Finite Group Databasebeta

← #13159262

Subgroups — #13159262

Order 1728.

740 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 158 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#11723450C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5#11722055C211yesmaximal, characteristic
6#11722054C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic
8#10592577—31nomaximal
9#10590417—34nomaximal
10C33×C4⋊C4C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13#88857—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(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—31no—
23#64099—34no—
24#64099—64no—
25#63872—34no—
26C32×#107—31no—
27#63645—31no—
28C8×C3⋊Dic3—31no—
29#63635—31no—
30#63635D311yescharacteristic
31C3×#808—34no—
32C3×#807D314yes—
33#63575—34no—
34#63573—34no—
35#63553—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—
42C2×C32×Dic3—21no—
43C3×C3⋊(C3⋊C8)Q812yes—
44C32×C3⋊C8Q812yes—
45#4878—94no—
46#4858—91no—
47C2×C32×C8—31no—
48C32×C4⋊C4—24no—
49C32×C4⋊C4—31no—
50C32×C4⋊C4C2×D314yes—
51C32×C4⋊C4C2×D311yescharacteristic
52C4⋊(C3⋊Dic3)—31no—
53C4⋊(C3⋊Dic3)—34no—
54C4⋊(C3⋊Dic3)—61no—
55C4⋊(C3⋊Dic3)—61no—
56C4⋊(C3⋊Dic3)—64no—
57C4⋊(C3⋊Dic3)—64no—
58C4⋊(C3⋊Dic3)—68no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—61no—
61#3750—61no—
62#3750—64no—
63#3750—68no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—64no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C4⋊Dic3—34no—
69C3×C4⋊Dic3C2×D314yes—
70C2×C3×C3⋊C8—34no—
71C2×C3×C3⋊C8C2×D314yes—
72Dic3⋊Dic3—64no—
73Dic3⋊Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3⋊(C3⋊Dic3)—21no—
79C3⋊(C3⋊Dic3)—21no—
80C3⋊(C3⋊Dic3)—41no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—41no—
83#890—94no—
84#890—94no—
85#890—91no—
86#890—91no—
87#890—184no—
88C3×#107—34no—
89C3×#106—91no—
90#818—94no—
91#817—91no—
92C8×Dic3—94no—
93C8×Dic3—91no—
94#808—34no—
95#808—94no—
96#807—91no—
97#807C3⋊D311yescharacteristic
98C2×C32×C4—28no—
99C2×C32×C4—24no—
100C2×C32×C4—21no—
101C2×C32×C4—24no—
102C2×C32×C4—61no—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—34no—
109C2×C3⋊Dic3—34no—
110C2×C3⋊Dic3—64no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—68no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—61no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—64no—
117C32×C8—32no—
118C3⋊(C3⋊C8)Dic612yes—
119C3×C3⋊C8—38no—
120C3×C3⋊C8Dic618yes—
121#499—271noSylow
122C2×C33C4⋊Q811yescharacteristic
123C2×C33#13511yescharacteristic
124C2×C33#13511yescharacteristic
125C2×C3×C8—34no—
126C2×C3×C8—91no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—34no—
129C3×C4⋊C4—91no—
130C3×C4⋊C4D3214yes—
131C3×C4⋊C4C2×C3⋊D311yescharacteristic
132C4⋊Dic3—34no—
133C4⋊Dic3—94no—
134C4⋊Dic3—91no—
135C4⋊Dic3—184no—
136C4⋊Dic3—181no—
137C4⋊Dic3—184no—
138C4⋊Dic3—184no—
139C4⋊Dic3—181no—
140C4⋊Dic3—188no—
141C4⋊Dic3C2×C3⋊D311yescharacteristic
142Dic3⋊C4—181no—
143Dic3⋊C4—184no—
144Dic3⋊C4—184no—
145Dic3⋊C4—181no—
146Dic3⋊C4—188no—
147C4×Dic3—94no—
148C4×Dic3—91no—
149C4×Dic3—184no—
150C2×C3⋊C8—34no—
151C2×C3⋊C8C2×C3⋊D311yescharacteristic
152C22×C32—24no—
153C22×C32D3×D414yes—
154C22×C32D3×D411yescharacteristic
155C32×C4—24no—
156C32×C4—24no—
157C32×C4—48no—
158C32×C4—44no—
159C32×C4—41no—
160C32×C4—121no—
161C32×C4C2×Dic614yes—
162C32×C4C2×Dic611yescharacteristic
163C32×C4D3×Q814yes—
164C32×C4D3×Q811yescharacteristic
165C3⋊Dic3—41no—
166C3⋊Dic3—61no—
167C3⋊Dic3—61no—
168C3⋊Dic3—64no—
169C3⋊Dic3—64no—
170C3⋊Dic3—124no—
171C3⋊Dic3—124no—
172C3⋊Dic3—128no—
173C3⋊Dic3—121no—
174C3⋊Dic3—124no—
175C3×Dic3—44no—
176C3×Dic3—124no—
177#125—271no—
178#125—271no—
179#107—31no—
180#107—271no—
181#106—91no—
182#106—271no—
183C4×C8—271no—
184C33#49911yescharacteristic, complemented, Sylow
185C2×C3×C4—28no—
186C2×C3×C4—24no—
187C2×C3×C4—21no—
188C2×C3×C4—24no—
189C2×C3×C4—64no—
190C2×C3×C4—181no—
191C2×C3×C4C2×D3214yes—
192C2×C3×C4C22×C3⋊D311yescharacteristic
193C2×Dic3—21no—
194C2×Dic3—64no—
195C2×Dic3—94no—
196C2×Dic3—94no—
197C2×Dic3—91no—
198C2×Dic3—91no—
199C2×Dic3—184no—
200C2×Dic3—184no—
201C2×Dic3—188no—
202C2×Dic3—184no—
203C2×Dic3—181no—
204C3×C8—38no—
205C3×C8—92no—
206C3⋊C8—38no—
207C3⋊C8#63412yes—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—24no—
211C2×C32#88814yes—
212C2×C32#88811yescharacteristic
213C2×C32#91214yes—
214C2×C32#91611yescharacteristic
215C2×C32#91611yescharacteristic
216C2×C32#91914yes—
217C2×C8—31no—
218C2×C8—91no—
219C4⋊C4—31no—
220C4⋊C4—91no—
221C4⋊C4—271no—
222C4⋊C4—541no—
223C4⋊C4—541no—
224C4⋊C4—541no—
225C4⋊C4—541no—
226C4⋊C4D3×C3⋊D311yescharacteristic
227C42—271no—
228C22×C3—24no—
229C22×C3#381114yes—
230C22×C3D4×C3⋊D311yescharacteristic
231C3×C4—24no—
232C3×C4—24no—
233C3×C4—48no—
234C3×C4—41no—
235C3×C4—44no—
236C3×C4—124no—
237C3×C4—361no—
238C3×C4D3×Dic614yes—
239C3×C4D3×Dic614yes—
240C3×C4C2×#63411yescharacteristic
241C3×C4Q8×C3⋊D311yescharacteristic
242Dic3—41no—
243Dic3—124no—
244Dic3—181no—
245Dic3—181no—
246Dic3—184no—
247Dic3—184no—
248Dic3—364no—
249Dic3—364no—
250Dic3—368no—
251Dic3—361no—
252Dic3—364no—
253C32—24no—
254C32#485811yescharacteristic, complemented
255C32#487814yescomplemented
256C2×C4—21no—
257C2×C4—61no—
258C2×C4—181no—
259C2×C4—271no—
260C2×C4—271no—
261C2×C4—541no—
262C2×C4C2×D3×C3⋊D311yescharacteristic
263C8—32no—
264C8—92no—
265C2×C3—24no—
266C2×C3—24no—
267C2×C3—24no—
268C2×C3#6384014yes—
269C2×C3#6393614yes—
270C2×C3#6395014yes—
271C2×C3#6409711yescharacteristic
272C2×C3#6412111yescharacteristic
273C2×C3#6412811yescharacteristic
274C22#8903911yescharacteristic
275C4—41no—
276C4—121no—
277C4—361no—
278C4—541no—
279C4—541no—
280C4—1081no—
281C4D3×#63411yescharacteristic
282C4C3⋊D3×Dic611yescharacteristic
283C3—24no—
284C3#1059041714yescomplemented
285C3#1059257711yescharacteristic, complemented
286C2#1172339511yescharacteristic
287C2#1172356311yescharacteristic
288C2#1172358911yescharacteristic
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#499
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
184. C33#499semidirect
254. C32#4858semidirect
255. C32#4878semidirect
284. C3#10590417semidirect
285. C3#10592577semidirect

Maximal subgroups

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

subgroupindex
#117239312
#117234502
C3×#636352
C32×#8072
#117220552
#117220542
#117220182
#105925773
#105904173