Finite Group Databasebeta

← #80656

Subgroups — #80656

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#916C211yesmaximal, characteristic, complemented
2C22×D3×C8C211yesmaximal, characteristic, complemented
3#5627C211yesmaximal, characteristic, complemented
4#5606C211yesmaximal, characteristic, complemented
5C3×#465C211yesmaximal, characteristic, complemented
6#5171C211yesmaximal, characteristic, complemented
7#5114C211yesmaximal, characteristic, complemented
8#5051C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4864C211yesmaximal, characteristic, complemented
11#4860C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4804C211yesmaximal, characteristic, complemented
14#4775C211yesmaximal, characteristic, complemented
15#4772C211yesmaximal, characteristic, complemented
16#3050—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic, complemented
26#938C2211yescharacteristic
27#937—21no—
28D4×Dic3—21no—
29C2×#213C2211yescharacteristic, complemented
30#930C2211yescharacteristic
31#924C2211yescharacteristic
32C22×C3⋊C8C2211yescharacteristic
33#916—24no—
34C2×#204C2211yescharacteristic, complemented
35C2×D3×C8—24no—
36C2×D3×C8C2211yescharacteristic
37C2×D3×C8C2211yescharacteristic
38C2×D3×C8C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic
40#897C2211yescharacteristic
41#895C2211yescharacteristic
42#894—21no—
43#893—21no—
44#892C2211yescharacteristic
45#892C2211yescharacteristic
46#883—21no—
47#882—21no—
48#67⋊C4—21no—
49#67⋊C4—21no—
50C3×#106C2211yescharacteristic, complemented
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#817C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#807C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×#135—31no—
63C23×C8—31no—
64#525—31no—
65#525—31no—
66#467—31no—
67#467—31no—
68#466—31no—
69#466—31no—
70#465—31no—
71#465—31no—
72#465—31no—
73#465D311yescharacteristic, complemented
74#427—31no—
75#416—31no—
76#416—31no—
77C23×D3D411yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86C22×Dic3D411yescharacteristic
87#239—22no—
88#239—22no—
89#237—22no—
90#237—22no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12C2311yescharacteristic
94C2×C4×D3—22no—
95C2×C4×D3—22no—
96C2×C4×D3—21no—
97C2×C4×D3D411yescharacteristic
98C2×C4×D3D411yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×C4×D3C2311yescharacteristic
101C2×Dic6C2311yescharacteristic
102C3×SD16—22no—
103C2×C3×C8—22no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111#217—21no—
112#217—21no—
113#216—22no—
114#213—22no—
115(C2×D3)⋊C4—21no—
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⋊Dic3—21no—
122C4⋊Dic3C2311yescharacteristic
123C4⋊Dic3C2311yescharacteristic
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131#204—22no—
132D3×C8—22no—
133D3×C8—22no—
134D3×C8—22no—
135D3×C8D414yescomplemented
136C2×#41—31no—
137C2×#41—31no—
138C23×C4—31no—
139#135—64no—
140C2×Q16—31no—
141C2×SD16—31no—
142C2×SD16C2×D311yescharacteristic, complemented
143C2×D8—31no—
144C22×C8—31no—
145C22×C8—31no—
146C22×C8—31no—
147C22×C8—31no—
148C22×C8—31no—
149C22×C8—64no—
150C22×C8C2×D311yescharacteristic
151#123—61no—
152#123—61no—
153C22⋊Q8—31no—
154C22⋊Q8—31no—
155C22⋊Q8—31no—
156C22⋊Q8C2×D311yescharacteristic
157#121—31no—
158#121—31no—
159#121—31no—
160#121C2×D311yescharacteristic
161#120—61no—
162#120—61no—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C4×D4—61no—
167#117—31no—
168#117—31no—
169#107—31no—
170#107—31no—
171#106—31no—
172#106C2×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, complemented
181C23×C3—21no—
182C23×C3C2×D411yescharacteristic
183C22×D3—22no—
184C22×D3—22no—
185C22×D3—22no—
186C22×D3—21no—
187C22×D3C2×D411yescharacteristic
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—42no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—42no—
203#67—42no—
204C2×Dic3—22no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—42no—
209C2×Dic3C2×D411yescharacteristic
210C2×Dic3C2×D411yescharacteristic
211D12—21no—
212D12—21no—
213D12—42no—
214D12—42no—
215C4×D3—22no—
216C4×D3—21no—
217C4×D3—21no—
218C4×D3—22no—
219C4×D3—42no—
220C4×D3C2×D412yes—
221C4×D3#4112yes—
222Dic6—21no—
223Dic6—21no—
224C3×C8—21no—
225C3×C8C2×D412yes—
226C3⋊C8—21no—
227C3⋊C8C2×D412yes—
228C24—31no—
229#41—62no—
230#41—62no—
231#41—62no—
232#41—62no—
233C2×Q8—31no—
234C2×Q8C22×D311yescharacteristic
235C2×D4—21no—
236C2×D4—31no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4C22×D311yescharacteristic
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4C22×D311yescharacteristic
253Q16—62no—
254SD16—22no—
255SD16—62no—
256D8—62no—
257C2×C8—22no—
258C2×C8—31no—
259C2×C8—34no—
260C2×C8—31no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8C22×D311yescharacteristic
266C2×C8C22×D311yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4C22×D311yescharacteristic
274C4⋊C4C22×D311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C42—61no—
286C42—61no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3#13512yescomplemented
293C22×C3C22×D411yescharacteristic
294C2×D3—28no—
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3#12112yes—
303C2×D3#13512yescomplemented
304C3×C4—21no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4C22×D411yescharacteristic
309C3×C4C2×#4111yescharacteristic
310Dic3—21no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314Dic3#12112yes—
315C23—21no—
316C23—31no—
317C23—31no—
318C23—62no—
319C23—62no—
320C23—62no—
321C23—61no—
322C23D3×D411yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—61no—
326Q8—61no—
327D4—21no—
328D4—21no—
329D4—42no—
330D4—61no—
331D4—61no—
332D4—122no—
333D4—122no—
334D4—122no—
335D4—122no—
336D4—122no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—31no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4D3×D411yescharacteristic
357C2×C4C23×D311yescharacteristic
358C8—21no—
359C8—32no—
360C8—61no—
361C8D3×D412yes—
362C2×C3—22no—
363C2×C3—81no—
364C2×C3C2×#12111yescharacteristic
365C2×C3C2×#13511yescharacteristic
366C2×C3C2×#13511yescharacteristic
367D3—22no—
368D3—22no—
369D3—81no—
370C22—21no—
371C22—21no—
372C22—32no—
373C22—32no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—68no—
378C22—61no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22#91612yescomplemented
386C22C2×D3×D411yescharacteristic
387C4—21no—
388C4—32no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—61no—
393C4—121no—
394C4—121no—
395C4—121no—
396C4C2×D3×D411yescharacteristic
397C4D3×#4111yescharacteristic
398C3#305011yescharacteristic, complemented, Sylow
399C2—22no—
400C2—62no—
401C2—62no—
402C2—81no—
403C2—241no—
404C2D3×#12111yescharacteristic
405C2C2×#91611yescharacteristic
406C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3050
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#916C2semidirect
2. C22×D3×C8C2semidirect
3. #5627C2semidirect
4. #5606C2semidirect
5. C3×#465C2semidirect
6. #5171C2semidirect
7. #5114C2semidirect
8. #5051C2semidirect
9. #5038C2semidirect
10. #4864C2semidirect
11. #4860C2semidirect
12. #4813C2semidirect
13. #4804C2semidirect
14. #4775C2semidirect
15. #4772C2semidirect
20. C2×C3×SD16C22semidirect
25. C2×#216C22semidirect
29. C2×#213C22semidirect
34. C2×#204C22semidirect
50. C3×#106C22semidirect
51. C3×#103C22semidirect
52. C3×#102C22semidirect
53. #835C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. #817C22semidirect
57. #816C22semidirect
58. #810C22semidirect
59. #809C22semidirect
60. #807C22semidirect
61. #807C22semidirect
73. #465D3semidirect
135. D3×C8D4semidirect
142. C2×SD16C2×D3semidirect
172. #106C2×D3semidirect
176. #103C2×D3semidirect
180. #102C2×D3semidirect
292. C22×C3#135semidirect
303. C2×D3#135semidirect
385. C22#916semidirect
398. C3#3050semidirect

Maximal subgroups

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

subgroupindex
C2×#9162
C22×D3×C82
#56272
#56062
C3×#4652
#51712
#51142
#50512
#50382
#48642
#48602
#48132
#48042
#47752
#47722
#30503