Finite Group Databasebeta

← #13159124

Subgroups — #13159124

Order 1728.

744 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C3×#64097C211yesmaximal, characteristic
2#11723406C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722051C211yesmaximal, characteristic
7#11722042C211yesmaximal, characteristic
8#10592553—31nomaximal
9#10590487—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634—21no—
12C33×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#64097D311yescharacteristic
22#64096—31no—
23C3×#888—34no—
24#63844—34no—
25C32×#106—31no—
26#63643—31no—
27#63641—31no—
28#63637—31no—
29#63637—34no—
30#63637—64no—
31#63635—31no—
32C3×#808D314yes—
33#63574—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63567—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39#6308—21no—
40#6308Q812yescomplemented
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—42no—
45C2×C32×Dic3—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48#5094—91no—
49#4854—94no—
50C2×#634—21no—
51C2×#634—31no—
52C2×#634—34no—
53C2×#634—64no—
54C2×C3×Dic6—64no—
55C2×C32×C8—31no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—31no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—21no—
62#3750—61no—
63C4×C3⋊Dic3—61no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C3×C4⋊Dic3—34no—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—34no—
72C2×C3×C3⋊C8C2×D314yes—
73#3724—64no—
74Dic32—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—22no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3SD1612yes—
83C32×Dic3—41no—
84C3×C4⋊Q8—91no—
85#888—34no—
86#887—94no—
87C4×Dic6—91no—
88C3×#106—34no—
89#816—94no—
90Dic3⋊C8—94no—
91#810—94no—
92#810—91no—
93#810—91no—
94#810—184no—
95#808—94no—
96#808—91no—
97#808C3⋊D311yescharacteristic
98C4⋊(C3⋊C8)—91no—
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4—44no—
103C2×C32×C4—61no—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—64no—
109C2×C3⋊Dic3—124no—
110C2×C3⋊Dic3#6711yescharacteristic
111C2×C3⋊Dic3#6711yescharacteristic
112#634—32no—
113#634—38no—
114#634—42no—
115#634—68no—
116#634—61no—
117#634—64no—
118#634—124no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—64no—
123C3×Dic6—128no—
124C32×C8—61no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—124no—
128C3×C3⋊C8—24no—
129#475—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33C2×SD1611yescharacteristic
132C2×C33#13711yescharacteristic
133C2×C3×Q8—181no—
134C2×Dic6—64no—
135C2×Dic6—94no—
136C2×Dic6—91no—
137C2×Dic6—184no—
138C2×C3×C8—34no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—34no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4D3214yes—
144C3×C4⋊C4C2×C3⋊D311yescharacteristic
145C3×C42—91no—
146C4⋊Dic3—34no—
147C4⋊Dic3C2×C3⋊D311yescharacteristic
148Dic3⋊C4—64no—
149Dic3⋊C4—181no—
150Dic3⋊C4—184no—
151C4×Dic3—34no—
152C4×Dic3—181no—
153C4×Dic3—184no—
154C2×C3⋊C8—94no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—184no—
157C2×C3⋊C8C2×C3⋊D311yescharacteristic
158C22×C32—24no—
159C22×C32D3×D414yes—
160C22×C32C2×#6711yescharacteristic
161C32×C4—24no—
162C32×C4—24no—
163C32×C4—41no—
164C32×C4—44no—
165C32×C4—84no—
166C32×C4—121no—
167C32×C4#23514yes—
168C32×C4D3×Q814yes—
169C32×C4D3×Q811yescharacteristic
170C32×C4#23911yescharacteristic
171C3⋊Dic3—21no—
172C3⋊Dic3—41no—
173C3⋊Dic3—62no—
174C3⋊Dic3—68no—
175C3⋊Dic3—128no—
176C3⋊Dic3#21412yes—
177C3×Dic3—38no—
178C3×Dic3—44no—
179C3×Dic3—64no—
180C3×Dic3—124no—
181C4⋊Q8—91no—
182C4×Q8—271no—
183#106—31no—
184#106—271no—
185C4⋊C8—271no—
186#103—271no—
187#103—271no—
188C33#47511yescharacteristic, complemented, Sylow
189C3×Q8—362no—
190C2×C3×C4—24no—
191C2×C3×C4—21no—
192C2×C3×C4—24no—
193C2×C3×C4—44no—
194C2×C3×C4—64no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—181no—
198C2×C3×C4C2×D3214yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—34no—
202C2×Dic3—34no—
203C2×Dic3—64no—
204C2×Dic3—184no—
205C2×Dic3—181no—
206C2×Dic3—364no—
207Dic6—98no—
208Dic6—92no—
209Dic6—128no—
210Dic6—188no—
211Dic6—184no—
212Dic6—181no—
213Dic6—364no—
214C3×C8—64no—
215C3⋊C8—21no—
216C3⋊C8—184no—
217C3⋊C8—181no—
218C3⋊C8—364no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32#89614yes—
223C2×C32D3×SD1614yes—
224C2×C32#91514yes—
225C2×C32C2×#21411yescharacteristic
226C2×C32#94211yescharacteristic
227C2×C32#94611yescharacteristic
228C2×Q8—181no—
229C2×Q8—271no—
230C2×C8—31no—
231C2×C8—271no—
232C4⋊C4—31no—
233C4⋊C4—91no—
234C4⋊C4—181no—
235C4⋊C4—541no—
236C4⋊C4D3×C3⋊D311yescharacteristic
237C42—91no—
238C42—541no—
239C22×C3—24no—
240C22×C3D3×#6714yes—
241C22×C3D4×C3⋊D311yescharacteristic
242C3×C4—24no—
243C3×C4—24no—
244C3×C4—41no—
245C3×C4—44no—
246C3×C4—84no—
247C3×C4—92no—
248C3×C4—124no—
249C3×C4—181no—
250C3×C4—361no—
251C3×C4#379714yes—
252C3×C4#379914yes—
253C3×C4#382811yescharacteristic
254C3×C4Q8×C3⋊D311yescharacteristic
255Dic3—38no—
256Dic3—41no—
257Dic3—64no—
258Dic3—124no—
259Dic3—182no—
260Dic3—188no—
261Dic3—368no—
262C32—24no—
263C32#485414yescomplemented
264C32#509411yescharacteristic, complemented
265Q8—272no—
266Q8—362no—
267Q8—541no—
268C2×C4—21no—
269C2×C4—61no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—181no—
273C2×C4—541no—
274C2×C4C2×D3×C3⋊D311yescharacteristic
275C8—61no—
276C8—541no—
277C2×C3—24no—
278C2×C3—24no—
279C2×C3—24no—
280C2×C3#6388814yes—
281C2×C3D3×#21414yes—
282C2×C3#6394614yes—
283C2×C3#6410511yescharacteristic
284C2×C3SD16×C3⋊D311yescharacteristic
285C2×C3#6412411yescharacteristic
286C22C3⋊D3×#6711yescharacteristic
287C4—41no—
288C4—92no—
289C4—121no—
290C4—181no—
291C4—361no—
292C4—542no—
293C4#8901911yescharacteristic
294C4#8902211yescharacteristic
295C3—24no—
296C3#1059048714yescomplemented
297C3#1059255311yescharacteristic, complemented
298C2#1172348911yescharacteristic
299C2C3⋊D3×#21411yescharacteristic
300C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
40. #6308Q8semidirect
188. C33#475semidirect
263. C32#4854semidirect
264. C32#5094semidirect
296. C3#10590487semidirect
297. C3#10592553semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
#117234062
#117221802
C32×#8082
#117220532
#117220512
#117220422
#105925533
#105904873