Finite Group Databasebeta

← #80922

Subgroups — #80922

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×#918C211yesmaximal, characteristic, complemented
2C2×#908C211yesmaximal, characteristic, complemented
3#5633C211yesmaximal, characteristic, complemented
4#5630C211yesmaximal, characteristic, complemented
5C3×#470C211yesmaximal, characteristic, complemented
6#5191C211yesmaximal, characteristic, complemented
7#5128C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5046C211yesmaximal, characteristic, complemented
10#4865C211yesmaximal, characteristic, complemented
11#4852C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4810C211yesmaximal, characteristic, complemented
14#4806C211yesmaximal, characteristic, complemented
15#4798C211yesmaximal, characteristic, complemented
16#3275—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×#235C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic, complemented
22C2×C3×M16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24C3×C22⋊Q8C2211yescharacteristic
25#947C2211yescharacteristic
26#947C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic
29C2×#215C2211yescharacteristic
30C2×#215C2211yescharacteristic, complemented
31C4×#67C2211yescharacteristic
32#918—24no—
33#908—24no—
34C2×D24C2211yescharacteristic
35C2×D24C2211yescharacteristic
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39#896—21no—
40#895—21no—
41#895C2211yescharacteristic
42#895C2211yescharacteristic, complemented
43#894—21no—
44#893—21no—
45#893C2211yescharacteristic
46#887C2211yescharacteristic, complemented
47#885—21no—
48#884—21no—
49#882—21no—
50D3×C22⋊C4—21no—
51C3×#106C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#103C2211yescharacteristic, complemented
54#830C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#809C2211yescharacteristic
61#809C2211yescharacteristic, complemented
62C2×#137—31no—
63C2×AGL(1,ℤ/8)—31no—
64#548—31no—
65#546—31no—
66#495—31no—
67#470D311yescharacteristic, complemented
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#436—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67D411yescharacteristic
83#239—22no—
84#239—41no—
85D3×Q8—22no—
86D3×Q8—41no—
87#235—22no—
88#235—41no—
89C2×D12—22no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—41no—
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3C2311yescharacteristic
103C2×Dic6—21no—
104C2×Dic6C2311yescharacteristic
105C3×Q16—22no—
106C3×M16—22no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4C2311yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C22⋊C4—21no—
114C3×C22⋊C4—21no—
115#217D411yescharacteristic
116#216—22no—
117#215—22no—
118#215—41no—
119#215—41no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128Dic3⋊C4—21no—
129Dic3⋊C4D411yescharacteristic
130C4×Dic3—21no—
131C4×Dic3C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134D24—22no—
135D24—22no—
136#204—22no—
137#204D414yescomplemented
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143#137—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16—31no—
146C2×Q16C2×D311yescharacteristic, complemented
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C2×M16C2×D311yescharacteristic
156#124—31no—
157#124—31no—
158#124—61no—
159#123—61no—
160C22⋊Q8—61no—
161C22⋊Q8C2×D311yescharacteristic
162C22⋊Q8C2×D311yescharacteristic
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#106C2×D311yescharacteristic, complemented
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183C4×C8—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—81no—
191C22×D3C2×D411yescharacteristic
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—41no—
205#67—42no—
206C2×Dic3—21no—
207C2×Dic3C2×D411yescharacteristic
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—21no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—42no—
216D12—42no—
217D12—42no—
218D12—81no—
219D12C2×D412yes—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—42no—
223C4×D3—42no—
224C4×D3—42no—
225C4×D3—81no—
226Dic6—21no—
227Dic6—42no—
228Dic6C2×D412yes—
229C3×C8—21no—
230C3×C8C2×D412yes—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C24—61no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238C2×Q8—31no—
239C2×Q8—61no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—62no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—121no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4C22×D311yescharacteristic
261Q16—22no—
262Q16—62no—
263SD16—34no—
264SD16—62no—
265SD16—62no—
266SD16—121no—
267SD16—121no—
268D8—62no—
269D8—62no—
270M16—22no—
271M16—62no—
272C2×C8—31no—
273C2×C8—31no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—31no—
279C4⋊C4—61no—
280C4⋊C4C22×D311yescharacteristic
281C4⋊C4C22×D311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—31no—
285C22⋊C4—31no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C42—31no—
294C42—61no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3C22×D411yescharacteristic
299C2×D3—21no—
300C2×D3—22no—
301C2×D3—21no—
302C2×D3—21no—
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—41no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4C22×D411yescharacteristic
318C3×C4H(5,𝔽2)11yescharacteristic
319Dic3—22no—
320Dic3—21no—
321Dic3—21no—
322Dic3—21no—
323C23—31no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—122no—
328C23—122no—
329C23—241no—
330C23D3×D411yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—32no—
336Q8—61no—
337Q8—122no—
338D4—32no—
339D4—61no—
340D4—61no—
341D4—61no—
342D4—61no—
343D4—61no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—121no—
348D4—122no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—241no—
366C2×C4D3×D411yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—61no—
370C8—61no—
371C8D3×D412yes—
372C2×C3—41no—
373C2×C3D4211yescharacteristic
374C2×C3#57511yescharacteristic
375C2×C3#57711yescharacteristic
376D3—41no—
377D3—42no—
378D3—81no—
379D3—81no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—61no—
403C4—61no—
404C4—61no—
405C4—62no—
406C4C2×D3×D411yescharacteristic
407C4#100911yescharacteristic
408C3#327511yescharacteristic, complemented, Sylow
409C2—41no—
410C2—121no—
411C2—122no—
412C2—241no—
413C2—241no—
414C2#560911yescharacteristic
415C2#577711yescharacteristic
416C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
hypercenter#470D3
hypocenterC3#3275
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#918C2semidirect
2. C2×#908C2semidirect
3. #5633C2semidirect
4. #5630C2semidirect
5. C3×#470C2semidirect
6. #5191C2semidirect
7. #5128C2semidirect
8. #5047C2semidirect
9. #5046C2semidirect
10. #4865C2semidirect
11. #4852C2semidirect
12. #4819C2semidirect
13. #4810C2semidirect
14. #4806C2semidirect
15. #4798C2semidirect
18. C2×D3×Q8C22semidirect
21. C2×C3×Q16C22semidirect
26. #947C22semidirect
27. C2×#216C22semidirect
30. C2×#215C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
42. #895C22semidirect
46. #887C22semidirect
51. C3×#106C22semidirect
52. C3×#103C22semidirect
53. C3×#103C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #810C22semidirect
61. #809C22semidirect
67. #470D3semidirect
137. #204D4semidirect
146. C2×Q16C2×D3semidirect
173. #106C2×D3semidirect
176. #103C2×D3semidirect
177. #103C2×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×#9182
C2×#9082
#56332
#56302
C3×#4702
#51912
#51282
#50472
#50462
#48652
#48522
#48192
#48102
#48062
#47982
#32753