Finite Group Databasebeta

← #11658980

Subgroups — #11658980

Order 768.

2042 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85731C211yesmaximal, characteristic
2C2×C3×#424C211yesmaximal, characteristic, complemented
3#83762C211yesmaximal, characteristic
4#83747C211yesmaximal, characteristic, complemented
5C2×#5110C211yesmaximal, characteristic
6C2×#5101C211yesmaximal, characteristic, complemented
7#82204C211yesmaximal, characteristic, complemented
8#79072C218yesmaximal, complemented
9#79071C214yesmaximal, complemented
10#79070C214yesmaximal, complemented
11#79000C218yesmaximal, complemented
12#60126—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#926C2211yescharacteristic
18C22×#208C2211yescharacteristic
19C22×#208C2211yescharacteristic
20#5523C414yes—
21#5522C2214yes—
22#5521C412yes—
23D3×#117C2214yes—
24#5516C412yes—
25#5504C414yes—
26#5486C414yes—
27C4×#235C2214yes—
28C3×#424C2214yes—
29C3×#424C2212yescomplemented
30C3×#424C2212yescomplemented
31C2×C3×C4⋊C8C2214yescomplemented
32#5129C2214yes—
33#5126C2214yes—
34#5122C2214yes—
35Dic3×M16C2214yescomplemented
36#5110C2214yes—
37#5110C2212yes—
38#5110C2212yes—
39#5110C2214yes—
40C2×(C2×D3)⋊C8C2214yes—
41#5101C2214yes—
42#5101C2214yescomplemented
43#5101C2214yescomplemented
44C2×#815C2212yescomplemented
45C2×C8×Dic3C2212yescomplemented
46#4999C2214yes—
47#4999C2214yescomplemented
48#4924C2214yescomplemented
49#4840C22116yescomplemented
50#4840C22116yescomplemented
51#4835C22116yescomplemented
52#4833C22116yescomplemented
53#3428—31no—
54#2925—38no—
55#2924—38no—
56#2923—34no—
57#2922—34no—
58#2905—31no—
59C2×#424D311yescharacteristic, complemented
60#2879—31no—
61C2×#408—31no—
62#2873—31no—
63C2×#405—31no—
64C23×C3×C4C2311yescharacteristic
65C22×#67C2×C412yes—
66C2×#235—24no—
67C2×#235—24no—
68C2×#235C2×C412yes—
69C2×#235C2×C412yes—
70C2×#235C2312yes—
71C2×#235C2312yes—
72C22×D12C2×C411yescharacteristic
73C22×C4×D3C2312yes—
74C22×Dic6C2×C411yescharacteristic
75C2×C3×M16—24no—
76C2×C3×M16C2312yes—
77C2×C3×M16C2312yes—
78C22×C3×C8—24no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2312yes—
84C3×#117C2×C412yes—
85C3×#117C2×C412yes—
86C3×#117C2312yes—
87C3×#117C2312yes—
88C2×C3×C4⋊C4C2×C412yes—
89C2×C3×C22⋊C4C2×C412yes—
90C2×C3×C42C2312yes—
91C2×#217C2×C412yes—
92C4×#67C2×C4116yes—
93#926C2×C412yes—
94#926C2×C412yes—
95#926C2312yes—
96#926C2312yes—
97C2×C4⋊Dic3C2×C412yes—
98C2×C4×Dic3C2312yes—
99C2×#208—24no—
100C2×#208—24no—
101C2×#208C2312yes—
102C2×#208C2312yes—
103C2×#208C2312yes—
104C2×#208C2312yes—
105C22×C3⋊C8C2312yes—
106C22×C3⋊C8C2312yes—
107#893C2×C418yes—
108#892C2×C418yes—
109D3×C4⋊C4C2×C418yes—
110#887C2×C418yes—
111#67⋊C4C2×C4116yes—
112D3×C22⋊C4C2×C418yes—
113#877C2×C418yes—
114C4×D12C2×C418yes—
115#872C2318yes—
116C42×D3C2318yes—
117C4×Dic6C2×C418yes—
118C3×C4⋊C8C2318yes—
119C3×C4⋊C8C2318yes—
120(C2×D3)⋊C8C23116yes—
121(C2×D3)⋊C8C2318yes—
122(C2×D3)⋊C8C2318yes—
123#815C2318yes—
124#815C2318yes—
125C8×Dic3C2318yes—
126C8×Dic3C2318yes—
127C4⋊(C3⋊C8)C23116yes—
128C22×#41—31no—
129C22×M16—31no—
130C22×M16—31no—
131C22×M16C2×D311yescharacteristic
132C23×C8C2×D311yescharacteristic
133#520—34no—
134#520—34no—
135#519—32no—
136#518—32no—
137#518—34no—
138C4×#41—34no—
139C4×#41—34no—
140C2×#117—31no—
141C2×#117—34no—
142C2×#117C2×D311yescharacteristic
143#433—316no—
144#433—316no—
145#432—316no—
146#431—316no—
147#424—34no—
148#424—34no—
149#424C2×D314yes—
150#424C2×D312yescomplemented
151#424C2×D312yescomplemented
152#423—34no—
153C2×C4⋊C8C2×D314yescomplemented
154#408—34no—
155#408—32no—
156#408—32no—
157#408—34no—
158#408—34no—
159#407—34no—
160C2×C22⋊C8—34no—
161#405—34no—
162#405—34no—
163#405—34no—
164#405—34no—
165C4×M16—34no—
166C2×#97—32no—
167C2×C4×C8—32no—
168C24×C3C22×C411yescharacteristic
169C23×D3C22×C412yes—
170C22×C3×C4C22×C414yes—
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4C22×C411yescharacteristic
174C22×C3×C4#4114yes—
175C22×C3×C4#4114yes—
176C22×C3×C4C2411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—28no—
180C2×#67C22×C414yes—
181C2×#67C22×C414yes—
182C22×Dic3C22×C412yes—
183C22×Dic3C22×C414yes—
184#235—28no—
185#235—28no—
186#235—24no—
187#235—24no—
188#235—24no—
189#235—24no—
190C2×D12—24no—
191C2×D12C22×C412yes—
192C2×D12C22×C412yes—
193C2×C4×D3—24no—
194C2×C4×D3—24no—
195C2×C4×D3C22×C414yes—
196C2×C4×D3C22×C4116yes—
197C2×C4×D3C2414yes—
198C2×Dic6—24no—
199C2×Dic6C22×C412yes—
200C2×Dic6C22×C412yes—
201C3×M16—24no—
202C3×M16—24no—
203C2×C3×C8—24no—
204C2×C3×C8—24no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—24no—
208C2×C3×C8C2412yes—
209C2×C3×C8C2414yes—
210C2×C3×C8C2412yes—
211C3×C4⋊C4C22×C414yes—
212C3×C4⋊C4C22×C414yes—
213C3×C22⋊C4C22×C414yes—
214C3×C22⋊C4C22×C414yes—
215C3×C42C22×C414yes—
216C3×C42C2414yes—
217#217C22×C414yes—
218#217C22×C414yes—
219(C2×D3)⋊C4C22×C4116yes—
220C4⋊Dic3C22×C414yes—
221C4⋊Dic3C22×C414yes—
222Dic3⋊C4C22×C4116yes—
223C4×Dic3C22×C4116yes—
224C4×Dic3C22×C414yes—
225C4×Dic3C2414yes—
226#208—24no—
227#208—24no—
228#208—24no—
229#208—24no—
230C2×C3⋊C8—24no—
231C2×C3⋊C8—24no—
232C2×C3⋊C8C2414yes—
233C2×C3⋊C8C2414yes—
234C2×#41—32no—
235C2×#41—32no—
236C2×#41—32no—
237C2×#41—32no—
238C2×#41—64no—
239C2×#41—64no—
240C22×Q8—31no—
241C22×D4—32no—
242C22×D4—31no—
243C23×C4—32no—
244C23×C4C22×D311yescharacteristic
245C2×M16—24no—
246C2×M16—32no—
247C2×M16—32no—
248C2×M16—32no—
249C2×M16—32no—
250C2×M16—64no—
251C2×M16—64no—
252C2×M16C22×D312yes—
253C2×M16C22×D312yes—
254C22×C8—24no—
255C22×C8—32no—
256C22×C8—32no—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D312yes—
260C22×C8C22×D312yes—
261C22×C8C22×D312yes—
262C4×Q8—38no—
263C4×Q8—38no—
264C4×D4—316no—
265C4×D4—38no—
266C4×D4—316no—
267C4×D4—38no—
268#117—32no—
269#117—32no—
270#117—32no—
271#117—32no—
272#117—38no—
273#117—38no—
274#117—38no—
275#117C4×D312yes—
276#117C4×D312yes—
277#117C22×D312yes—
278#117C22×D312yes—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—38no—
281C2×C4⋊C4C4×D312yes—
282C2×C22⋊C4—32no—
283C2×C22⋊C4—38no—
284C2×C22⋊C4C4×D312yes—
285C2×C42—32no—
286C2×C42—38no—
287C2×C42C22×D312yes—
288C4⋊C8—316no—
289C4⋊C8C22×D318yes—
290C4⋊C8C22×D318yes—
291C22⋊C8—38no—
292C22⋊C8—38no—
293C22⋊C8—316no—
294#97—38no—
295#97—38no—
296C4×C8—38no—
297C4×C8—38no—
298C23×C3—42no—
299C23×C3#11714yes—
300C23×C3C23×C411yescharacteristic
301C23×C3C23×C411yescharacteristic
302C23×C3C23×C411yescharacteristic
303C22×D3—24no—
304C22×D3—44no—
305C22×D3C23×C414yes—
306C2×C3×C4—44no—
307C2×C3×C4#11714yes—
308C2×C3×C4#11714yes—
309C2×C3×C4#11714yes—
310C2×C3×C4C23×C418yes—
311C2×C3×C4C23×C411yescharacteristic
312C2×C3×C4C23×C411yescharacteristic
313C2×C3×C4C23×C411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2×#4112yes—
317C2×C3×C4C2×#4112yes—
318C2×C3×C4C2×#4112yes—
319C2×C3×C4C2×#4112yes—
320C2×C3×C4C2511yescharacteristic
321#67—28no—
322#67—28no—
323C2×Dic3—24no—
324C2×Dic3—44no—
325C2×Dic3C23×C414yes—
326C2×Dic3C23×C418yes—
327D12—24no—
328D12—24no—
329C4×D3—28no—
330C4×D3—24no—
331C4×D3—24no—
332C4×D3—416no—
333Dic6—24no—
334Dic6—24no—
335C3×C8—22no—
336C3×C8—22no—
337C3×C8—24no—
338C3⋊C8—24no—
339C3⋊C8—24no—
340C24—32no—
341C24C2×C4×D311yescharacteristic
342#41—68no—
343#41—68no—
344#41—64no—
345#41—64no—
346#41—64no—
347#41—64no—
348C2×Q8—32no—
349C2×Q8—32no—
350C2×Q8—64no—
351C2×D4—34no—
352C2×D4—34no—
353C2×D4—32no—
354C2×D4—32no—
355C2×D4—68no—
356C2×D4—64no—
357C22×C4—32no—
358C22×C4—34no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—316no—
362C22×C4—64no—
363C22×C4—64no—
364C22×C4C2×C4×D314yes—
365C22×C4C2×C4×D311yescharacteristic
366C22×C4C2×C4×D311yescharacteristic
367C22×C4C2×C4×D311yescharacteristic
368C22×C4#23714yes—
369C22×C4#23914yes—
370C22×C4C23×D311yescharacteristic
371C22×C4C23×D311yescharacteristic
372C22×C4C23×D311yescharacteristic
373M16—24no—
374M16—24no—
375M16—64no—
376M16—64no—
377M16—64no—
378M16—64no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—34no—
385C2×C8—34no—
386C2×C8—64no—
387C2×C8—64no—
388C2×C8C23×D312yes—
389C2×C8C23×D314yes—
390C2×C8C23×D312yes—
391C4⋊C4—34no—
392C4⋊C4—316no—
393C4⋊C4—34no—
394C4⋊C4C2×C4×D314yes—
395C4⋊C4C2×C4×D314yes—
396C22⋊C4—34no—
397C22⋊C4—34no—
398C22⋊C4—316no—
399C22⋊C4C2×C4×D314yes—
400C22⋊C4C2×C4×D314yes—
401C42—34no—
402C42—316no—
403C42—34no—
404C42C2×C4×D314yes—
405C42C23×D314yes—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—44no—
410C22×C3C2×#11712yes—
411C22×C3C2×#11712yes—
412C22×C3C2×#11712yes—
413C22×C3C24×C411yescharacteristic
414C2×D3—24no—
415C2×D3—28no—
416C2×D3—84no—
417C3×C4—28no—
418C3×C4C2×#11712yes—
419C3×C4C2×#11712yes—
420C3×C4C2×#11712yes—
421C3×C4C22×#4111yescharacteristic
422C3×C4C22×#4111yescharacteristic
423Dic3—24no—
424Dic3—28no—
425C23—34no—
426C23—42no—
427C23—64no—
428C23—124no—
429C23#89214yes—
430C23C22×C4×D311yescharacteristic
431C23C22×C4×D311yescharacteristic
432C23C22×C4×D311yescharacteristic
433Q8—64no—
434Q8—64no—
435D4—68no—
436D4—68no—
437D4—64no—
438D4—64no—
439C2×C4—38no—
440C2×C4—34no—
441C2×C4—44no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—68no—
446C2×C4—1216no—
447C2×C4—124no—
448C2×C4#89214yes—
449C2×C4#89214yes—
450C2×C4#89214yes—
451C2×C4C22×C4×D318yes—
452C2×C4C22×C4×D311yescharacteristic
453C2×C4C22×C4×D311yescharacteristic
454C2×C4C22×C4×D311yescharacteristic
455C2×C4C2×#23712yes—
456C2×C4C2×#23712yes—
457C2×C4C2×#23712yes—
458C2×C4C2×#23912yes—
459C2×C4C2×#23912yes—
460C2×C4C2×#23912yes—
461C2×C4C24×D311yescharacteristic
462C8—22no—
463C8—22no—
464C8—24no—
465C8—64no—
466C8—64no—
467C2×C3—22no—
468C2×C3—22no—
469C2×C3—22no—
470C2×C3C22×#11711yescharacteristic
471C2×C3#357312yes—
472D3—44no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—44no—
477C22—68no—
478C22—64no—
479C22—244no—
480C22C2×#89212yes—
481C22C2×#89212yes—
482C22C2×#89212yes—
483C22C23×C4×D311yescharacteristic
484C4—28no—
485C4—68no—
486C4—64no—
487C4C2×#89212yes—
488C4C2×#89212yes—
489C4C2×#89212yes—
490C4C22×#23711yescharacteristic
491C4C22×#23911yescharacteristic
492C3#6012611yescharacteristic, complemented, Sylow
493C2—22no—
494C2—22no—
495C2—22no—
496C2—124no—
497C2C22×#89211yescharacteristic
498C2#8716012yes—
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC2×C3×#424C2
prime radicalC22×#235C22
cosocleC2×C3×C4C25
hypercenterC2×#424D3
hypocenterC3#60126
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
2. C2×C3×#424C2semidirect
4. #83747C2semidirect
6. C2×#5101C2semidirect
7. #82204C2semidirect
8. #79072C2semidirect
9. #79071C2semidirect
10. #79070C2semidirect
11. #79000C2semidirect
29. C3×#424C22semidirect
30. C3×#424C22semidirect
31. C2×C3×C4⋊C8C22semidirect
35. Dic3×M16C22semidirect
42. #5101C22semidirect
43. #5101C22semidirect
44. C2×#815C22semidirect
45. C2×C8×Dic3C22semidirect
47. #4999C22semidirect
48. #4924C22semidirect
49. #4840C22semidirect
50. #4840C22semidirect
51. #4835C22semidirect
52. #4833C22semidirect
59. C2×#424D3semidirect
150. #424C2×D3semidirect
151. #424C2×D3semidirect
153. C2×C4⋊C8C2×D3semidirect
492. C3#60126semidirect

Maximal subgroups

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

subgroupindex
#857312
C2×C3×#4242
#837622
#837472
C2×#51102
C2×#51012
#822042
#790722
#790712
#790702
#790002
#601263