Finite Group Databasebeta

← #79795

Subgroups — #79795

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
3Q8×D12C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic
5C3×#440C211yesmaximal, characteristic, complemented
6#5028C211yesmaximal, characteristic, complemented
7#5019C211yesmaximal, characteristic, complemented
8#4887C211yesmaximal, characteristic, complemented
9#4886C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4715C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×#237C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23C3×C4×D4C2211yescharacteristic
24C2×#216C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic
26C2×#214C2211yescharacteristic
27C2×#214C2211yescharacteristic
28#930—21no—
29C2×(C2×D3)⋊C4—21no—
30#915—24no—
31D3×SD16—24no—
32C2×Dic12C2211yescharacteristic, complemented
33C2×Dic12C2211yescharacteristic, complemented
34C2×#204C2211yescharacteristic
35C2×#204C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37C2×D3×C8C2211yescharacteristic
38#897—21no—
39#897—21no—
40#897C2211yescharacteristic
41#897C2211yescharacteristic
42#897C2211yescharacteristic, complemented
43#895C2211yescharacteristic
44#886—21no—
45#882—21no—
46#875C2211yescharacteristic
47C4×D12—21no—
48C4×D12C2211yescharacteristic
49#869—21no—
50#869C2211yescharacteristic, complemented
51C3×#107C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic
54C3×#97C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic, 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#461—31no—
71#452—31no—
72#451—31no—
73#450—31no—
74#448—31no—
75#440D311yescharacteristic, complemented
76#435—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84D3×Q8—22no—
85D3×Q8—22no—
86D3×Q8—41no—
87D3×Q8—41no—
88#237—22no—
89#237—41no—
90D3×D4—22no—
91D3×D4—41no—
92C2×D12—21no—
93C2×D12D411yescharacteristic
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C2×C4×D3C2311yescharacteristic
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×SD16D414yescomplemented
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4—21no—
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C42—21no—
112C3×C42C2311yescharacteristic
113#216—22no—
114#215—22no—
115#214—22no—
116#214—22no—
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)⋊C4D411yescharacteristic
124(C2×D3)⋊C4D411yescharacteristic
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133Dic12—22no—
134Dic12—41no—
135#204—22no—
136#204—41no—
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—31no—
152C2×SD16—64no—
153C2×SD16C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
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×Q8C2×D311yescharacteristic, complemented
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
172C2×C22⋊C4—61no—
173#107C2×D311yescharacteristic, complemented
174C4⋊C8—31no—
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#97C2×D311yescharacteristic, complemented
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×Q8C2×D412yes—
193C3×D4—21no—
194C3×D4C2×D412yes—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—81no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—42no—
203#67—42no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—22no—
211D12—21no—
212D12—21no—
213D12—81no—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—21no—
217C4×D3—21no—
218C4×D3—42no—
219C4×D3—42no—
220Dic6—21no—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—42no—
227Dic6—42no—
228C3×C8—21no—
229C3×C8C2×D412yes—
230C3⋊C8—21no—
231C3⋊C8—21no—
232C24—61no—
233#41—62no—
234#41—121no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—62no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8—121no—
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—62no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—121no—
267SD16D1214yescomplemented
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—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4C22×D311yescharacteristic
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—21no—
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—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—61no—
329Q8—61no—
330Q8—61no—
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—122no—
335Q8—122no—
336Q8C2×D1212yes—
337D4—21no—
338D4—62no—
339D4—61no—
340D4—61no—
341D4—122no—
342D4—122no—
343D4—241no—
344D4C2×D1212yes—
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—61no—
366C8—61no—
367C8C2×D1212yes—
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—22no—
392C4—41no—
393C4—41no—
394C4—61no—
395C4—61no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4C22×D1211yescharacteristic
400C4#100911yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—42no—
403C2—62no—
404C2—121no—
405C2—241no—
406C2D4×D1211yescharacteristic
407C2#577511yescharacteristic
408C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radical#5554C2
hypercenter#440D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
2. C2×D3×SD16C2semidirect
3. Q8×D12C2semidirect
5. C3×#440C2semidirect
6. #5028C2semidirect
7. #5019C2semidirect
8. #4887C2semidirect
9. #4886C2semidirect
10. #4809C2semidirect
11. #4806C2semidirect
12. #4783C2semidirect
13. #4776C2semidirect
14. #4715C2semidirect
15. #4706C2semidirect
18. C2×D3×Q8C22semidirect
22. C3×C4×Q8C22semidirect
24. C2×#216C22semidirect
32. C2×Dic12C22semidirect
33. C2×Dic12C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
42. #897C22semidirect
50. #869C22semidirect
51. C3×#107C22semidirect
52. C3×#103C22semidirect
54. C3×#97C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #810C22semidirect
58. #810C22semidirect
59. #810C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
75. #440D3semidirect
104. C3×SD16D4semidirect
168. C4×Q8C2×D3semidirect
173. #107C2×D3semidirect
178. #103C2×D3semidirect
183. #97C2×D3semidirect
267. SD16D12semidirect
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
Q8×D122
#55542
C3×#4402
#50282
#50192
#48872
#48862
#48092
#48062
#47832
#47762
#47152
#47062
#32663