Finite Group Databasebeta

← #80293

Subgroups — #80293

Order 384.

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

normal subgroups only

This group has 704 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
2D3×#137C212yesmaximal, complemented
3#5675C211yesmaximal, characteristic, complemented
4C3×#455C211yesmaximal, characteristic, complemented
5#5242C211yesmaximal, characteristic, complemented
6#5205C212yesmaximal, complemented
7#5061C211yesmaximal, characteristic, complemented
8#4826C212yesmaximal, complemented
9#4821C212yesmaximal, complemented
10#4746C211yesmaximal, characteristic, complemented
11#4745C211yesmaximal, characteristic, complemented
12#3022—31nomaximal, Sylow
13C3×H(5,𝔽2)C2211yescharacteristic
14#1010—22no—
15#1010C2212yes—
16D3×#41—22no—
17D3×#41C2212yes—
18C2×D3×Q8C2211yescharacteristic, complemented
19#1004—22no—
20#1004—22no—
21#1004C2211yescharacteristic
22C2×#237—22no—
23C2×#237—22no—
24C2×#237C2211yescharacteristic
25C3×#137C2212yescomplemented
26C3×#124C2211yescharacteristic, complemented
27#951C2212yescomplemented
28#944C2211yescharacteristic, complemented
29#942C2212yescomplemented
30#936C2211yescharacteristic, complemented
31#918—22no—
32D3×Q16—22no—
33#915—22no—
34D3×SD16—22no—
35#909C2212yescomplemented
36D3×M16C2212yescomplemented
37#883—22no—
38#878—22no—
39#872C2211yescharacteristic, complemented
40#869C2211yescharacteristic, complemented
41C3×#104C2212yescomplemented
42C3×#100C2211yescharacteristic, complemented
43#837C2212yescomplemented
44#833C2211yescharacteristic, complemented
45#825C2212yescomplemented
46#824C2212yescomplemented
47#805C2212yescomplemented
48#585—31no—
49C2×#137—32no—
50#536—31no—
51#456—32no—
52#456—32no—
53#455—31no—
54#455—31no—
55#455—31no—
56#455D311yescharacteristic, complemented
57#421—32no—
58#413—31no—
59C3×#41—22no—
60C3×#41C2312yes—
61C2×C3×Q8C2311yescharacteristic
62C2×C3×D4—22no—
63C2×C3×D4—22no—
64C2×C3×D4C2311yescharacteristic
65C2×#67—24no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67D412yes—
69C22×Dic3—22no—
70C22×Dic3D412yes—
71#239D412yes—
72D3×Q8—22no—
73D3×Q8—22no—
74D3×Q8—22no—
75D3×Q8D412yes—
76#237—24no—
77#237—24no—
78#237—24no—
79#237—22no—
80#237—22no—
81#237—24no—
82#237—22no—
83#237—22no—
84#237—22no—
85#237D412yes—
86D3×D4—24no—
87D3×D4—22no—
88D3×D4—22no—
89D3×D4D412yes—
90#235—24no—
91#235—22no—
92#235—22no—
93#235C2312yes—
94C2×C4×D3—22no—
95C2×C4×D3—22no—
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—22no—
99C2×Dic6—22no—
100C2×Dic6—21no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C3×Q16—22no—
104C3×SD16—22no—
105C3×M16C2312yes—
106C3×C22⋊C4—22no—
107C3×C42C2311yescharacteristic
108#217—22no—
109#216—22no—
110#216—22no—
111#215—22no—
112#214—22no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3—22no—
115Dic3⋊C4—21no—
116Dic3⋊C4—22no—
117C4×Dic3C2311yescharacteristic
118#208C2312yes—
119Dic12—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
129C2×#41—31no—
130C2×#41—32no—
131C2×#41—62no—
132C2×#41—62no—
133C2×#41—62no—
134C22×Q8—31no—
135#137—32no—
136#137—32no—
137#137—32no—
138#137—62no—
139#137—62no—
140#137C2×D312yescomplemented
141C2×Q16—62no—
142C2×SD16—62no—
143C2×M16—32no—
144C4⋊Q8—31no—
145C4⋊Q8—31no—
146#124—31no—
147#124C2×D311yescharacteristic, complemented
148#123—62no—
149C22⋊Q8—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×C3C2×D412yes—
160C22×D3—22no—
161C22×D3—22no—
162C22×D3C2×D411yescharacteristic
163C3×Q8—22no—
164C3×Q8C2×D412yes—
165C3×D4—24no—
166C3×D4—22no—
167C3×D4—22no—
168C3×D4C2×D412yes—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4C2411yescharacteristic
174#67—24no—
175#67—24no—
176#67—22no—
177#67—24no—
178#67—22no—
179#67—22no—
180C2×Dic3—24no—
181C2×Dic3—22no—
182C2×Dic3—22no—
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3C2×D412yes—
188C2×Dic3C2×D411yescharacteristic
189D12—22no—
190D12C2×D412yes—
191C4×D3—24no—
192C4×D3—22no—
193C4×D3—22no—
194C4×D3—22no—
195C4×D3—42no—
196C4×D3C2×D412yes—
197C4×D3C2×D412yes—
198Dic6—24no—
199Dic6—22no—
200Dic6—22no—
201Dic6—22no—
202Dic6—41no—
203Dic6—41no—
204Dic6C2×D412yes—
205C3×C8—22no—
206C3⋊C8—22no—
207#41—22no—
208#41—32no—
209#41—32no—
210#41—32no—
211#41—62no—
212#41—62no—
213#41—64no—
214#41—62no—
215#41—62no—
216#41—62no—
217#41—64no—
218#41—62no—
219#41—64no—
220#41—64no—
221#41—64no—
222#41—62no—
223#41C22×D312yes—
224C2×Q8—31no—
225C2×Q8—32no—
226C2×Q8—31no—
227C2×Q8—61no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—62no—
231C2×Q8—62no—
232C2×Q8—62no—
233C2×Q8C22×D311yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—32no—
237C2×D4—32no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—64no—
241C2×D4—62no—
242C2×D4—64no—
243C2×D4—62no—
244C2×D4C22×D311yescharacteristic
245C22×C4—31no—
246C22×C4—32no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—62no—
251Q16—22no—
252Q16—62no—
253Q16—62no—
254Q16—62no—
255SD16—22no—
256SD16—62no—
257SD16—62no—
258SD16—62no—
259M16—32no—
260M16—62no—
261M16C22×D312yes—
262C2×C8—62no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C22⋊C4—22no—
267C22⋊C4—61no—
268C22⋊C4—62no—
269C42—31no—
270C42C22×D311yescharacteristic
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—42no—
274C22×C3—82no—
275C22×C3C22×D411yescharacteristic
276C2×D3—22no—
277C2×D3—41no—
278C2×D3—82no—
279C2×D3—82no—
280C2×D3#12012yes—
281C3×C4—22no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4C22×D412yes—
285Dic3—22no—
286Dic3—22no—
287Dic3—41no—
288Dic3—41no—
289Dic3#12011yescharacteristic
290Dic3#12011yescharacteristic
291C23—22no—
292C23—31no—
293C23—62no—
294C23—62no—
295C23D3×D412yes—
296Q8—22no—
297Q8—32no—
298Q8—62no—
299Q8—64no—
300Q8—62no—
301Q8—62no—
302Q8—121no—
303Q8—121no—
304Q8D3×D412yes—
305D4—24no—
306D4—22no—
307D4—22no—
308D4—32no—
309D4—62no—
310D4—62no—
311D4—64no—
312D4—62no—
313D4—64no—
314D4—64no—
315D4—62no—
316D4D3×D412yes—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—31no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—64no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—64no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—122no—
337C2×C4C23×D311yescharacteristic
338C8—22no—
339C8—62no—
340C2×C3—21no—
341C2×C3—42no—
342C2×C3—42no—
343C2×C3C2×#12011yescharacteristic
344D3—22no—
345D3—42no—
346C22—22no—
347C22—22no—
348C22—32no—
349C22—42no—
350C22—62no—
351C22—82no—
352C22—121no—
353C22—242no—
354C22—242no—
355C22C2×D3×D411yescharacteristic
356C4—22no—
357C4—31no—
358C4—31no—
359C4—41no—
360C4—41no—
361C4—62no—
362C4—62no—
363C4—121no—
364C4—121no—
365C4C2×D3×D412yes—
366C3#302211yescharacteristic, complemented, Sylow
367C2—21no—
368C2—42no—
369C2—42no—
370C2—62no—
371C2—122no—
372C2D3×#12011yescharacteristic
subgroupG/H
centerC2D3×#120
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#455C2
socleC2×C3C2×#120
prime radical#5966C2
hypercenter#455D3
hypocenterC3#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
2. D3×#137C2semidirect
3. #5675C2semidirect
4. C3×#455C2semidirect
5. #5242C2semidirect
6. #5205C2semidirect
7. #5061C2semidirect
8. #4826C2semidirect
9. #4821C2semidirect
10. #4746C2semidirect
11. #4745C2semidirect
18. C2×D3×Q8C22semidirect
25. C3×#137C22semidirect
26. C3×#124C22semidirect
27. #951C22semidirect
28. #944C22semidirect
29. #942C22semidirect
30. #936C22semidirect
35. #909C22semidirect
36. D3×M16C22semidirect
39. #872C22semidirect
40. #869C22semidirect
41. C3×#104C22semidirect
42. C3×#100C22semidirect
43. #837C22semidirect
44. #833C22semidirect
45. #825C22semidirect
46. #824C22semidirect
47. #805C22semidirect
56. #455D3semidirect
140. #137C2×D3semidirect
147. #124C2×D3semidirect
154. #104C2×D3semidirect
157. #100C2×D3semidirect
366. C3#3022semidirect

Maximal subgroups

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

subgroupindex
#59662
D3×#1372
#56752
C3×#4552
#52422
#52052
#50612
#48262
#48212
#47462
#47452
#30223