Finite Group Databasebeta

← #13159131

Subgroups — #13159131

Order 1728.

708 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 154 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
2#11723399C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7#11722041C211yesmaximal, characteristic
8#10592762—31nomaximal
9#10590492—34nomaximal
10C2×#6308C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21C32×#125—31no—
22C4×#634—31no—
23C3×#890D314yes—
24#63844—34no—
25#63637—31no—
26#63637—31no—
27#63637—34no—
28#63637—64no—
29#63635—31no—
30#63634D311yescharacteristic
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#807—34no—
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#6308Q812yes—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3×C3⋊(C3⋊C8)—21no—
47#5063—94no—
48#4877—91no—
49C2×#634—31no—
50C2×#634—34no—
51C2×#634—64no—
52C32×C4⋊C4—24no—
53C32×C4⋊C4—31no—
54C32×C4⋊C4—61no—
55C32×C4⋊C4—61no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C42—31no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—61no—
61C4×C3⋊Dic3—61no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×C3⋊(C3⋊C8)—34no—
64C2×C3⋊(C3⋊C8)—64no—
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4⋊Dic3—24no—
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3C2×D314yes—
69C3×Dic3⋊C4—24no—
70C3×C4×Dic3C2×D314yes—
71C2×C3×C3⋊C8—34no—
72#3724—64no—
73Dic32—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—21no—
81C32×Dic3—21no—
82C32×Dic3—41no—
83C3×#125—34no—
84#890C3⋊D311yescharacteristic
85#887—91no—
86C4×Dic6—94no—
87C3×#107—91no—
88#816—91no—
89Dic3⋊C8—91no—
90#810—94no—
91#810—94no—
92#810—91no—
93#810—184no—
94#808—94no—
95#808—91no—
96#807—34no—
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—32no—
112#634—38no—
113#634—68no—
114#634—61no—
115#634—64no—
116#634—124no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3#6714yes—
120C2×C3×Dic3#6714yes—
121C3⋊(C3⋊C8)—21no—
122C3⋊(C3⋊C8)—61no—
123C3⋊(C3⋊C8)—64no—
124C3⋊(C3⋊C8)—124no—
125C3×C3⋊C8—64no—
126#479—271noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33#13711yescharacteristic
130C2×Dic6—94no—
131C2×Dic6—91no—
132C2×Dic6—184no—
133C2×C3×C8—91no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—34no—
136C3×C4⋊C4—64no—
137C3×C4⋊C4—64no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4D3214yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141C3×C42—34no—
142C4⋊Dic3—21no—
143C4⋊Dic3—34no—
144C4⋊Dic3C2×C3⋊D311yescharacteristic
145Dic3⋊C4—21no—
146Dic3⋊C4—184no—
147Dic3⋊C4—181no—
148C4×Dic3—184no—
149C4×Dic3—181no—
150C4×Dic3C2×C3⋊D311yescharacteristic
151C2×C3⋊C8—34no—
152C2×C3⋊C8—94no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—184no—
155C22×C32—24no—
156C22×C32D3×D411yescharacteristic
157C22×C32C2×#6714yes—
158C32×C4—24no—
159C32×C4—24no—
160C32×C4—41no—
161C32×C4—44no—
162C32×C4—61no—
163C32×C4—61no—
164C32×C4—84no—
165C32×C4—121no—
166C32×C4#23511yescharacteristic
167C32×C4D3×Q814yes—
168C32×C4D3×Q811yescharacteristic
169C32×C4#23914yes—
170C3⋊Dic3—41no—
171C3⋊Dic3—62no—
172C3⋊Dic3—68no—
173C3⋊Dic3—128no—
174C3×Dic3—24no—
175C3×Dic3—24no—
176C3×Dic3—44no—
177C3×Dic3—124no—
178#125—31no—
179C4×Q8—271no—
180#107—91no—
181#106—271no—
182C4⋊C8—271no—
183#103—271no—
184#103—271no—
185C33#47911yescharacteristic, complemented, Sylow
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—98no—
204Dic6—92no—
205Dic6—188no—
206Dic6—184no—
207Dic6—181no—
208Dic6—364no—
209C3×C8—181no—
210C3⋊C8—64no—
211C3⋊C8—184no—
212C3⋊C8—181no—
213C3⋊C8—364no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32#89611yescharacteristic
218C2×C32#91211yescharacteristic
219C2×C32#91811yescharacteristic
220C2×C32#94614yes—
221C2×C32#95014yes—
222C2×C32#95114yes—
223C2×Q8—271no—
224C2×C8—91no—
225C2×C8—271no—
226C4⋊C4—31no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—91no—
230C4⋊C4—541no—
231C4⋊C4D3×C3⋊D311yescharacteristic
232C42—31no—
233C42—541no—
234C22×C3—24no—
235C22×C3D3×#6714yes—
236C22×C3C2×#63811yescharacteristic
237C3×C4—24no—
238C3×C4—24no—
239C3×C4—41no—
240C3×C4—44no—
241C3×C4—64no—
242C3×C4—64no—
243C3×C4—84no—
244C3×C4—124no—
245C3×C4—361no—
246C3×C4#379714yes—
247C3×C4#379914yes—
248C3×C4Q8×C3⋊D311yescharacteristic
249C3×C4#383211yescharacteristic
250Dic3—21no—
251Dic3—21no—
252Dic3—41no—
253Dic3—124no—
254Dic3—188no—
255Dic3—182no—
256Dic3—368no—
257C32—24no—
258C32#487711yescharacteristic, complemented
259C32#506314yescomplemented
260Q8—272no—
261Q8—541no—
262C2×C4—21no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—61no—
266C2×C4—181no—
267C2×C4—541no—
268C2×C4C2×D3×C3⋊D311yescharacteristic
269C8—181no—
270C8—541no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3#6388814yes—
275C2×C3#6393314yes—
276C2×C3#6394314yes—
277C2×C3#6415511yescharacteristic
278C2×C3#6415911yescharacteristic
279C2×C3#6416011yescharacteristic
280C22D3×#63811yescharacteristic
281C4—41no—
282C4—61no—
283C4—61no—
284C4—121no—
285C4—361no—
286C4—542no—
287C4#8901811yescharacteristic
288C4#8901911yescharacteristic
289C3—24no—
290C3#1059049214yescomplemented
291C3#1059276211yescharacteristic, complemented
292C2#1172347711yescharacteristic
293C2#1172355611yescharacteristic
294C2#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
185. C33#479semidirect
258. C32#4877semidirect
259. C32#5063semidirect
290. C3#10590492semidirect
291. C3#10592762semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
#117233992
#117221802
C3×#636342
#117220532
Dic3⋊(C3⋊(C3⋊C8))2
#117220412
#105927623
#105904923