Finite Group Databasebeta

← #80278

Subgroups — #80278

Order 384.

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

normal subgroups only

This group has 762 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×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#5773C212yesmaximal, complemented
3#5715C211yesmaximal, characteristic, complemented
4C3×#453C211yesmaximal, characteristic, complemented
5#5242C211yesmaximal, characteristic, complemented
6#5198C212yesmaximal, complemented
7#5077C211yesmaximal, characteristic, complemented
8#4825C212yesmaximal, complemented
9#4821C212yesmaximal, complemented
10#4746C211yesmaximal, characteristic, complemented
11D3×#100C211yesmaximal, characteristic, complemented
12#3019—31nomaximal, Sylow
13C3×H(5,𝔽2)C2211yescharacteristic
14#1009—22no—
15#1009C2212yes—
16D3×#41—22no—
17D3×#41C2212yes—
18#1004—22no—
19#1004—22no—
20#1004C2211yescharacteristic
21C2×#237C2211yescharacteristic, complemented
22C2×D3×D4—22no—
23C2×D3×D4—22no—
24C2×D3×D4C2211yescharacteristic
25C3×AGL(1,ℤ/8)C2212yescomplemented
26C3×#127C2211yescharacteristic, complemented
27#951C2212yescomplemented
28#940C2211yescharacteristic, complemented
29#939—22no—
30#937—22no—
31#936C2211yescharacteristic, complemented
32#932C2212yescomplemented
33#916—22no—
34#915—22no—
35#912—22no—
36#911—22no—
37#909C2212yescomplemented
38D3×M16C2212yescomplemented
39#875C2211yescharacteristic, complemented
40#872C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#100C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#833C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#823C2212yescomplemented
47#805C2212yescomplemented
48C2×H(5,𝔽2)—31no—
49#575—32no—
50#535—31no—
51#455—31no—
52#455—32no—
53#455—31no—
54#453—32no—
55#453—31no—
56#453D311yescharacteristic, complemented
57#421—32no—
58C2×#100—31no—
59C23×D3—22no—
60C23×D3D412yes—
61C3×#41—22no—
62C3×#41C2312yes—
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4—22no—
66C2×C3×D4C2311yescharacteristic
67C2×C3×D4C2311yescharacteristic, complemented
68C2×#67—22no—
69C2×#67—24no—
70C2×#67—22no—
71C2×#67—21no—
72C2×#67—22no—
73C2×#67D412yes—
74C22×Dic3—21no—
75#239—22no—
76#239D412yescomplemented
77D3×Q8D412yescomplemented
78#237—24no—
79#237—22no—
80#237—22no—
81#237—22no—
82#237—22no—
83#237D412yescomplemented
84D3×D4—24no—
85D3×D4—24no—
86D3×D4—24no—
87D3×D4—22no—
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×D12—22no—
99C2×D12—22no—
100C2×D12C2311yescharacteristic
101C2×C4×D3—22no—
102C2×C4×D3—22no—
103C2×C4×D3C2311yescharacteristic
104C2×Dic6C2311yescharacteristic, complemented
105C3×SD16—22no—
106C3×D8—22no—
107C3×M16C2312yescomplemented
108C3×C42C2311yescharacteristic, complemented
109#217—22no—
110#216—22no—
111#214—22no—
112#214—22no—
113#213—22no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—22no—
116Dic3⋊C4—21no—
117C4×Dic3C2311yescharacteristic, complemented
118#208C2312yescomplemented
119Dic12—22no—
120#204—22no—
121ΓL(1,𝔽25)—22no—
122D3×C8—22no—
123H(5,𝔽2)—32no—
124H(5,𝔽2)—31no—
125H(5,𝔽2)—62no—
126H(5,𝔽2)—62no—
127H(5,𝔽2)—62no—
128H(5,𝔽2)C2×D311yescharacteristic
129C2×#41—32no—
130C2×#41—31no—
131C2×#41—62no—
132C22×D4—31no—
133C22×D4—62no—
134C22×D4—62no—
135#137—32no—
136#137—32no—
137#137—62no—
138AGL(1,ℤ/8)—32no—
139AGL(1,ℤ/8)—62no—
140AGL(1,ℤ/8)C2×D312yescomplemented
141#135—62no—
142#135—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#100—31no—
156#100—32no—
157#100C2×D311yescharacteristic, complemented
158C23×C3—22no—
159C23×C3—21no—
160C23×C3C2×D412yes—
161C22×D3—24no—
162C22×D3—22no—
163C22×D3—22no—
164C22×D3—22no—
165C22×D3—22no—
166C22×D3—82no—
167C22×D3—84no—
168C22×D3C2×D412yes—
169C22×D3C2×D411yescharacteristic
170C3×Q8C2×D412yes—
171C3×D4—24no—
172C3×D4—22no—
173C3×D4—22no—
174C3×D4—22no—
175C3×D4—82no—
176C3×D4C2×D412yes—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4C2411yescharacteristic
181#67—24no—
182#67—24no—
183#67—22no—
184#67—22no—
185#67—24no—
186#67—22no—
187#67—41no—
188#67—41no—
189#67—82no—
190C2×Dic3—22no—
191C2×Dic3—22no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—42no—
195C2×Dic3C2×D411yescharacteristic
196D12—24no—
197D12—22no—
198D12—22no—
199D12C2×D412yes—
200C4×D3—24no—
201C4×D3—22no—
202C4×D3—22no—
203C4×D3—22no—
204C4×D3C2×D412yes—
205C4×D3C2×D412yes—
206Dic6—22no—
207Dic6—22no—
208Dic6C2×D412yes—
209C3×C8—22no—
210C3⋊C8—22no—
211C24—32no—
212C24—62no—
213#41—22no—
214#41—32no—
215#41—32no—
216#41—32no—
217#41—62no—
218#41—62no—
219#41—64no—
220#41—62no—
221#41—62no—
222#41—62no—
223#41—62no—
224#41—64no—
225#41—62no—
226#41C22×D312yes—
227C2×Q8—32no—
228C2×Q8—31no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—32no—
233C2×D4—32no—
234C2×D4—31no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—64no—
242C2×D4—64no—
243C2×D4—64no—
244C2×D4—64no—
245C2×D4—64no—
246C2×D4—62no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4—61no—
250C2×D4—62no—
251C2×D4C22×D311yescharacteristic
252C2×D4C22×D311yescharacteristic, complemented
253C22×C4—31no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4—61no—
257Q16—62no—
258Q16—62no—
259SD16—22no—
260SD16—62no—
261SD16—62no—
262SD16—62no—
263D8—22no—
264D8—62no—
265M16—32no—
266M16—62no—
267M16C22×D312yescomplemented
268C2×C8—62no—
269C4⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C42—31no—
274C42C22×D311yescharacteristic, complemented
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—41no—
280C22×C3—82no—
281C22×C3C22×D411yescharacteristic
282C2×D3—22no—
283C2×D3—22no—
284C2×D3—41no—
285C2×D3—44no—
286C2×D3—44no—
287C2×D3—42no—
288C2×D3—82no—
289C2×D3—84no—
290C2×D3—82no—
291C2×D3#12011yescharacteristic
292C2×D3#12011yescharacteristic
293C3×C4—22no—
294C3×C4—41no—
295C3×C4C22×D412yes—
296Dic3—22no—
297Dic3—41no—
298Dic3—41no—
299Dic3#12012yes—
300C23—22no—
301C23—21no—
302C23—31no—
303C23—32no—
304C23—62no—
305C23—62no—
306C23—62no—
307C23—62no—
308C23—64no—
309C23—242no—
310C23—244no—
311C23D3×D412yes—
312Q8—32no—
313Q8—62no—
314Q8—62no—
315Q8D3×D412yes—
316D4—24no—
317D4—22no—
318D4—22no—
319D4—22no—
320D4—32no—
321D4—62no—
322D4—64no—
323D4—64no—
324D4—62no—
325D4—62no—
326D4—64no—
327D4—64no—
328D4—62no—
329D4—62no—
330D4—82no—
331D4—121no—
332D4—121no—
333D4—242no—
334D4D3×D412yes—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—31no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—64no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—122no—
350C2×C4C23×D311yescharacteristic
351C8—22no—
352C8—62no—
353C2×C3—21no—
354C2×C3—42no—
355C2×C3—42no—
356C2×C3—81no—
357C2×C3C2×#12011yescharacteristic
358D3—21no—
359D3—21no—
360D3—42no—
361D3—42no—
362C22—22no—
363C22—22no—
364C22—31no—
365C22—31no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—62no—
370C22—62no—
371C22—82no—
372C22—121no—
373C22—122no—
374C22—124no—
375C22—124no—
376C22—242no—
377C22—242no—
378C22—244no—
379C22C2×D3×D411yescharacteristic
380C4—22no—
381C4—32no—
382C4—41no—
383C4—62no—
384C4—121no—
385C4—121no—
386C4C2×D3×D412yes—
387C3#301911yescharacteristic, complemented, Sylow
388C2—21no—
389C2—42no—
390C2—42no—
391C2—61no—
392C2—61no—
393C2—81no—
394C2—122no—
395C2—122no—
396C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#453C2
socleC2×C3C2×#120
hypercenter#453D3
hypocenterC3#3019
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. D3×H(5,𝔽2)C2semidirect
2. #5773C2semidirect
3. #5715C2semidirect
4. C3×#453C2semidirect
5. #5242C2semidirect
6. #5198C2semidirect
7. #5077C2semidirect
8. #4825C2semidirect
9. #4821C2semidirect
10. #4746C2semidirect
11. D3×#100C2semidirect
21. C2×#237C22semidirect
25. C3×AGL(1,ℤ/8)C22semidirect
26. C3×#127C22semidirect
27. #951C22semidirect
28. #940C22semidirect
31. #936C22semidirect
32. #932C22semidirect
37. #909C22semidirect
38. D3×M16C22semidirect
39. #875C22semidirect
40. #872C22semidirect
41. C3×#104C22semidirect
42. C3×#100C22semidirect
43. #837C22semidirect
44. #833C22semidirect
45. #825C22semidirect
46. #823C22semidirect
47. #805C22semidirect
56. #453D3semidirect
67. C2×C3×D4C23semidirect
76. #239D4semidirect
77. D3×Q8D4semidirect
83. #237D4semidirect
93. D3×D4D4semidirect
104. C2×Dic6C23semidirect
107. C3×M16C23semidirect
108. C3×C42C23semidirect
117. C4×Dic3C23semidirect
118. #208C23semidirect
140. AGL(1,ℤ/8)C2×D3semidirect
145. #127C2×D3semidirect
154. #104C2×D3semidirect
157. #100C2×D3semidirect
252. C2×D4C22×D3semidirect
267. M16C22×D3semidirect
274. C42C22×D3semidirect
387. C3#3019semidirect

Maximal subgroups

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

subgroupindex
D3×H(5,𝔽2)2
#57732
#57152
C3×#4532
#52422
#51982
#50772
#48252
#48212
#47462
D3×#1002
#30193