Finite Group Databasebeta

← #79495

Subgroups — #79495

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×#909C211yesmaximal, characteristic, complemented
2C22×#204C211yesmaximal, characteristic, complemented
3#5550C211yesmaximal, characteristic
4D4×Dic6C211yesmaximal, characteristic, complemented
5C3×#435C211yesmaximal, characteristic, complemented
6#5134C211yesmaximal, characteristic, complemented
7#5116C211yesmaximal, characteristic, complemented
8#4845C211yesmaximal, characteristic, complemented
9#4841C211yesmaximal, characteristic, complemented
10#4739C211yesmaximal, characteristic, complemented
11#4738C211yesmaximal, characteristic, complemented
12#4733C211yesmaximal, characteristic, complemented
13#4731C211yesmaximal, characteristic, complemented
14#4715C211yesmaximal, characteristic, complemented
15#4710C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic, complemented
22C22×C3×C8C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24#937—21no—
25D4×Dic3—21no—
26#930C2211yescharacteristic
27#924C2211yescharacteristic
28#924C2211yescharacteristic
29#924C2211yescharacteristic, complemented
30#909—24no—
31C2×Dic12C2211yescharacteristic, complemented
32C2×Dic12C2211yescharacteristic, complemented
33C2×Dic12C2211yescharacteristic, complemented
34C2×#204—24no—
35C2×#204C2211yescharacteristic
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×#204C2211yescharacteristic
39C2×#204C2211yescharacteristic
40C2×#204C2211yescharacteristic
41#894—21no—
42#888—21no—
43#884—21no—
44D3×C22⋊C4—21no—
45#878—21no—
46#878—21no—
47#875C2211yescharacteristic
48C4×D12C2211yescharacteristic
49#869C2211yescharacteristic, complemented
50C4×Dic6C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×C22⋊C8C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic, complemented
54C3×#97C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57#818C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#816C2211yescharacteristic, complemented
61#816C2211yescharacteristic, 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#440—31no—
76#435D311yescharacteristic, complemented
77C23×D3—21no—
78C2×C3×D4D411yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85#235—22no—
86#235—41no—
87C2×D12—22no—
88C2×D12—21no—
89C2×D12—41no—
90C2×D12C2311yescharacteristic
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—21no—
97C2×Dic6—21no—
98C2×Dic6—41no—
99C2×Dic6—41no—
100C2×Dic6C2311yescharacteristic
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×M16—22no—
105C2×C3×C8—22no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4D411yescharacteristic
111C3×C22⋊C4D411yescharacteristic
112C3×C22⋊C4D411yescharacteristic
113C3×C42C2311yescharacteristic
114#217—21no—
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—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3C2311yescharacteristic
124C4⋊Dic3C2311yescharacteristic
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129C4×Dic3—21no—
130Dic12—22no—
131Dic12—22no—
132Dic12—41no—
133#204—22no—
134#204—22no—
135#204—22no—
136#204—22no—
137#204—41no—
138#204D414yescomplemented
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—31no—
153C2×SD16—64no—
154C2×M16C2×D311yescharacteristic, complemented
155C22×C8C2×D311yescharacteristic
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×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#103—31no—
178#103—31no—
179#102—31no—
180#102—31no—
181C22⋊C8C2×D311yescharacteristic
182C22⋊C8C2×D311yescharacteristic, complemented
183#97C2×D311yescharacteristic, complemented
184C23×C3C2×D411yescharacteristic
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—81no—
191C3×D4—22no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—42no—
215D12C2×D412yes—
216C4×D3—42no—
217C4×D3—81no—
218Dic6—21no—
219Dic6—21no—
220Dic6—21no—
221Dic6—21no—
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—31no—
238C2×Q8—31no—
239C2×Q8—62no—
240C2×Q8—62no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—62no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4D1211yescharacteristic
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4C22×D311yescharacteristic
258C22×C4C22×D311yescharacteristic
259Q16—62no—
260Q16—62no—
261Q16—121no—
262SD16—34no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268M16—22no—
269C2×C8—22no—
270C2×C8C22×D311yescharacteristic
271C2×C8C22×D311yescharacteristic
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4D1211yescharacteristic
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4D1211yescharacteristic
291C22⋊C4D1211yescharacteristic
292C42—61no—
293C42C22×D311yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3#13712yescomplemented
300C22×C3C22×D411yescharacteristic
301C2×D3—22no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—84no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4C22×D411yescharacteristic
313C3×C4H(5,𝔽2)11yescharacteristic
314Dic3—22no—
315Dic3—41no—
316Dic3—41no—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320C23—61no—
321C23—61no—
322C23—122no—
323C23—122no—
324C23—241no—
325C23C2×D1211yescharacteristic
326C23C2×D1211yescharacteristic
327Q8—32no—
328Q8—61no—
329Q8—61no—
330Q8—61no—
331Q8—61no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8—122no—
337D4—22no—
338D4—32no—
339D4—61no—
340D4—61no—
341D4—61no—
342D4—122no—
343D4—122no—
344D4—241no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—241no—
359C2×C4C2×D1211yescharacteristic
360C2×C4C2×D1211yescharacteristic
361C2×C4C2×D1211yescharacteristic
362C2×C4C2×D1211yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8D3×D412yes—
368C2×C3—22no—
369C2×C3—41no—
370C2×C3D4211yescharacteristic
371C2×C3C2×#13711yescharacteristic
372C2×C3#57711yescharacteristic
373D3—42no—
374D3—81no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—62no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22—244no—
387C22#90912yescomplemented
388C22C22×D1211yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—62no—
394C4—121no—
395C4—121no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4C2×D3×D411yescharacteristic
400C4#100911yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—22no—
403C2—41no—
404C2—122no—
405C2—241no—
406C2D4×D1211yescharacteristic
407C2C2×#90911yescharacteristic
408C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
prime radical#5550C2
hypercenter#435D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
2. C22×#204C2semidirect
4. D4×Dic6C2semidirect
5. C3×#435C2semidirect
6. #5134C2semidirect
7. #5116C2semidirect
8. #4845C2semidirect
9. #4841C2semidirect
10. #4739C2semidirect
11. #4738C2semidirect
12. #4733C2semidirect
13. #4731C2semidirect
14. #4715C2semidirect
15. #4710C2semidirect
20. C22×Dic6C22semidirect
21. C2×C3×M16C22semidirect
29. #924C22semidirect
31. C2×Dic12C22semidirect
32. C2×Dic12C22semidirect
33. C2×Dic12C22semidirect
49. #869C22semidirect
50. C4×Dic6C22semidirect
51. C3×C4⋊C8C22semidirect
53. C3×C22⋊C8C22semidirect
54. C3×#97C22semidirect
57. #818C22semidirect
58. #816C22semidirect
59. #816C22semidirect
60. #816C22semidirect
61. #816C22semidirect
76. #435D3semidirect
138. #204D4semidirect
154. C2×M16C2×D3semidirect
174. C4⋊C8C2×D3semidirect
182. C22⋊C8C2×D3semidirect
183. #97C2×D3semidirect
299. C22×C3#137semidirect
387. C22#909semidirect
401. C3#3266semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
C22×#2042
#55502
D4×Dic62
C3×#4352
#51342
#51162
#48452
#48412
#47392
#47382
#47332
#47312
#47152
#47102
#32663