Finite Group Databasebeta

← #79551

Subgroups — #79551

Order 384.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#909C211yesmaximal, characteristic, complemented
2C2×#908C211yesmaximal, characteristic, complemented
3#5552C211yesmaximal, characteristic, complemented
4#5550C211yesmaximal, characteristic, complemented
5C3×#436C211yesmaximal, characteristic, complemented
6#5137C211yesmaximal, characteristic, complemented
7#5135C211yesmaximal, characteristic, complemented
8#4845C211yesmaximal, characteristic, complemented
9#4844C211yesmaximal, characteristic, complemented
10#4738C211yesmaximal, characteristic, complemented
11#4734C211yesmaximal, characteristic, complemented
12#4733C211yesmaximal, characteristic, complemented
13#4731C211yesmaximal, characteristic, complemented
14#4696C211yesmaximal, characteristic, complemented
15C4×#204C211yesmaximal, characteristic, complemented
16#3275—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×Dic6C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic
22C2×C3×M16C2211yescharacteristic, complemented
23C3×C4×D4C2211yescharacteristic
24#939—21no—
25#937—21no—
26#930C2211yescharacteristic
27#930C2211yescharacteristic, complemented
28#924C2211yescharacteristic
29#924C2211yescharacteristic
30#909—24no—
31#908—24no—
32C2×Dic12C2211yescharacteristic, complemented
33C2×Dic12C2211yescharacteristic, complemented
34C2×D24C2211yescharacteristic
35C2×D24C2211yescharacteristic
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×#204C2211yescharacteristic
39C2×#204C2211yescharacteristic
40C2×#204C2211yescharacteristic, complemented
41#896—21no—
42#894—21no—
43#885—21no—
44#884—21no—
45#67⋊C4—21no—
46D3×C22⋊C4—21no—
47#875C2211yescharacteristic
48#875C2211yescharacteristic, complemented
49C4×D12C2211yescharacteristic
50C4×Dic6C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×C22⋊C8C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic, complemented
54C3×C4×C8C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57#821C2211yescharacteristic, complemented
58#817C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#816C2211yescharacteristic, complemented
61#816C2211yescharacteristic, complemented
62C2×#137—31no—
63C2×AGL(1,ℤ/8)—31no—
64#548—31no—
65#546—31no—
66#495—31no—
67#470—31no—
68#469—31no—
69#464—31no—
70#463—31no—
71#452—31no—
72#450—31no—
73#449—31no—
74#448—31no—
75C4×SD16—31no—
76#436D311yescharacteristic, complemented
77C23×D3—21no—
78C2×C3×D4D411yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#235—22no—
87#235—22no—
88#235—41no—
89#235—41no—
90C2×D12—22no—
91C2×D12—21no—
92C2×D12—41no—
93C2×D12C2311yescharacteristic
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×Dic6—22no—
100C2×Dic6—21no—
101C2×Dic6—41no—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×M16—22no—
106C3×M16—22no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C3×C4⋊C4D411yescharacteristic
112C3×C22⋊C4D411yescharacteristic
113C3×C22⋊C4D411yescharacteristic
114C3×C42C2311yescharacteristic
115#217—21no—
116#217—21no—
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—
123(C2×D3)⋊C4—21no—
124C4⋊Dic3C2311yescharacteristic
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129C4×Dic3—21no—
130Dic12—22no—
131Dic12—22no—
132D24—22no—
133D24—22no—
134#204—22no—
135#204—22no—
136#204—41no—
137#204—41no—
138#204D414yescomplemented
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143#137—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16C2×D311yescharacteristic
155C2×M16C2×D311yescharacteristic, complemented
156#124—31no—
157#124—31no—
158#124—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8C2×D311yescharacteristic
182C22⋊C8C2×D311yescharacteristic, complemented
183C4×C8C2×D311yescharacteristic, complemented
184C23×C3C2×D411yescharacteristic
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—81no—
192C3×D4—41no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—42no—
203#67—42no—
204#67—42no—
205#67—81no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—42no—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—21no—
216D12—21no—
217D12—42no—
218D12C2×D412yes—
219C4×D3—42no—
220C4×D3—42no—
221C4×D3—81no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—42no—
228Dic6C2×D412yes—
229C3×C8—21no—
230C3×C8—21no—
231C3×C8—21no—
232C3×C8C2×D412yes—
233C24—61no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238C2×Q8—31no—
239C2×Q8—31no—
240C2×Q8—31no—
241C2×Q8—62no—
242C2×Q8—61no—
243C2×Q8—121no—
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—62no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—121no—
254C2×D4D1211yescharacteristic
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4C22×D311yescharacteristic
260C22×C4C22×D311yescharacteristic
261Q16—62no—
262Q16—62no—
263SD16—34no—
264SD16—62no—
265SD16—62no—
266SD16—121no—
267SD16—121no—
268D8—62no—
269D8—62no—
270M16—22no—
271M16—22no—
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4D1211yescharacteristic
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4D1211yescharacteristic
292C22⋊C4D1211yescharacteristic
293C42—61no—
294C42C22×D311yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3C22×D411yescharacteristic
302C2×D3—22no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—84no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4C22×D411yescharacteristic
317C3×C4H(5,𝔽2)11yescharacteristic
318Dic3—22no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—122no—
327C23—122no—
328C23—241no—
329C23C2×D1211yescharacteristic
330C23C2×D1211yescharacteristic
331Q8—32no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337Q8—122no—
338D4—32no—
339D4—41no—
340D4—61no—
341D4—61no—
342D4—61no—
343D4—61no—
344D4—61no—
345D4—122no—
346D4—122no—
347D4—122no—
348D4—122no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—241no—
363C2×C4C2×D1211yescharacteristic
364C2×C4C2×D1211yescharacteristic
365C2×C4C2×D1211yescharacteristic
366C2×C4C2×D1211yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—21no—
370C8—21no—
371C8D3×D412yes—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3D4211yescharacteristic
375C2×C3#57511yescharacteristic
376C2×C3#57711yescharacteristic
377D3—42no—
378D3—81no—
379D3—81no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C22×D1211yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—62no—
402C4—121no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100911yescharacteristic
408C3#327511yescharacteristic, complemented, Sylow
409C2—41no—
410C2—41no—
411C2—122no—
412C2—241no—
413C2—241no—
414C2D4×D1211yescharacteristic
415C2#574811yescharacteristic
416C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
hypercenter#436D3
hypocenterC3#3275
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
2. C2×#908C2semidirect
3. #5552C2semidirect
4. #5550C2semidirect
5. C3×#436C2semidirect
6. #5137C2semidirect
7. #5135C2semidirect
8. #4845C2semidirect
9. #4844C2semidirect
10. #4738C2semidirect
11. #4734C2semidirect
12. #4733C2semidirect
13. #4731C2semidirect
14. #4696C2semidirect
15. C4×#204C2semidirect
20. C22×Dic6C22semidirect
22. C2×C3×M16C22semidirect
27. #930C22semidirect
32. C2×Dic12C22semidirect
33. C2×Dic12C22semidirect
40. C2×#204C22semidirect
48. #875C22semidirect
50. C4×Dic6C22semidirect
51. C3×C4⋊C8C22semidirect
53. C3×C22⋊C8C22semidirect
54. C3×C4×C8C22semidirect
57. #821C22semidirect
58. #817C22semidirect
59. #816C22semidirect
60. #816C22semidirect
61. #816C22semidirect
76. #436D3semidirect
138. #204D4semidirect
155. C2×M16C2×D3semidirect
174. C4⋊C8C2×D3semidirect
182. C22⋊C8C2×D3semidirect
183. C4×C8C2×D3semidirect
408. C3#3275semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
C2×#9082
#55522
#55502
C3×#4362
#51372
#51352
#48452
#48442
#47382
#47342
#47332
#47312
#46962
C4×#2042
#32753