Finite Group Databasebeta

← #80021

Subgroups — #80021

Order 384.

460 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#67C211yesmaximal, characteristic, complemented
2#5616C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5215C211yesmaximal, characteristic, complemented
5#5159C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic, complemented
8Dic3×D8C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#4784C211yesmaximal, characteristic, complemented
11#4770C211yesmaximal, characteristic, complemented
12#4761C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4737C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3325—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C2×D3×D4C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic, complemented
22C3×#121C2211yescharacteristic, complemented
23#953—21no—
24#940C2211yescharacteristic
25#939—21no—
26#939—21no—
27#938C2211yescharacteristic
28#938C2211yescharacteristic, complemented
29#937—21no—
30#936—21no—
31D4×Dic3C2211yescharacteristic
32D4×Dic3C2211yescharacteristic, complemented
33C2×#213C2211yescharacteristic, complemented
34#930C2211yescharacteristic
35C4×#67C2211yescharacteristic
36#926C2211yescharacteristic, complemented
37C2×#208C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39#892C2211yescharacteristic, complemented
40#888C2211yescharacteristic, complemented
41#885—21no—
42#877—21no—
43C3×#102C2211yescharacteristic
44C3×#102C2211yescharacteristic, complemented
45C3×C22⋊C8C2211yescharacteristic, complemented
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic, complemented
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#808C2211yescharacteristic, complemented
60#555—31no—
61D42—31no—
62#486—31no—
63#482—31no—
64#482—31no—
65#476—31no—
66#469—31no—
67#469—31no—
68#459—31no—
69#447—31no—
70#447D311yescharacteristic, complemented
71C4×D8—31no—
72#436—31no—
73#418—31no—
74#418—31no—
75C24×C3—21no—
76C23×D3—21no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4—41no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—44no—
92C2×#67D411yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95D3×D4—22no—
96D3×D4—41no—
97C2×D12C2311yescharacteristic
98C2×C4×D3C2311yescharacteristic
99C2×Dic6—21no—
100C3×D8—22no—
101C3×D8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#217D411yescharacteristic
112#213—41no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
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×D8—31no—
132C2×D8C2×D311yescharacteristic
133C2×D8C2×D311yescharacteristic, complemented
134C2×M16—31no—
135C2×M16—31no—
136C4⋊Q8—31no—
137#127—31no—
138#124—61no—
139#124—61no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121—61no—
145#121C2×D311yescharacteristic, complemented
146#120—61no—
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—31no—
151#117—31no—
152#117—31no—
153#117—61no—
154#107—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#102—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—21no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—81no—
177C23×C3C2×D411yescharacteristic
178C22×D3—21no—
179C22×D3—42no—
180C22×D3—42no—
181C22×D3—81no—
182C22×D3C2×D411yescharacteristic
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—42no—
189C3×D4—81no—
190C3×D4#4112yes—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196#67—22no—
197#67—42no—
198#67—42no—
199#67—42no—
200#67—42no—
201#67—81no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211D12—21no—
212D12—21no—
213C4×D3—21no—
214C4×D3—21no—
215Dic6—81no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—21no—
221C24—61no—
222C2×Q8—61no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—41no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—62no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—121no—
237C2×D4—124no—
238C2×D4C22×D311yescharacteristic
239C2×D4C22×D311yescharacteristic
240C2×D4C22×D311yescharacteristic
241C22×C4—31no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4C22×D311yescharacteristic
245D8—22no—
246D8—41no—
247D8—121no—
248M16—121no—
249M16—121no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8C22×D311yescharacteristic
253C2×C8C22×D311yescharacteristic
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C42—31no—
271C42—61no—
272C42—61no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—84no—
286C22×C3C22×D411yescharacteristic
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—84no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4C2×#4111yescharacteristic
297C3×C4H(5,𝔽2)11yescharacteristic
298Dic3—21no—
299Dic3—21no—
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303Dic3AGL(1,ℤ/8)12yes—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—31no—
308C23—42no—
309C23—42no—
310C23—61no—
311C23—81no—
312C23—122no—
313C23—122no—
314C23—241no—
315C23D3×D411yescharacteristic
316Q8—241no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—62no—
324D4—61no—
325D4—61no—
326D4—81no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—241no—
332D4—241no—
333D4#23712yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—61no—
352C8—61no—
353C2×C3—42no—
354C2×C3—41no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3#54611yescharacteristic
358C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
359C2×C3#57611yescharacteristic
360D3—41no—
361D3—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—84no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—244no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—32no—
385C4—41no—
386C4—61no—
387C4—61no—
388C4—121no—
389C4—121no—
390C4—121no—
391C4C2×#23711yescharacteristic
392C4#100411yescharacteristic
393C3#332511yescharacteristic, complemented, Sylow
394C2—41no—
395C2—42no—
396C2—81no—
397C2—81no—
398C2—121no—
399C2—241no—
400C2#559111yescharacteristic
401C2#575711yescharacteristic
402C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3325
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. D4×#67C2semidirect
2. #5616C2semidirect
3. C3×#447C2semidirect
4. #5215C2semidirect
5. #5159C2semidirect
6. #5156C2semidirect
7. #5150C2semidirect
8. Dic3×D8C2semidirect
9. #5039C2semidirect
10. #4784C2semidirect
11. #4770C2semidirect
12. #4761C2semidirect
13. #4760C2semidirect
14. #4737C2semidirect
15. #4729C2semidirect
21. C2×C3×D8C22semidirect
22. C3×#121C22semidirect
28. #938C22semidirect
32. D4×Dic3C22semidirect
33. C2×#213C22semidirect
36. #926C22semidirect
37. C2×#208C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. #892C22semidirect
40. #888C22semidirect
44. C3×#102C22semidirect
45. C3×C22⋊C8C22semidirect
48. #832C22semidirect
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. #808C22semidirect
70. #447D3semidirect
133. C2×D8C2×D3semidirect
145. #121C2×D3semidirect
167. #102C2×D3semidirect
169. C22⋊C8C2×D3semidirect
393. C3#3325semidirect

Maximal subgroups

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

subgroupindex
D4×#672
#56162
C3×#4472
#52152
#51592
#51562
#51502
Dic3×D82
#50392
#47842
#47702
#47612
#47602
#47372
#47292
#33253