Finite Group Databasebeta

← #78910

Subgroups — #78910

Order 384.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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#5832C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C2×#908C211yesmaximal, characteristic, complemented
4C2×#893C211yesmaximal, characteristic, complemented
5C3×#417C211yesmaximal, characteristic, complemented
6#5118C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic, complemented
8#4819C212yesmaximal, complemented
9#4818C212yesmaximal, complemented
10#4787C212yesmaximal, complemented
11#4786C212yesmaximal, complemented
12#3036—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C22×C4×D3—21no—
17C2×C3×M16C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19#949—24no—
20#947C2211yescharacteristic, complemented
21#944C2211yescharacteristic, complemented
22C2×#215C2212yes—
23#940C2211yescharacteristic, complemented
24#939—21no—
25#939—21no—
26#936C2211yescharacteristic, complemented
27C2×#213C2212yes—
28C4×#67—21no—
29C4×#67—21no—
30C2×(C2×D3)⋊C4—21no—
31C2×C4×Dic3C2211yescharacteristic
32C2×#208C2211yescharacteristic
33#908—24no—
34C2×D24C2212yes—
35C2×#204C2212yes—
36#893—21no—
37#893—21no—
38#893C2212yes—
39#893C2212yes—
40C3×#103C2212yescomplemented
41C3×#102C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43Dic3⋊C8C2212yescomplemented
44#810C2212yescomplemented
45#809C2212yescomplemented
46C2×AGL(1,ℤ/8)—31no—
47C2×AGL(1,ℤ/8)—31no—
48#532—31no—
49C2×C4×D4—31no—
50#463—32no—
51#463—32no—
52#461—32no—
53#459—32no—
54#423—31no—
55#417—31no—
56#417D311yescharacteristic, complemented
57C23×D3—21no—
58C3×#41—22no—
59C3×#41—22no—
60C2×C3×Q8C2311yescharacteristic
61C2×C3×D4—21no—
62C2×C3×D4C2311yescharacteristic
63C22×C3×C4—21no—
64C22×C3×C4—21no—
65C22×C3×C4C2311yescharacteristic
66C2×#67—21no—
67C2×#67—21no—
68C2×#67—21no—
69C22×Dic3D411yescharacteristic
70C22×Dic3D411yescharacteristic
71#235—22no—
72#235—22no—
73C2×D12—21no—
74C2×D12—22no—
75C2×D12D414yescomplemented
76C2×D12C2311yescharacteristic
77C2×D12C2312yes—
78C2×C4×D3—21no—
79C2×C4×D3—22no—
80C2×C4×D3—22no—
81C2×C4×D3—44no—
82C2×Dic6C2311yescharacteristic
83C3×M16—22no—
84C2×C3×C8C2312yes—
85C3×C4⋊C4—21no—
86C3×C4⋊C4C2312yes—
87#217—21no—
88#217—21no—
89#215—24no—
90#213—24no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—21no—
94(C2×D3)⋊C4—21no—
95Dic3⋊C4—21no—
96Dic3⋊C4—21no—
97C4×Dic3D411yescharacteristic
98C4×Dic3D411yescharacteristic
99C4×Dic3C2311yescharacteristic
100C4×Dic3C2311yescharacteristic
101#208—22no—
102C2×C3⋊C8C2312yes—
103D24—24no—
104#204—24no—
105C2×#41—31no—
106C2×#41C2×D311yescharacteristic
107C22×D4—31no—
108C23×C4—61no—
109AGL(1,ℤ/8)—64no—
110AGL(1,ℤ/8)—64no—
111C2×SD16—32no—
112C2×SD16—32no—
113C2×D8—32no—
114C2×D8—32no—
115C2×M16—31no—
116C2×M16C2×D311yescharacteristic
117C4⋊Q8—31no—
118#127—31no—
119#124—31no—
120#124—31no—
121#121—61no—
122#121—61no—
123C4×D4—32no—
124C4×D4—32no—
125C4×D4—61no—
126C4×D4—61no—
127C4×D4—61no—
128C4×D4—61no—
129C2×C4⋊C4C2×D311yescharacteristic
130C2×C22⋊C4—61no—
131C2×C42—31no—
132C4⋊C8—32no—
133C4⋊C8—32no—
134#103—32no—
135#103C2×D312yescomplemented
136#102—32no—
137#102C2×D312yescomplemented
138C23×C3—21no—
139C23×C3C2×D411yescharacteristic
140C22×D3—24no—
141C22×D3—22no—
142C22×D3—21no—
143C22×D3—42no—
144C22×D3—42no—
145C3×Q8—21no—
146C3×Q8—21no—
147C3×D4—21no—
148C3×D4—21no—
149C3×D4—42no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4C2×D411yescharacteristic
155C2×C3×C4C2×D412yes—
156C2×C3×C4#4112yes—
157C2×C3×C4C2411yescharacteristic
158#67—42no—
159#67—42no—
160#67—42no—
161C2×Dic3—22no—
162C2×Dic3—21no—
163C2×Dic3—41no—
164C2×Dic3C2×D411yescharacteristic
165C2×Dic3C2×D411yescharacteristic
166C2×Dic3C2×D411yescharacteristic
167C2×Dic3C2×D411yescharacteristic
168D12—24no—
169D12—21no—
170D12—22no—
171D12—21no—
172D12C2×D414yes—
173C4×D3—44no—
174C4×D3—42no—
175C4×D3—44no—
176Dic6—21no—
177Dic6—21no—
178C3×C8—22no—
179C3⋊C8—22no—
180C24—61no—
181#41—22no—
182#41—22no—
183#41—62no—
184#41—62no—
185C2×Q8—31no—
186C2×Q8C22×D311yescharacteristic
187C2×D4—21no—
188C2×D4—34no—
189C2×D4—32no—
190C2×D4—31no—
191C2×D4—61no—
192C2×D4—61no—
193C2×D4—61no—
194C2×D4—62no—
195C2×D4—61no—
196C2×D4C22×D311yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4—61no—
204C22×C4—124no—
205C22×C4C22×D311yescharacteristic
206SD16—64no—
207SD16—64no—
208D8—64no—
209D8—64no—
210M16—22no—
211M16—62no—
212C2×C8—32no—
213C2×C8C22×D312yes—
214C4⋊C4—21no—
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4C22×D312yes—
218C22⋊C4—62no—
219C22⋊C4—62no—
220C22⋊C4—61no—
221C22⋊C4—61no—
222C22⋊C4—61no—
223C22⋊C4—61no—
224C42—31no—
225C42—31no—
226C42—31no—
227C42—31no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3#12112yes—
234C22×C3C22×D411yescharacteristic
235C2×D3—24no—
236C2×D3—48no—
237C2×D3—42no—
238C2×D3—42no—
239C2×D3—41no—
240C2×D3—41no—
241C2×D3—41no—
242C3×C4—41no—
243C3×C4—42no—
244C3×C4#12112yes—
245C3×C4C22×D411yescharacteristic
246C3×C4C2×#4111yescharacteristic
247Dic3—21no—
248Dic3—21no—
249Dic3—21no—
250Dic3—41no—
251Dic3AGL(1,ℤ/8)12yescomplemented
252C23—21no—
253C23—64no—
254C23—62no—
255C23—61no—
256C23—122no—
257C23—122no—
258C23D3×D411yescharacteristic
259Q8—21no—
260Q8—21no—
261Q8—61no—
262Q8—61no—
263D4—21no—
264D4—21no—
265D4—34no—
266D4—42no—
267D4—64no—
268D4—62no—
269D4—61no—
270D4—61no—
271D4—122no—
272D4—122no—
273D4—122no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—62no—
283C2×C4—61no—
284C2×C4—124no—
285C2×C4—124no—
286C2×C4—121no—
287C2×C4—122no—
288C2×C4#23512yes—
289C2×C4D3×D411yescharacteristic
290C2×C4D3×D412yes—
291C2×C4C23×D311yescharacteristic
292C8—22no—
293C8—62no—
294C2×C3—22no—
295C2×C3—81no—
296C2×C3C2×#12111yescharacteristic
297C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
298C2×C3#57511yescharacteristic
299D3—44no—
300D3—81no—
301C22—21no—
302C22—21no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—64no—
307C22—128no—
308C22—122no—
309C22—122no—
310C22—121no—
311C22—121no—
312C22—121no—
313C22#88412yes—
314C22C2×D3×D411yescharacteristic
315C4—32no—
316C4—41no—
317C4—42no—
318C4—61no—
319C4—61no—
320C4—61no—
321C4—121no—
322C4#88412yes—
323C4C2×#23511yescharacteristic
324C4C2×D3×D411yescharacteristic
325C3#303611yescharacteristic, complemented, Sylow
326C2—22no—
327C2—81no—
328C2—124no—
329C2—241no—
330C2C2×#88411yescharacteristic
331C2D3×AGL(1,ℤ/8)11yescharacteristic
332C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
hypercenter#417D3
hypocenterC3#3036
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #5832C2semidirect
2. C2×#949C2semidirect
3. C2×#908C2semidirect
4. C2×#893C2semidirect
5. C3×#417C2semidirect
6. #5118C2semidirect
7. #4966C2semidirect
8. #4819C2semidirect
9. #4818C2semidirect
10. #4787C2semidirect
11. #4786C2semidirect
20. #947C22semidirect
21. #944C22semidirect
23. #940C22semidirect
26. #936C22semidirect
40. C3×#103C22semidirect
41. C3×#102C22semidirect
42. Dic3⋊C8C22semidirect
43. Dic3⋊C8C22semidirect
44. #810C22semidirect
45. #809C22semidirect
56. #417D3semidirect
75. C2×D12D4semidirect
135. #103C2×D3semidirect
137. #102C2×D3semidirect
251. Dic3AGL(1,ℤ/8)semidirect
325. C3#3036semidirect

Maximal subgroups

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

subgroupindex
#58322
C2×#9492
C2×#9082
C2×#8932
C3×#4172
#51182
#49662
#48192
#48182
#47872
#47862
#30363