Finite Group Databasebeta

← #80298

Subgroups — #80298

Order 384.

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

normal subgroups only

This group has 724 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
2#5777C212yesmaximal, complemented
3#5725C211yesmaximal, characteristic, complemented
4C3×#456C211yesmaximal, characteristic, complemented
5#5245C211yesmaximal, characteristic, complemented
6#5202C212yesmaximal, complemented
7#5095C211yesmaximal, characteristic, complemented
8#4822C212yesmaximal, complemented
9D3×#104C212yesmaximal, complemented
10#4753C211yesmaximal, characteristic, complemented
11#4751C211yesmaximal, characteristic, complemented
12#3018—31nomaximal, Sylow
13C3×2−1+4C2211yescharacteristic
14#1009—22no—
15#1009—22no—
16#1009C2212yes—
17D3×#41—22no—
18D3×#41—22no—
19D3×#41C2212yes—
20#1007—22no—
21#1007C2211yescharacteristic
22C2×#239—22no—
23C2×#239C2211yescharacteristic
24C2×#239C2211yescharacteristic, complemented
25C3×#137C2212yescomplemented
26C3×C4⋊Q8C2211yescharacteristic, complemented
27#949C2212yescomplemented
28#947C2211yescharacteristic, complemented
29#947C2211yescharacteristic, complemented
30#942C2212yescomplemented
31#919—22no—
32#918—22no—
33#916—22no—
34#914—22no—
35#908C2212yescomplemented
36D3×M16C2212yescomplemented
37#895—22no—
38#893—22no—
39#874C2211yescharacteristic, complemented
40C42×D3C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#101C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#836C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#823C2212yescomplemented
47#805C2212yescomplemented
48#585—31no—
49#575—32no—
50#532—31no—
51#456—31no—
52#456D311yescharacteristic, complemented
53#455—32no—
54#454—31no—
55#454—31no—
56#453—32no—
57C2×#104—32no—
58#413—31no—
59C3×#41—22no—
60C3×#41—22no—
61C3×#41C2312yes—
62C2×C3×Q8—22no—
63C2×C3×Q8C2311yescharacteristic
64C2×C3×Q8C2311yescharacteristic
65#239—22no—
66#239—24no—
67#239—22no—
68#239—22no—
69#239—22no—
70#239—22no—
71#239—22no—
72#239—22no—
73#239D412yescomplemented
74D3×Q8—22no—
75D3×Q8—22no—
76D3×Q8D412yescomplemented
77#237—22no—
78#237—22no—
79#237D412yescomplemented
80D3×D4—24no—
81D3×D4—22no—
82D3×D4—24no—
83D3×D4—22no—
84D3×D4—22no—
85D3×D4D412yescomplemented
86#235—24no—
87#235—24no—
88#235—22no—
89#235—22no—
90#235—22no—
91#235C2312yes—
92C2×D12—24no—
93C2×D12—22no—
94C2×D12—22no—
95C2×D12—21no—
96C2×D12D412yes—
97C2×D12C2311yescharacteristic
98C2×D12C2311yescharacteristic
99C2×C4×D3—24no—
100C2×C4×D3—22no—
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3D412yes—
104C2×C4×D3C2311yescharacteristic
105C3×Q16—22no—
106C3×SD16—22no—
107C3×M16C2312yes—
108C3×C4⋊C4—22no—
109C3×C42C2311yescharacteristic
110#216—22no—
111#215—22no—
112#215—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—31no—
1242−1+4—62no—
1252−1+4C2×D311yescharacteristic
126H(5,𝔽2)—32no—
127H(5,𝔽2)—62no—
128H(5,𝔽2)—62no—
129C2×#41—31no—
130C2×#41—32no—
131C2×#41—31no—
132C2×#41—62no—
133C2×#41—62no—
134C2×#41—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⋊Q8C2×D311yescharacteristic, complemented
145#127—31no—
146#124—31no—
147#124—31no—
148#121—62no—
149C4×D4—62no—
150C2×C42—31no—
151#104—32no—
152#104—32no—
153#104—32no—
154#104C2×D312yescomplemented
155#101—31no—
156#101C2×D311yescharacteristic, complemented
157#100—32no—
158C22×D3—24no—
159C22×D3—22no—
160C22×D3—21no—
161C22×D3C2×D412yes—
162C22×D3C2×D411yescharacteristic
163C3×Q8—22no—
164C3×Q8—22no—
165C3×Q8—22no—
166C3×Q8C2×D412yes—
167C3×D4—22no—
168C3×D4—22no—
169C3×D4C2×D412yes—
170C2×C3×C4—24no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4C2×D412yes—
175C2×C3×C4C2411yescharacteristic
176#67—24no—
177#67—22no—
178#67—22no—
179#67—22no—
180C2×Dic3—22no—
181C2×Dic3—21no—
182C2×Dic3C2×D411yescharacteristic
183D12—24no—
184D12—22no—
185D12—22no—
186D12—24no—
187D12—22no—
188D12—22no—
189D12—44no—
190D12—42no—
191D12C2×D412yes—
192C4×D3—24no—
193C4×D3—24no—
194C4×D3—22no—
195C4×D3—22no—
196C4×D3—22no—
197C4×D3—22no—
198C4×D3—22no—
199C4×D3—42no—
200C4×D3—41no—
201C4×D3—41no—
202C4×D3C2×D412yes—
203C4×D3C2×D412yes—
204Dic6—22no—
205Dic6—22no—
206Dic6C2×D412yes—
207C3×C8—22no—
208C3⋊C8—22no—
209#41—22no—
210#41—22no—
211#41—32no—
212#41—32no—
213#41—32no—
214#41—62no—
215#41—62no—
216#41—64no—
217#41—64no—
218#41—62no—
219#41—62no—
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×Q8C22×D311yescharacteristic
236C2×D4—32no—
237C2×D4—32no—
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—62no—
241C2×D4—64no—
242C2×D4—62no—
243C2×D4—61no—
244C2×D4—62no—
245C2×D4—64no—
246C2×D4—62no—
247C2×D4—64no—
248C2×D4—62no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—64no—
254C22×C4—61no—
255Q16—22no—
256Q16—62no—
257SD16—22no—
258SD16—62no—
259SD16—62no—
260SD16—62no—
261D8—62no—
262D8—62no—
263M16—32no—
264M16—62no—
265M16C22×D312yes—
266C2×C8—62no—
267C4⋊C4—22no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C42—31no—
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—22no—
274C22×C3C22×D411yescharacteristic
275C2×D3—22no—
276C2×D3—22no—
277C2×D3—41no—
278C2×D3—42no—
279C2×D3—42no—
280C2×D3—41no—
281C2×D3—82no—
282C2×D3—84no—
283C2×D3#12012yes—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—41no—
288C3×C4C22×D412yes—
289Dic3—22no—
290Dic3—41no—
291Dic3#12011yescharacteristic
292Dic3#12011yescharacteristic
293C23—31no—
294C23—32no—
295C23—62no—
296C23—64no—
297C23—61no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—32no—
302Q8—62no—
303Q8—62no—
304Q8D3×D412yes—
305D4—22no—
306D4—22no—
307D4—32no—
308D4—64no—
309D4—62no—
310D4—64no—
311D4—62no—
312D4—62no—
313D4—62no—
314D4—64no—
315D4—62no—
316D4—62no—
317D4—62no—
318D4—124no—
319D4—122no—
320D4D3×D412yes—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—31no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—64no—
333C2×C4—64no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4D3×D412yes—
341C2×C4C23×D311yescharacteristic
342C8—22no—
343C8—62no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3C2×#12011yescharacteristic
347D3—22no—
348D3—42no—
349D3—42no—
350D3—81no—
351C22—22no—
352C22—32no—
353C22—62no—
354C22—62no—
355C22—121no—
356C22—122no—
357C22—122no—
358C22—121no—
359C22—242no—
360C22—244no—
361C22C2×D3×D411yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—31no—
366C4—31no—
367C4—41no—
368C4—62no—
369C4—121no—
370C4C2×D3×D412yes—
371C3#301811yescharacteristic, complemented, Sylow
372C2—21no—
373C2—42no—
374C2—62no—
375C2—122no—
376C2—122no—
377C2—241no—
378C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#456C2
socleC2×C3C2×#120
hypercenter#456D3
hypocenterC3#3018
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. #5968C2semidirect
2. #5777C2semidirect
3. #5725C2semidirect
4. C3×#456C2semidirect
5. #5245C2semidirect
6. #5202C2semidirect
7. #5095C2semidirect
8. #4822C2semidirect
9. D3×#104C2semidirect
10. #4753C2semidirect
11. #4751C2semidirect
24. C2×#239C22semidirect
25. C3×#137C22semidirect
26. C3×C4⋊Q8C22semidirect
27. #949C22semidirect
28. #947C22semidirect
29. #947C22semidirect
30. #942C22semidirect
35. #908C22semidirect
36. D3×M16C22semidirect
39. #874C22semidirect
40. C42×D3C22semidirect
41. C3×#104C22semidirect
42. C3×#101C22semidirect
43. #837C22semidirect
44. #836C22semidirect
45. #825C22semidirect
46. #823C22semidirect
47. #805C22semidirect
52. #456D3semidirect
73. #239D4semidirect
76. D3×Q8D4semidirect
79. #237D4semidirect
85. D3×D4D4semidirect
137. #137C2×D3semidirect
144. C4⋊Q8C2×D3semidirect
154. #104C2×D3semidirect
156. #101C2×D3semidirect
371. C3#3018semidirect

Maximal subgroups

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

subgroupindex
#59682
#57772
#57252
C3×#4562
#52452
#52022
#50952
#48222
D3×#1042
#47532
#47512
#30183