Finite Group Databasebeta

← #80599

Subgroups — #80599

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#918C211yesmaximal, characteristic, complemented
2C2×#914C211yesmaximal, characteristic
3D3×#124C211yesmaximal, characteristic, complemented
4C4×#239C211yesmaximal, characteristic, complemented
5C3×#464C211yesmaximal, characteristic, complemented
6#5185C211yesmaximal, characteristic, complemented
7#5166C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5028C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4837C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4802C211yesmaximal, characteristic, complemented
14#4787C211yesmaximal, characteristic, complemented
15#4770C211yesmaximal, characteristic, complemented
16#3040—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic, complemented
21C2×C3×SD16C2211yescharacteristic
22C3×#124C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic, complemented
24#947C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic, complemented
29#936C2211yescharacteristic
30C2×#213C2211yescharacteristic
31#918—24no—
32#914—24no—
33C2×D24C2211yescharacteristic
34C2×#204C2211yescharacteristic, complemented
35C2×ΓL(1,𝔽25)C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#893—21no—
38#893C2211yescharacteristic, complemented
39#892—21no—
40#892C2211yescharacteristic
41#885—21no—
42#885—21no—
43D3×C22⋊C4—21no—
44D3×C22⋊C4—21no—
45#875C2211yescharacteristic
46C4×D12—21no—
47C4×D12C2211yescharacteristic, complemented
48C42×D3—21no—
49C42×D3C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic, complemented
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56#816C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×#137—31no—
63C2×AGL(1,ℤ/8)—31no—
64C2×#124—31no—
65C4×#41—31no—
66#464—31no—
67#464—31no—
68#464—31no—
69#464D311yescharacteristic, complemented
70#463—31no—
71#463—31no—
72#463—31no—
73#463—31no—
74#423—31no—
75#419—31no—
76#418—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82#239—22no—
83#239D414yes—
84D3×Q8—22no—
85D3×Q8—22no—
86D3×D4—22no—
87D3×D4—22no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3D411yescharacteristic
96C2×C4×D3D411yescharacteristic
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—21no—
99C2×Dic6C2311yescharacteristic
100C3×Q16—22no—
101C3×SD16—22no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4—21no—
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C42—21no—
109C3×C42C2311yescharacteristic
110#217—21no—
111#217—21no—
112#216—22no—
113#215—22no—
114#215—22no—
115#213—22no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
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—
123C4⋊Dic3—21no—
124C4⋊Dic3C2311yescharacteristic
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3D411yescharacteristic
128C4×Dic3D411yescharacteristic
129C4×Dic3C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132D24—22no—
133#204—22no—
134ΓL(1,𝔽25)—22no—
135ΓL(1,𝔽25)—22no—
136C2×#41—31no—
137C22×Q8—31no—
138C22×D4—31no—
139#137—64no—
140AGL(1,ℤ/8)—64no—
141C2×Q16—31no—
142C2×Q16C2×D311yescharacteristic, complemented
143C2×SD16—31no—
144C2×SD16—31no—
145C2×SD16—31no—
146C2×SD16C2×D311yescharacteristic
147C2×D8—31no—
148C2×D8—31no—
149C2×M16—31no—
150C2×M16—31no—
151#124—31no—
152#124—31no—
153#124—31no—
154#124—61no—
155#124—61no—
156#124C2×D311yescharacteristic
157C4×Q8—31no—
158C4×Q8C2×D311yescharacteristic, complemented
159C4×D4—31no—
160C4×D4—31no—
161C4×D4—61no—
162C4×D4—61no—
163#117—31no—
164#117—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—31no—
168C2×C42—61no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8C2×D311yescharacteristic, complemented
173#103—31no—
174#103—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181C23×C3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—42no—
185C22×D3—42no—
186C22×D3—42no—
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8C2×D412yes—
192C3×D4—21no—
193C3×D4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2411yescharacteristic
200#67—42no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206D12—24no—
207D12—21no—
208D12—21no—
209D12—21no—
210D12C2×D412yes—
211C4×D3—24no—
212C4×D3—41no—
213C4×D3—41no—
214C4×D3—42no—
215C4×D3—42no—
216C4×D3—42no—
217C4×D3C2×D412yes—
218C4×D3#4112yes—
219Dic6—21no—
220Dic6—21no—
221Dic6—42no—
222C3×C8—21no—
223C3×C8—21no—
224C3⋊C8—21no—
225C3⋊C8—21no—
226C24—61no—
227#41—34no—
228#41—62no—
229C2×Q8—31no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—62no—
233C2×Q8C22×D311yescharacteristic
234C2×Q8C22×D311yescharacteristic
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—61no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—61no—
241C2×D4C22×D311yescharacteristic
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249Q16—22no—
250Q16—62no—
251SD16—22no—
252SD16—62no—
253SD16—62no—
254SD16—62no—
255D8—62no—
256D8—62no—
257M16—62no—
258M16—62no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8C22×D311yescharacteristic
262C2×C8C22×D311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—31no—
265C4⋊C4—61no—
266C4⋊C4C22×D311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C42—21no—
279C42—31no—
280C42—31no—
281C42—31no—
282C42—61no—
283C42—61no—
284C42C22×D311yescharacteristic
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3C22×D411yescharacteristic
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—22no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—84no—
298C2×D3#12112yes—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4C22×D411yescharacteristic
305C3×C4C2×#4111yescharacteristic
306Dic3—21no—
307Dic3—21no—
308Dic3—41no—
309Dic3—41no—
310Dic3#12112yes—
311C23—21no—
312C23—31no—
313C23—61no—
314C23—61no—
315C23—122no—
316C23—122no—
317C23—122no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—61no—
322Q8—61no—
323Q8—122no—
324Q8D3×D412yes—
325D4—21no—
326D4—21no—
327D4—32no—
328D4—64no—
329D4—61no—
330D4—61no—
331D4—61no—
332D4—122no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—64no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4D3×D411yescharacteristic
350C2×C4D3×D411yescharacteristic
351C2×C4C23×D311yescharacteristic
352C8—21no—
353C8—21no—
354C8—61no—
355C8—61no—
356C2×C3—81no—
357C2×C3C2×#12111yescharacteristic
358C2×C3#57511yescharacteristic
359C2×C3#57511yescharacteristic
360D3—22no—
361D3—42no—
362D3—81no—
363C22—32no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—61no—
368C22—61no—
369C22—62no—
370C22—121no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—244no—
376C22C2×D3×D411yescharacteristic
377C4—21no—
378C4—21no—
379C4—22no—
380C4—32no—
381C4—41no—
382C4—41no—
383C4—61no—
384C4—61no—
385C4—121no—
386C4—121no—
387C4C2×D3×D411yescharacteristic
388C4D3×#4111yescharacteristic
389C3#304011yescharacteristic, complemented, Sylow
390C2—62no—
391C2—81no—
392C2—122no—
393C2—241no—
394C2D3×#12111yescharacteristic
395C2#576811yescharacteristic
396C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
prime radicalC2×#914C2
hypercenter#464D3
hypocenterC3#3040
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×#918C2semidirect
3. D3×#124C2semidirect
4. C4×#239C2semidirect
5. C3×#464C2semidirect
6. #5185C2semidirect
7. #5166C2semidirect
8. #5057C2semidirect
9. #5028C2semidirect
10. #4844C2semidirect
11. #4837C2semidirect
12. #4819C2semidirect
13. #4802C2semidirect
14. #4787C2semidirect
15. #4770C2semidirect
20. C2×C3×Q16C22semidirect
23. C3×C4×Q8C22semidirect
25. Q8×Dic3C22semidirect
26. C2×#216C22semidirect
28. C2×#215C22semidirect
34. C2×#204C22semidirect
38. #893C22semidirect
47. C4×D12C22semidirect
50. C3×C4⋊C8C22semidirect
51. C3×#103C22semidirect
53. #835C22semidirect
56. #816C22semidirect
57. Dic3⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. #810C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
69. #464D3semidirect
142. C2×Q16C2×D3semidirect
158. C4×Q8C2×D3semidirect
172. C4⋊C8C2×D3semidirect
176. #103C2×D3semidirect
389. C3#3040semidirect

Maximal subgroups

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

subgroupindex
C2×#9182
C2×#9142
D3×#1242
C4×#2392
C3×#4642
#51852
#51662
#50572
#50282
#48442
#48372
#48192
#48022
#47872
#47702
#30403