Finite Group Databasebeta

← #83962

Subgroups — #83962

Order 384.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#5831C211yesmaximal, characteristic, complemented
3#5806C211yesmaximal, characteristic, complemented
4C2×#915C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5218C211yesmaximal, characteristic, complemented
7#5170C211yesmaximal, characteristic, complemented
8#5166C211yesmaximal, characteristic, complemented
9Dic3×SD16C211yesmaximal, characteristic, complemented
10#5159C211yesmaximal, characteristic, complemented
11#5158C211yesmaximal, characteristic, complemented
12#5155C211yesmaximal, characteristic, complemented
13#5153C211yesmaximal, characteristic, complemented
14#5137C211yesmaximal, characteristic, complemented
15#5128C211yesmaximal, characteristic, complemented
16#3275—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic, complemented
20C2×#237C2211yescharacteristic
21C3×AGL(1,ℤ/8)—24no—
22C2×C3×SD16C2211yescharacteristic
23C2×C3×SD16C2211yescharacteristic
24C2×C3×D8C2211yescharacteristic
25C2×C3×D8C2211yescharacteristic
26C2×C3×M16C2211yescharacteristic
27#953—21no—
28C2×#217—21no—
29Q8×Dic3C2211yescharacteristic, complemented
30C2×#216C2211yescharacteristic, complemented
31#939—21no—
32#938—21no—
33#938C2211yescharacteristic
34#938C2211yescharacteristic, complemented
35#936—21no—
36#936C2211yescharacteristic
37#936C2211yescharacteristic, complemented
38#935—21no—
39D4×Dic3C2211yescharacteristic
40C2×#214C2211yescharacteristic
41C2×#214C2211yescharacteristic
42C2×#214C2211yescharacteristic, complemented
43C4×#67—21no—
44C4×#67C2211yescharacteristic
45#924—21no—
46#924C2211yescharacteristic
47#924C2211yescharacteristic
48#915—24no—
49C2×Dic12C2211yescharacteristic, complemented
50C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
51#835C2211yescharacteristic, complemented
52#832C2211yescharacteristic
53#832C2211yescharacteristic
54#832C2211yescharacteristic, complemented
55#830C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#817C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60Dic3⋊C8C2211yescharacteristic, complemented
61C8×Dic3C2211yescharacteristic, complemented
62C2×#137—31no—
63C2×AGL(1,ℤ/8)D311yescharacteristic, complemented
64#548—31no—
65#546—31no—
66#495—31no—
67#470—31no—
68#469—31no—
69#464—31no—
70#463—31no—
71#452—31no—
72#450—31no—
73#449—31no—
74#448—31no—
75C4×SD16—31no—
76#436—31no—
77C24×C3—21no—
78C3×#41—22no—
79C3×#41—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—22no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67D411yescharacteristic
94C22×Dic3—21no—
95C22×Dic3—21no—
96D3×Q8—22no—
97D3×Q8—41no—
98#237—22no—
99#237—41no—
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×SD16—22no—
106C3×SD16D414yescomplemented
107C3×D8—22no—
108C3×D8—22no—
109C3×M16—22no—
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#217—21no—
117#217—21no—
118#217—21no—
119#217—21no—
120#217—21no—
121#217D411yescharacteristic
122#216—22no—
123#214—22no—
124#214—41no—
125#214—41no—
126(C2×D3)⋊C4D411yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3C2311yescharacteristic
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132Dic3⋊C4D411yescharacteristic
133C4×Dic3—21no—
134C4×Dic3C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137Dic12—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic
141C22×Q8—31no—
142C22×D4C2×D311yescharacteristic
143#137—64no—
144AGL(1,ℤ/8)—24no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16C2×D311yescharacteristic
151C2×SD16C2×D311yescharacteristic
152C2×D8C2×D311yescharacteristic
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C2×M16C2×D311yescharacteristic
156#124—31no—
157#124—31no—
158#124—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183C4×C8—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—81no—
190C23×C3C2×D411yescharacteristic
191C22×D3C2×D411yescharacteristic
192C3×Q8—21no—
193C3×Q8C2×D412yes—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—42no—
200C3×D4—42no—
201C3×D4C2×D412yes—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2411yescharacteristic
208#67—41no—
209#67—42no—
210#67—42no—
211#67—81no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—42no—
217C2×Dic3—81no—
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220C2×Dic3C2×D411yescharacteristic
221C4×D3—21no—
222C4×D3—21no—
223C4×D3—42no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—21no—
228Dic6—42no—
229C3×C8—21no—
230C3×C8C2×D412yes—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C24—21no—
234#41—22no—
235#41—41no—
236#41—62no—
237#41—121no—
238C2×Q8—31no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—31no—
248C2×D4—41no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4C22×D311yescharacteristic
253C2×D4C22×D311yescharacteristic
254C2×D4C22×D311yescharacteristic
255C22×C4—21no—
256C22×C4—31no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4C22×D311yescharacteristic
261Q16—62no—
262Q16—62no—
263SD16—22no—
264SD16—62no—
265SD16—121no—
266SD16—121no—
267SD16#6714yescomplemented
268D8—22no—
269D8—22no—
270M16—22no—
271M16—62no—
272C2×C8—31no—
273C2×C8—31no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—31no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C22⋊C4—31no—
283C22⋊C4—31no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C42—31no—
294C42—61no—
295C22×C3—22no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—84no—
308C22×C3C22×D411yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—21no—
312C3×C4—22no—
313C3×C4—21no—
314C3×C4C22×D411yescharacteristic
315C3×C4H(5,𝔽2)11yescharacteristic
316Dic3—21no—
317Dic3—21no—
318Dic3—21no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—31no—
327C23—42no—
328C23—42no—
329C23—81no—
330C23D3×D411yescharacteristic
331Q8—21no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—122no—
337Q8C2×#6712yes—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—42no—
345D4—121no—
346D4—122no—
347D4—122no—
348D4—241no—
349D4C2×#6712yes—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—42no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—241no—
366C2×C4D3×D411yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—61no—
370C8—61no—
371C8C2×#6712yes—
372C2×C3—41no—
373C2×C3—42no—
374C2×C3—81no—
375C2×C3—81no—
376C2×C3D4211yescharacteristic
377C2×C3#57511yescharacteristic
378C2×C3#57711yescharacteristic
379D3—41no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—61no—
393C22—61no—
394C22—61no—
395C22—84no—
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—22no—
399C4—61no—
400C4—61no—
401C4—61no—
402C4—121no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4#100411yescharacteristic
407C4C22×#6711yescharacteristic
408C3#327511yescharacteristic, complemented, Sylow
409C2—41no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—121no—
414C2#577311yescharacteristic
415C2#577511yescharacteristic
416C2D4×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×AGL(1,ℤ/8)C2
socleC22×C3C22×D4
hypercenterC2×AGL(1,ℤ/8)D3
hypocenterC3#3275
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×AGL(1,ℤ/8)C2semidirect
2. #5831C2semidirect
3. #5806C2semidirect
4. C2×#915C2semidirect
5. #5239C2semidirect
6. #5218C2semidirect
7. #5170C2semidirect
8. #5166C2semidirect
9. Dic3×SD16C2semidirect
10. #5159C2semidirect
11. #5158C2semidirect
12. #5155C2semidirect
13. #5153C2semidirect
14. #5137C2semidirect
15. #5128C2semidirect
19. C2×D3×Q8C22semidirect
29. Q8×Dic3C22semidirect
30. C2×#216C22semidirect
34. #938C22semidirect
37. #936C22semidirect
42. C2×#214C22semidirect
49. C2×Dic12C22semidirect
50. C2×ΓL(1,𝔽25)C22semidirect
51. #835C22semidirect
54. #832C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #817C22semidirect
58. #816C22semidirect
59. #816C22semidirect
60. Dic3⋊C8C22semidirect
61. C8×Dic3C22semidirect
63. C2×AGL(1,ℤ/8)D3semidirect
106. C3×SD16D4semidirect
267. SD16#67semidirect
408. C3#3275semidirect

Maximal subgroups

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

subgroupindex
C2×C3×AGL(1,ℤ/8)2
#58312
#58062
C2×#9152
#52392
#52182
#51702
#51662
Dic3×SD162
#51592
#51582
#51552
#51532
#51372
#51282
#32753