Finite Group Databasebeta

← #80627

Subgroups — #80627

Order 384.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#918C211yesmaximal, characteristic, complemented
2C2×D3×SD16C211yesmaximal, characteristic, complemented
3#5676C211yesmaximal, characteristic
4Q8×D12C211yesmaximal, characteristic, complemented
5C3×#464C211yesmaximal, characteristic, complemented
6#5186C211yesmaximal, characteristic, complemented
7#5170C211yesmaximal, characteristic, complemented
8#5060C211yesmaximal, characteristic, complemented
9#5026C211yesmaximal, characteristic, complemented
10#4841C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4780C211yesmaximal, characteristic, complemented
15#4776C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic, complemented
22C2×C3×SD16C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×C4×Q8C2211yescharacteristic, complemented
25#946C2211yescharacteristic
26C2×#216C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic
29#938C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×#214C2211yescharacteristic
32#918—24no—
33D3×SD16—24no—
34C2×#204C2211yescharacteristic
35C2×#204C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37C2×D3×C8C2211yescharacteristic
38#897—21no—
39#897—21no—
40#897C2211yescharacteristic
41#897C2211yescharacteristic, complemented
42#884—21no—
43#883—21no—
44#882—21no—
45D3×C22⋊C4—21no—
46C4×D12—21no—
47C4×D12C2211yescharacteristic
48C4×D12C2211yescharacteristic
49#869—21no—
50#869C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic
54#835C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#807C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×#137—31no—
63C22×SD16—31no—
64D4×Q8—31no—
65#546—31no—
66#497—31no—
67#468—31no—
68#467—31no—
69#464D311yescharacteristic, complemented
70#461—31no—
71#452—31no—
72#451—31no—
73#450—31no—
74#448—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83#239—22no—
84#239—41no—
85D3×Q8—22no—
86D3×Q8—22no—
87D3×Q8—41no—
88D3×Q8—41no—
89D3×D4—22no—
90D3×D4—41no—
91C2×D12—21no—
92C2×D12D411yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×Q16—22no—
105C3×SD16—22no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C42—21no—
113C3×C42C2311yescharacteristic
114#217—21no—
115#216—22no—
116#216—41no—
117#215D414yescomplemented
118#214—22no—
119#214—41no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4D411yescharacteristic
125(C2×D3)⋊C4D411yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C4⋊Dic3C2311yescharacteristic
132Dic3⋊C4—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135#204—22no—
136#204—22no—
137ΓL(1,𝔽25)—22no—
138D3×C8—22no—
139C2×#41—31no—
140C22×Q8—31no—
141C22×Q8—31no—
142C22×D4—31no—
143#137—64no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×Q16C2×D311yescharacteristic, complemented
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16—64no—
153C2×SD16C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#124C2×D311yescharacteristic
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165#121—31no—
166#121—61no—
167#120—61no—
168C4×Q8C2×D311yescharacteristic, complemented
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103—31no—
177#103—31no—
178#103C2×D311yescharacteristic, complemented
179#102—31no—
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C22×D3—21no—
186C22×D3—42no—
187C22×D3—42no—
188C22×D3—81no—
189C22×D3C2×D411yescharacteristic
190C22×D3C2×D411yescharacteristic
191C3×Q8—21no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8C2×D412yes—
195C3×D4—21no—
196C3×D4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2411yescharacteristic
203#67—42no—
204#67—81no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—22no—
211D12—21no—
212D12—42no—
213D12C2×D412yes—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—21no—
217C4×D3—21no—
218C4×D3—42no—
219C4×D3—42no—
220C4×D3—42no—
221C4×D3—81no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—42no—
227Dic6—42no—
228C3×C8—21no—
229C3×C8—21no—
230C3⋊C8—21no—
231C3⋊C8C2×D412yes—
232C24—61no—
233#41—62no—
234#41—121no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4C22×D311yescharacteristic
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259Q16—22no—
260Q16—62no—
261Q16—121no—
262SD16—22no—
263SD16—34no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268M16—62no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—62no—
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—31no—
276C4⋊C4—31no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—31no—
286C22⋊C4—31no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—21no—
293C42C22×D311yescharacteristic
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3C22×D411yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—22no—
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—84no—
307C2×D3#13712yescomplemented
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4C22×D411yescharacteristic
314C3×C4H(5,𝔽2)11yescharacteristic
315Dic3—21no—
316Dic3—21no—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320C23—21no—
321C23—31no—
322C23—31no—
323C23—61no—
324C23—122no—
325C23—122no—
326C23—241no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—61no—
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—122no—
335Q8—122no—
336Q8D3×D412yes—
337D4—21no—
338D4—21no—
339D4—32no—
340D4—62no—
341D4—61no—
342D4—122no—
343D4—122no—
344D4—241no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—241no—
361C2×C4D3×D411yescharacteristic
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—21no—
366C8—32no—
367C8—61no—
368C2×C3—81no—
369C2×C3D4211yescharacteristic
370C2×C3C2×#13711yescharacteristic
371C2×C3#57711yescharacteristic
372D3—22no—
373D3—41no—
374D3—42no—
375C22—32no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—61no—
384C22—62no—
385C22—121no—
386C22—121no—
387C22—244no—
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—61no—
395C4—61no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4C2×D3×D411yescharacteristic
400C4#100911yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—62no—
403C2—81no—
404C2—121no—
405C2—122no—
406C2#560911yescharacteristic
407C2#577011yescharacteristic
408C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
prime radical#5676C2
hypercenter#464D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#918C2semidirect
2. C2×D3×SD16C2semidirect
4. Q8×D12C2semidirect
5. C3×#464C2semidirect
6. #5186C2semidirect
7. #5170C2semidirect
8. #5060C2semidirect
9. #5026C2semidirect
10. #4841C2semidirect
11. #4836C2semidirect
12. #4813C2semidirect
13. #4808C2semidirect
14. #4780C2semidirect
15. #4776C2semidirect
19. C2×D3×Q8C22semidirect
21. C2×C3×Q16C22semidirect
24. C3×C4×Q8C22semidirect
26. C2×#216C22semidirect
27. C2×#216C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
41. #897C22semidirect
50. #869C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
54. #835C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #816C22semidirect
58. #816C22semidirect
60. #807C22semidirect
61. #803C22semidirect
69. #464D3semidirect
117. #215D4semidirect
146. C2×Q16C2×D3semidirect
168. C4×Q8C2×D3semidirect
174. C4⋊C8C2×D3semidirect
178. #103C2×D3semidirect
307. C2×D3#137semidirect
401. C3#3266semidirect

Maximal subgroups

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

subgroupindex
C2×#9182
C2×D3×SD162
#56762
Q8×D122
C3×#4642
#51862
#51702
#50602
#50262
#48412
#48362
#48132
#48082
#47802
#47762
#32663