Finite Group Databasebeta

← #80809

Subgroups — #80809

Order 384.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#914C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5629C211yesmaximal, characteristic, complemented
4#5607C211yesmaximal, characteristic, complemented
5C3×#468C211yesmaximal, characteristic, complemented
6#5171C211yesmaximal, characteristic, complemented
7#5135C211yesmaximal, characteristic, complemented
8#5048C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4883C211yesmaximal, characteristic, complemented
11#4881C211yesmaximal, characteristic, complemented
12#4812C211yesmaximal, characteristic, complemented
13#4804C211yesmaximal, characteristic, complemented
14#4782C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3062—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic
26#940—21no—
27#938C2211yescharacteristic
28#937—21no—
29C2×#213C2211yescharacteristic
30#930C2211yescharacteristic
31#930C2211yescharacteristic, complemented
32C22×C3⋊C8C2211yescharacteristic, complemented
33#914—24no—
34#907—24no—
35C2×D24C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
38C2×D3×C8C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic, complemented
40#896—21no—
41#895—21no—
42#895C2211yescharacteristic
43#895C2211yescharacteristic, complemented
44#892C2211yescharacteristic
45D3×C4⋊C4C2211yescharacteristic
46#885—21no—
47#884—21no—
48#884—21no—
49#882—21no—
50C3×#107C2211yescharacteristic, complemented
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56#821C2211yescharacteristic, complemented
57#817C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59#809C2211yescharacteristic, complemented
60#808C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#131—31no—
64#537—31no—
65#535—31no—
66#469—31no—
67#469—31no—
68#468—31no—
69#468D311yescharacteristic, complemented
70#466—31no—
71#466—31no—
72#465—31no—
73#465—31no—
74#428—31no—
75#416—31no—
76C2×#102—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67D411yescharacteristic
87C2×#67D411yescharacteristic
88#239—22no—
89#239—22no—
90D3×D4—22no—
91D3×D4—22no—
92#235—22no—
93#235—22no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12D411yescharacteristic
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3C2311yescharacteristic
103C2×C4×D3C2311yescharacteristic
104C2×Dic6D411yescharacteristic
105C3×SD16—22no—
106C3×M16—22no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4—21no—
110C3×C4⋊C4C2311yescharacteristic
111C3×C4⋊C4C2311yescharacteristic
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114#217—21no—
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)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123C4⋊Dic3C2311yescharacteristic
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127C4×Dic3—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131D24—22no—
132ΓL(1,𝔽25)—22no—
133ΓL(1,𝔽25)D414yescomplemented
134D3×C8—22no—
135D3×C8—22no—
136C2×#41—31no—
137C2×#41—31no—
138C22×D4—31no—
139AGL(1,ℤ/8)—64no—
140C2×SD16—31no—
141C2×SD16C2×D311yescharacteristic
142C2×D8—31no—
143C2×D8—31no—
144#131—64no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16C2×D311yescharacteristic, complemented
148C22×C8—31no—
149C22×C8—31no—
150C22×C8—31no—
151#127—61no—
152#124—61no—
153C22⋊Q8—31no—
154C22⋊Q8—61no—
155C22⋊Q8C2×D311yescharacteristic, complemented
156#121—31no—
157#121—31no—
158#121—31no—
159#121—31no—
160#121—31no—
161#121—61no—
162#121—61no—
163#121—61no—
164#121C2×D311yescharacteristic
165#120—61no—
166#120—61no—
167#117—31no—
168C2×C4⋊C4—31no—
169#107—31no—
170#107C2×D311yescharacteristic, complemented
171#106—31no—
172#106—31no—
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—
182C23×C3C2×D411yescharacteristic
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—42no—
186C22×D3—81no—
187C22×D3—81no—
188C22×D3C2×D411yescharacteristic
189C22×D3C2×D411yescharacteristic
190C3×Q8—21no—
191C3×Q8—21no—
192C3×D4—21no—
193C3×D4—21no—
194C3×D4—81no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—22no—
203#67—22no—
204#67—42no—
205#67—81no—
206#67—81no—
207#67—81no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3C2×D411yescharacteristic
211C2×Dic3C2×D411yescharacteristic
212D12—22no—
213D12—21no—
214D12—21no—
215D12—21no—
216D12—21no—
217D12—42no—
218D12—81no—
219D12—81no—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—42no—
223C4×D3—81no—
224C4×D3C2×D412yes—
225C4×D3#4112yes—
226Dic6—22no—
227C3×C8—21no—
228C3×C8C2×D412yes—
229C3⋊C8—21no—
230C3⋊C8C2×D412yes—
231C24—61no—
232#41—62no—
233#41—62no—
234#41—62no—
235#41—62no—
236C2×Q8—31no—
237C2×Q8C22×D311yescharacteristic
238C2×D4—21no—
239C2×D4—31no—
240C2×D4—31no—
241C2×D4—31no—
242C2×D4—31no—
243C2×D4—31no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4C22×D311yescharacteristic
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4C22×D311yescharacteristic
259SD16—22no—
260SD16—62no—
261D8—62no—
262D8—62no—
263M16—22no—
264M16—34no—
265M16—62no—
266C2×C8—31no—
267C2×C8—31no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—62no—
271C2×C8C22×D311yescharacteristic
272C2×C8C22×D311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4C22×D311yescharacteristic
279C4⋊C4C22×D311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C42—61no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3C22×D411yescharacteristic
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—84no—
309C2×D3#12112yes—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4C22×D411yescharacteristic
315C3×C4C2×#4111yescharacteristic
316Dic3—21no—
317Dic3—41no—
318Dic3—41no—
319Dic3#12112yes—
320C23—21no—
321C23—31no—
322C23—31no—
323C23—61no—
324C23—61no—
325C23—122no—
326C23—241no—
327C23—241no—
328C23D3×D411yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—62no—
332D4—21no—
333D4—21no—
334D4—62no—
335D4—62no—
336D4—62no—
337D4—61no—
338D4—61no—
339D4—61no—
340D4—61no—
341D4—81no—
342D4—122no—
343D4—122no—
344D4—241no—
345D4—241no—
346D4—241no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—31no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—31no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—122no—
363C2×C4—241no—
364C2×C4D3×D411yescharacteristic
365C2×C4C23×D311yescharacteristic
366C8—21no—
367C8—32no—
368C8—61no—
369C8D3×D412yes—
370C2×C3—41no—
371C2×C3—81no—
372C2×C3C2×#12111yescharacteristic
373C2×C3#57611yescharacteristic
374C2×C3#57711yescharacteristic
375D3—22no—
376D3—41no—
377D3—81no—
378D3—81no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—32no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—61no—
387C22—61no—
388C22—61no—
389C22—61no—
390C22—61no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22—121no—
397C22—244no—
398C22C2×D3×D411yescharacteristic
399C4—21no—
400C4—32no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—61no—
405C4—121no—
406C4—121no—
407C4C2×D3×D411yescharacteristic
408C4D3×#4111yescharacteristic
409C3#306211yescharacteristic, complemented, Sylow
410C2—41no—
411C2—62no—
412C2—81no—
413C2—121no—
414C2—241no—
415C2—241no—
416C2D3×#12111yescharacteristic
417C2#577411yescharacteristic
418C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
hypercenter#468D3
hypocenterC3#3062
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. C2×#907C2semidirect
3. #5629C2semidirect
4. #5607C2semidirect
5. C3×#468C2semidirect
6. #5171C2semidirect
7. #5135C2semidirect
8. #5048C2semidirect
9. #5038C2semidirect
10. #4883C2semidirect
11. #4881C2semidirect
12. #4812C2semidirect
13. #4804C2semidirect
14. #4782C2semidirect
15. D3×#102C2semidirect
21. C2×C3×M16C22semidirect
22. C3×C22⋊Q8C22semidirect
24. #946C22semidirect
31. #930C22semidirect
32. C22×C3⋊C8C22semidirect
37. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
43. #895C22semidirect
50. C3×#107C22semidirect
51. C3×#103C22semidirect
53. #835C22semidirect
56. #821C22semidirect
57. #817C22semidirect
59. #809C22semidirect
60. #808C22semidirect
61. #807C22semidirect
69. #468D3semidirect
133. ΓL(1,𝔽25)D4semidirect
147. C2×M16C2×D3semidirect
155. C22⋊Q8C2×D3semidirect
170. #107C2×D3semidirect
174. #103C2×D3semidirect
409. C3#3062semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
C2×#9072
#56292
#56072
C3×#4682
#51712
#51352
#50482
#50382
#48832
#48812
#48122
#48042
#47822
D3×#1022
#30623