Finite Group Databasebeta

← #80165

Subgroups — #80165

Order 384.

448 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#67C211yesmaximal, characteristic
2#5641C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5215C211yesmaximal, characteristic, complemented
5#5174C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5163C211yesmaximal, characteristic, complemented
8Dic3×SD16C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4779C211yesmaximal, characteristic, complemented
11#4770C211yesmaximal, characteristic, complemented
12#4760C211yesmaximal, characteristic, complemented
13#4759C211yesmaximal, characteristic, complemented
14#4737C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3324—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C2×D3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23#953—21no—
24#946C2211yescharacteristic, complemented
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic, complemented
27#940C2211yescharacteristic
28#939—21no—
29#938C2211yescharacteristic
30#937—21no—
31D4×Dic3C2211yescharacteristic
32#930C2211yescharacteristic
33C4×#67C2211yescharacteristic
34#926C2211yescharacteristic, complemented
35C2×#208C2211yescharacteristic, complemented
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37#898—21no—
38#892C2211yescharacteristic, complemented
39#888C2211yescharacteristic, complemented
40#887—21no—
41#885—21no—
42#879—21no—
43C3×#102C2211yescharacteristic
44C3×#102C2211yescharacteristic, complemented
45C3×C22⋊C8C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#818C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#556—31no—
61D42—31no—
62#486—31no—
63#482—31no—
64#482—31no—
65#474—31no—
66#468—31no—
67#468—31no—
68#460—31no—
69#450—31no—
70#450D311yescharacteristic, complemented
71C4×SD16—31no—
72#436—31no—
73#418—31no—
74#418—31no—
75C24×C3—21no—
76C23×D3—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—22no—
79C2×C3×D4—21no—
80C2×C3×D4—41no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—44no—
90C2×#67D411yescharacteristic
91C22×Dic3—21no—
92D3×D4—22no—
93D3×D4—41no—
94C2×D12C2311yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C2×Dic6—21no—
97C3×SD16—22no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217D411yescharacteristic
108#215—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124#208—41no—
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127ΓL(1,𝔽25)—41no—
128C22×D4—31no—
129C22×D4—61no—
130C22×D4C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16C2×D311yescharacteristic
133C2×SD16C2×D311yescharacteristic, complemented
134C2×M16—31no—
135C2×M16—31no—
136C4⋊Q8—31no—
137#127—31no—
138#126—61no—
139#126—61no—
140#124—61no—
141C22⋊Q8—31no—
142C22⋊Q8C2×D311yescharacteristic, complemented
143#121—31no—
144#121—31no—
145#121—61no—
146#120—61no—
147#120—61no—
148C4×Q8—31no—
149C4×Q8—61no—
150C4×D4—31no—
151C4×D4—31no—
152#117—31no—
153#117—31no—
154#107—31no—
155#107—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8C2×D311yescharacteristic, complemented
170C4×C8—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—81no—
176C23×C3C2×D411yescharacteristic
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—42no—
188C3×D4#4112yes—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—22no—
196#67—42no—
197#67—42no—
198#67—42no—
199#67—42no—
200#67—81no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—42no—
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—21no—
209D12—21no—
210C4×D3—21no—
211C4×D3—21no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—21no—
218C24—61no—
219C2×Q8—61no—
220C2×Q8C22×D311yescharacteristic
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—31no—
224C2×D4—31no—
225C2×D4—41no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—62no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—121no—
233C2×D4—124no—
234C2×D4C22×D311yescharacteristic
235C2×D4C22×D311yescharacteristic
236C22×C4—31no—
237C22×C4—61no—
238C22×C4C22×D311yescharacteristic
239SD16—22no—
240SD16—41no—
241SD16—121no—
242M16—121no—
243M16—121no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8C22×D311yescharacteristic
247C2×C8C22×D311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4C22×D311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—31no—
259C22⋊C4—31no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C42—31no—
267C42—61no—
268C42—61no—
269C42—61no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—84no—
280C22×C3C22×D411yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—84no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4C2×#4111yescharacteristic
292C3×C4H(5,𝔽2)11yescharacteristic
293Dic3—21no—
294Dic3—21no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3AGL(1,ℤ/8)12yes—
299C23—21no—
300C23—21no—
301C23—31no—
302C23—42no—
303C23—42no—
304C23—61no—
305C23—81no—
306C23—122no—
307C23—122no—
308C23—241no—
309C23D3×D411yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—122no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—62no—
318D4—61no—
319D4—61no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—122no—
324D4—241no—
325D4#23712yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—122no—
339C2×C4D3×D411yescharacteristic
340C2×C4C23×D311yescharacteristic
341C8—21no—
342C8—21no—
343C8—61no—
344C8—61no—
345C2×C3—42no—
346C2×C3—41no—
347C2×C3—81no—
348C2×C3#54611yescharacteristic
349C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
350C2×C3#57711yescharacteristic
351D3—41no—
352D3—81no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—84no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—244no—
370C22C2×D3×D411yescharacteristic
371C4—21no—
372C4—32no—
373C4—41no—
374C4—41no—
375C4—61no—
376C4—61no—
377C4—121no—
378C4—121no—
379C4—121no—
380C4C2×#23711yescharacteristic
381C4#100411yescharacteristic
382C3#332411yescharacteristic, complemented, Sylow
383C2—41no—
384C2—42no—
385C2—81no—
386C2—121no—
387C2—241no—
388C2#559111yescharacteristic
389C2#576211yescharacteristic
390C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalD4×#67C2
hypercenter#450D3
hypocenterC3#3324
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #5641C2semidirect
3. C3×#450C2semidirect
4. #5215C2semidirect
5. #5174C2semidirect
6. #5169C2semidirect
7. #5163C2semidirect
8. Dic3×SD16C2semidirect
9. #5049C2semidirect
10. #4779C2semidirect
11. #4770C2semidirect
12. #4760C2semidirect
13. #4759C2semidirect
14. #4737C2semidirect
15. #4729C2semidirect
21. C2×C3×SD16C22semidirect
22. C3×C22⋊Q8C22semidirect
24. #946C22semidirect
25. Q8×Dic3C22semidirect
26. C2×#215C22semidirect
34. #926C22semidirect
35. C2×#208C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
38. #892C22semidirect
39. #888C22semidirect
44. C3×#102C22semidirect
45. C3×C22⋊C8C22semidirect
46. #835C22semidirect
49. #832C22semidirect
50. #832C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #818C22semidirect
55. #817C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #809C22semidirect
59. #807C22semidirect
70. #450D3semidirect
133. C2×SD16C2×D3semidirect
142. C22⋊Q8C2×D3semidirect
167. #102C2×D3semidirect
169. C22⋊C8C2×D3semidirect
382. C3#3324semidirect

Maximal subgroups

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

subgroupindex
D4×#672
#56412
C3×#4502
#52152
#51742
#51692
#51632
Dic3×SD162
#50492
#47792
#47702
#47602
#47592
#47372
#47292
#33243