Finite Group Databasebeta

← #80299

Subgroups — #80299

Order 384.

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

normal subgroups only

This group has 698 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
1D3×2−1+4C211yesmaximal, characteristic, complemented
2#5777C212yesmaximal, complemented
3#5724C211yesmaximal, characteristic, complemented
4C3×#456C211yesmaximal, characteristic, complemented
5#5245C211yesmaximal, characteristic, complemented
6#5205C212yesmaximal, complemented
7#5095C211yesmaximal, characteristic, complemented
8#4827C212yesmaximal, complemented
9#4821C212yesmaximal, complemented
10#4753C211yesmaximal, characteristic, complemented
11D3×#101C211yesmaximal, characteristic, complemented
12#3020—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×#239C2211yescharacteristic
23C2×D3×Q8—22no—
24C2×D3×Q8C2211yescharacteristic
25C3×#137C2212yescomplemented
26C3×C4⋊Q8C2211yescharacteristic, complemented
27#949C2212yescomplemented
28#947C2211yescharacteristic, complemented
29#944C2211yescharacteristic, complemented
30#942C2212yescomplemented
31#919—22no—
32#918—22no—
33#916—22no—
34#914—22no—
35#908C2212yescomplemented
36D3×M16C2212yes—
37#896—22no—
38#894—22no—
39#875C2211yescharacteristic, complemented
40#872C2211yescharacteristic
41C3×#104C2212yescomplemented
42C3×#101C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#836C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#824C2212yescomplemented
47#805C2212yescomplemented
48C2×2−1+4—31no—
49#575—32no—
50#536—31no—
51#456—32no—
52#456—31no—
53#456D311yescharacteristic, complemented
54#454—31no—
55#454—32no—
56#454—31no—
57#421—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#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×D4D412yes—
86#235—24no—
87#235—24no—
88#235—22no—
89#235—22no—
90#235—22no—
91#235C2312yes—
92C2×D12—21no—
93C2×D12C2311yescharacteristic
94C2×C4×D3—24no—
95C2×C4×D3—22no—
96C2×C4×D3—21no—
97C2×C4×D3D412yes—
98C2×C4×D3C2311yescharacteristic
99C2×Dic6—24no—
100C2×Dic6—22no—
101C2×Dic6D412yes—
102C2×Dic6C2311yescharacteristic
103C3×Q16—22no—
104C3×SD16—22no—
105C3×M16C2312yes—
106C3×C4⋊C4—22no—
107C3×C42C2311yescharacteristic
108#216—22no—
109#215—22no—
110#215—22no—
111#213—22no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—22no—
115Dic3⋊C4—21no—
116Dic3⋊C4—22no—
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
129C2×#41—32no—
130C2×#41—31no—
131C2×#41—62no—
132C2×#41—62no—
133C22×Q8—31no—
134C22×Q8—62no—
135#137—32no—
136#137—62no—
137#137C2×D312yescomplemented
138AGL(1,ℤ/8)—32no—
139AGL(1,ℤ/8)—32no—
140AGL(1,ℤ/8)—62no—
141#135—62no—
142#135—62no—
143C2×M16—32no—
144C4⋊Q8—31no—
145C4⋊Q8C2×D311yescharacteristic, complemented
146#124—31no—
147#124—31no—
148#123—62no—
149C22⋊Q8—62no—
150#117—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×D3—21no—
160C22×D3C2×D411yescharacteristic
161C3×Q8—22no—
162C3×Q8—22no—
163C3×Q8—22no—
164C3×Q8C2×D412yes—
165C3×D4—22no—
166C3×D4—22no—
167C3×D4C2×D412yes—
168C2×C3×C4—24no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4C2×D412yes—
173C2×C3×C4C2411yescharacteristic
174#67—24no—
175#67—22no—
176#67—22no—
177#67—22no—
178C2×Dic3—24no—
179C2×Dic3—22no—
180C2×Dic3—21no—
181C2×Dic3C2×D412yes—
182C2×Dic3C2×D411yescharacteristic
183D12—22no—
184D12—22no—
185D12—22no—
186D12—41no—
187D12—41no—
188D12C2×D412yes—
189C4×D3—24no—
190C4×D3—24no—
191C4×D3—22no—
192C4×D3—22no—
193C4×D3—22no—
194C4×D3—22no—
195C4×D3—22no—
196C4×D3—42no—
197C4×D3C2×D412yes—
198C4×D3C2×D412yes—
199Dic6—24no—
200Dic6—22no—
201Dic6—22no—
202Dic6—22no—
203Dic6—24no—
204Dic6C2×D412yes—
205C3×C8—22no—
206C3⋊C8—22no—
207#41—22no—
208#41—22no—
209#41—32no—
210#41—32no—
211#41—32no—
212#41—62no—
213#41—62no—
214#41—64no—
215#41—64no—
216#41—62no—
217#41—64no—
218#41—62no—
219#41—62no—
220#41—62no—
221#41—64no—
222#41—62no—
223#41—62no—
224#41—62no—
225#41—62no—
226#41C22×D312yes—
227C2×Q8—22no—
228C2×Q8—32no—
229C2×Q8—31no—
230C2×Q8—32no—
231C2×Q8—62no—
232C2×Q8—64no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—64no—
238C2×Q8—62no—
239C2×Q8C22×D311yescharacteristic
240C2×Q8C22×D311yescharacteristic
241C2×D4—32no—
242C2×D4—31no—
243C2×D4—61no—
244C2×D4—62no—
245C2×D4—62no—
246C22×C4—31no—
247C22×C4—32no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—64no—
251Q16—22no—
252Q16—62no—
253SD16—22no—
254SD16—62no—
255SD16—62no—
256SD16—62no—
257D8—62no—
258D8—62no—
259M16—32no—
260M16—62no—
261M16C22×D312yes—
262C2×C8—62no—
263C4⋊C4—22no—
264C4⋊C4—61no—
265C4⋊C4—62no—
266C22⋊C4—62no—
267C22⋊C4—61no—
268C22⋊C4—62no—
269C42—31no—
270C42C22×D311yescharacteristic
271C22×C3—22no—
272C22×C3C22×D411yescharacteristic
273C2×D3—22no—
274C2×D3—41no—
275C2×D3—44no—
276C2×D3—42no—
277C2×D3—41no—
278C2×D3#12011yescharacteristic
279C2×D3#12011yescharacteristic
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4C22×D412yes—
285Dic3—22no—
286Dic3—22no—
287Dic3—41no—
288Dic3#12012yes—
289C23—31no—
290C23—62no—
291C23—61no—
292Q8—22no—
293Q8—22no—
294Q8—22no—
295Q8—32no—
296Q8—62no—
297Q8—64no—
298Q8—64no—
299Q8—62no—
300Q8—62no—
301Q8D3×D412yes—
302D4—22no—
303D4—22no—
304D4—32no—
305D4—62no—
306D4—62no—
307D4—62no—
308D4—62no—
309D4—64no—
310D4—62no—
311D4—62no—
312D4—121no—
313D4—121no—
314D4D3×D412yes—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—64no—
328C2×C4—64no—
329C2×C4—62no—
330C2×C4—64no—
331C2×C4—62no—
332C2×C4—61no—
333C2×C4—122no—
334C2×C4D3×D412yes—
335C2×C4C23×D311yescharacteristic
336C8—22no—
337C8—62no—
338C2×C3—21no—
339C2×C3—42no—
340C2×C3C2×#12011yescharacteristic
341D3—21no—
342D3—21no—
343D3—42no—
344D3—81no—
345C22—22no—
346C22—31no—
347C22—31no—
348C22—62no—
349C22—121no—
350C22—122no—
351C22—124no—
352C22—121no—
353C22C2×D3×D411yescharacteristic
354C4—22no—
355C4—22no—
356C4—32no—
357C4—41no—
358C4—41no—
359C4—62no—
360C4—62no—
361C4—121no—
362C4C2×D3×D412yes—
363C3#302011yescharacteristic, complemented, Sylow
364C2—21no—
365C2—42no—
366C2—61no—
367C2—61no—
368C2—122no—
369C2—241no—
370C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#456C2
socleC2×C3C2×#120
hypercenter#456D3
hypocenterC3#3020
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
1. D3×2−1+4C2semidirect
2. #5777C2semidirect
3. #5724C2semidirect
4. C3×#456C2semidirect
5. #5245C2semidirect
6. #5205C2semidirect
7. #5095C2semidirect
8. #4827C2semidirect
9. #4821C2semidirect
10. #4753C2semidirect
11. D3×#101C2semidirect
25. C3×#137C22semidirect
26. C3×C4⋊Q8C22semidirect
27. #949C22semidirect
28. #947C22semidirect
29. #944C22semidirect
30. #942C22semidirect
35. #908C22semidirect
39. #875C22semidirect
41. C3×#104C22semidirect
42. C3×#101C22semidirect
43. #837C22semidirect
44. #836C22semidirect
45. #825C22semidirect
46. #824C22semidirect
47. #805C22semidirect
53. #456D3semidirect
137. #137C2×D3semidirect
145. C4⋊Q8C2×D3semidirect
154. #104C2×D3semidirect
157. #101C2×D3semidirect
363. C3#3020semidirect

Maximal subgroups

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

subgroupindex
D3×2−1+42
#57772
#57242
C3×#4562
#52452
#52052
#50952
#48272
#48212
#47532
D3×#1012
#30203