Finite Group Databasebeta

← #80867

Subgroups — #80867

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×#911C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5608C211yesmaximal, characteristic, complemented
4#5607C211yesmaximal, characteristic, complemented
5C3×#469C211yesmaximal, characteristic, complemented
6#5157C211yesmaximal, characteristic, complemented
7#5134C211yesmaximal, characteristic, complemented
8#5041C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4864C211yesmaximal, characteristic, complemented
11#4861C211yesmaximal, characteristic, complemented
12#4782C211yesmaximal, characteristic, complemented
13#4780C211yesmaximal, characteristic, complemented
14#4772C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3062—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×#121C2211yescharacteristic
23C3×#121C2211yescharacteristic, complemented
24#940—21no—
25#939—21no—
26#938—21no—
27#938C2211yescharacteristic
28#938C2211yescharacteristic, complemented
29#937—21no—
30C2×#214C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#924C2211yescharacteristic, complemented
34C22×C3⋊C8C2211yescharacteristic, complemented
35#911—24no—
36#907—24no—
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
40C2×D3×C8C2211yescharacteristic
41C2×D3×C8C2211yescharacteristic, complemented
42#897C2211yescharacteristic, complemented
43#895C2211yescharacteristic
44#892C2211yescharacteristic
45D3×C4⋊C4C2211yescharacteristic
46#885—21no—
47#884—21no—
48#882—21no—
49#878—21no—
50C3×#106C2211yescharacteristic, complemented
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic, complemented
53#832C2211yescharacteristic
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#818C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic
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#469D311yescharacteristic, complemented
68#468—31no—
69#468—31no—
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×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67D411yescharacteristic
89C2×#67D411yescharacteristic
90C22×Dic3—21no—
91#237—22no—
92#237—22no—
93D3×D4—22no—
94D3×D4—22no—
95#235—22no—
96#235—22no—
97C2×D12—21no—
98C2×D12D411yescharacteristic
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×C4×D3C2311yescharacteristic
103C2×Dic6D411yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×D8—22no—
106C3×M16—22no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#214—22no—
117#213—22no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3C2311yescharacteristic
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127C4×Dic3—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131#204—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×SD16—31no—
142C2×D8—31no—
143C2×D8C2×D311yescharacteristic
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—31no—
155C22⋊Q8—61no—
156#121—31no—
157#121—31no—
158#121—31no—
159#121—31no—
160#121—61no—
161#121—61no—
162#121—61no—
163#121C2×D311yescharacteristic
164#121C2×D311yescharacteristic, complemented
165#120—61no—
166#120—61no—
167#117—31no—
168C2×C4⋊C4—31no—
169#107—31no—
170#107—31no—
171#106—31no—
172#106C2×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—
183C23×C3C2×D411yescharacteristic
184C22×D3—21no—
185C22×D3—42no—
186C22×D3—81no—
187C22×D3—81no—
188C22×D3C2×D411yescharacteristic
189C22×D3C2×D411yescharacteristic
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×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—22no—
203#67—22no—
204#67—42no—
205#67—42no—
206#67—81no—
207#67—81no—
208#67—81no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—42no—
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215D12—22no—
216D12—21no—
217D12—21no—
218D12—81no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—81no—
222C4×D3C2×D412yes—
223C4×D3#4112yes—
224Dic6—22no—
225Dic6—21no—
226Dic6—21no—
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×Q8—31no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—31no—
241C2×D4—31no—
242C2×D4—31no—
243C2×D4—31no—
244C2×D4—61no—
245C2×D4—62no—
246C2×D4—62no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4C22×D311yescharacteristic
253C2×D4C22×D311yescharacteristic
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4C22×D311yescharacteristic
259SD16—62no—
260SD16—62no—
261D8—22no—
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—31no—
274C4⋊C4—31no—
275C4⋊C4—61no—
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×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3C22×D411yescharacteristic
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—21no—
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×C4C22×D411yescharacteristic
314C3×C4C2×#4111yescharacteristic
315Dic3—21no—
316Dic3—41no—
317Dic3—41no—
318Dic3—41no—
319Dic3#12112yes—
320C23—21no—
321C23—21no—
322C23—31no—
323C23—31no—
324C23—61no—
325C23—122no—
326C23—241no—
327C23—241no—
328C23D3×D411yescharacteristic
329Q8—62no—
330Q8—61no—
331Q8—61no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—62no—
337D4—62no—
338D4—62no—
339D4—61no—
340D4—61no—
341D4—81no—
342D4—81no—
343D4—122no—
344D4—122no—
345D4—241no—
346D4—241no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—31no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—31no—
357C2×C4—61no—
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×C3—81no—
373C2×C3C2×#12111yescharacteristic
374C2×C3#57611yescharacteristic
375C2×C3#57711yescharacteristic
376D3—22no—
377D3—41no—
378D3—81no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—32no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22—121no—
395C22—121no—
396C22—121no—
397C22—244no—
398C22C2×D3×D411yescharacteristic
399C4—21no—
400C4—32no—
401C4—41no—
402C4—41no—
403C4—61no—
404C4—121no—
405C4—121no—
406C4—121no—
407C4C2×D3×D411yescharacteristic
408C4D3×#4111yescharacteristic
409C3#306211yescharacteristic, complemented, Sylow
410C2—41no—
411C2—62no—
412C2—81no—
413C2—81no—
414C2—121no—
415C2—241no—
416C2D3×#12111yescharacteristic
417C2#577411yescharacteristic
418C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3062
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×#907C2semidirect
3. #5608C2semidirect
4. #5607C2semidirect
5. C3×#469C2semidirect
6. #5157C2semidirect
7. #5134C2semidirect
8. #5041C2semidirect
9. #5038C2semidirect
10. #4864C2semidirect
11. #4861C2semidirect
12. #4782C2semidirect
13. #4780C2semidirect
14. #4772C2semidirect
15. D3×#102C2semidirect
21. C2×C3×M16C22semidirect
23. C3×#121C22semidirect
28. #938C22semidirect
33. #924C22semidirect
34. C22×C3⋊C8C22semidirect
39. C2×ΓL(1,𝔽25)C22semidirect
41. C2×D3×C8C22semidirect
42. #897C22semidirect
50. C3×#106C22semidirect
52. C3×#102C22semidirect
54. #832C22semidirect
56. #818C22semidirect
57. #816C22semidirect
58. #810C22semidirect
60. #808C22semidirect
61. #807C22semidirect
67. #469D3semidirect
133. ΓL(1,𝔽25)D4semidirect
147. C2×M16C2×D3semidirect
164. #121C2×D3semidirect
172. #106C2×D3semidirect
180. #102C2×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×#9112
C2×#9072
#56082
#56072
C3×#4692
#51572
#51342
#50412
#50382
#48642
#48612
#47822
#47802
#47722
D3×#1022
#30623