Finite Group Databasebeta

← #80444

Subgroups — #80444

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×#915C211yesmaximal, characteristic, complemented
2C2×D3×SD16C211yesmaximal, characteristic, complemented
3#5727C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic
5C3×#461C211yesmaximal, characteristic, complemented
6#5173C211yesmaximal, characteristic, complemented
7#5169C211yesmaximal, characteristic, complemented
8#5086C211yesmaximal, characteristic, complemented
9#5018C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4812C211yesmaximal, characteristic, complemented
14#4809C211yesmaximal, characteristic, complemented
15#4806C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×#237C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×C4⋊Q8C2211yescharacteristic, complemented
24C3×C4×D4C2211yescharacteristic
25#946C2211yescharacteristic
26#946C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28C2×#216C2211yescharacteristic, complemented
29C2×#215C2211yescharacteristic
30C2×#215C2211yescharacteristic
31C2×#214C2211yescharacteristic
32#930—21no—
33C2×(C2×D3)⋊C4—21no—
34#915—24no—
35D3×SD16—24no—
36C2×Dic12C2211yescharacteristic, complemented
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×D3×C8C2211yescharacteristic
40#897C2211yescharacteristic
41#896—21no—
42#896—21no—
43#895C2211yescharacteristic
44#893—21no—
45#888—21no—
46#886—21no—
47#882—21no—
48#875C2211yescharacteristic
49C4×D12C2211yescharacteristic
50C4×Dic6C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#103C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
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#464—31no—
70#461D311yescharacteristic, complemented
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
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84C22×Dic3—21no—
85D3×Q8—22no—
86D3×Q8—22no—
87D3×Q8—41no—
88D3×Q8—41no—
89#237—22no—
90#237—41no—
91D3×D4—22no—
92D3×D4—41no—
93C2×D12—21no—
94C2×D12D411yescharacteristic
95C2×D12C2311yescharacteristic
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×SD16—22no—
105C3×SD16—22no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112C3×C42C2311yescharacteristic
113#216—22no—
114#216—41no—
115#215—22no—
116#215—41no—
117#214D414yescomplemented
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—
124(C2×D3)⋊C4D411yescharacteristic
125(C2×D3)⋊C4D411yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3D411yescharacteristic
128C4⋊Dic3C2311yescharacteristic
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132C4×Dic3—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135Dic12—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×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—64no—
152C2×SD16C2×D311yescharacteristic
153C2×SD16C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—61no—
157C4⋊Q8C2×D311yescharacteristic, complemented
158#124—31no—
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×Q8—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#103C2×D311yescharacteristic, complemented
179#102—31no—
180#102—31no—
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×Q8—21no—
195C3×D4—21no—
196C3×D4C2×D412yes—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—81no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—42no—
205#67—42no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—42no—
210C2×Dic3C2×D411yescharacteristic
211C2×Dic3C2×D411yescharacteristic
212D12—22no—
213D12—21no—
214D12—21no—
215D12—81no—
216C4×D3—21no—
217C4×D3—21no—
218C4×D3—21no—
219C4×D3—21no—
220C4×D3—42no—
221C4×D3—42no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—42no—
226Dic6—42no—
227Dic6C2×D412yes—
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—21no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259Q16—62no—
260Q16—62no—
261Q16—121no—
262SD16—22no—
263SD16—22no—
264SD16—34no—
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—21no—
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
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—
292C42—61no—
293C42C22×D311yescharacteristic
294C22×C3—22no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3C22×D411yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—84no—
307C2×D3#13712yescomplemented
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4C22×D411yescharacteristic
314C3×C4H(5,𝔽2)11yescharacteristic
315Dic3—22no—
316Dic3—21no—
317Dic3—21no—
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—21no—
331Q8—32no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8—122no—
337D4—21no—
338D4—62no—
339D4—61no—
340D4—61no—
341D4—122no—
342D4—122no—
343D4—241no—
344D4D3×D412yes—
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—81no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4D3×D411yescharacteristic
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—21no—
366C8—32no—
367C8—61no—
368C2×C3—42no—
369C2×C3D4211yescharacteristic
370C2×C3C2×#13711yescharacteristic
371C2×C3#57711yescharacteristic
372D3—22no—
373D3—41no—
374D3—81no—
375C22—22no—
376C22—32no—
377C22—41no—
378C22—41no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—61no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—244no—
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—61no—
395C4—61no—
396C4—62no—
397C4—121no—
398C4—121no—
399C4C2×D3×D411yescharacteristic
400C4#100911yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—42no—
403C2—62no—
404C2—121no—
405C2—241no—
406C2#560911yescharacteristic
407C2#576211yescharacteristic
408C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radical#5554C2
hypercenter#461D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
2. C2×D3×SD16C2semidirect
3. #5727C2semidirect
5. C3×#461C2semidirect
6. #5173C2semidirect
7. #5169C2semidirect
8. #5086C2semidirect
9. #5018C2semidirect
10. #4845C2semidirect
11. #4836C2semidirect
12. #4813C2semidirect
13. #4812C2semidirect
14. #4809C2semidirect
15. #4806C2semidirect
18. C2×D3×Q8C22semidirect
23. C3×C4⋊Q8C22semidirect
26. #946C22semidirect
27. C2×#216C22semidirect
28. C2×#216C22semidirect
36. C2×Dic12C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
50. C4×Dic6C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
53. C3×#103C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #816C22semidirect
59. #810C22semidirect
60. #807C22semidirect
61. #803C22semidirect
70. #461D3semidirect
117. #214D4semidirect
157. C4⋊Q8C2×D3semidirect
174. C4⋊C8C2×D3semidirect
177. #103C2×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×#9152
C2×D3×SD162
#57272
#55542
C3×#4612
#51732
#51692
#50862
#50182
#48452
#48362
#48132
#48122
#48092
#48062
#32663