Finite Group Databasebeta

← #80334

Subgroups — #80334

Order 384.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic, complemented
2C2×D3×D8C211yesmaximal, characteristic, complemented
3#5715C211yesmaximal, characteristic, complemented
4#5545C211yesmaximal, characteristic, complemented
5C3×#459C211yesmaximal, characteristic, complemented
6#5153C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic, complemented
8#5072C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4787C211yesmaximal, characteristic, complemented
13#4786C211yesmaximal, characteristic, complemented
14#4772C211yesmaximal, characteristic, complemented
15#4770C211yesmaximal, characteristic, complemented
16#3041—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic, complemented
22C3×#127C2211yescharacteristic, complemented
23C3×C4×D4C2211yescharacteristic, complemented
24#940C2211yescharacteristic, complemented
25#939—21no—
26#939—21no—
27#937—21no—
28#937—21no—
29#936C2211yescharacteristic, complemented
30D4×Dic3C2211yescharacteristic, complemented
31C2×#214C2211yescharacteristic, complemented
32C2×#213C2211yescharacteristic
33C2×#213C2211yescharacteristic
34C2×#213C2211yescharacteristic, complemented
35C4×#67—21no—
36C2×(C2×D3)⋊C4—21no—
37#911—24no—
38D3×D8—24no—
39C2×D24C2211yescharacteristic
40C2×#204C2211yescharacteristic, complemented
41C2×ΓL(1,𝔽25)C2211yescharacteristic
42C2×D3×C8C2211yescharacteristic
43#893C2211yescharacteristic, complemented
44#892C2211yescharacteristic
45#67⋊C4—21no—
46D3×C22⋊C4—21no—
47#875C2211yescharacteristic, complemented
48C4×D12C2211yescharacteristic, complemented
49#872C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic, complemented
51C3×#102C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53#832C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×D8—31no—
64#535—31no—
65#518—31no—
66#463—31no—
67#463—31no—
68#463—31no—
69#463—31no—
70#459—31no—
71#459—31no—
72#459—31no—
73#459D311yescharacteristic, complemented
74#424—31no—
75#418—31no—
76#416—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92#237—22no—
93#237—22no—
94D3×D4—22no—
95D3×D4—22no—
96D3×D4—22no—
97D3×D4D414yescomplemented
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×D8—22no—
105C3×D8—22no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C42C2311yescharacteristic
111#217—21no—
112#217—21no—
113#217—21no—
114#214—22no—
115#213—22no—
116#213—22no—
117#213—22no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4D411yescharacteristic
124(C2×D3)⋊C4D411yescharacteristic
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4D411yescharacteristic
127Dic3⋊C4D411yescharacteristic
128C4×Dic3—21no—
129C4×Dic3C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132D24—22no—
133#204—22no—
134ΓL(1,𝔽25)—22no—
135D3×C8—22no—
136C2×#41—31no—
137C22×D4—31no—
138C22×D4—31no—
139AGL(1,ℤ/8)—64no—
140C2×SD16—31no—
141C2×SD16—31no—
142C2×D8—31no—
143C2×D8—31no—
144C2×D8—31no—
145C2×D8—31no—
146C2×D8—64no—
147C2×D8C2×D311yescharacteristic
148C2×D8C2×D311yescharacteristic, complemented
149C2×M16—31no—
150C22×C8—31no—
151#127—31no—
152#127C2×D311yescharacteristic, complemented
153#124—31no—
154#124—31no—
155#121—61no—
156#121—61no—
157#120—61no—
158#120—61no—
159C4×D4—31no—
160C4×D4—31no—
161C4×D4—31no—
162C4×D4—61no—
163C4×D4—61no—
164C4×D4C2×D311yescharacteristic, complemented
165#117—31no—
166#117—31no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8C2×D311yescharacteristic, complemented
173#103—31no—
174#103—31no—
175#102—31no—
176#102—31no—
177#102—31no—
178#102—31no—
179#102C2×D311yescharacteristic, complemented
180#102C2×D311yescharacteristic, complemented
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C22×D3—21no—
185C22×D3—24no—
186C22×D3—21no—
187C22×D3—42no—
188C22×D3—81no—
189C22×D3—81no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3C2×D411yescharacteristic
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—81no—
199C3×D4—81no—
200C3×D4C2×D412yes—
201C2×C3×C4—21no—
202C2×C3×C4—81no—
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2411yescharacteristic
206#67—24no—
207#67—42no—
208#67—42no—
209#67—81no—
210#67—81no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—42no—
214C2×Dic3—81no—
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218D12—21no—
219D12—21no—
220D12—21no—
221D12C2×D412yes—
222C4×D3—81no—
223C4×D3—81no—
224C4×D3C2×D412yes—
225C4×D3#4112yes—
226Dic6—21no—
227Dic6—21no—
228C3×C8—21no—
229C3×C8—21no—
230C3⋊C8—21no—
231C3⋊C8—21no—
232C24—61no—
233C24—61no—
234#41—62no—
235#41—62no—
236C2×Q8—31no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—34no—
240C2×D4—31no—
241C2×D4—31no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4C22×D311yescharacteristic
251C2×D4C22×D311yescharacteristic
252C2×D4C22×D311yescharacteristic
253C22×C4—21no—
254C22×C4—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259SD16—62no—
260SD16—62no—
261D8—22no—
262D8—22no—
263D8—62no—
264D8—62no—
265D8—62no—
266D8—62no—
267M16—62no—
268C2×C8—31no—
269C2×C8—31no—
270C2×C8—62no—
271C2×C8C22×D311yescharacteristic
272C2×C8C22×D311yescharacteristic
273C4⋊C4—31no—
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4C22×D311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—31no—
279C22⋊C4—31no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C42—31no—
289C42—61no—
290C42C22×D311yescharacteristic
291C22×C3—22no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C22×D411yescharacteristic
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—22no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—84no—
310C2×D3—84no—
311C2×D3#12112yes—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4C22×D411yescharacteristic
316C3×C4C2×#4111yescharacteristic
317Dic3—21no—
318Dic3—21no—
319Dic3—41no—
320Dic3—41no—
321Dic3#12112yes—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—31no—
326C23—64no—
327C23—61no—
328C23—61no—
329C23—122no—
330C23—241no—
331C23—241no—
332C23—241no—
333C23—241no—
334Q8—61no—
335Q8—61no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—32no—
342D4—64no—
343D4—61no—
344D4—61no—
345D4—61no—
346D4—81no—
347D4—81no—
348D4—122no—
349D4—122no—
350D4—241no—
351D4—241no—
352D4D3×D412yes—
353C2×C4—21no—
354C2×C4—31no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—81no—
362C2×C4—122no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4D3×D411yescharacteristic
367C2×C4D3×D411yescharacteristic
368C2×C4C23×D311yescharacteristic
369C8—21no—
370C8—21no—
371C8—61no—
372C8—61no—
373C2×C3—42no—
374C2×C3—81no—
375C2×C3—81no—
376C2×C3C2×#12111yescharacteristic
377C2×C3#57611yescharacteristic
378C2×C3#57611yescharacteristic
379D3—22no—
380D3—42no—
381D3—81no—
382C22—22no—
383C22—32no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—61no—
393C22—61no—
394C22—62no—
395C22—121no—
396C22—121no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—244no—
401C22—244no—
402C22C2×D3×D411yescharacteristic
403C4—21no—
404C4—21no—
405C4—32no—
406C4—41no—
407C4—61no—
408C4—61no—
409C4—121no—
410C4—121no—
411C4C2×D3×D411yescharacteristic
412C4D3×#4111yescharacteristic
413C3#304111yescharacteristic, complemented, Sylow
414C2—42no—
415C2—62no—
416C2—81no—
417C2—81no—
418C2—122no—
419C2—241no—
420C2D3×#12111yescharacteristic
421C2#575711yescharacteristic
422C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3041
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×D3×D8C2semidirect
3. #5715C2semidirect
4. #5545C2semidirect
5. C3×#459C2semidirect
6. #5153C2semidirect
7. #5150C2semidirect
8. #5072C2semidirect
9. #5016C2semidirect
10. #4844C2semidirect
11. #4839C2semidirect
12. #4787C2semidirect
13. #4786C2semidirect
14. #4772C2semidirect
15. #4770C2semidirect
21. C2×C3×D8C22semidirect
22. C3×#127C22semidirect
23. C3×C4×D4C22semidirect
24. #940C22semidirect
29. #936C22semidirect
30. D4×Dic3C22semidirect
31. C2×#214C22semidirect
34. C2×#213C22semidirect
40. C2×#204C22semidirect
43. #893C22semidirect
47. #875C22semidirect
48. C4×D12C22semidirect
50. C3×C4⋊C8C22semidirect
51. C3×#102C22semidirect
52. C3×#102C22semidirect
53. #832C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. #816C22semidirect
57. Dic3⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. #810C22semidirect
60. #809C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
73. #459D3semidirect
97. D3×D4D4semidirect
148. C2×D8C2×D3semidirect
152. #127C2×D3semidirect
164. C4×D4C2×D3semidirect
172. C4⋊C8C2×D3semidirect
179. #102C2×D3semidirect
180. #102C2×D3semidirect
413. C3#3041semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C2×D3×D82
#57152
#55452
C3×#4592
#51532
#51502
#50722
#50162
#48442
#48392
#47872
#47862
#47722
#47702
#30413