Finite Group Databasebeta

← #80943

Subgroups — #80943

Order 384.

452 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 123 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#5777C212yesmaximal, complemented
2C2×#919C211yesmaximal, characteristic, complemented
3D3×#131C211yesmaximal, characteristic, complemented
4C3×#473C211yesmaximal, characteristic, complemented
5#5203C212yesmaximal, complemented
6#5192C211yesmaximal, characteristic, complemented
7#5142C211yesmaximal, characteristic, complemented
8#4897C211yesmaximal, characteristic, complemented
9D3×#108C211yesmaximal, characteristic, complemented
10#4754C212yesmaximal, complemented
11#4751C212yesmaximal, complemented
12#3067—31nomaximal, Sylow
13D3×#41C2211yescharacteristic
14C2×#239C2212yes—
15C3×#137C2212yes—
16C2×C3×Q16C2211yescharacteristic, complemented
17C3×#131C2211yescharacteristic
18#949C2212yes—
19#948C2211yescharacteristic
20#942C2212yes—
21C2×#215C2212yescomplemented
22#919—22no—
23#919—21no—
24#919—21no—
25#919—21no—
26#919—21no—
27#918—22no—
28#916—22no—
29#914—22no—
30#908C2212yes—
31#907—21no—
32#907C2211yescharacteristic
33D3×M16—21no—
34D3×M16C2212yes—
35D3×M16C2211yescharacteristic
36C2×D24C2211yescharacteristic, complemented
37#901—21no—
38#901C2211yescharacteristic
39C2×D3×C8—21no—
40C2×D3×C8C2211yescharacteristic
41C3×#108C2211yescharacteristic, complemented
42C3×#101C2212yescomplemented
43#836C2212yescomplemented
44#823C2212yescomplemented
45#823C2212yescomplemented
46#822C2212yescomplemented
47#819C2211yescharacteristic, complemented
48#575—32no—
49C2×#135—31no—
50C2×#131—31no—
51#473—31no—
52#473D311yescharacteristic, complemented
53#472—31no—
54#472—31no—
55#471—32no—
56#471—32no—
57C2×#108—31no—
58#413—32no—
59C3×#41C2311yescharacteristic
60C2×C3×Q8C2312yes—
61#239—22no—
62#239—22no—
63#239—22no—
64#239—22no—
65#239D411yescharacteristic
66D3×Q8D411yescharacteristic
67#237—21no—
68#237D411yescharacteristic
69D3×D4—21no—
70D3×D4D411yescharacteristic
71#235—21no—
72#235C2311yescharacteristic
73C2×D12—22no—
74C2×D12C2312yes—
75C2×C4×D3—21no—
76C2×C4×D3—22no—
77C2×C4×D3C2311yescharacteristic
78C3×Q16—22no—
79C3×Q16—21no—
80C3×Q16—21no—
81C3×SD16—22no—
82C3×M16—21no—
83C3×M16C2312yes—
84C3×M16C2311yescharacteristic
85C2×C3×C8—21no—
86C2×C3×C8C2311yescharacteristic
87#216—22no—
88#215—22no—
89#215—22no—
90#215—22no—
91#215—22no—
92#213—22no—
93#208—21no—
94#208C2312yes—
95#208C2311yescharacteristic
96C2×C3⋊C8—21no—
97C2×C3⋊C8C2311yescharacteristic
98D24—22no—
99D24—21no—
100D24—21no—
101#204—22no—
102ΓL(1,𝔽25)—21no—
103ΓL(1,𝔽25)—22no—
104ΓL(1,𝔽25)—21no—
105ΓL(1,𝔽25)—21no—
106D3×C8—21no—
107D3×C8—22no—
108D3×C8—21no—
109D3×C8—21no—
110D3×C8D411yescharacteristic
111D3×C8D411yescharacteristic
112D3×C8D411yescharacteristic
113D3×C8D411yescharacteristic
114C2×#41—31no—
115C2×#41—32no—
116#137—32no—
117#137—62no—
118#137C2×D312yes—
119AGL(1,ℤ/8)—32no—
120AGL(1,ℤ/8)—32no—
121AGL(1,ℤ/8)—62no—
122#135—61no—
123#135—61no—
124#135—61no—
125#135—61no—
126#135—62no—
127#135—62no—
128C2×Q16C2×D311yescharacteristic, complemented
129C2×SD16—32no—
130C2×D8—31no—
131#131—31no—
132#131—31no—
133#131—31no—
134#131—61no—
135#131—61no—
136#131C2×D311yescharacteristic
137C2×M16—31no—
138C2×M16—32no—
139C2×M16—61no—
140C22×C8—31no—
141C22×C8—61no—
142#108—32no—
143#108—31no—
144#108C2×D311yescharacteristic, complemented
145#101—32no—
146#101C2×D312yescomplemented
147#100—32no—
148#100—32no—
149C22×D3—21no—
150C22×D3—22no—
151C22×D3#4111yescharacteristic
152C3×Q8—22no—
153C3×Q8—22no—
154C3×Q8C2×D411yescharacteristic
155C3×D4—21no—
156C3×D4C2×D411yescharacteristic
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4C2411yescharacteristic
160#67—21no—
161#67—21no—
162#67—21no—
163C2×Dic3—21no—
164C2×Dic3#4111yescharacteristic
165D12—21no—
166D12—22no—
167D12—22no—
168D12—42no—
169D12—42no—
170D12C2×D411yescharacteristic
171C4×D3—21no—
172C4×D3—21no—
173C4×D3—21no—
174C4×D3—42no—
175C4×D3—42no—
176C4×D3C2×D411yescharacteristic
177C4×D3C2×D411yescharacteristic
178C4×D3C2×D411yescharacteristic
179C4×D3C2×D411yescharacteristic
180Dic6—21no—
181Dic6C2×D411yescharacteristic
182C3×C8—21no—
183C3×C8—22no—
184C3×C8C2×D411yescharacteristic
185C3×C8C2×D411yescharacteristic
186C3⋊C8—21no—
187C3⋊C8—22no—
188C3⋊C8C2×D411yescharacteristic
189C3⋊C8C2×D411yescharacteristic
190#41—31no—
191#41—31no—
192#41—31no—
193#41—61no—
194#41—61no—
195#41—62no—
196#41—62no—
197#41—62no—
198#41—62no—
199#41C22×D311yescharacteristic
200C2×Q8—31no—
201C2×Q8C22×D312yes—
202C2×D4—31no—
203C2×D4—32no—
204C2×D4—62no—
205C2×D4—61no—
206C22×C4—31no—
207C22×C4—61no—
208C22×C4—62no—
209Q16—22no—
210Q16—21no—
211Q16—21no—
212Q16—62no—
213SD16—22no—
214SD16—62no—
215SD16—62no—
216SD16—62no—
217SD16—62no—
218SD16—62no—
219D8—62no—
220D8—62no—
221D8—61no—
222D8—61no—
223M16—21no—
224M16—31no—
225M16—32no—
226M16—61no—
227M16—61no—
228M16—62no—
229M16—61no—
230M16—61no—
231M16C22×D311yescharacteristic
232M16C22×D312yes—
233C2×C8—21no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—61no—
240C2×C8—61no—
241C2×C8—62no—
242C2×C8—61no—
243C2×C8—61no—
244C2×C8C22×D311yescharacteristic
245C22×C3—21no—
246C22×C3C2×#4111yescharacteristic
247C2×D3—21no—
248C2×D3—41no—
249C2×D3—42no—
250C2×D3—42no—
251C2×D3—82no—
252C2×D3#12111yescharacteristic
253C2×D3#12111yescharacteristic
254C3×C4—21no—
255C3×C4—42no—
256C3×C4C22×D411yescharacteristic
257C3×C4C22×D411yescharacteristic
258Dic3—21no—
259Dic3#12111yescharacteristic
260Dic3#12111yescharacteristic
261C23—31no—
262C23—61no—
263C23—62no—
264Q8—22no—
265Q8—22no—
266Q8—31no—
267Q8—61no—
268Q8D3×D411yescharacteristic
269D4—21no—
270D4—31no—
271D4—61no—
272D4—61no—
273D4—61no—
274D4—61no—
275D4—62no—
276D4—62no—
277D4—122no—
278D4—122no—
279D4D3×D411yescharacteristic
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4C23×D311yescharacteristic
294C8—21no—
295C8—22no—
296C8—31no—
297C8—31no—
298C8—61no—
299C8—62no—
300C8D3×D411yescharacteristic
301C8D3×D411yescharacteristic
302C2×C3—21no—
303C2×C3—41no—
304C2×C3C2×#12111yescharacteristic
305D3—21no—
306D3—21no—
307D3—41no—
308D3—82no—
309C22—21no—
310C22—31no—
311C22—31no—
312C22—61no—
313C22—121no—
314C22—122no—
315C22—122no—
316C22—242no—
317C22D3×#4111yescharacteristic
318C4—21no—
319C4—31no—
320C4—31no—
321C4—42no—
322C4—61no—
323C4C2×D3×D411yescharacteristic
324C4C2×D3×D411yescharacteristic
325C3#306711yescharacteristic, complemented, Sylow
326C2—21no—
327C2—41no—
328C2—61no—
329C2—61no—
330C2—121no—
331C2—242no—
332C2D3×#12111yescharacteristic
subgroupG/H
centerC2D3×#121
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#473C2
socleC2×C3C2×#121
hypercenter#473D3
hypocenterC3#3067
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #5777C2semidirect
2. C2×#919C2semidirect
3. D3×#131C2semidirect
4. C3×#473C2semidirect
5. #5203C2semidirect
6. #5192C2semidirect
7. #5142C2semidirect
8. #4897C2semidirect
9. D3×#108C2semidirect
10. #4754C2semidirect
11. #4751C2semidirect
16. C2×C3×Q16C22semidirect
21. C2×#215C22semidirect
36. C2×D24C22semidirect
41. C3×#108C22semidirect
42. C3×#101C22semidirect
43. #836C22semidirect
44. #823C22semidirect
45. #823C22semidirect
46. #822C22semidirect
47. #819C22semidirect
52. #473D3semidirect
128. C2×Q16C2×D3semidirect
144. #108C2×D3semidirect
146. #101C2×D3semidirect
325. C3#3067semidirect

Maximal subgroups

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

subgroupindex
#57772
C2×#9192
D3×#1312
C3×#4732
#52032
#51922
#51422
#48972
D3×#1082
#47542
#47512
#30673