Finite Group Databasebeta

← #13159142

Subgroups — #13159142

Order 1728.

764 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723931C211yesmaximal, characteristic
2#11723410C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6#11722050C211yesmaximal, characteristic
7#11722040C211yesmaximal, characteristic
8#10592576—31nomaximal
9#10590494—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21#64099—31no—
22#64099—34no—
23#64099—64no—
24#64096—31no—
25#63843—34no—
26C32×#107—31no—
27#63643—31no—
28#63641—31no—
29#63637D311yescharacteristic
30#63635—31no—
31C3×#810—34no—
32C3×#807D314yes—
33#63573—34no—
34#63572—34no—
35#63566—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634Q812yes—
44C2×C32×Dic3—21no—
45C3×C3⋊(C3⋊C8)—21no—
46C32×C3⋊C8—21no—
47#4877—94no—
48#4799—91no—
49C2×#634C2×D311yescharacteristic
50C2×C3×Dic6—34no—
51C2×C32×C8—31no—
52C32×C4⋊C4—24no—
53C32×C4⋊C4—31no—
54C32×C4⋊C4C2×D314yes—
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—34no—
58C4⋊(C3⋊Dic3)—61no—
59C4⋊(C3⋊Dic3)—64no—
60C4⋊(C3⋊Dic3)—64no—
61C4⋊(C3⋊Dic3)—68no—
62#3750—61no—
63#3750—61no—
64#3750—64no—
65#3750—68no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—61no—
69C4×C3⋊Dic3—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3C2×D314yes—
72C2×C3×C3⋊C8—34no—
73C2×C3×C3⋊C8C2×D314yes—
74Dic3⋊Dic3—64no—
75Dic32—64no—
76C22×C33C2×D411yescharacteristic
77C33×C4—41no—
78C33×C4C2×Q811yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)—21no—
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—22no—
84C32×Dic3—41no—
85Q8×Dic3—91no—
86#890—94no—
87#890—91no—
88#890—184no—
89#887—94no—
90C3×#107—34no—
91C3×#103—91no—
92#835—91no—
93#817—91no—
94#816—94no—
95Dic3⋊C8—94no—
96Dic3⋊C8—91no—
97#810—34no—
98#808—94no—
99#807C3⋊D311yescharacteristic
100C2×C32×C4—28no—
101C2×C32×C4—24no—
102C2×C32×C4—21no—
103C2×C32×C4—24no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—34no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—68no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—61no—
117#634—21no—
118#634Dic612yes—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—64no—
121C3×Dic6—38no—
122C3×Dic6—64no—
123C32×C8—61no—
124C3⋊(C3⋊C8)—21no—
125C3×C3⋊C8—24no—
126C3×C3⋊C8—64no—
127#479—271noSylow
128C2×C33C22⋊Q811yescharacteristic
129C2×C33#13511yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—91no—
132C2×Dic6—34no—
133C2×C3×C8—34no—
134C2×C3×C8—91no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—34no—
137C3×C4⋊C4D3214yes—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C4⋊Dic3—94no—
140C4⋊Dic3—91no—
141C4⋊Dic3—184no—
142C4⋊Dic3—181no—
143C4⋊Dic3—184no—
144C4⋊Dic3—181no—
145C4⋊Dic3—188no—
146C4⋊Dic3C2×C3⋊D311yescharacteristic
147Dic3⋊C4—184no—
148Dic3⋊C4—184no—
149Dic3⋊C4—181no—
150Dic3⋊C4—188no—
151C4×Dic3—94no—
152C4×Dic3—91no—
153C4×Dic3—184no—
154C4×Dic3—181no—
155C4×Dic3—184no—
156C2×C3⋊C8—34no—
157C2×C3⋊C8C2×C3⋊D311yescharacteristic
158C22×C32—24no—
159C22×C32D3×D414yes—
160C22×C32D3×D411yescharacteristic
161C32×C4—24no—
162C32×C4—24no—
163C32×C4—48no—
164C32×C4—44no—
165C32×C4—41no—
166C32×C4—121no—
167C32×C4C2×Dic611yescharacteristic
168C32×C4#23514yes—
169C32×C4#23711yescharacteristic
170C32×C4D3×Q814yes—
171C3⋊Dic3—22no—
172C3⋊Dic3—61no—
173C3⋊Dic3—61no—
174C3⋊Dic3—64no—
175C3⋊Dic3—64no—
176C3⋊Dic3—124no—
177C3⋊Dic3—124no—
178C3⋊Dic3—128no—
179C3⋊Dic3—124no—
180C3⋊Dic3—121no—
181C3×Dic3—44no—
182C3×Dic3—68no—
183#125—271no—
184C4×Q8—271no—
185#107—31no—
186#106—271no—
187C4⋊C8—271no—
188#103—91no—
189#103—271no—
190C33#47911yescharacteristic, complemented, Sylow
191C3×Q8—92no—
192C3×Q8—181no—
193C2×C3×C4—28no—
194C2×C3×C4—24no—
195C2×C3×C4—21no—
196C2×C3×C4—24no—
197C2×C3×C4—64no—
198C2×C3×C4—181no—
199C2×C3×C4C2×D3214yes—
200C2×C3×C4C22×C3⋊D311yescharacteristic
201C2×Dic3—21no—
202C2×Dic3—64no—
203C2×Dic3—94no—
204C2×Dic3—94no—
205C2×Dic3—91no—
206C2×Dic3—91no—
207C2×Dic3—184no—
208C2×Dic3—184no—
209C2×Dic3—188no—
210C2×Dic3—184no—
211C2×Dic3—181no—
212Dic6—38no—
213Dic6—64no—
214C3×C8—64no—
215C3×C8—181no—
216C3⋊C8—21no—
217C3⋊C8—64no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32#87811yescharacteristic
222C2×C32#89614yes—
223C2×C32#91214yes—
224C2×C32#91611yescharacteristic
225C2×C32#91814yes—
226C2×C32#91811yescharacteristic
227C2×Q8—91no—
228C2×C8—31no—
229C2×C8—91no—
230C4⋊C4—31no—
231C4⋊C4—271no—
232C4⋊C4—541no—
233C4⋊C4—541no—
234C4⋊C4—541no—
235C4⋊C4D3×C3⋊D311yescharacteristic
236C42—271no—
237C42—541no—
238C22×C3—24no—
239C22×C3#381114yes—
240C22×C3D4×C3⋊D311yescharacteristic
241C3×C4—24no—
242C3×C4—24no—
243C3×C4—48no—
244C3×C4—41no—
245C3×C4—44no—
246C3×C4—124no—
247C3×C4—182no—
248C3×C4D3×Dic614yes—
249C3×C4#379514yes—
250C3×C4#382811yescharacteristic
251C3×C4Q8×C3⋊D311yescharacteristic
252Dic3—41no—
253Dic3—68no—
254Dic3—181no—
255Dic3—181no—
256Dic3—184no—
257Dic3—184no—
258Dic3—364no—
259Dic3—364no—
260Dic3—368no—
261Dic3—361no—
262Dic3—364no—
263C32—24no—
264C32#479911yescharacteristic, complemented
265C32#487714yescomplemented
266Q8—92no—
267Q8—181no—
268C2×C4—21no—
269C2×C4—61no—
270C2×C4—181no—
271C2×C4—271no—
272C2×C4—271no—
273C2×C4—541no—
274C2×C4C2×D3×C3⋊D311yescharacteristic
275C8—61no—
276C8—181no—
277C2×C3—24no—
278C2×C3—24no—
279C2×C3—24no—
280C2×C3#6389614yes—
281C2×C3#6393614yes—
282C2×C3#6394514yes—
283C2×C3#6410511yescharacteristic
284C2×C3#6412111yescharacteristic
285C2×C3#6412711yescharacteristic
286C22#8903911yescharacteristic
287C4—41no—
288C4—121no—
289C4—182no—
290C4—541no—
291C4—541no—
292C4—1081no—
293C4C3⋊D3×Dic611yescharacteristic
294C4#8902111yescharacteristic
295C3—24no—
296C3#1059049414yescomplemented
297C3#1059257611yescharacteristic, complemented
298C2#1172349711yescharacteristic
299C2#1172356311yescharacteristic
300C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#479
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
190. C33#479semidirect
264. C32#4799semidirect
265. C32#4877semidirect
296. C3#10590494semidirect
297. C3#10592576semidirect

Maximal subgroups

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

subgroupindex
#117239312
#117234102
C3×#636372
C32×#8072
#117220542
#117220502
#117220402
#105925763
#105904943