Finite Group Databasebeta

← #80602

Subgroups — #80602

Order 384.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#919C211yesmaximal, characteristic, complemented
2C2×#914C211yesmaximal, characteristic, complemented
3#5674C211yesmaximal, characteristic, complemented
4#5574C211yesmaximal, characteristic, complemented
5C3×#464C211yesmaximal, characteristic, complemented
6#5185C211yesmaximal, characteristic, complemented
7#5163C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5027C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4803C211yesmaximal, characteristic, complemented
14#4786C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3045—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic, complemented
21C2×C3×SD16C2211yescharacteristic
22C3×#124C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic, complemented
24#947C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29#940C2211yescharacteristic
30C2×#213C2211yescharacteristic
31#919—24no—
32#914—24no—
33C2×D24C2211yescharacteristic
34C2×D24C2211yescharacteristic, complemented
35C2×ΓL(1,𝔽25)C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#893—21no—
38#893C2211yescharacteristic, complemented
39D3×C4⋊C4—21no—
40D3×C4⋊C4C2211yescharacteristic
41#884—21no—
42#884—21no—
43#882—21no—
44#882—21no—
45#874C2211yescharacteristic
46C4×D12—21no—
47C4×D12C2211yescharacteristic, complemented
48#872—21no—
49#872C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic, complemented
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56#821C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#809C2211yescharacteristic, complemented
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#535—31no—
65#520—31no—
66#464—31no—
67#464D311yescharacteristic, complemented
68#463—31no—
69#463—31no—
70#460—31no—
71#460—31no—
72#459—31no—
73#459—31no—
74#424—31no—
75#417—31no—
76C2×#102—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84#239—22no—
85#239—22no—
86#239—22no—
87#239D414yescomplemented
88D3×D4—22no—
89D3×D4—22no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3C2311yescharacteristic
103C3×Q16—22no—
104C3×SD16—22no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4—21no—
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C42—21no—
112C3×C42C2311yescharacteristic
113#215—22no—
114#215—22no—
115#215—22no—
116#213—22no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4D411yescharacteristic
122(C2×D3)⋊C4D411yescharacteristic
123C4⋊Dic3—21no—
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4—21no—
126Dic3⋊C4D411yescharacteristic
127Dic3⋊C4D411yescharacteristic
128C4×Dic3—21no—
129C4×Dic3C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132D24—22no—
133D24—22no—
134ΓL(1,𝔽25)—22no—
135D3×C8—22no—
136C2×#41—31no—
137C2×#41—31no—
138C22×D4—31no—
139AGL(1,ℤ/8)—64no—
140#135—64no—
141C2×Q16C2×D311yescharacteristic, complemented
142C2×SD16—31no—
143C2×SD16—31no—
144C2×SD16—31no—
145C2×SD16C2×D311yescharacteristic
146C2×D8—31no—
147C2×D8—31no—
148C2×D8—31no—
149C2×M16—31no—
150C22×C8—31no—
151#127—31no—
152#127—31no—
153#124—31no—
154#124C2×D311yescharacteristic
155#121—61no—
156#121—61no—
157#120—61no—
158#120—61no—
159C4×Q8—31no—
160C4×Q8C2×D311yescharacteristic, complemented
161C4×D4—31no—
162C4×D4—31no—
163C4×D4—61no—
164C4×D4—61no—
165#117—31no—
166#117—61no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8C2×D311yescharacteristic, complemented
173#103—31no—
174#103C2×D311yescharacteristic, complemented
175#102—31no—
176#102—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181C23×C3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—42no—
186C22×D3—81no—
187C22×D3—81no—
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8C2×D412yes—
193C3×D4—21no—
194C3×D4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—81no—
203#67—81no—
204C2×Dic3—21no—
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—24no—
209D12—21no—
210D12—21no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—42no—
215D12—81no—
216D12—81no—
217D12C2×D412yes—
218C4×D3—24no—
219C4×D3—42no—
220C4×D3—81no—
221C4×D3—81no—
222C4×D3—81no—
223C4×D3C2×D412yes—
224C4×D3#4112yes—
225C3×C8—21no—
226C3×C8—21no—
227C3⋊C8—21no—
228C3⋊C8—21no—
229C24—61no—
230#41—34no—
231#41—62no—
232#41—62no—
233#41—62no—
234C2×Q8C22×D311yescharacteristic
235C2×Q8C22×D311yescharacteristic
236C2×D4—31no—
237C2×D4—31no—
238C2×D4—31no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—62no—
242C2×D4—62no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4C22×D311yescharacteristic
249C22×C4—31no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255Q16—22no—
256SD16—22no—
257SD16—62no—
258SD16—62no—
259SD16—62no—
260D8—62no—
261D8—62no—
262D8—62no—
263M16—62no—
264C2×C8—31no—
265C2×C8—31no—
266C2×C8—62no—
267C2×C8C22×D311yescharacteristic
268C2×C8C22×D311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4C22×D311yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—31no—
279C22⋊C4—31no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C42—21no—
285C42—31no—
286C42—61no—
287C42C22×D311yescharacteristic
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3C22×D411yescharacteristic
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—22no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—84no—
304C2×D3#12112yes—
305C3×C4—22no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4C22×D411yescharacteristic
311C3×C4C2×#4111yescharacteristic
312Dic3—21no—
313Dic3—21no—
314Dic3—41no—
315Dic3#12112yes—
316C23—21no—
317C23—31no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—122no—
322C23—241no—
323C23—241no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8D3×D412yes—
328D4—21no—
329D4—21no—
330D4—32no—
331D4—64no—
332D4—61no—
333D4—61no—
334D4—61no—
335D4—61no—
336D4—61no—
337D4—122no—
338D4—122no—
339D4—241no—
340D4—241no—
341D4—241no—
342D4—241no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—64no—
352C2×C4—61no—
353C2×C4—122no—
354C2×C4—241no—
355C2×C4—241no—
356C2×C4—241no—
357C2×C4D3×D411yescharacteristic
358C2×C4D3×D411yescharacteristic
359C2×C4C23×D311yescharacteristic
360C8—21no—
361C8—21no—
362C8—61no—
363C8—61no—
364C2×C3—81no—
365C2×C3C2×#12111yescharacteristic
366C2×C3#57611yescharacteristic
367C2×C3#57711yescharacteristic
368D3—22no—
369D3—42no—
370D3—81no—
371D3—81no—
372C22—32no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—61no—
377C22—61no—
378C22—62no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—244no—
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—32no—
393C4—41no—
394C4—41no—
395C4—61no—
396C4—61no—
397C4—121no—
398C4C2×D3×D411yescharacteristic
399C4D3×#4111yescharacteristic
400C3#304511yescharacteristic, complemented, Sylow
401C2—62no—
402C2—81no—
403C2—122no—
404C2—241no—
405C2—241no—
406C2D3×#12111yescharacteristic
407C2#576911yescharacteristic
408C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
hypercenter#464D3
hypocenterC3#3045
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#919C2semidirect
2. C2×#914C2semidirect
3. #5674C2semidirect
4. #5574C2semidirect
5. C3×#464C2semidirect
6. #5185C2semidirect
7. #5163C2semidirect
8. #5057C2semidirect
9. #5027C2semidirect
10. #4843C2semidirect
11. #4839C2semidirect
12. #4819C2semidirect
13. #4803C2semidirect
14. #4786C2semidirect
15. D3×#102C2semidirect
20. C2×C3×Q16C22semidirect
23. C3×C4×Q8C22semidirect
25. Q8×Dic3C22semidirect
27. C2×#215C22semidirect
28. C2×#215C22semidirect
34. C2×D24C22semidirect
38. #893C22semidirect
47. C4×D12C22semidirect
50. C3×C4⋊C8C22semidirect
51. C3×#103C22semidirect
53. #835C22semidirect
56. #821C22semidirect
57. Dic3⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
60. #809C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
67. #464D3semidirect
87. #239D4semidirect
141. C2×Q16C2×D3semidirect
160. C4×Q8C2×D3semidirect
172. C4⋊C8C2×D3semidirect
174. #103C2×D3semidirect
400. C3#3045semidirect

Maximal subgroups

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

subgroupindex
C2×#9192
C2×#9142
#56742
#55742
C3×#4642
#51852
#51632
#50572
#50272
#48432
#48392
#48192
#48032
#47862
D3×#1022
#30453