Finite Group Databasebeta

← #80385

Subgroups — #80385

Order 384.

490 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×D3×SD16C211yesmaximal, characteristic, complemented
3#5715C211yesmaximal, characteristic
4#5572C211yesmaximal, characteristic, complemented
5C3×#460C211yesmaximal, characteristic, complemented
6#5166C211yesmaximal, characteristic, complemented
7#5163C211yesmaximal, characteristic, complemented
8#5075C211yesmaximal, characteristic, complemented
9#5028C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4839C211yesmaximal, characteristic, complemented
12#4772C211yesmaximal, characteristic, complemented
13#4770C211yesmaximal, characteristic, complemented
14#4767C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3043—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic, complemented
22C3×#127C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic, complemented
24Q8×Dic3C2211yescharacteristic, complemented
25C2×#216C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27#940C2211yescharacteristic
28#939—21no—
29#939—21no—
30#937—21no—
31#937—21no—
32#936C2211yescharacteristic
33C2×#214C2211yescharacteristic
34C2×#214C2211yescharacteristic, complemented
35#915—24no—
36D3×SD16—24no—
37C2×Dic12C2211yescharacteristic, complemented
38C2×#204C2211yescharacteristic
39C2×ΓL(1,𝔽25)C2211yescharacteristic
40C2×D3×C8C2211yescharacteristic
41#892—21no—
42#892C2211yescharacteristic
43#887—21no—
44#887C2211yescharacteristic, complemented
45#875C2211yescharacteristic
46#872—21no—
47#872C2211yescharacteristic
48C4×Dic6—21no—
49C4×Dic6C2211yescharacteristic, complemented
50C3×C4⋊C8C2211yescharacteristic, complemented
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic, complemented
53#832C2211yescharacteristic
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#816C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×#137—31no—
63C22×SD16—31no—
64#535—31no—
65#519—31no—
66#464—31no—
67#464—31no—
68#464—31no—
69#464—31no—
70#460—31no—
71#460—31no—
72#460—31no—
73#460D311yescharacteristic, complemented
74#424—31no—
75#418—31no—
76#416—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88D3×Q8—22no—
89D3×Q8D414yescomplemented
90#237—22no—
91#237—22no—
92D3×D4—22no—
93D3×D4—22no—
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6C2311yescharacteristic
99C2×Dic6C2311yescharacteristic
100C3×SD16—22no—
101C3×SD16—22no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4—21no—
105C3×C4⋊C4C2311yescharacteristic
106C3×C42—21no—
107C3×C42C2311yescharacteristic
108#217—21no—
109#217—21no—
110#216—22no—
111#215—22no—
112#214—22no—
113#214—22no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4D411yescharacteristic
118(C2×D3)⋊C4D411yescharacteristic
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124Dic3⋊C4D411yescharacteristic
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132Dic12—22no—
133#204—22no—
134ΓL(1,𝔽25)—22no—
135D3×C8—22no—
136C2×#41—31no—
137C22×Q8—31no—
138C22×D4—31no—
139#137—64no—
140C2×Q16—31no—
141C2×Q16—31no—
142C2×SD16—31no—
143C2×SD16—31no—
144C2×SD16—31no—
145C2×SD16—31no—
146C2×SD16—64no—
147C2×SD16C2×D311yescharacteristic
148C2×SD16C2×D311yescharacteristic, complemented
149C2×M16—31no—
150C22×C8—31no—
151#127—31no—
152#127C2×D311yescharacteristic
153#124—31no—
154#124—31no—
155#121—61no—
156#121—61no—
157#120—61no—
158#120—61no—
159C4×Q8—31no—
160C4×Q8—31no—
161C4×Q8—31no—
162C4×Q8—61no—
163C4×Q8—61no—
164C4×Q8C2×D311yescharacteristic, complemented
165#117—31no—
166#117—31no—
167#117—61no—
168#117—61no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8C2×D311yescharacteristic, complemented
173#103—31no—
174#103—31no—
175#102—31no—
176#102—31no—
177#102—31no—
178#102—31no—
179#102C2×D311yescharacteristic
180#102C2×D311yescharacteristic, complemented
181C23×C3—21no—
182C23×C3—21no—
183C22×D3—21no—
184C22×D3—42no—
185C22×D3—81no—
186C22×D3—81no—
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8C2×D412yes—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—21no—
194C3×D4—81no—
195C3×D4—81no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—42no—
203#67—81no—
204#67—81no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—42no—
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211C2×Dic3C2×D411yescharacteristic
212D12—21no—
213D12—21no—
214C4×D3—24no—
215C4×D3C2×D412yes—
216C4×D3#4112yes—
217Dic6—21no—
218Dic6—24no—
219Dic6—21no—
220Dic6—21no—
221Dic6C2×D412yes—
222C3×C8—21no—
223C3×C8—21no—
224C3⋊C8—21no—
225C3⋊C8—21no—
226C24—61no—
227#41—62no—
228#41—62no—
229C2×Q8—34no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8C22×D311yescharacteristic
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—31no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4C22×D311yescharacteristic
245C2×D4C22×D311yescharacteristic
246C22×C4—31no—
247C22×C4—61no—
248C22×C4—61no—
249Q16—62no—
250Q16—62no—
251SD16—22no—
252SD16—22no—
253SD16—62no—
254SD16—62no—
255SD16—62no—
256SD16—62no—
257M16—62no—
258C2×C8—31no—
259C2×C8—31no—
260C2×C8—62no—
261C2×C8C22×D311yescharacteristic
262C2×C8C22×D311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—31no—
265C4⋊C4—31no—
266C4⋊C4—31no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4C22×D311yescharacteristic
271C22⋊C4—31no—
272C22⋊C4—31no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C42—21no—
279C42—31no—
280C42—61no—
281C42—61no—
282C42—61no—
283C42—61no—
284C42C22×D311yescharacteristic
285C22×C3—41no—
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—22no—
300C3×C4—21no—
301C3×C4—21no—
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—241no—
317C23—241no—
318Q8—21no—
319Q8—32no—
320Q8—64no—
321Q8—61no—
322Q8—61no—
323Q8—61no—
324Q8D3×D412yes—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—61no—
330D4—61no—
331D4—81no—
332D4—81no—
333D4—122no—
334D4—122no—
335D4—241no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—31no—
340C2×C4—32no—
341C2×C4—32no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—64no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—122no—
349C2×C4D3×D411yescharacteristic
350C2×C4D3×D411yescharacteristic
351C2×C4C23×D311yescharacteristic
352C8—21no—
353C8—21no—
354C8—61no—
355C8—61no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3C2×#12111yescharacteristic
359C2×C3#57711yescharacteristic
360C2×C3#57711yescharacteristic
361D3—22no—
362D3—81no—
363C22—32no—
364C22—41no—
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—22no—
380C4—32no—
381C4—41no—
382C4—61no—
383C4—61no—
384C4—62no—
385C4—121no—
386C4—121no—
387C4C2×D3×D411yescharacteristic
388C4D3×#4111yescharacteristic
389C3#304311yescharacteristic, complemented, Sylow
390C2—62no—
391C2—81no—
392C2—81no—
393C2—241no—
394C2D3×#12111yescharacteristic
395C2#576211yescharacteristic
396C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
prime radical#5715C2
hypercenter#460D3
hypocenterC3#3043
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
2. C2×D3×SD16C2semidirect
4. #5572C2semidirect
5. C3×#460C2semidirect
6. #5166C2semidirect
7. #5163C2semidirect
8. #5075C2semidirect
9. #5028C2semidirect
10. #4845C2semidirect
11. #4839C2semidirect
12. #4772C2semidirect
13. #4770C2semidirect
14. #4767C2semidirect
15. #4762C2semidirect
21. C2×C3×SD16C22semidirect
23. C3×C4×Q8C22semidirect
24. Q8×Dic3C22semidirect
25. C2×#216C22semidirect
34. C2×#214C22semidirect
37. C2×Dic12C22semidirect
44. #887C22semidirect
49. C4×Dic6C22semidirect
50. C3×C4⋊C8C22semidirect
52. C3×#102C22semidirect
54. #832C22semidirect
56. #816C22semidirect
57. Dic3⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. #810C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
73. #460D3semidirect
89. D3×Q8D4semidirect
148. C2×SD16C2×D3semidirect
164. C4×Q8C2×D3semidirect
172. C4⋊C8C2×D3semidirect
180. #102C2×D3semidirect
389. C3#3043semidirect

Maximal subgroups

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

subgroupindex
C2×#9152
C2×D3×SD162
#57152
#55722
C3×#4602
#51662
#51632
#50752
#50282
#48452
#48392
#47722
#47702
#47672
#47622
#30433