Finite Group Databasebeta

← #80286

Subgroups — #80286

Order 384.

736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5968C211yesmaximal, characteristic, complemented
2D3×AGL(1,ℤ/8)C212yesmaximal, complemented
3#5674C211yesmaximal, characteristic, complemented
4C3×#454C211yesmaximal, characteristic, complemented
5#5245C211yesmaximal, characteristic, complemented
6#5198C212yesmaximal, complemented
7#5062C211yesmaximal, characteristic, complemented
8#4822C212yesmaximal, complemented
9#4821C212yesmaximal, complemented
10#4753C211yesmaximal, characteristic, complemented
11#4751C211yesmaximal, characteristic, complemented
12#3021—31nomaximal, Sylow
13C3×2−1+4C2211yescharacteristic, complemented
14#1009—22no—
15#1009—22no—
16#1009C2212yescomplemented
17D3×#41—22no—
18D3×#41—22no—
19D3×#41C2212yes—
20#1007—22no—
21#1007C2211yescharacteristic, complemented
22C2×#239—22no—
23C2×#239C2211yescharacteristic
24C2×D3×D4C2211yescharacteristic, complemented
25C3×AGL(1,ℤ/8)C2212yescomplemented
26C3×#124C2211yescharacteristic, complemented
27#949C2212yescomplemented
28#947C2211yescharacteristic, complemented
29#940C2211yescharacteristic, complemented
30#932C2212yescomplemented
31#914—22no—
32D3×SD16—22no—
33#911—22no—
34D3×D8—22no—
35#908C2212yescomplemented
36D3×M16C2212yescomplemented
37#884—22no—
38#882—22no—
39#874C2211yescharacteristic, complemented
40#872C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#101C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#836C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#823C2212yescomplemented
47#805C2212yescomplemented
48#585—31no—
49C2×AGL(1,ℤ/8)—32no—
50#535—31no—
51#454—31no—
52#454—31no—
53#454—31no—
54#454D311yescharacteristic, complemented
55#453—32no—
56#453—32no—
57#421—32no—
58#413—31no—
59C23×D3—21no—
60C3×#41—22no—
61C3×#41—22no—
62C3×#41C2312yes—
63C2×C3×Q8—22no—
64C2×C3×Q8C2311yescharacteristic
65C2×C3×D4C2311yescharacteristic
66C2×#67—22no—
67C2×#67—21no—
68#239—22no—
69#239—24no—
70#239—22no—
71#239—22no—
72#239—22no—
73#239—22no—
74#239D412yescomplemented
75D3×Q8—22no—
76D3×Q8—22no—
77D3×Q8D412yescomplemented
78#237—22no—
79#237—22no—
80#237D412yescomplemented
81D3×D4—24no—
82D3×D4—22no—
83D3×D4—24no—
84D3×D4—22no—
85D3×D4—22no—
86D3×D4—22no—
87D3×D4—22no—
88D3×D4D412yescomplemented
89#235—24no—
90#235—24no—
91#235—22no—
92#235—22no—
93#235—22no—
94#235C2312yes—
95C2×D12—24no—
96C2×D12—22no—
97C2×D12—22no—
98C2×D12D412yes—
99C2×D12C2311yescharacteristic
100C2×D12C2311yescharacteristic
101C2×C4×D3—24no—
102C2×C4×D3—22no—
103C2×C4×D3D412yes—
104C2×C4×D3C2311yescharacteristic
105C3×SD16—22no—
106C3×D8—22no—
107C3×M16C2312yes—
108C3×C22⋊C4—22no—
109C3×C42C2311yescharacteristic
110#215—22no—
111#214—22no—
112#213—22no—
113#213—22no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—22no—
116Dic3⋊C4—21no—
117C4×Dic3C2311yescharacteristic
118#208C2312yes—
119D24—22no—
120#204—22no—
121ΓL(1,𝔽25)—22no—
122D3×C8—22no—
1232−1+4—31no—
1242−1+4—62no—
1252−1+4C2×D311yescharacteristic, complemented
126H(5,𝔽2)—32no—
127H(5,𝔽2)—62no—
128H(5,𝔽2)—62no—
129C2×#41—31no—
130C2×#41—32no—
131C2×#41—62no—
132C2×#41—62no—
133C2×#41—62no—
134C22×D4—31no—
135AGL(1,ℤ/8)—32no—
136AGL(1,ℤ/8)—32no—
137AGL(1,ℤ/8)—32no—
138AGL(1,ℤ/8)—62no—
139AGL(1,ℤ/8)—62no—
140AGL(1,ℤ/8)C2×D312yescomplemented
141C2×SD16—62no—
142C2×D8—62no—
143C2×M16—32no—
144#127—31no—
145#127—31no—
146#124—31no—
147#124C2×D311yescharacteristic, complemented
148#121—62no—
149#120—62no—
150#117—31no—
151#104—32no—
152#104—32no—
153#104—32no—
154#104C2×D312yescomplemented
155#101—31no—
156#101C2×D311yescharacteristic, complemented
157#100—32no—
158C23×C3—21no—
159C22×D3—24no—
160C22×D3—22no—
161C22×D3—21no—
162C22×D3—42no—
163C22×D3—82no—
164C22×D3C2×D412yes—
165C22×D3C2×D411yescharacteristic
166C3×Q8—22no—
167C3×Q8—22no—
168C3×Q8C2×D412yes—
169C3×D4—22no—
170C3×D4—22no—
171C3×D4—22no—
172C3×D4C2×D412yes—
173C2×C3×C4—24no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4C2×D412yes—
177C2×C3×C4C2411yescharacteristic
178#67—24no—
179#67—22no—
180#67—22no—
181#67—22no—
182#67—41no—
183#67—41no—
184#67—82no—
185C2×Dic3—22no—
186C2×Dic3—21no—
187C2×Dic3C2×D411yescharacteristic
188D12—24no—
189D12—22no—
190D12—22no—
191D12—24no—
192D12—22no—
193D12—22no—
194D12—82no—
195D12C2×D412yes—
196C4×D3—24no—
197C4×D3—24no—
198C4×D3—22no—
199C4×D3—22no—
200C4×D3—22no—
201C4×D3—22no—
202C4×D3—22no—
203C4×D3C2×D412yes—
204C4×D3C2×D412yes—
205Dic6—22no—
206Dic6—22no—
207Dic6C2×D412yes—
208C3×C8—22no—
209C3⋊C8—22no—
210C24—61no—
211#41—22no—
212#41—22no—
213#41—32no—
214#41—32no—
215#41—32no—
216#41—62no—
217#41—62no—
218#41—64no—
219#41—64no—
220#41—62no—
221#41—62no—
222#41—62no—
223#41—62no—
224#41—62no—
225#41—62no—
226#41—64no—
227#41—62no—
228#41—62no—
229#41C22×D312yes—
230C2×Q8—22no—
231C2×Q8—32no—
232C2×Q8—62no—
233C2×Q8—62no—
234C2×Q8C22×D311yescharacteristic
235C2×D4—32no—
236C2×D4—32no—
237C2×D4—31no—
238C2×D4—31no—
239C2×D4—62no—
240C2×D4—64no—
241C2×D4—62no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—61no—
245C2×D4—62no—
246C2×D4—64no—
247C2×D4—62no—
248C2×D4—64no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4C22×D311yescharacteristic
252C22×C4—31no—
253C22×C4—32no—
254C22×C4—62no—
255C22×C4—64no—
256SD16—22no—
257SD16—62no—
258SD16—62no—
259SD16—62no—
260D8—22no—
261D8—62no—
262D8—62no—
263D8—62no—
264M16—32no—
265M16—62no—
266M16C22×D312yes—
267C2×C8—62no—
268C4⋊C4—61no—
269C22⋊C4—22no—
270C22⋊C4—61no—
271C22⋊C4—62no—
272C42—31no—
273C42C22×D311yescharacteristic
274C22×C3—22no—
275C22×C3—42no—
276C22×C3—41no—
277C22×C3C22×D411yescharacteristic
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—41no—
281C2×D3—42no—
282C2×D3—42no—
283C2×D3—41no—
284C2×D3—82no—
285C2×D3—84no—
286C2×D3—84no—
287C2×D3#12012yes—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—41no—
291C3×C4C22×D412yes—
292Dic3—22no—
293Dic3—41no—
294Dic3#12011yescharacteristic
295Dic3#12011yescharacteristic
296C23—21no—
297C23—31no—
298C23—32no—
299C23—62no—
300C23—64no—
301C23—61no—
302C23—122no—
303C23—242no—
304Q8—22no—
305Q8—22no—
306Q8—32no—
307Q8—62no—
308Q8—62no—
309Q8D3×D412yes—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—32no—
314D4—64no—
315D4—62no—
316D4—64no—
317D4—62no—
318D4—62no—
319D4—62no—
320D4—64no—
321D4—62no—
322D4—62no—
323D4—62no—
324D4—121no—
325D4—121no—
326D4—242no—
327D4—242no—
328D4D3×D412yes—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—64no—
340C2×C4—64no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4D3×D412yes—
345C2×C4C23×D311yescharacteristic
346C8—22no—
347C8—62no—
348C2×C3—21no—
349C2×C3—42no—
350C2×C3—81no—
351C2×C3C2×#12011yescharacteristic
352D3—22no—
353D3—42no—
354D3—42no—
355D3—81no—
356C22—22no—
357C22—32no—
358C22—42no—
359C22—41no—
360C22—62no—
361C22—62no—
362C22—121no—
363C22—122no—
364C22—122no—
365C22—121no—
366C22—242no—
367C22—244no—
368C22—244no—
369C22C2×D3×D411yescharacteristic
370C4—22no—
371C4—22no—
372C4—31no—
373C4—31no—
374C4—41no—
375C4—62no—
376C4—121no—
377C4C2×D3×D412yes—
378C3#302111yescharacteristic, complemented, Sylow
379C2—21no—
380C2—42no—
381C2—62no—
382C2—81no—
383C2—122no—
384C2—122no—
385C2—241no—
386C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#454C2
socleC2×C3C2×#120
hypercenter#454D3
hypocenterC3#3021
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #5968C2semidirect
2. D3×AGL(1,ℤ/8)C2semidirect
3. #5674C2semidirect
4. C3×#454C2semidirect
5. #5245C2semidirect
6. #5198C2semidirect
7. #5062C2semidirect
8. #4822C2semidirect
9. #4821C2semidirect
10. #4753C2semidirect
11. #4751C2semidirect
13. C3×2−1+4C22semidirect
16. #1009C22semidirect
21. #1007C22semidirect
24. C2×D3×D4C22semidirect
25. C3×AGL(1,ℤ/8)C22semidirect
26. C3×#124C22semidirect
27. #949C22semidirect
28. #947C22semidirect
29. #940C22semidirect
30. #932C22semidirect
35. #908C22semidirect
36. D3×M16C22semidirect
39. #874C22semidirect
40. #872C22semidirect
41. C3×#104C22semidirect
42. C3×#101C22semidirect
43. #837C22semidirect
44. #836C22semidirect
45. #825C22semidirect
46. #823C22semidirect
47. #805C22semidirect
54. #454D3semidirect
74. #239D4semidirect
77. D3×Q8D4semidirect
80. #237D4semidirect
88. D3×D4D4semidirect
125. 2−1+4C2×D3semidirect
140. AGL(1,ℤ/8)C2×D3semidirect
147. #124C2×D3semidirect
154. #104C2×D3semidirect
156. #101C2×D3semidirect
378. C3#3021semidirect

Maximal subgroups

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

subgroupindex
#59682
D3×AGL(1,ℤ/8)2
#56742
C3×#4542
#52452
#51982
#50622
#48222
#48212
#47532
#47512
#30213