Finite Group Databasebeta

← D3×#454 #80283

Subgroups — D3×#454 #80283

Order 384.

728 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 135 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
1D3×2−1+4C211yesmaximal, characteristic, complemented
2D3×AGL(1,ℤ/8)C212yesmaximal, complemented
3D3×#124C211yesmaximal, characteristic, complemented
4C3×#454C211yesmaximal, characteristic, complemented
5#5245C211yesmaximal, characteristic, complemented
6#5201C212yesmaximal, complemented
7#5062C211yesmaximal, characteristic, complemented
8#4827C212yesmaximal, complemented
9D3×#104C212yesmaximal, complemented
10#4753C211yesmaximal, characteristic, complemented
11D3×#101C211yesmaximal, characteristic, complemented
12C2×#454—31nomaximal, Sylow
13C3×2−1+4C2211yescharacteristic, complemented
14#1010—22no—
15#1010—22no—
16#1010C2212yescomplemented
17D3×#41—22no—
18D3×#41—22no—
19D3×#41C2212yes—
20#1007—22no—
21#1007C2211yescharacteristic, complemented
22C2×D3×Q8—22no—
23C2×D3×Q8C2211yescharacteristic
24C2×D3×D4C2211yescharacteristic
25C3×AGL(1,ℤ/8)C2212yescomplemented
26C3×#124C2211yescharacteristic, complemented
27#949C2212yescomplemented
28#947C2211yescharacteristic, complemented
29#936C2211yescharacteristic, complemented
30#932C2212yescomplemented
31#914—22no—
32D3×SD16—22no—
33#911—22no—
34D3×D8—22no—
35#908C2212yescomplemented
36D3×M16C2212yes—
37#885—22no—
38D3×C22⋊C4—22no—
39#875C2211yescharacteristic, complemented
40C42×D3C2211yescharacteristic
41C3×#104C2212yescomplemented
42C3×#101C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#836C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#824C2212yescomplemented
47#805C2212yescomplemented
48C2×2−1+4—31no—
49C2×AGL(1,ℤ/8)—32no—
50C2×#124—31no—
51#454—31no—
52#454—32no—
53#454—31no—
54#454—32no—
55#454—31no—
56#454D311yescharacteristic, direct product factor
57C2×#104—32no—
58C2×#101—31no—
59C23×D3—21no—
60C3×#41—22no—
61C3×#41—22no—
62C3×#41C2312yes—
63C2×C3×Q8—22no—
64C2×C3×Q8C2311yescharacteristic
65C2×C3×D4C2311yescharacteristic
66C2×#67—21no—
67#239—22no—
68#239—22no—
69#239D412yes—
70D3×Q8—22no—
71D3×Q8—24no—
72D3×Q8—22no—
73D3×Q8—22no—
74D3×Q8—22no—
75D3×Q8—22no—
76D3×Q8D412yes—
77#237—22no—
78#237—24no—
79#237—22no—
80#237—24no—
81#237—22no—
82#237D412yes—
83D3×D4—22no—
84D3×D4—22no—
85D3×D4—22no—
86D3×D4—22no—
87D3×D4D412yes—
88#235—24no—
89#235—24no—
90#235—22no—
91#235—22no—
92#235—22no—
93#235C2312yes—
94C2×D12C2311yescharacteristic
95C2×C4×D3—24no—
96C2×C4×D3—22no—
97C2×C4×D3—21no—
98C2×C4×D3D412yes—
99C2×C4×D3C2311yescharacteristic
100C2×Dic6—24no—
101C2×Dic6—22no—
102C2×Dic6D412yes—
103C2×Dic6C2311yescharacteristic
104C3×SD16—22no—
105C3×D8—22no—
106C3×M16C2312yes—
107C3×C22⋊C4—22no—
108C3×C42C2311yescharacteristic
109#217—22no—
110#215—22no—
111#214—22no—
112#213—22no—
113#213—22no—
114(C2×D3)⋊C4—22no—
115(C2×D3)⋊C4—22no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118#208C2312yes—
119D24—22no—
120#204—22no—
121ΓL(1,𝔽25)—22no—
122D3×C8—22no—
1232−1+4—32no—
1242−1+4—31no—
1252−1+4—62no—
1262−1+4—62no—
1272−1+4—62no—
1282−1+4C2×D311yescharacteristic, complemented
129C2×#41—32no—
130C2×#41—62no—
131C2×#41—62no—
132C22×Q8—31no—
133C22×Q8—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#124—31no—
145#124—31no—
146#124—31no—
147#124—62no—
148#124C2×D311yescharacteristic, complemented
149C2×C22⋊C4—62no—
150C2×C42—31no—
151#104—32no—
152#104—32no—
153#104—32no—
154#104C2×D312yescomplemented
155#101—31no—
156#101—32no—
157#101C2×D311yescharacteristic, complemented
158C23×C3—21no—
159C22×D3—22no—
160C22×D3—21no—
161C22×D3—44no—
162C22×D3—41no—
163C22×D3—41no—
164C22×D3C2×D411yescharacteristic
165C3×Q8—22no—
166C3×Q8—22no—
167C3×Q8C2×D412yes—
168C3×D4—22no—
169C3×D4—22no—
170C3×D4—22no—
171C3×D4C2×D412yes—
172C2×C3×C4—24no—
173C2×C3×C4—22no—
174C2×C3×C4—21no—
175C2×C3×C4C2×D412yes—
176C2×C3×C4C2411yescharacteristic
177#67—24no—
178#67—22no—
179#67—22no—
180#67—22no—
181#67—42no—
182C2×Dic3—24no—
183C2×Dic3—22no—
184C2×Dic3—21no—
185C2×Dic3C2×D412yes—
186C2×Dic3C2×D411yescharacteristic
187D12—22no—
188D12—22no—
189D12—22no—
190D12C2×D412yes—
191C4×D3—24no—
192C4×D3—24no—
193C4×D3—22no—
194C4×D3—22no—
195C4×D3—22no—
196C4×D3—22no—
197C4×D3—22no—
198C4×D3—42no—
199C4×D3C2×D412yes—
200C4×D3C2×D412yes—
201Dic6—24no—
202Dic6—22no—
203Dic6—22no—
204Dic6—22no—
205Dic6—24no—
206Dic6C2×D412yes—
207C3×C8—22no—
208C3⋊C8—22no—
209C24—61no—
210#41—22no—
211#41—22no—
212#41—32no—
213#41—32no—
214#41—32no—
215#41—62no—
216#41—62no—
217#41—64no—
218#41—64no—
219#41—62no—
220#41—64no—
221#41—62no—
222#41—64no—
223#41—62no—
224#41—62no—
225#41—62no—
226#41—62no—
227#41C22×D312yes—
228C2×Q8—22no—
229C2×Q8—32no—
230C2×Q8—31no—
231C2×Q8—32no—
232C2×Q8—62no—
233C2×Q8—64no—
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—64no—
239C2×Q8—62no—
240C2×Q8C22×D311yescharacteristic
241C2×D4—32no—
242C2×D4—31no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—61no—
246C2×D4—62no—
247C2×D4—62no—
248C2×D4C22×D311yescharacteristic
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—62no—
252C22×C4—64no—
253C22×C4—61no—
254SD16—22no—
255SD16—62no—
256SD16—62no—
257SD16—62no—
258D8—22no—
259D8—62no—
260D8—62no—
261D8—62no—
262M16—32no—
263M16—62no—
264M16C22×D312yes—
265C2×C8—62no—
266C22⋊C4—22no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C42—31no—
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—22no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—44no—
280C2×D3—42no—
281C2×D3—41no—
282C2×D3—82no—
283C2×D3—82no—
284C2×D3#12011yescharacteristic
285C2×D3#12011yescharacteristic
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—41no—
289C3×C4C22×D412yes—
290Dic3—22no—
291Dic3—22no—
292Dic3—41no—
293Dic3#12012yes—
294C23—21no—
295C23—31no—
296C23—62no—
297C23—61no—
298C23—121no—
299C23—121no—
300C23—124no—
301Q8—22no—
302Q8—22no—
303Q8—32no—
304Q8—62no—
305Q8—64no—
306Q8—64no—
307Q8—62no—
308Q8—62no—
309Q8D3×D412yes—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—32no—
314D4—62no—
315D4—62no—
316D4—62no—
317D4—62no—
318D4—64no—
319D4—62no—
320D4—62no—
321D4—122no—
322D4D3×D412yes—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—31no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4—62no—
337C2×C4—64no—
338C2×C4—62no—
339C2×C4—61no—
340C2×C4—122no—
341C2×C4D3×D412yes—
342C2×C4C23×D311yescharacteristic
343C8—22no—
344C8—62no—
345C2×C3—21no—
346C2×C3—42no—
347C2×C3—81no—
348C2×C3C2×#12011yescharacteristic
349D3—21no—
350D3—42no—
351D3—81no—
352D3#45412yesdirect product factor
353C22—22no—
354C22—31no—
355C22—31no—
356C22—42no—
357C22—41no—
358C22—62no—
359C22—62no—
360C22—122no—
361C22—124no—
362C22—121no—
363C22—242no—
364C22—242no—
365C22C2×D3×D411yescharacteristic
366C4—22no—
367C4—22no—
368C4—32no—
369C4—41no—
370C4—62no—
371C4—62no—
372C4—121no—
373C4C2×D3×D412yes—
374C3C2×#45411yescharacteristic, complemented, Sylow
375C2—21no—
376C2—32no—
377C2—42no—
378C2—61no—
379C2—81no—
380C2—122no—
381C2—241no—
382C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#454C2
socleC2×C3C2×#120
hypercenter#454D3
hypocenterC3C2×#454
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. D3×2−1+4C2semidirect
2. D3×AGL(1,ℤ/8)C2semidirect
3. D3×#124C2semidirect
4. C3×#454C2semidirect
5. #5245C2semidirect
6. #5201C2semidirect
7. #5062C2semidirect
8. #4827C2semidirect
9. D3×#104C2semidirect
10. #4753C2semidirect
11. D3×#101C2semidirect
13. C3×2−1+4C22semidirect
16. #1010C22semidirect
21. #1007C22semidirect
25. C3×AGL(1,ℤ/8)C22semidirect
26. C3×#124C22semidirect
27. #949C22semidirect
28. #947C22semidirect
29. #936C22semidirect
30. #932C22semidirect
35. #908C22semidirect
39. #875C22semidirect
41. C3×#104C22semidirect
42. C3×#101C22semidirect
43. #837C22semidirect
44. #836C22semidirect
45. #825C22semidirect
46. #824C22semidirect
47. #805C22semidirect
56. #454D3direct (N × H)
128. 2−1+4C2×D3semidirect
140. AGL(1,ℤ/8)C2×D3semidirect
148. #124C2×D3semidirect
154. #104C2×D3semidirect
157. #101C2×D3semidirect
352. D3#454direct (N × H)
374. C3C2×#454semidirect

Maximal subgroups

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

subgroupindex
D3×2−1+42
D3×AGL(1,ℤ/8)2
D3×#1242
C3×#4542
#52452
#52012
#50622
#48272
D3×#1042
#47532
D3×#1012
C2×#4543