Finite Group Databasebeta

← #13159286

Subgroups — #13159286

Order 1728.

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

normal subgroups only

This group has 684 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
1C3×#64099C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5#11722056C211yesmaximal, characteristic
6#11722055C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic
8#10592555—31nomaximal
9#10590419—34nomaximal
10C2×#6293—21no—
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64099D311yescharacteristic
22#64097—31no—
23C3×#890—34no—
24#63918—34no—
25C32×#106—31no—
26#63645—31no—
27#63642—31no—
28#63635—31no—
29#63635—34no—
30#63635—64no—
31#63634—31no—
32C3×#808D314yes—
33#63576—34no—
34#63575—34no—
35#63554—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C2×C32×Dic3—21no—
43#6293—42no—
44C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
45C32×C3⋊C8Q812yescomplemented
46#5064—91no—
47#4856—94no—
48C2×#634—61no—
49C2×#627—64no—
50C2×C32×C8—31no—
51C32×C4⋊C4—24no—
52C32×C4⋊C4—31no—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—34no—
58C4⋊(C3⋊Dic3)—64no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—21no—
61#3750—61no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—64no—
68C3×C4⋊Dic3C2×D314yes—
69C3×Dic3⋊C4—64no—
70C3×C4×Dic3—34no—
71C2×C3×C3⋊C8C2×D314yes—
72Dic3⋊Dic3—64no—
73C22×C33C2×D411yescharacteristic
74C33×C4—41no—
75C33×C4C2×Q811yescharacteristic
76C33×C4C2×Q811yescharacteristic
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—41no—
81C32×Dic3—41no—
82C3×#125—91no—
83#890—34no—
84#888—94no—
85#869—91no—
86C3×#106—34no—
87#818—94no—
88#815—94no—
89#808—94no—
90#808—91no—
91#808—184no—
92#808C3⋊D311yescharacteristic
93#807—94no—
94#807—91no—
95#807—91no—
96#803—91no—
97C2×C32×C4—21no—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4—44no—
101C2×C32×C4—61no—
102C2×C32×C4C22×D314yes—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—64no—
107C2×C3⋊Dic3—124no—
108C2×C3⋊Dic3#6711yescharacteristic
109C2×C3⋊Dic3#6711yescharacteristic
110#634—122no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—64no—
115#627—128no—
116C32×C8—32no—
117C3⋊(C3⋊C8)—32no—
118C3⋊(C3⋊C8)—38no—
119C3⋊(C3⋊C8)—68no—
120C3×C3⋊C8Dic618yescomplemented
121#501—271noSylow
122C2×C33C4⋊Q811yescharacteristic
123C2×C33AGL(1,ℤ/8)11yescharacteristic
124C2×C33#13711yescharacteristic
125C2×Dic6—181no—
126C2×Dic6—184no—
127C2×C3×C8—34no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—34no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—181no—
132C3×C4⋊C4—181no—
133C3×C4⋊C4D3214yes—
134C3×C4⋊C4C2×C3⋊D311yescharacteristic
135C3×C42—91no—
136C4⋊Dic3—34no—
137C4⋊Dic3—64no—
138C4⋊Dic3—94no—
139C4⋊Dic3—91no—
140C4⋊Dic3—184no—
141C4⋊Dic3—181no—
142C4⋊Dic3C2×C3⋊D311yescharacteristic
143Dic3⋊C4—64no—
144Dic3⋊C4—184no—
145C4×Dic3—34no—
146C2×C3⋊C8—94no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—184no—
149C2×C3⋊C8C2×C3⋊D311yescharacteristic
150C22×C32—24no—
151C22×C32D3×D414yes—
152C22×C32C2×#6711yescharacteristic
153C32×C4—24no—
154C32×C4—24no—
155C32×C4—41no—
156C32×C4—44no—
157C32×C4—84no—
158C32×C4—121no—
159C32×C4C2×Dic614yes—
160C32×C4D3×Q814yes—
161C32×C4D3×Q811yescharacteristic
162C32×C4D3×Q811yescharacteristic
163C3⋊Dic3—21no—
164C3⋊Dic3—21no—
165C3⋊Dic3—41no—
166C3⋊Dic3—121no—
167C3⋊Dic3—124no—
168C3⋊Dic3—244no—
169C3×Dic3—44no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C3×Dic3—124no—
173C4⋊Q8—271no—
174#125—91no—
175#107—271no—
176#107—271no—
177#106—31no—
178#106—271no—
179#97—271no—
180C33#50111yescharacteristic, complemented, Sylow
181C2×C3×C4—24no—
182C2×C3×C4—21no—
183C2×C3×C4—24no—
184C2×C3×C4—44no—
185C2×C3×C4—64no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—181no—
189C2×C3×C4C2×D3214yes—
190C2×C3×C4C22×C3⋊D311yescharacteristic
191C2×Dic3—21no—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—64no—
195C2×Dic3—184no—
196C2×Dic3—181no—
197C2×Dic3—364no—
198Dic6—362no—
199Dic6—368no—
200C3×C8—38no—
201C3⋊C8—98no—
202C3⋊C8—92no—
203C3⋊C8—188no—
204C3⋊C8#63412yescomplemented
205C2×C32—24no—
206C2×C32—24no—
207C2×C32—24no—
208C2×C32#88814yes—
209C2×C32#91414yes—
210C2×C32#91514yes—
211C2×C32#94411yescharacteristic
212C2×C32#94911yescharacteristic
213C2×C32#95111yescharacteristic
214C2×Q8—541no—
215C2×C8—31no—
216C2×C8—271no—
217C4⋊C4—31no—
218C4⋊C4—91no—
219C4⋊C4—181no—
220C4⋊C4—181no—
221C4⋊C4—271no—
222C4⋊C4—541no—
223C4⋊C4D3×C3⋊D311yescharacteristic
224C42—91no—
225C22×C3—24no—
226C22×C3D3×#6714yes—
227C22×C3D4×C3⋊D311yescharacteristic
228C3×C4—24no—
229C3×C4—24no—
230C3×C4—41no—
231C3×C4—44no—
232C3×C4—84no—
233C3×C4—124no—
234C3×C4—181no—
235C3×C4—181no—
236C3×C4—361no—
237C3×C4D3×Dic614yes—
238C3×C4#379714yes—
239C3×C4C2×#63411yescharacteristic
240C3×C4Q8×C3⋊D311yescharacteristic
241Dic3—41no—
242Dic3—64no—
243Dic3—64no—
244Dic3—124no—
245Dic3—361no—
246Dic3—364no—
247Dic3—724no—
248C32—24no—
249C32#485614yescomplemented
250C32#506411yescharacteristic, complemented
251Q8—1082no—
252C2×C4—21no—
253C2×C4—61no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—181no—
257C2×C4—541no—
258C2×C4C2×D3×C3⋊D311yescharacteristic
259C8—32no—
260C8—272no—
261C2×C3—24no—
262C2×C3—24no—
263C2×C3—24no—
264C2×C3#6383914yes—
265C2×C3#6392914yes—
266C2×C3#6393914yes—
267C2×C3#6409711yescharacteristic
268C2×C3#6412311yescharacteristic
269C2×C3#6412411yescharacteristic
270C22C3⋊D3×#6711yescharacteristic
271C4—41no—
272C4—121no—
273C4—181no—
274C4—181no—
275C4—361no—
276C4—1081no—
277C4D3×#63411yescharacteristic
278C4#8901911yescharacteristic
279C3—24no—
280C3#1059041914yescomplemented
281C3#1059255511yescharacteristic, complemented
282C2#1172339411yescharacteristic
283C2#1172355211yescharacteristic
284C2#1172357011yescharacteristic
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#501
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
44. C3⋊(C3⋊(C3⋊C8))Q8semidirect
45. C32×C3⋊C8Q8semidirect
120. C3×C3⋊C8Dic6semidirect
180. C33#501semidirect
204. C3⋊C8#634semidirect
249. C32#4856semidirect
250. C32#5064semidirect
280. C3#10590419semidirect
281. C3#10592555semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
#117235352
#117221782
C32×#8082
#117220562
#117220552
#117220202
#105925553
#105904193