Finite Group Databasebeta

← #83902

Subgroups — #83902

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#135C211yesmaximal, characteristic
2#5827C211yesmaximal, characteristic, complemented
3#5825C211yesmaximal, characteristic, complemented
4C2×#951C211yesmaximal, characteristic, complemented
5#5232C211yesmaximal, characteristic, complemented
6#5182C212yesmaximal, complemented
7#5166C212yesmaximal, complemented
8#5162C212yesmaximal, complemented
9#5153C212yesmaximal, complemented
10C2×#816C211yesmaximal, characteristic, complemented
11#5102C211yesmaximal, characteristic, complemented
12#3046—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C2×C3×#41C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C3×#135—24no—
17C2×C3×Q16C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C22×C3×C8C2211yescharacteristic
22#951—24no—
23Q8×Dic3C2212yes—
24#944C2212yes—
25C2×#216C2212yes—
26#936C2212yes—
27#935—22no—
28D4×Dic3—22no—
29D4×Dic3C2212yes—
30C2×#214C2212yes—
31#926—21no—
32#926C2211yescharacteristic
33C2×C4⋊Dic3—21no—
34C2×C4⋊Dic3C2211yescharacteristic
35#924—21no—
36#924—21no—
37C2×#208C2211yescharacteristic
38#835C2212yes—
39#832C2212yes—
40#816C2212yes—
41#816C2212yes—
42Dic3⋊C8C2212yes—
43Dic3⋊C8C2212yes—
44C2×#137—31no—
45C2×#135D311yescharacteristic
46#536—31no—
47#520—31no—
48#464—32no—
49#463—32no—
50#462—32no—
51#461—32no—
52#424—31no—
53#417—31no—
54C2×#103—31no—
55C3×#41—22no—
56C3×#41—22no—
57C3×#41—22no—
58C3×#41D414yes—
59C2×C3×Q8C2311yescharacteristic
60C2×C3×Q8C2311yescharacteristic
61C2×C3×D4—21no—
62C2×C3×D4—21no—
63C2×C3×D4C2311yescharacteristic
64C2×C3×D4C2311yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4—21no—
67C22×C3×C4C2311yescharacteristic
68C22×Dic3—21no—
69C22×Dic3—21no—
70C22×Dic3—22no—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6—21no—
74C2×Dic6C2312yes—
75C3×Q16—22no—
76C3×SD16—22no—
77C3×SD16—22no—
78C3×D8—22no—
79C2×C3×C8—22no—
80C2×C3×C8C2311yescharacteristic
81C2×C3×C8C2311yescharacteristic
82#217—22no—
83#217—22no—
84#217—22no—
85#217D412yes—
86#216—24no—
87#214—24no—
88C4⋊Dic3—21no—
89C4⋊Dic3—22no—
90C4⋊Dic3D411yescharacteristic
91C4⋊Dic3D411yescharacteristic
92C4⋊Dic3C2312yes—
93Dic3⋊C4—22no—
94Dic3⋊C4—22no—
95C4×Dic3—22no—
96C4×Dic3C2312yes—
97#208—22no—
98C2×C3⋊C8C2312yes—
99C2×#41C2×D311yescharacteristic
100C2×#41C2×D311yescharacteristic
101C22×Q8—31no—
102#137—64no—
103#135—24no—
104C2×Q16—32no—
105C2×Q16C2×D311yescharacteristic
106C2×SD16—32no—
107C2×SD16C2×D311yescharacteristic
108C2×SD16C2×D311yescharacteristic
109C2×D8C2×D311yescharacteristic
110C2×M16—31no—
111C22×C8C2×D311yescharacteristic
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—31no—
123C2×C4⋊C4—61no—
124C4⋊C8—32no—
125C4⋊C8—32no—
126#103—32no—
127#103—32no—
128#103—32no—
129#102—32no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3C2×D411yescharacteristic
133C3×Q8—21no—
134C3×Q8—21no—
135C3×Q8—21no—
136C3×Q8C2×D412yes—
137C3×D4—24no—
138C3×D4—21no—
139C3×D4—21no—
140C3×D4—21no—
141C3×D4—42no—
142C3×D4C2×D412yes—
143C2×C3×C4—21no—
144C2×C3×C4—24no—
145C2×C3×C4—21no—
146C2×C3×C4—42no—
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2×D412yes—
149C2×C3×C4#4112yes—
150C2×C3×C4C2411yescharacteristic
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—42no—
154C2×Dic3—82no—
155C2×Dic3—81no—
156C2×Dic3C2×D412yes—
157C2×Dic3C2×D412yes—
158Dic6—22no—
159Dic6—22no—
160Dic6—42no—
161C3×C8—21no—
162C3×C8—21no—
163C3⋊C8—22no—
164#41—22no—
165#41—22no—
166#41—22no—
167#41#6714yes—
168C2×Q8—32no—
169C2×Q8—61no—
170C2×Q8—62no—
171C2×Q8—62no—
172C2×Q8C22×D311yescharacteristic
173C2×Q8C22×D311yescharacteristic
174C2×D4—21no—
175C2×D4—21no—
176C2×D4C22×D311yescharacteristic
177C2×D4C22×D311yescharacteristic
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4C22×D311yescharacteristic
184Q16—22no—
185Q16—64no—
186SD16—22no—
187SD16—22no—
188SD16—64no—
189D8—22no—
190M16—62no—
191C2×C8—22no—
192C2×C8—32no—
193C2×C8C22×D311yescharacteristic
194C2×C8C22×D311yescharacteristic
195C4⋊C4—31no—
196C4⋊C4—31no—
197C4⋊C4—32no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C4⋊C4—61no—
202C22⋊C4—32no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C42—32no—
207C42—62no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—22no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3#12112yes—
217C22×C3C22×D411yescharacteristic
218C3×C4—22no—
219C3×C4—41no—
220C3×C4#12112yes—
221C3×C4C22×D411yescharacteristic
222C3×C4C2×#4111yescharacteristic
223Dic3—22no—
224Dic3—22no—
225Dic3—42no—
226Dic3—42no—
227C23—21no—
228C23—21no—
229C23D3×D411yescharacteristic
230Q8—21no—
231Q8—21no—
232Q8—21no—
233Q8—62no—
234Q8—62no—
235Q8—122no—
236Q8C2×#6712yes—
237D4—24no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—42no—
242D4C2×#6712yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—21no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—42no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—122no—
252C2×C4—242no—
253C2×C4—241no—
254C2×C4D3×D411yescharacteristic
255C2×C4#23712yes—
256C2×C4C2×#6712yes—
257C2×C4C23×D311yescharacteristic
258C8—21no—
259C8—21no—
260C8—62no—
261C2×C3—22no—
262C2×C3—42no—
263C2×C3—81no—
264C2×C3C2×#12111yescharacteristic
265C2×C3#57711yescharacteristic
266C2×C3#57811yescharacteristic
267C22—21no—
268C22—21no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22#93912yes—
276C22C2×D3×D411yescharacteristic
277C4—22no—
278C4—41no—
279C4—62no—
280C4—62no—
281C4—122no—
282C4—122no—
283C4#93912yes—
284C4C2×#23711yescharacteristic
285C4C22×#6711yescharacteristic
286C3#304611yescharacteristic, complemented, Sylow
287C2—22no—
288C2—42no—
289C2—81no—
290C2#577011yescharacteristic
291C2#577111yescharacteristic
292C2C2×#93911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC2×C3×#135C2
socleC22×C3C22×D4
hypercenterC2×#135D3
hypocenterC3#3046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
2. #5827C2semidirect
3. #5825C2semidirect
4. C2×#951C2semidirect
5. #5232C2semidirect
6. #5182C2semidirect
7. #5166C2semidirect
8. #5162C2semidirect
9. #5153C2semidirect
10. C2×#816C2semidirect
11. #5102C2semidirect
286. C3#3046semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1352
#58272
#58252
C2×#9512
#52322
#51822
#51662
#51622
#51532
C2×#8162
#51022
#30463