Finite Group Databasebeta

← D3×#456 #80297

Subgroups — D3×#456 #80297

Order 384.

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

normal subgroups only

This group has 708 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
2D3×#137C212yesmaximal, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4C3×#456C211yesmaximal, characteristic, complemented
5#5246C211yesmaximal, characteristic, complemented
6#5205C212yesmaximal, complemented
7#5095C211yesmaximal, characteristic, complemented
8#4826C212yesmaximal, complemented
9D3×#104C212yesmaximal, complemented
10#4752C211yesmaximal, characteristic, complemented
11D3×#101C211yesmaximal, characteristic, complemented
12C2×#456—31nomaximal, Sylow
13C3×2−1+4C2211yescharacteristic
14#1010—22no—
15#1010—22no—
16#1010C2212yes—
17D3×#41—22no—
18D3×#41—22no—
19D3×#41C2212yes—
20#1007—22no—
21#1007C2211yescharacteristic
22C2×D3×Q8—22no—
23C2×D3×Q8C2211yescharacteristic
24C2×D3×Q8C2211yescharacteristic
25C3×#137C2212yescomplemented
26C3×C4⋊Q8C2211yescharacteristic, complemented
27#951C2212yescomplemented
28#944C2211yescharacteristic, complemented
29#944C2211yescharacteristic, complemented
30#942C2212yescomplemented
31#918—22no—
32D3×Q16—22no—
33#915—22no—
34D3×SD16—22no—
35#909C2212yescomplemented
36D3×M16C2212yes—
37D3×C4⋊C4—22no—
38#888—22no—
39C42×D3C2211yescharacteristic
40#869C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#101C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#836C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#824C2212yescomplemented
47#805C2212yescomplemented
48C2×2−1+4—31no—
49C2×#137—32no—
50C2×C4⋊Q8—31no—
51#456—31no—
52#456—32no—
53#456—31no—
54#456—32no—
55#456—31no—
56#456D311yescharacteristic, direct product factor
57C2×#104—32no—
58C2×#101—31no—
59C3×#41—22no—
60C3×#41—22no—
61C3×#41C2312yes—
62C2×C3×Q8—22no—
63C2×C3×Q8C2311yescharacteristic
64C2×C3×Q8C2311yescharacteristic
65#239—22no—
66#239—22no—
67#239D412yes—
68D3×Q8—22no—
69D3×Q8—24no—
70D3×Q8—22no—
71D3×Q8—22no—
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×D4D412yes—
86#235—24no—
87#235—24no—
88#235—22no—
89#235—22no—
90#235—22no—
91#235C2312yes—
92C2×C4×D3—24no—
93C2×C4×D3—21no—
94C2×C4×D3—22no—
95C2×C4×D3—21no—
96C2×C4×D3D412yes—
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—24no—
99C2×Dic6—22no—
100C2×Dic6—21no—
101C2×Dic6D412yes—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×Q16—22no—
105C3×SD16—22no—
106C3×M16C2312yes—
107C3×C4⋊C4—22no—
108C3×C42C2311yescharacteristic
109#216—22no—
110#216—22no—
111#215—22no—
112#214—22no—
113C4⋊Dic3—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118#208C2312yes—
119Dic12—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
129C2×#41—32no—
130C2×#41—62no—
131C2×#41—62no—
132C22×Q8—31no—
133C22×Q8—31no—
134C22×Q8—62no—
135#137—32no—
136#137—32no—
137#137—32no—
138#137—62no—
139#137—62no—
140#137C2×D312yescomplemented
141C2×Q16—62no—
142C2×SD16—62no—
143C2×M16—32no—
144C4⋊Q8—31no—
145C4⋊Q8—31no—
146C4⋊Q8—31no—
147C4⋊Q8—62no—
148C4⋊Q8C2×D311yescharacteristic, complemented
149C2×C4⋊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
158C22×D3—22no—
159C22×D3C2×D411yescharacteristic
160C3×Q8—22no—
161C3×Q8—22no—
162C3×Q8—22no—
163C3×Q8C2×D412yes—
164C3×D4—22no—
165C3×D4—22no—
166C3×D4C2×D412yes—
167C2×C3×C4—24no—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4C2×D412yes—
172C2×C3×C4C2411yescharacteristic
173#67—24no—
174#67—22no—
175#67—22no—
176#67—22no—
177C2×Dic3—24no—
178C2×Dic3—21no—
179C2×Dic3—22no—
180C2×Dic3—21no—
181C2×Dic3C2×D412yes—
182C2×Dic3C2×D411yescharacteristic
183D12—22no—
184D12—22no—
185D12C2×D412yes—
186C4×D3—24no—
187C4×D3—24no—
188C4×D3—24no—
189C4×D3—22no—
190C4×D3—22no—
191C4×D3—22no—
192C4×D3—22no—
193C4×D3—22no—
194C4×D3—41no—
195C4×D3—41no—
196C4×D3C2×D412yes—
197C4×D3C2×D412yes—
198Dic6—24no—
199Dic6—22no—
200Dic6—22no—
201Dic6—22no—
202Dic6—24no—
203Dic6—22no—
204Dic6—42no—
205Dic6C2×D412yes—
206C3×C8—22no—
207C3⋊C8—22no—
208#41—22no—
209#41—22no—
210#41—32no—
211#41—32no—
212#41—32no—
213#41—62no—
214#41—62no—
215#41—64no—
216#41—64no—
217#41—62no—
218#41—64no—
219#41—62no—
220#41—64no—
221#41—62no—
222#41—62no—
223#41—62no—
224#41—62no—
225#41C22×D312yes—
226C2×Q8—22no—
227C2×Q8—32no—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8—31no—
231C2×Q8—61no—
232C2×Q8—62no—
233C2×Q8—64no—
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—64no—
241C2×Q8—62no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—32no—
245C2×D4—62no—
246C2×D4—62no—
247C22×C4—31no—
248C22×C4—32no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—64no—
252C22×C4—61no—
253Q16—22no—
254Q16—62no—
255Q16—62no—
256Q16—62no—
257SD16—22no—
258SD16—62no—
259SD16—62no—
260SD16—62no—
261M16—32no—
262M16—62no—
263M16C22×D312yes—
264C2×C8—62no—
265C4⋊C4—22no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C42—31no—
270C42—61no—
271C42C22×D311yescharacteristic
272C22×C3—22no—
273C22×C3C22×D411yescharacteristic
274C2×D3—22no—
275C2×D3—22no—
276C2×D3—44no—
277C2×D3#12011yescharacteristic
278C2×D3#12011yescharacteristic
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—41no—
283C3×C4C22×D412yes—
284Dic3—22no—
285Dic3—22no—
286Dic3—22no—
287Dic3—41no—
288Dic3#12012yes—
289C23—31no—
290C23—62no—
291Q8—22no—
292Q8—22no—
293Q8—22no—
294Q8—32no—
295Q8—62no—
296Q8—64no—
297Q8—64no—
298Q8—62no—
299Q8—62no—
300Q8—62no—
301Q8—122no—
302Q8D3×D412yes—
303D4—22no—
304D4—22no—
305D4—32no—
306D4—62no—
307D4—62no—
308D4—62no—
309D4—62no—
310D4—64no—
311D4—62no—
312D4D3×D412yes—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—64no—
326C2×C4—64no—
327C2×C4—64no—
328C2×C4—62no—
329C2×C4—64no—
330C2×C4—61no—
331C2×C4—62no—
332C2×C4—61no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4D3×D412yes—
336C2×C4C23×D311yescharacteristic
337C8—22no—
338C8—62no—
339C2×C3—21no—
340C2×C3—42no—
341C2×C3C2×#12011yescharacteristic
342D3—21no—
343D3—42no—
344D3#45612yesdirect product factor
345C22—22no—
346C22—31no—
347C22—31no—
348C22—62no—
349C22—62no—
350C22—124no—
351C22C2×D3×D411yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—32no—
356C4—41no—
357C4—62no—
358C4—62no—
359C4—62no—
360C4—121no—
361C4C2×D3×D412yes—
362C3C2×#45611yescharacteristic, complemented, Sylow
363C2—21no—
364C2—32no—
365C2—42no—
366C2—61no—
367C2—122no—
368C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#456C2
socleC2×C3C2×#120
prime radicalD3×2−1+4C2
hypercenter#456D3
hypocenterC3C2×#456
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. D3×#137C2semidirect
3. D3×C4⋊Q8C2semidirect
4. C3×#456C2semidirect
5. #5246C2semidirect
6. #5205C2semidirect
7. #5095C2semidirect
8. #4826C2semidirect
9. D3×#104C2semidirect
10. #4752C2semidirect
11. D3×#101C2semidirect
25. C3×#137C22semidirect
26. C3×C4⋊Q8C22semidirect
27. #951C22semidirect
28. #944C22semidirect
29. #944C22semidirect
30. #942C22semidirect
35. #909C22semidirect
40. #869C22semidirect
41. C3×#104C22semidirect
42. C3×#101C22semidirect
43. #837C22semidirect
44. #836C22semidirect
45. #825C22semidirect
46. #824C22semidirect
47. #805C22semidirect
56. #456D3direct (N × H)
140. #137C2×D3semidirect
148. C4⋊Q8C2×D3semidirect
154. #104C2×D3semidirect
157. #101C2×D3semidirect
344. D3#456direct (N × H)
362. C3C2×#456semidirect

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×#1372
D3×C4⋊Q82
C3×#4562
#52462
#52052
#50952
#48262
D3×#1042
#47522
D3×#1012
C2×#4563