Finite Group Databasebeta

← #84005

Subgroups — #84005

Order 384.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#1009C211yesmaximal, characteristic, complemented
2C3×#575C211yesmaximal, characteristic, complemented
3#5830C211yesmaximal, characteristic, complemented
4#5822C211yesmaximal, characteristic, complemented
5#5236C211yesmaximal, characteristic, complemented
6#5202C214yesmaximal, complemented
7#5198C214yesmaximal, complemented
8#5139C211yesmaximal, characteristic, complemented
9C2×#823C211yesmaximal, characteristic, complemented
10#3019—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C2×C3×#41C2211yescharacteristic, complemented
13#1009—22no—
14#1009—22no—
15#1009—22no—
16#1009C2214yes—
17C2×#239—21no—
18C2×D3×D4—22no—
19C2×D3×D4—21no—
20C2×#235—21no—
21C2×#235C2211yescharacteristic
22C22×D12—21no—
23C22×D12C2211yescharacteristic
24C3×#137—21no—
25C3×#137C2212yescomplemented
26C3×AGL(1,ℤ/8)—21no—
27C3×AGL(1,ℤ/8)C2212yescomplemented
28C3×#135—22no—
29C2×C3×M16C2211yescharacteristic, complemented
30#951—21no—
31#950—22no—
32#949—21no—
33#947C2212yescomplemented
34#942C2212yescomplemented
35#940C2212yescomplemented
36#937—22no—
37#932C2212yescomplemented
38#930—22no—
39#926C2211yescharacteristic, complemented
40C2×#208C2211yescharacteristic, complemented
41#837C2214yescomplemented
42#825C2214yescomplemented
43#823C2214yescomplemented
44C2×H(5,𝔽2)—31no—
45#575—31no—
46#575D311yescharacteristic, complemented
47#535—31no—
48#455—34no—
49#453—34no—
50#421—31no—
51#421—31no—
52C2×#100—31no—
53C23×D3—22no—
54C23×D3D412yes—
55C3×#41—21no—
56C3×#41—21no—
57C3×#41—21no—
58C3×#41—21no—
59C3×#41—21no—
60C3×#41—21no—
61C3×#41D412yescomplemented
62C3×#41C2312yes—
63C2×C3×Q8D411yescharacteristic, complemented
64C2×C3×Q8C2311yescharacteristic, complemented
65C2×C3×D4—21no—
66C2×C3×D4—21no—
67C2×C3×D4D411yescharacteristic, complemented
68C2×C3×D4C2311yescharacteristic, complemented
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4C2311yescharacteristic
72C2×#67—22no—
73C2×#67—21no—
74C2×#67—24no—
75#239—24no—
76#239—24no—
77D3×D4—28no—
78D3×D4—24no—
79D3×D4—28no—
80D3×D4—24no—
81#235—24no—
82#235—21no—
83#235—21no—
84#235—24no—
85#235D412yescomplemented
86#235C2312yes—
87C2×D12—22no—
88C2×D12—24no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—24no—
94C2×D12D412yes—
95C2×D12D411yescharacteristic, complemented
96C2×D12C2312yes—
97C2×C4×D3—22no—
98C2×C4×D3—21no—
99C2×Dic6D411yescharacteristic, complemented
100C3×Q16—22no—
101C3×SD16—22no—
102C3×SD16—22no—
103C3×D8—22no—
104C3×M16—21no—
105C3×M16C2312yescomplemented
106C2×C3×C8—21no—
107#217—21no—
108#216—22no—
109#215—22no—
110#214—22no—
111#213—22no—
112(C2×D3)⋊C4—24no—
113C4⋊Dic3—21no—
114C4×Dic3C2312yescomplemented
115#208—21no—
116#208C2312yescomplemented
117C2×C3⋊C8—21no—
118H(5,𝔽2)—34no—
119H(5,𝔽2)—62no—
120H(5,𝔽2)—62no—
121H(5,𝔽2)—62no—
122C2×#41—31no—
123C2×#41—61no—
124C2×#41—61no—
125C2×#41C2×D311yescharacteristic
126C2×#41C2×D311yescharacteristic, complemented
127C22×D4—31no—
128C22×D4—61no—
129C22×D4—62no—
130C22×D4—61no—
131#137—21no—
132#137—32no—
133#137—61no—
134#137C2×D312yescomplemented
135AGL(1,ℤ/8)—21no—
136AGL(1,ℤ/8)—32no—
137AGL(1,ℤ/8)—61no—
138AGL(1,ℤ/8)C2×D312yescomplemented
139#135—22no—
140#135—62no—
141C2×M16—31no—
142C2×M16C2×D311yescharacteristic, complemented
143#127—32no—
144#124—32no—
145#121—62no—
146#120—62no—
147#117—31no—
148#104—34no—
149#104—34no—
150#100—34no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3C2×D411yescharacteristic
154C22×D3—22no—
155C22×D3—21no—
156C22×D3—28no—
157C22×D3—22no—
158C22×D3—82no—
159C22×D3—84no—
160C22×D3C2×D414yes—
161C3×Q8—21no—
162C3×Q8—21no—
163C3×Q8—21no—
164C3×Q8C2×D412yes—
165C3×D4—22no—
166C3×D4—22no—
167C3×D4—21no—
168C3×D4—21no—
169C3×D4—21no—
170C3×D4—41no—
171C3×D4—41no—
172C3×D4C2×D412yes—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—28no—
186#67—22no—
187#67—22no—
188#67—84no—
189C2×Dic3—21no—
190C2×Dic3—22no—
191D12—28no—
192D12—22no—
193D12—21no—
194D12—22no—
195D12—22no—
196D12—22no—
197D12C2×D412yes—
198C4×D3—28no—
199C4×D3—22no—
200C4×D3—22no—
201Dic6—21no—
202Dic6C2×D412yes—
203C3×C8—22no—
204C3⋊C8—22no—
205C24—32no—
206C24—62no—
207#41—21no—
208#41—21no—
209#41—21no—
210#41—21no—
211#41—21no—
212#41—21no—
213#41—32no—
214#41—32no—
215#41—64no—
216#41—64no—
217#41—64no—
218#41—64no—
219#41—61no—
220#41—61no—
221#41#6712yescomplemented
222#41C22×D312yes—
223C2×Q8—31no—
224C2×Q8#6711yescharacteristic, complemented
225C2×Q8C22×D311yescharacteristic, complemented
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—31no—
229C2×D4—32no—
230C2×D4—32no—
231C2×D4—64no—
232C2×D4—62no—
233C2×D4—64no—
234C2×D4—68no—
235C2×D4—68no—
236C2×D4—64no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4—64no—
240C2×D4—64no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4#6711yescharacteristic, complemented
246C2×D4C22×D311yescharacteristic, complemented
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4C22×D311yescharacteristic
252Q16—22no—
253Q16—62no—
254SD16—22no—
255SD16—22no—
256SD16—62no—
257SD16—62no—
258D8—22no—
259D8—62no—
260M16—21no—
261M16—32no—
262M16—61no—
263M16C22×D312yescomplemented
264C2×C8—21no—
265C2×C8—61no—
266C4⋊C4—61no—
267C22⋊C4—64no—
268C22⋊C4—61no—
269C42—32no—
270C22×C3—22no—
271C22×C3—41no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—41no—
275C22×C3#12011yescharacteristic
276C22×C3#12011yescharacteristic
277C22×C3C22×D411yescharacteristic
278C2×D3—24no—
279C2×D3—22no—
280C2×D3—44no—
281C2×D3—42no—
282C2×D3—44no—
283C2×D3—82no—
284C2×D3—88no—
285C3×C4—22no—
286C3×C4—41no—
287C3×C4#12011yescharacteristic
288C3×C4#12011yescharacteristic
289C3×C4C22×D411yescharacteristic
290C3×C4C22×D411yescharacteristic
291Dic3—22no—
292Dic3—42no—
293C23—21no—
294C23—21no—
295C23—34no—
296C23—68no—
297C23—61no—
298C23—62no—
299C23—62no—
300C23—242no—
301C23—244no—
302C23D3×D411yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—32no—
307Q8—61no—
308Q8C2×#6712yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—32no—
315D4—41no—
316D4—41no—
317D4—68no—
318D4—68no—
319D4—62no—
320D4—62no—
321D4—61no—
322D4—62no—
323D4—62no—
324D4—62no—
325D4—62no—
326D4—244no—
327D4C2×#6712yes—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—68no—
335C2×C4—61no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4D3×D411yescharacteristic
340C2×C4D3×D411yescharacteristic
341C2×C4D3×D411yescharacteristic
342C2×C4C2×#6711yescharacteristic
343C2×C4C2×#6711yescharacteristic
344C2×C4C23×D311yescharacteristic
345C8—22no—
346C8—62no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—42no—
351C2×C3—81no—
352C2×C3C2×#12011yescharacteristic
353D3—44no—
354D3—42no—
355C22—22no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—62no—
361C22—64no—
362C22—124no—
363C22—124no—
364C22—122no—
365C22—242no—
366C22—248no—
367C22#93711yescharacteristic
368C22#93711yescharacteristic
369C22C2×D3×D411yescharacteristic
370C4—22no—
371C4—41no—
372C4—62no—
373C4—122no—
374C4#93711yescharacteristic
375C4#93711yescharacteristic
376C4C2×D3×D411yescharacteristic
377C4C22×#6711yescharacteristic
378C3#301911yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—124no—
385C2—122no—
386C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#575C2
socleC2×C3C2×#120
hypercenter#575D3
hypocenterC3#3019
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C3×#575C2semidirect
3. #5830C2semidirect
4. #5822C2semidirect
5. #5236C2semidirect
6. #5202C2semidirect
7. #5198C2semidirect
8. #5139C2semidirect
9. C2×#823C2semidirect
12. C2×C3×#41C22semidirect
25. C3×#137C22semidirect
27. C3×AGL(1,ℤ/8)C22semidirect
29. C2×C3×M16C22semidirect
33. #947C22semidirect
34. #942C22semidirect
35. #940C22semidirect
37. #932C22semidirect
39. #926C22semidirect
40. C2×#208C22semidirect
41. #837C22semidirect
42. #825C22semidirect
43. #823C22semidirect
46. #575D3semidirect
61. C3×#41D4semidirect
63. C2×C3×Q8D4semidirect
64. C2×C3×Q8C23semidirect
67. C2×C3×D4D4semidirect
68. C2×C3×D4C23semidirect
85. #235D4semidirect
95. C2×D12D4semidirect
99. C2×Dic6D4semidirect
105. C3×M16C23semidirect
114. C4×Dic3C23semidirect
116. #208C23semidirect
126. C2×#41C2×D3semidirect
134. #137C2×D3semidirect
138. AGL(1,ℤ/8)C2×D3semidirect
142. C2×M16C2×D3semidirect
221. #41#67semidirect
224. C2×Q8#67semidirect
225. C2×Q8C22×D3semidirect
245. C2×D4#67semidirect
246. C2×D4C22×D3semidirect
263. M16C22×D3semidirect
378. C3#3019semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C3×#5752
#58302
#58222
#52362
#52022
#51982
#51392
C2×#8232
#30193