Finite Group Databasebeta

← #13159136

Subgroups — #13159136

Order 1728.

684 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C32×#890C211yesmaximal, characteristic
2Dic3×#634C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7#11722039C211yesmaximal, characteristic
8#10592762—31nomaximal
9#10590493—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21C32×#125—31no—
22C4×#634—31no—
23C3×#890D314yes—
24Dic3×Dic6—34no—
25#63637—31no—
26#63637D311yescharacteristic
27#63635—31no—
28#63634—31no—
29#63634—34no—
30#63634—64no—
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#810—34no—
33#63573—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63565—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C3×#634—21no—
41C3×#634Q812yes—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3×C3⋊(C3⋊C8)—21no—
47#5063—94no—
48#4799—91no—
49C2×#634C2×D311yescharacteristic
50C2×C3×Dic6—34no—
51C32×C4⋊C4—24no—
52C32×C4⋊C4—31no—
53C32×C4⋊C4—61no—
54C32×C4⋊C4—61no—
55C32×C4⋊C4C2×D314yes—
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C42—31no—
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—34no—
60C4⋊(C3⋊Dic3)—64no—
61#3750—61no—
62C4×C3⋊Dic3—61no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—24no—
68C3×C4⋊Dic3C2×D314yes—
69C3×Dic3⋊C4—24no—
70C3×C4×Dic3C2×D314yes—
71C2×C3×C3⋊C8—34no—
72Dic3⋊Dic3—64no—
73Dic32—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—22no—
80C32×Dic3—21no—
81C32×Dic3—21no—
82C32×Dic3—41no—
83C3×#125—34no—
84Q8×Dic3—91no—
85#890C3⋊D311yescharacteristic
86C4×Dic6—94no—
87C3×#103—91no—
88#835—91no—
89#817—91no—
90Dic3⋊C8—91no—
91#810—34no—
92#810—94no—
93#808—94no—
94#807—94no—
95#807—91no—
96#807—184no—
97C4⋊(C3⋊C8)—94no—
98C2×C32×C4—21no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—31no—
102C2×C32×C4—31no—
103C2×C32×C4—44no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—124no—
111#634—21no—
112#634Dic612yes—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3#6714yes—
116C2×C3×Dic3#6714yes—
117C3×Dic6—38no—
118C3×Dic6—64no—
119C3⋊(C3⋊C8)—21no—
120C3⋊(C3⋊C8)—68no—
121C3⋊(C3⋊C8)—61no—
122C3⋊(C3⋊C8)—64no—
123C3×C3⋊C8—64no—
124#479—271noSylow
125C2×C33C22⋊Q811yescharacteristic
126C2×C33#13511yescharacteristic
127C2×C33#13711yescharacteristic
128C2×C3×Q8—91no—
129C2×Dic6—34no—
130C2×C3×C8—91no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—34no—
133C3×C4⋊C4—64no—
134C3×C4⋊C4—64no—
135C3×C4⋊C4D3214yes—
136C3×C4⋊C4C2×C3⋊D311yescharacteristic
137C3×C42—34no—
138C4⋊Dic3—21no—
139C4⋊Dic3—94no—
140C4⋊Dic3—91no—
141C4⋊Dic3—184no—
142C4⋊Dic3—181no—
143C4⋊Dic3C2×C3⋊D311yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—184no—
146C4×Dic3—184no—
147C4×Dic3—181no—
148C4×Dic3C2×C3⋊D311yescharacteristic
149C2×C3⋊C8—34no—
150C2×C3⋊C8—94no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—184no—
153C22×C32—24no—
154C22×C32D3×D411yescharacteristic
155C22×C32C2×#6714yes—
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—41no—
159C32×C4—44no—
160C32×C4—61no—
161C32×C4—61no—
162C32×C4—84no—
163C32×C4—121no—
164C32×C4C2×Dic611yescharacteristic
165C32×C4#23711yescharacteristic
166C32×C4D3×Q814yes—
167C32×C4#23914yes—
168C3⋊Dic3—22no—
169C3⋊Dic3—121no—
170C3⋊Dic3—124no—
171C3⋊Dic3—244no—
172C3×Dic3—24no—
173C3×Dic3—24no—
174C3×Dic3—44no—
175C3×Dic3—68no—
176#125—31no—
177C4×Q8—271no—
178#107—271no—
179#106—271no—
180C4⋊C8—271no—
181#103—91no—
182#103—271no—
183C33#47911yescharacteristic, complemented, Sylow
184C3×Q8—92no—
185C3×Q8—181no—
186C2×C3×C4—24no—
187C2×C3×C4—21no—
188C2×C3×C4—24no—
189C2×C3×C4—34no—
190C2×C3×C4—34no—
191C2×C3×C4—44no—
192C2×C3×C4—64no—
193C2×C3×C4—181no—
194C2×C3×C4C2×D3214yes—
195C2×C3×C4C22×C3⋊D311yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—64no—
198C2×Dic3—184no—
199C2×Dic3—181no—
200C2×Dic3—364no—
201C2×Dic3#63811yescharacteristic
202C2×Dic3#63811yescharacteristic
203Dic6—38no—
204Dic6—64no—
205C3×C8—181no—
206C3⋊C8—64no—
207C3⋊C8—188no—
208C3⋊C8—184no—
209C3⋊C8—181no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32#87811yescharacteristic
214C2×C32#91611yescharacteristic
215C2×C32#91811yescharacteristic
216C2×C32#94614yes—
217C2×C32#95014yes—
218C2×C32#95114yes—
219C2×Q8—91no—
220C2×C8—91no—
221C2×C8—271no—
222C4⋊C4—31no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—271no—
226C4⋊C4—541no—
227C4⋊C4D3×C3⋊D311yescharacteristic
228C42—31no—
229C42—541no—
230C22×C3—24no—
231C22×C3D3×#6714yes—
232C22×C3C2×#63811yescharacteristic
233C3×C4—24no—
234C3×C4—24no—
235C3×C4—41no—
236C3×C4—44no—
237C3×C4—64no—
238C3×C4—64no—
239C3×C4—84no—
240C3×C4—124no—
241C3×C4—182no—
242C3×C4D3×Dic614yes—
243C3×C4#379614yes—
244C3×C4Q8×C3⋊D311yescharacteristic
245C3×C4#383211yescharacteristic
246Dic3—21no—
247Dic3—21no—
248Dic3—41no—
249Dic3—68no—
250Dic3—361no—
251Dic3—364no—
252Dic3—724no—
253C32—24no—
254C32#479911yescharacteristic, complemented
255C32#506314yescomplemented
256Q8—92no—
257Q8—181no—
258C2×C4—21no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—61no—
262C2×C4—181no—
263C2×C4—541no—
264C2×C4C2×D3×C3⋊D311yescharacteristic
265C8—181no—
266C8—541no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3#6388714yes—
271C2×C3#6393514yes—
272C2×C3#6394314yes—
273C2×C3#6415511yescharacteristic
274C2×C3#6415911yescharacteristic
275C2×C3#6416011yescharacteristic
276C22D3×#63811yescharacteristic
277C4—41no—
278C4—61no—
279C4—61no—
280C4—121no—
281C4—182no—
282C4—1081no—
283C4C3⋊D3×Dic611yescharacteristic
284C4#8901711yescharacteristic
285C3—24no—
286C3#1059049314yescomplemented
287C3#1059276211yescharacteristic, complemented
288C2#1172347611yescharacteristic
289C2#1172355811yescharacteristic
290C2#1172357411yescharacteristic
subgroupG/H
centerC22D3×#638
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
183. C33#479semidirect
254. C32#4799semidirect
255. C32#5063semidirect
286. C3#10590493semidirect
287. C3#10592762semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
Dic3×#6342
#117221772
C3×#636372
#117220542
Dic3⋊(C3⋊(C3⋊C8))2
#117220392
#105927623
#105904933