458C2×C3—41no—
459C2×C3—82no—
460C2×C3C22×#9911yescharacteristic
461D3—42no—
462D3—81no—
463C22—22no—
464C22—21no—
465C22—41no—
466C22—42no—
467C22—44no—
468C22—42no—
469C22—62no—
470C22—122no—
471C22—122no—
472C22—121no—
473C22—162no—
474C22—484no—
475C22C2×D3×C22⋊C411yescharacteristic
476C4—21no—
477C4—22no—
478C4—34no—
479C4—42no—
480C4—64no—
481C4—84no—
482C4—121no—
483C4—121no—
484C4—121no—
485C4—244no—
486C4D3×#9912yes—
487C3#3336211yescharacteristic, complemented, Sylow
488C2—21no—
489C2—42no—
490C2—41no—
491C2—82no—
492C2—122no—
493C2—241no—
494C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2128C2
socleC2×C3C22×#99
prime radical#86317C2
cosocleC2×C3×D4C24
hypercenter#2128D3
hypocenterC3#33362
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2128

Complements

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

NH ≅ G/Ntype
2. #78689C2semidirect
3. C3×#2128C2semidirect
4. #71861C2semidirect
5. #70060C2semidirect
6. #69152C2semidirect
7. #69151C2semidirect
10. C2×#1004C4semidirect
13. C2×#936C4semidirect
16. #5675C4semidirect
17. D3×#124C4semidirect
21. #5526C4semidirect
22. #5522C4semidirect
24. C3×#410C22semidirect
25. #5240C22semidirect
28. #4740C22semidirect
29. C3×#353C22semidirect
30. #4545C22semidirect
31. #4474C22semidirect
32. #4473C22semidirect
37. #2128D3semidirect
119. #410C2×D3semidirect
123. #353C2×D3semidirect
487. C3#33362semidirect

Maximal subgroups

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

subgroupindex
#863172
#786892
C3×#21282
#718612
#700602
#691522
#691512
#333623