Finite Group Databasebeta

← #10816048

Subgroups — #10816048

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#453C211yesmaximal, characteristic
2#84099C212yesmaximal, complemented
3#83930C212yesmaximal, complemented
4#72091C212yesmaximal, complemented
5#71764C211yesmaximal, characteristic, complemented
6#71763C211yesmaximal, characteristic, complemented
7#70369C212yesmaximal, complemented
8#70363C212yesmaximal, complemented
9#70072C212yesmaximal, complemented
10#22897—31nomaximal, Sylow
11C2×C3×H(5,𝔽2)C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C2×C3×#127C2211yescharacteristic
14#5827C2212yes—
15C2×#948C2212yes—
16#5798C2211yescharacteristic, complemented
17C3×#453C418yescomplemented
18C2×C3×#104C2212yes—
19C2×C3×#100C2211yescharacteristic
20#5233—22no—
21#5233C2212yes—
22#5232—22no—
23#5232C2212yes—
24C2×#833C2211yescharacteristic, complemented
25#5215—22no—
26#5215C2212yes—
27C2×#832—22no—
28C2×#832C2212yes—
29C2×#832C2212yescomplemented
30#5164—22no—
31Dic3×SD16—22no—
32#5152—22no—
33Dic3×D8—22no—
34#5122C2212yescomplemented
35#5121C2212yescomplemented
36#4935C2211yescharacteristic, complemented
37C2×C4×C3⋊C8C2211yescharacteristic, complemented
38#4558C2212yescomplemented
39#4554C2212yescomplemented
40#4523C2212yescomplemented
41C2×#453D311yescharacteristic
42#2946—32no—
43#2897—32no—
44#1957—31no—
45#1931—32no—
46#1907—32no—
47#1887—31no—
48#1859—32no—
49#1764—32no—
50C3×H(5,𝔽2)—24no—
51C3×H(5,𝔽2)—41no—
52C3×H(5,𝔽2)C2×C414yescomplemented
53C2×C3×#41—22no—
54C2×C3×#41C2312yes—
55C22×C3×D4—22no—
56C22×C3×D4—41no—
57C22×C3×D4—41no—
58C22×C3×D4C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C3×AGL(1,ℤ/8)—24no—
61C3×AGL(1,ℤ/8)C2×C418yescomplemented
62C2×C3×SD16—22no—
63C2×C3×D8—22no—
64C2×C3×M16C2312yes—
65C3×#127—41no—
66C3×#127C2×C414yescomplemented
67C2×C3×C42C2311yescharacteristic
68C2×#217—41no—
69#948—24no—
70#948—42no—
71Q8×Dic3—22no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74D4×Dic3—22no—
75D4×Dic3—42no—
76#926—22no—
77#926C2312yes—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—41no—
80C2×C4⋊Dic3C2312yes—
81C2×#208—22no—
82C2×#208C2312yes—
83C22×C3⋊C8—22no—
84C22×C3⋊C8C2312yes—
85C3×#104C2×C418yescomplemented
86C3×#100C2×C414yescomplemented
87#835—22no—
88#835D412yescomplemented
89#835D412yescomplemented
90#833—41no—
91#832—22no—
92#832—22no—
93#832—22no—
94#832—22no—
95#832—22no—
96#832—22no—
97#832—22no—
98#832D412yescomplemented
99#832D412yescomplemented
100#818—22no—
101#817—22no—
102#815—22no—
103C8×Dic3—22no—
104C4×C3⋊C8—41no—
105C2×H(5,𝔽2)C2×D311yescharacteristic
106C2×AGL(1,ℤ/8)C2×D312yes—
107C2×#131—32no—
108C2×#127C2×D311yescharacteristic
109#520—32no—
110#518—31no—
111#453Dic318yescomplemented
112#442—62no—
113#440—62no—
114C4×SD16—62no—
115C4×D8—62no—
116#428—32no—
117C2×#104C2×D312yes—
118#418—32no—
119#418—62no—
120#417—32no—
121#417—62no—
122#416—32no—
123#416—62no—
124C2×#102—32no—
125C2×#102—32no—
126C2×#102—62no—
127C2×#100—31no—
128C2×#100C2×D311yescharacteristic
129#405—32no—
130C2×C4×C8—31no—
131#384—31no—
132#343—32no—
133#341—32no—
134#339—32no—
135C24×C3—21no—
136C24×C3—21no—
137C24×C3—41no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—28no—
141C3×#41—44no—
142C3×#41C22×C414yes—
143C3×#41#4114yes—
144C2×C3×Q8C2×D412yes—
145C2×C3×D4—22no—
146C2×C3×D4—28no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—44no—
154C2×C3×D4—42no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4—44no—
158C2×C3×D4—82no—
159C2×C3×D4—82no—
160C2×C3×D4C22×C412yes—
161C2×C3×D4C22×C412yes—
162C2×C3×D4C2×D412yes—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—41no—
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—22no—
168C22×Dic3—41no—
169C22×Dic3—41no—
170C22×Dic3—42no—
171C3×SD16—28no—
172C3×D8—28no—
173C3×M16—22no—
174C3×M16C22×C414yes—
175C2×C3×C8—22no—
176C3×C42—21no—
177C3×C42C22×C412yes—
178#217—22no—
179#217—42no—
180#217—42no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—42no—
184C4⋊Dic3—82no—
185C4⋊Dic3C2×D412yes—
186C4⋊Dic3C2×D412yes—
187C4×Dic3—22no—
188C4×Dic3—42no—
189#208—22no—
190#208—22no—
191#208—44no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—44no—
196C2×C3⋊C8C2×D412yes—
197C2×C3⋊C8C2×D412yes—
198H(5,𝔽2)—24no—
199H(5,𝔽2)—41no—
200H(5,𝔽2)C2×Dic314yescomplemented
201C2×#41—22no—
202C2×#41C22×D312yes—
203C22×D4—22no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4C22×D311yescharacteristic
207C22×D4C22×D311yescharacteristic
208AGL(1,ℤ/8)—24no—
209AGL(1,ℤ/8)C2×Dic318yescomplemented
210C2×SD16—22no—
211C2×D8—22no—
212#131—64no—
213#131—122no—
214C2×M16—32no—
215C2×M16—62no—
216C2×M16C22×D312yes—
217C22×C8—32no—
218C22×C8—62no—
219#127—41no—
220#127C2×Dic314yescomplemented
221C4×Q8—62no—
222C4×D4—62no—
223C4×D4—62no—
224C4×D4—62no—
225C4×D4—122no—
226#117—32no—
227#117—62no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—121no—
231C2×C22⋊C4—121no—
232C2×C42C22×D311yescharacteristic
233#107—62no—
234#106—62no—
235#104C2×Dic318yescomplemented
236#103—32no—
237#103—32no—
238#103—62no—
239#102—32no—
240#102—32no—
241#102—62no—
242#102—62no—
243#102—62no—
244#102—62no—
245#102—62no—
246#102—62no—
247#102—62no—
248#100—121no—
249#100C2×Dic314yescomplemented
250#97—62no—
251C4×C8—62no—
252C4×C8—121no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—44no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—81no—
262C23×C3—81no—
263C23×C3—81no—
264C23×C3—81no—
265C23×C3—161no—
266C23×C3—161no—
267C23×C3H(5,𝔽2)11yescharacteristic
268C3×Q8—22no—
269C3×Q8C4×D414yes—
270C3×D4—24no—
271C3×D4—24no—
272C3×D4—24no—
273C3×D4—24no—
274C3×D4—22no—
275C3×D4—48no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—88no—
279C3×D4C4×D414yes—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—44no—
287C2×C3×C4C23×C411yescharacteristic
288C2×C3×C4C22×D412yes—
289C2×C3×C4C2×#4112yes—
290C2×C3×C4H(5,𝔽2)11yescharacteristic
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—42no—
294C2×Dic3—82no—
295C2×Dic3—161no—
296C2×Dic3—161no—
297C2×Dic3—162no—
298C3×C8—24no—
299C3⋊C8—22no—
300C3⋊C8—22no—
301C3⋊C8—42no—
302C24—21no—
303C24—21no—
304C24—41no—
305#41—22no—
306#41—22no—
307#41—28no—
308#41—44no—
309#41#23714yes—
310#41C22×Dic314yes—
311C2×Q8D3×D412yes—
312C2×D4—22no—
313C2×D4—28no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4D3×D412yes—
328C2×D4C22×Dic312yes—
329C2×D4C22×Dic312yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—62no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—28no—
339D8—28no—
340M16—22no—
341M16—62no—
342M16—62no—
343M16—124no—
344M16C22×Dic314yes—
345C2×C8—22no—
346C2×C8—32no—
347C2×C8—32no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—124no—
352C4⋊C4—32no—
353C4⋊C4—32no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C4⋊C4—242no—
358C22⋊C4—62no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C42—21no—
362C42—62no—
363C42—122no—
364C42C22×Dic312yes—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—24no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—161no—
379C22×C3—161no—
380C22×C3—164no—
381C22×C3#51811yescharacteristic
382C22×C3D4211yescharacteristic
383C22×C3#55111yescharacteristic
384C3×C4—24no—
385C3×C4—24no—
386C3×C4C2×C4×D412yes—
387C3×C4#54612yes—
388Dic3—42no—
389Dic3—42no—
390Dic3—82no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—44no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—161no—
404C23—161no—
405C23#100411yescharacteristic
406Q8—22no—
407Q8D4×Dic314yes—
408D4—24no—
409D4—24no—
410D4—24no—
411D4—24no—
412D4—22no—
413D4—48no—
414D4—42no—
415D4—42no—
416D4—88no—
417D4D4×Dic314yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—44no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—122no—
428C2×C4—242no—
429C2×C4—481no—
430C2×C4—481no—
431C2×C4—482no—
432C2×C4C2×D3×D412yes—
433C2×C4C2×#23712yes—
434C2×C4#100411yescharacteristic
435C2×C4C23×Dic311yescharacteristic
436C8—24no—
437C8—62no—
438C8—62no—
439C8—122no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—44no—
443C2×C3—82no—
444C2×C3—82no—
445C2×C3#235811yescharacteristic
446C2×C3#329111yescharacteristic
447C2×C3#329211yescharacteristic
448C22—21no—
449C22—21no—
450C22—24no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—82no—
456C22—82no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—161no—
462C22—161no—
463C22—164no—
464C22#558711yescharacteristic
465C22#559011yescharacteristic
466C22#579811yescharacteristic
467C4—24no—
468C4—24no—
469C4—122no—
470C4—122no—
471C4—242no—
472C4#559112yes—
473C4C2×D4×Dic312yes—
474C3#2289711yescharacteristic, complemented, Sylow
475C2—21no—
476C2—21no—
477C2—44no—
478C2—82no—
479C2—82no—
480C2#8028111yescharacteristic
481C2#8028211yescharacteristic
482C2#8257911yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#453C2
socleC22×C3D42
cosocleC22×C3×C4C24
hypercenterC2×#453D3
hypocenterC3#22897
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#453

Complements

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

NH ≅ G/Ntype
2. #84099C2semidirect
3. #83930C2semidirect
4. #72091C2semidirect
5. #71764C2semidirect
6. #71763C2semidirect
7. #70369C2semidirect
8. #70363C2semidirect
9. #70072C2semidirect
16. #5798C22semidirect
17. C3×#453C4semidirect
24. C2×#833C22semidirect
29. C2×#832C22semidirect
34. #5122C22semidirect
35. #5121C22semidirect
36. #4935C22semidirect
37. C2×C4×C3⋊C8C22semidirect
38. #4558C22semidirect
39. #4554C22semidirect
40. #4523C22semidirect
52. C3×H(5,𝔽2)C2×C4semidirect
61. C3×AGL(1,ℤ/8)C2×C4semidirect
66. C3×#127C2×C4semidirect
85. C3×#104C2×C4semidirect
86. C3×#100C2×C4semidirect
88. #835D4semidirect
89. #835D4semidirect
98. #832D4semidirect
99. #832D4semidirect
111. #453Dic3semidirect
200. H(5,𝔽2)C2×Dic3semidirect
209. AGL(1,ℤ/8)C2×Dic3semidirect
220. #127C2×Dic3semidirect
235. #104C2×Dic3semidirect
249. #100C2×Dic3semidirect
474. C3#22897semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4532
#840992
#839302
#720912
#717642
#717632
#703692
#703632
#700722
#228973