Finite Group Databasebeta

← #78912

Subgroups — #78912

Order 384.

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

normal subgroups only

This group has 472 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#5825C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic
3C2×#909C211yesmaximal, characteristic, complemented
4#5504C211yesmaximal, characteristic, complemented
5C3×#417C211yesmaximal, characteristic, complemented
6#5119C211yesmaximal, characteristic, complemented
7C2×#810C211yesmaximal, characteristic, complemented
8#4818C212yesmaximal, complemented
9#4795C212yesmaximal, complemented
10#4787C212yesmaximal, complemented
11#4767C212yesmaximal, complemented
12#3046—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×M16C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18#950—24no—
19#944C2212yes—
20C2×#216C2211yescharacteristic
21C2×#215C2211yescharacteristic
22#936C2212yes—
23#935—22no—
24C2×#214C2211yescharacteristic
25C2×#213C2211yescharacteristic
26C4×#67—22no—
27#926C2211yescharacteristic
28#924—21no—
29#924—21no—
30C22×C3⋊C8C2211yescharacteristic
31#909—24no—
32C2×Dic12C2212yes—
33C2×#204C2212yes—
34#893C2212yes—
35#892—21no—
36D3×C4⋊C4—21no—
37#887C2212yes—
38C3×#103C2212yes—
39C3×#102C2212yes—
40Dic3⋊C8C2212yes—
41Dic3⋊C8C2212yes—
42#810C2212yes—
43#810C2212yes—
44C2×#137—31no—
45C2×#135—31no—
46#536—31no—
47#520—31no—
48#464—32no—
49#463—32no—
50#462—32no—
51#461—32no—
52#424—31no—
53#417D311yescharacteristic, complemented
54C2×#103—31no—
55C3×#41—22no—
56C3×#41—22no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4—21no—
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C22×Dic3—21no—
65#235—22no—
66#235D414yes—
67C2×D12C2311yescharacteristic
68C2×C4×D3—21no—
69C2×C4×D3—22no—
70C2×Dic6—22no—
71C2×Dic6—22no—
72C2×Dic6—21no—
73C2×Dic6C2312yes—
74C2×Dic6C2311yescharacteristic
75C3×M16—22no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2312yes—
79#217—22no—
80#217—22no—
81#217D412yes—
82#216—22no—
83#215—22no—
84#214—22no—
85#213—22no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3D411yescharacteristic
88C4⋊Dic3D411yescharacteristic
89Dic3⋊C4—22no—
90Dic3⋊C4—22no—
91Dic3⋊C4—22no—
92C4×Dic3—22no—
93C4×Dic3C2312yes—
94C2×C3⋊C8—22no—
95C2×C3⋊C8C2311yescharacteristic
96C2×C3⋊C8C2311yescharacteristic
97Dic12—24no—
98#204—24no—
99C2×#41—31no—
100C2×#41C2×D311yescharacteristic
101C22×Q8—31no—
102#137—64no—
103#135—64no—
104C2×Q16—31no—
105C2×Q16—32no—
106C2×SD16—31no—
107C2×SD16—31no—
108C2×SD16—32no—
109C2×D8—31no—
110C2×M16C2×D311yescharacteristic
111C22×C8—31no—
112C4⋊Q8—32no—
113#124—32no—
114#123—62no—
115C22⋊Q8—61no—
116C22⋊Q8—61no—
117C4×Q8—32no—
118C4×D4—32no—
119C4×D4—62no—
120#117—31no—
121#117—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4C2×D311yescharacteristic
124C4⋊C8—32no—
125C4⋊C8—32no—
126#103—32no—
127#103—32no—
128#103C2×D312yes—
129#102C2×D312yes—
130C23×C3—21no—
131C23×C3C2×D411yescharacteristic
132C22×D3—21no—
133C3×Q8—21no—
134C3×Q8—21no—
135C3×D4—21no—
136C3×D4—21no—
137C3×D4—42no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—42no—
141C2×C3×C4—81no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yes—
145C2×C3×C4C2411yescharacteristic
146#67—24no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—42no—
150C2×Dic3C2×D412yes—
151C2×Dic3C2×D412yes—
152D12—21no—
153D12C2×D412yes—
154C4×D3—24no—
155C4×D3—82no—
156Dic6—21no—
157Dic6—22no—
158Dic6—22no—
159Dic6—42no—
160Dic6C2×D412yes—
161C3×C8—22no—
162C3⋊C8—21no—
163C3⋊C8—21no—
164#41—22no—
165#41—22no—
166#41—34no—
167#41—62no—
168C2×Q8—31no—
169C2×Q8—32no—
170C2×Q8—61no—
171C2×Q8—62no—
172C2×Q8—62no—
173C2×Q8C22×D311yescharacteristic
174C2×D4—21no—
175C2×D4—31no—
176C2×D4—61no—
177C2×D4C22×D311yescharacteristic
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4C22×D311yescharacteristic
184Q16—64no—
185Q16—62no—
186SD16—64no—
187SD16—62no—
188SD16—62no—
189D8—62no—
190M16—22no—
191C2×C8—31no—
192C2×C8—31no—
193C2×C8—62no—
194C2×C8C22×D312yes—
195C4⋊C4—21no—
196C4⋊C4—31no—
197C4⋊C4—31no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C4⋊C4C22×D312yes—
202C22⋊C4—32no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C42—32no—
207C42—62no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3#12112yes—
214C22×C3C22×D411yescharacteristic
215C2×D3—22no—
216C2×D3—41no—
217C2×D3—41no—
218C3×C4—41no—
219C3×C4—42no—
220C3×C4#12112yes—
221C3×C4C22×D411yescharacteristic
222C3×C4C2×#4111yescharacteristic
223Dic3—22no—
224Dic3—22no—
225Dic3—22no—
226Dic3—42no—
227C23—21no—
228C23—61no—
229C23D3×D411yescharacteristic
230Q8—21no—
231Q8—21no—
232Q8—32no—
233Q8—61no—
234Q8—62no—
235Q8—62no—
236Q8—122no—
237D4—21no—
238D4—21no—
239D4—32no—
240D4—42no—
241D4—64no—
242D4—61no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—32no—
246C2×C4—32no—
247C2×C4—42no—
248C2×C4—61no—
249C2×C4—64no—
250C2×C4—62no—
251C2×C4—81no—
252C2×C4—122no—
253C2×C4—242no—
254C2×C4#23512yes—
255C2×C4D3×D411yescharacteristic
256C2×C4D3×D412yes—
257C2×C4C23×D311yescharacteristic
258C8—22no—
259C8—61no—
260C8—61no—
261C2×C3—22no—
262C2×C3—81no—
263C2×C3C2×#12111yescharacteristic
264C2×C3#57711yescharacteristic
265C2×C3#57811yescharacteristic
266D3—42no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—62no—
273C22—121no—
274C22—121no—
275C22#88412yes—
276C22C2×D3×D411yescharacteristic
277C4—41no—
278C4—42no—
279C4—62no—
280C4—62no—
281C4—62no—
282C4—122no—
283C4#88412yes—
284C4C2×#23511yescharacteristic
285C4C2×D3×D411yescharacteristic
286C3#304611yescharacteristic, complemented, Sylow
287C2—22no—
288C2—81no—
289C2—122no—
290C2C2×#88411yescharacteristic
291C2#577511yescharacteristic
292C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
prime radicalC2×#950C2
hypercenter#417D3
hypocenterC3#3046
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. #5825C2semidirect
3. C2×#909C2semidirect
4. #5504C2semidirect
5. C3×#417C2semidirect
6. #5119C2semidirect
7. C2×#810C2semidirect
8. #4818C2semidirect
9. #4795C2semidirect
10. #4787C2semidirect
11. #4767C2semidirect
53. #417D3semidirect
286. C3#3046semidirect

Maximal subgroups

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

subgroupindex
#58252
C2×#9502
C2×#9092
#55042
C3×#4172
#51192
C2×#8102
#48182
#47952
#47872
#47672
#30463