Finite Group Databasebeta

← #80279

Subgroups — #80279

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#5966C211yesmaximal, characteristic, complemented
2D3×AGL(1,ℤ/8)C212yesmaximal, complemented
3#5715C211yesmaximal, characteristic, complemented
4C3×#453C211yesmaximal, characteristic, complemented
5#5241C211yesmaximal, characteristic, complemented
6#5201C212yesmaximal, complemented
7#5077C211yesmaximal, characteristic, complemented
8#4827C212yesmaximal, complemented
9#4821C212yesmaximal, complemented
10#4747C211yesmaximal, characteristic, complemented
11#4745C211yesmaximal, characteristic, complemented
12#3021—31nomaximal, Sylow
13C3×H(5,𝔽2)C2211yescharacteristic, complemented
14#1010—22no—
15#1010C2212yescomplemented
16D3×#41—22no—
17D3×#41C2212yes—
18#1004—22no—
19#1004—22no—
20#1004C2211yescharacteristic, complemented
21C2×#237—22no—
22C2×#237—22no—
23C2×#237C2211yescharacteristic
24C2×D3×D4C2211yescharacteristic, complemented
25C3×AGL(1,ℤ/8)C2212yescomplemented
26C3×#127C2211yescharacteristic, complemented
27#949C2212yescomplemented
28#940C2211yescharacteristic, complemented
29#939—22no—
30#937—22no—
31#936C2211yescharacteristic, complemented
32#932C2212yescomplemented
33#914—22no—
34D3×SD16—22no—
35#911—22no—
36D3×D8—22no—
37#908C2212yescomplemented
38D3×M16C2212yescomplemented
39#875C2211yescharacteristic, complemented
40#872C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#100C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#833C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#824C2212yescomplemented
47#805C2212yescomplemented
48#585—31no—
49C2×AGL(1,ℤ/8)—32no—
50#535—31no—
51#454—32no—
52#454—32no—
53#453—31no—
54#453—31no—
55#453—31no—
56#453D311yescharacteristic, complemented
57#421—32no—
58#413—31no—
59C23×D3—21no—
60C3×#41—22no—
61C3×#41C2312yes—
62C2×C3×D4—22no—
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4C2311yescharacteristic
66C2×C3×D4C2311yescharacteristic
67C2×#67—24no—
68C2×#67—22no—
69C2×#67—22no—
70C2×#67—22no—
71C2×#67—21no—
72C2×#67D412yes—
73C22×Dic3—22no—
74C22×Dic3D412yes—
75#239D412yescomplemented
76D3×Q8—22no—
77D3×Q8D412yescomplemented
78#237—24no—
79#237—24no—
80#237—24no—
81#237—22no—
82#237—22no—
83#237—24no—
84#237—22no—
85#237—22no—
86#237—22no—
87#237D412yescomplemented
88D3×D4—24no—
89D3×D4—22no—
90D3×D4—22no—
91D3×D4—22no—
92D3×D4—22no—
93D3×D4D412yescomplemented
94#235—24no—
95#235—22no—
96#235—22no—
97#235C2312yes—
98C2×D12C2311yescharacteristic
99C2×C4×D3—22no—
100C2×C4×D3—22no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—22no—
103C2×Dic6—22no—
104C2×Dic6C2311yescharacteristic
105C3×SD16—22no—
106C3×D8—22no—
107C3×M16C2312yes—
108C3×C42C2311yescharacteristic
109#217—22no—
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—32no—
1242−1+4—62no—
125H(5,𝔽2)—31no—
126H(5,𝔽2)—62no—
127H(5,𝔽2)—62no—
128H(5,𝔽2)C2×D311yescharacteristic, complemented
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#127C2×D311yescharacteristic, complemented
146#124—31no—
147#124—31no—
148#121—62no—
149#120—62no—
150#117—31no—
151#104—32no—
152#104—32no—
153#104—32no—
154#104C2×D312yescomplemented
155#101—32no—
156#100—31no—
157#100C2×D311yescharacteristic, complemented
158C23×C3—22no—
159C23×C3—21no—
160C23×C3C2×D412yes—
161C22×D3—22no—
162C22×D3—22no—
163C22×D3—21no—
164C22×D3—42no—
165C22×D3—82no—
166C22×D3C2×D411yescharacteristic
167C3×Q8C2×D412yes—
168C3×D4—24no—
169C3×D4—22no—
170C3×D4—22no—
171C3×D4—22no—
172C3×D4—82no—
173C3×D4C2×D412yes—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4C2411yescharacteristic
178#67—24no—
179#67—24no—
180#67—22no—
181#67—24no—
182#67—22no—
183#67—22no—
184#67—41no—
185#67—41no—
186#67—82no—
187C2×Dic3—24no—
188C2×Dic3—22no—
189C2×Dic3—22no—
190C2×Dic3—22no—
191C2×Dic3—22no—
192C2×Dic3—21no—
193C2×Dic3C2×D412yes—
194C2×Dic3C2×D411yescharacteristic
195D12—22no—
196D12—22no—
197D12C2×D412yes—
198C4×D3—24no—
199C4×D3—22no—
200C4×D3—22no—
201C4×D3—22no—
202C4×D3C2×D412yes—
203C4×D3C2×D412yes—
204Dic6—24no—
205Dic6—22no—
206Dic6—22no—
207Dic6C2×D412yes—
208C3×C8—22no—
209C3⋊C8—22no—
210C24—61no—
211#41—22no—
212#41—32no—
213#41—32no—
214#41—32no—
215#41—62no—
216#41—62no—
217#41—64no—
218#41—62no—
219#41—62no—
220#41—62no—
221#41—64no—
222#41—62no—
223#41—64no—
224#41—64no—
225#41—64no—
226#41—62no—
227#41C22×D312yes—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8—62no—
231C2×Q8—62no—
232C2×Q8—62no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—32no—
237C2×D4—32no—
238C2×D4—31no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—61no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—64no—
245C2×D4—62no—
246C2×D4—64no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4C22×D311yescharacteristic
250C2×D4C22×D311yescharacteristic
251C22×C4—31no—
252C22×C4—32no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4—62no—
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—62no—
270C22⋊C4—61no—
271C22⋊C4—62no—
272C42—31no—
273C42C22×D311yescharacteristic
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—42no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—82no—
280C22×C3C22×D411yescharacteristic
281C2×D3—22no—
282C2×D3—41no—
283C2×D3—42no—
284C2×D3—41no—
285C2×D3—82no—
286C2×D3—84no—
287C2×D3—82no—
288C2×D3#12012yes—
289C3×C4—22no—
290C3×C4—41no—
291C3×C4C22×D412yes—
292Dic3—22no—
293Dic3—22no—
294Dic3—41no—
295Dic3#12011yescharacteristic
296Dic3#12011yescharacteristic
297C23—22no—
298C23—21no—
299C23—31no—
300C23—62no—
301C23—62no—
302C23—61no—
303C23—122no—
304C23—242no—
305C23D3×D412yes—
306Q8—32no—
307Q8—62no—
308Q8—64no—
309Q8—62no—
310Q8D3×D412yes—
311D4—24no—
312D4—22no—
313D4—22no—
314D4—22no—
315D4—32no—
316D4—62no—
317D4—62no—
318D4—64no—
319D4—62no—
320D4—64no—
321D4—64no—
322D4—62no—
323D4—62no—
324D4—82no—
325D4—121no—
326D4—121no—
327D4—242no—
328D4D3×D412yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—64no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—64no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4C23×D311yescharacteristic
347C8—22no—
348C8—62no—
349C2×C3—21no—
350C2×C3—42no—
351C2×C3—42no—
352C2×C3—81no—
353C2×C3C2×#12011yescharacteristic
354D3—22no—
355D3—42no—
356D3—81no—
357C22—22no—
358C22—22no—
359C22—32no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—62no—
364C22—82no—
365C22—121no—
366C22—122no—
367C22—121no—
368C22—242no—
369C22—242no—
370C22—244no—
371C22C2×D3×D411yescharacteristic
372C4—22no—
373C4—31no—
374C4—31no—
375C4—41no—
376C4—62no—
377C4—62no—
378C4—121no—
379C4C2×D3×D412yes—
380C3#302111yescharacteristic, complemented, Sylow
381C2—21no—
382C2—42no—
383C2—42no—
384C2—62no—
385C2—81no—
386C2—122no—
387C2—241no—
388C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#453C2
socleC2×C3C2×#120
hypercenter#453D3
hypocenterC3#3021
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #5966C2semidirect
2. D3×AGL(1,ℤ/8)C2semidirect
3. #5715C2semidirect
4. C3×#453C2semidirect
5. #5241C2semidirect
6. #5201C2semidirect
7. #5077C2semidirect
8. #4827C2semidirect
9. #4821C2semidirect
10. #4747C2semidirect
11. #4745C2semidirect
13. C3×H(5,𝔽2)C22semidirect
15. #1010C22semidirect
20. #1004C22semidirect
24. C2×D3×D4C22semidirect
25. C3×AGL(1,ℤ/8)C22semidirect
26. C3×#127C22semidirect
27. #949C22semidirect
28. #940C22semidirect
31. #936C22semidirect
32. #932C22semidirect
37. #908C22semidirect
38. D3×M16C22semidirect
39. #875C22semidirect
40. #872C22semidirect
41. C3×#104C22semidirect
42. C3×#100C22semidirect
43. #837C22semidirect
44. #833C22semidirect
45. #825C22semidirect
46. #824C22semidirect
47. #805C22semidirect
56. #453D3semidirect
75. #239D4semidirect
77. D3×Q8D4semidirect
87. #237D4semidirect
93. D3×D4D4semidirect
128. H(5,𝔽2)C2×D3semidirect
140. AGL(1,ℤ/8)C2×D3semidirect
145. #127C2×D3semidirect
154. #104C2×D3semidirect
157. #100C2×D3semidirect
380. C3#3021semidirect

Maximal subgroups

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

subgroupindex
#59662
D3×AGL(1,ℤ/8)2
#57152
C3×#4532
#52412
#52012
#50772
#48272
#48212
#47472
#47452
#30213