Finite Group Databasebeta

← #79587

Subgroups — #79587

Order 384.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic
2#5546C211yesmaximal, characteristic, complemented
3#5545C211yesmaximal, characteristic, complemented
4C3×C4×D8C211yesmaximal, characteristic, complemented
5#5152C211yesmaximal, characteristic, complemented
6#5018C211yesmaximal, characteristic, complemented
7#5014C211yesmaximal, characteristic, complemented
8#4889C211yesmaximal, characteristic, complemented
9#4881C211yesmaximal, characteristic, complemented
10#4772C211yesmaximal, characteristic, complemented
11D3×#102C211yesmaximal, characteristic, complemented
12#4758C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14C4×#204C211yesmaximal, characteristic, complemented
15#4689C211yesmaximal, characteristic, complemented
16#2946—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic, complemented
22D4×Dic3C2211yescharacteristic
23D4×Dic3C2211yescharacteristic, complemented
24C2×#214C2211yescharacteristic
25C2×#213C2211yescharacteristic
26C4×#67—21no—
27C4×#67—21no—
28C2×(C2×D3)⋊C4—21no—
29C2×Dic3⋊C4—21no—
30#911C418yescomplemented
31C2×#204C2211yescharacteristic
32C2×ΓL(1,𝔽25)C2211yescharacteristic
33C2×D3×C8C2211yescharacteristic
34#893C2211yescharacteristic
35#892C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic
37#887C2211yescharacteristic, complemented
38#67⋊C4—21no—
39#67⋊C4—21no—
40D3×C22⋊C4—21no—
41#877—21no—
42C4×D12C2211yescharacteristic
43#872C2211yescharacteristic
44C4×Dic6C2211yescharacteristic, complemented
45C3×#107C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic, complemented
48C3×C4×C8C2211yescharacteristic, complemented
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52#817C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#803C2211yescharacteristic, complemented
61C2×AGL(1,ℤ/8)—31no—
62#520—31no—
63#518—31no—
64#442—31no—
65#442—31no—
66#440—31no—
67#440—31no—
68C4×SD16—31no—
69C4×SD16—31no—
70C4×D8—31no—
71C4×D8D311yescharacteristic, complemented
72#428—31no—
73#416—31no—
74C2×#102—31no—
75#405—31no—
76C23×D3—21no—
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86#237—22no—
87#237C2×C414yescomplemented
88D3×D4—22no—
89D3×D4C2×C414yescomplemented
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×Dic6C2311yescharacteristic
96C3×D8C2×C414yescomplemented
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C42C2311yescharacteristic
104#217—21no—
105#217—21no—
106#214C2×C414yescomplemented
107#213C2×C414yescomplemented
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4D411yescharacteristic
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204C2×C414yescomplemented
127ΓL(1,𝔽25)C2×C414yescomplemented
128D3×C8—22no—
129C2×#41—31no—
130C22×D4—31no—
131AGL(1,ℤ/8)—38no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×D8—31no—
135C2×D8C2×D311yescharacteristic
136C2×M16—31no—
137C22×C8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—31no—
144C4×D4—61no—
145C4×D4—61no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4C2×D311yescharacteristic
149C4×D4C2×D311yescharacteristic, complemented
150#117—31no—
151#117—31no—
152#117—61no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic, complemented
169#97—31no—
170#97—31no—
171C4×C8—31no—
172C4×C8C2×D311yescharacteristic, complemented
173C23×C3—21no—
174C23×C3—21no—
175C22×D3—21no—
176C22×D3—24no—
177C22×D3—81no—
178C22×D3—81no—
179C22×D3C2×D411yescharacteristic
180C3×D4—21no—
181C3×D4—21no—
182C3×D4C22×C412yes—
183C3×D4C22×C412yes—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—24no—
192#67—24no—
193C2×Dic3—21no—
194C2×Dic3—24no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202D12—21no—
203D12C22×C412yes—
204C4×D3—81no—
205C4×D3—81no—
206C4×D3—81no—
207C4×D3C22×C412yes—
208C4×D3#4112yes—
209Dic6—21no—
210Dic6C22×C412yes—
211C3×C8—21no—
212C3×C8C22×C412yes—
213C3⋊C8—21no—
214C3⋊C8C22×C412yes—
215C24—61no—
216#41—34no—
217#41—62no—
218C2×Q8—31no—
219C2×D4—34no—
220C2×D4—31no—
221C2×D4—62no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4C22×D311yescharacteristic
225C2×D4C22×D311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235SD16—34no—
236SD16—34no—
237D8—34no—
238D8C4×D314yescomplemented
239M16—34no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8—62no—
243C2×C8C22×D311yescharacteristic
244C2×C8C22×D311yescharacteristic
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4C22×D311yescharacteristic
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—31no—
257C22⋊C4—31no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C42—31no—
264C42—61no—
265C42—61no—
266C42—61no—
267C42C22×D311yescharacteristic
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C22×D411yescharacteristic
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—22no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—84no—
281C2×D3C4×D412yes—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4C23×C411yescharacteristic
287C3×C4C2×#4111yescharacteristic
288Dic3—22no—
289Dic3—21no—
290Dic3—21no—
291Dic3—41no—
292Dic3—41no—
293Dic3C4×D412yes—
294C23—21no—
295C23—21no—
296C23—31no—
297C23—64no—
298C23—61no—
299C23—241no—
300C23—241no—
301Q8—32no—
302Q8—61no—
303D4—21no—
304D4—21no—
305D4—32no—
306D4—64no—
307D4—64no—
308D4—61no—
309D4C2×C4×D312yes—
310D4C2×C4×D312yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—32no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—64no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—241no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4D3×D411yescharacteristic
330C2×C4D3×D411yescharacteristic
331C2×C4C23×D311yescharacteristic
332C8—21no—
333C8—32no—
334C8—61no—
335C8C2×C4×D312yes—
336C2×C3—42no—
337C2×C3—42no—
338C2×C3C2×C4×D411yescharacteristic
339C2×C3#57611yescharacteristic
340C2×C3#57711yescharacteristic
341D3—22no—
342D3—42no—
343C22—22no—
344C22—22no—
345C22—32no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—61no—
351C22—61no—
352C22—62no—
353C22—121no—
354C22—121no—
355C22—244no—
356C22C2×D3×D411yescharacteristic
357C4—21no—
358C4—21no—
359C4—32no—
360C4—41no—
361C4—41no—
362C4—61no—
363C4—61no—
364C4—62no—
365C4—121no—
366C4—121no—
367C4C22×C4×D311yescharacteristic
368C4D3×#4111yescharacteristic
369C3#294611yescharacteristic, complemented, Sylow
370C2—42no—
371C2—42no—
372C2—62no—
373C2—122no—
374C2C4×D3×D411yescharacteristic
375C2#575711yescharacteristic
376C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
prime radicalC2×#911C2
cosocleC2×C3×C4C24
hypercenterC4×D8D3
hypocenterC3#2946
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
2. #5546C2semidirect
3. #5545C2semidirect
4. C3×C4×D8C2semidirect
5. #5152C2semidirect
6. #5018C2semidirect
7. #5014C2semidirect
8. #4889C2semidirect
9. #4881C2semidirect
10. #4772C2semidirect
11. D3×#102C2semidirect
12. #4758C2semidirect
13. #4756C2semidirect
14. C4×#204C2semidirect
15. #4689C2semidirect
21. C3×C4×D4C22semidirect
23. D4×Dic3C22semidirect
30. #911C4semidirect
37. #887C22semidirect
44. C4×Dic6C22semidirect
45. C3×#107C22semidirect
47. C3×#102C22semidirect
48. C3×C4×C8C22semidirect
50. #832C22semidirect
52. #817C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. C8×Dic3C22semidirect
56. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
60. #803C22semidirect
71. C4×D8D3semidirect
87. #237C2×C4semidirect
89. D3×D4C2×C4semidirect
96. C3×D8C2×C4semidirect
106. #214C2×C4semidirect
107. #213C2×C4semidirect
126. #204C2×C4semidirect
127. ΓL(1,𝔽25)C2×C4semidirect
149. C4×D4C2×D3semidirect
158. #107C2×D3semidirect
168. #102C2×D3semidirect
172. C4×C8C2×D3semidirect
238. D8C4×D3semidirect
369. C3#2946semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
#55462
#55452
C3×C4×D82
#51522
#50182
#50142
#48892
#48812
#47722
D3×#1022
#47582
#47562
C4×#2042
#46892
#29463