Finite Group Databasebeta

← #80535

Subgroups — #80535

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×#915C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic
3D3×#124C211yesmaximal, characteristic, complemented
4C4×#237C211yesmaximal, characteristic, complemented
5C3×#463C211yesmaximal, characteristic, complemented
6#5165C211yesmaximal, characteristic, complemented
7#5153C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4837C211yesmaximal, characteristic, complemented
12#4802C211yesmaximal, characteristic, complemented
13#4798C211yesmaximal, characteristic, complemented
14#4770C211yesmaximal, characteristic, complemented
15#4767C211yesmaximal, characteristic, complemented
16#3040—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×D8C2211yescharacteristic
22C3×#124C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic, complemented
24#947C2211yescharacteristic
25C2×#216C2211yescharacteristic, complemented
26#936C2211yescharacteristic
27D4×Dic3C2211yescharacteristic, complemented
28C2×#214C2211yescharacteristic
29C2×#214C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C4×#67—21no—
32C2×C4×Dic3—21no—
33#915—24no—
34#911—24no—
35C2×Dic12C2211yescharacteristic, complemented
36C2×#204C2211yescharacteristic
37C2×ΓL(1,𝔽25)C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39#892C2211yescharacteristic
40#887C2211yescharacteristic, complemented
41#885—21no—
42#885—21no—
43#67⋊C4—21no—
44D3×C22⋊C4—21no—
45D3×C22⋊C4—21no—
46#877—21no—
47#875C2211yescharacteristic
48C42×D3C2211yescharacteristic
49C4×Dic6C2211yescharacteristic, complemented
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#464—31no—
70#463—31no—
71#463—31no—
72#463—31no—
73#463D311yescharacteristic, complemented
74#423—31no—
75#419—31no—
76#418—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86D3×Q8—22no—
87D3×Q8—22no—
88#237—22no—
89#237D414yes—
90D3×D4—22no—
91D3×D4—22no—
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3D411yescharacteristic
95C2×C4×D3D411yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6C2311yescharacteristic
99C2×Dic6C2311yescharacteristic
100C3×SD16—22no—
101C3×D8—22no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C42C2311yescharacteristic
109#217—21no—
110#217—21no—
111#217—21no—
112#216—22no—
113#214—22no—
114#214—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—
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C4×Dic3—21no—
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3D411yescharacteristic
128C4×Dic3D411yescharacteristic
129C4×Dic3C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132Dic12—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×Q16—31no—
143C2×SD16—31no—
144C2×SD16—31no—
145C2×SD16—31no—
146C2×SD16C2×D311yescharacteristic, complemented
147C2×D8—31no—
148C2×D8C2×D311yescharacteristic
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×Q8—31no—
159C4×D4—31no—
160C4×D4—61no—
161C4×D4—61no—
162C4×D4C2×D311yescharacteristic, complemented
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—
182C23×C3—21no—
183C22×D3—21no—
184C22×D3—42no—
185C22×D3—42no—
186C22×D3—42no—
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4C2×D412yes—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2411yescharacteristic
200#67—24no—
201#67—42no—
202C2×Dic3—21no—
203C2×Dic3—24no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—42no—
207C2×Dic3C2×D411yescharacteristic
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—21no—
211D12—21no—
212C4×D3—41no—
213C4×D3—41no—
214C4×D3—42no—
215C4×D3C2×D412yes—
216C4×D3#4112yes—
217Dic6—21no—
218Dic6—21no—
219Dic6—21no—
220Dic6—42no—
221Dic6C2×D412yes—
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—31no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—62no—
234C2×Q8C22×D311yescharacteristic
235C2×D4—31no—
236C2×D4—62no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4C22×D311yescharacteristic
241C2×D4C22×D311yescharacteristic
242C22×C4—21no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249Q16—62no—
250Q16—62no—
251SD16—22no—
252SD16—62no—
253SD16—62no—
254SD16—62no—
255D8—22no—
256D8—62no—
257M16—62no—
258M16—62no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8C22×D311yescharacteristic
262C2×C8C22×D311yescharacteristic
263C4⋊C4—31no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4C22×D311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
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—31no—
279C42—31no—
280C42—31no—
281C42—61no—
282C42—61no—
283C42—61no—
284C42C22×D311yescharacteristic
285C22×C3—22no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3C22×D411yescharacteristic
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—84no—
298C2×D3#12112yes—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4C22×D411yescharacteristic
304C3×C4C2×#4111yescharacteristic
305Dic3—22no—
306Dic3—21no—
307Dic3—21no—
308Dic3—41no—
309Dic3—41no—
310Dic3#12112yes—
311C23—21no—
312C23—21no—
313C23—31no—
314C23—61no—
315C23—122no—
316C23—122no—
317C23—122no—
318Q8—21no—
319Q8—21no—
320Q8—32no—
321Q8—61no—
322Q8—61no—
323Q8—61no—
324Q8—122no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—64no—
329D4—61no—
330D4—61no—
331D4—122no—
332D4D3×D412yes—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—42no—
341C2×C4—64no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—121no—
346C2×C4—121no—
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—42no—
357C2×C3—81no—
358C2×C3C2×#12111yescharacteristic
359C2×C3#57511yescharacteristic
360C2×C3#57511yescharacteristic
361D3—22no—
362D3—81no—
363C22—22no—
364C22—32no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—61no—
371C22—61no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—244no—
376C22C2×D3×D411yescharacteristic
377C4—21no—
378C4—21no—
379C4—32no—
380C4—41no—
381C4—41no—
382C4—61no—
383C4—61no—
384C4—62no—
385C4—121no—
386C4—121no—
387C4C2×D3×D411yescharacteristic
388C4D3×#4111yescharacteristic
389C3#304011yescharacteristic, complemented, Sylow
390C2—42no—
391C2—62no—
392C2—81no—
393C2—241no—
394C2D3×#12111yescharacteristic
395C2#576811yescharacteristic
396C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
prime radicalC2×#911C2
hypercenter#463D3
hypocenterC3#3040
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
3. D3×#124C2semidirect
4. C4×#237C2semidirect
5. C3×#463C2semidirect
6. #5165C2semidirect
7. #5153C2semidirect
8. #5054C2semidirect
9. #5016C2semidirect
10. #4845C2semidirect
11. #4837C2semidirect
12. #4802C2semidirect
13. #4798C2semidirect
14. #4770C2semidirect
15. #4767C2semidirect
20. C2×C3×SD16C22semidirect
23. C3×C4×D4C22semidirect
25. C2×#216C22semidirect
27. D4×Dic3C22semidirect
29. C2×#214C22semidirect
35. C2×Dic12C22semidirect
40. #887C22semidirect
49. C4×Dic6C22semidirect
50. C3×C4⋊C8C22semidirect
51. C3×#103C22semidirect
53. #835C22semidirect
56. #816C22semidirect
57. Dic3⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. #810C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
73. #463D3semidirect
146. C2×SD16C2×D3semidirect
162. C4×D4C2×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×#9152
C2×#9112
D3×#1242
C4×#2372
C3×#4632
#51652
#51532
#50542
#50162
#48452
#48372
#48022
#47982
#47702
#47672
#30403