Finite Group Databasebeta

← #13159282

Subgroups — #13159282

Order 1728.

732 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 166 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×#64097C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722052C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
8#10592554—31nomaximal
9#10590412—34nomaximal
10C2×C3×#634—21no—
11C2×#6293—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64097—31no—
22#64097D311yescharacteristic
23C3×#888—34no—
24#63918—34no—
25C32×#106—31no—
26#63644—31no—
27C8×C3⋊Dic3—31no—
28#63635—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32C3×#808D314yes—
33#63574—34no—
34#63573—34no—
35Dic3×C3⋊C8—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
42C3×#634—42no—
43C2×C32×Dic3—21no—
44#6293—42no—
45C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
46C32×C3⋊C8Q812yescomplemented
47#5080—91no—
48#4855—94no—
49C2×#634—21no—
50C2×#634—61no—
51C2×C3×Dic6—64no—
52C2×#627—64no—
53C2×C32×C8—31no—
54C32×C4⋊C4—24no—
55C32×C4⋊C4—31no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—34no—
60C4⋊(C3⋊Dic3)—64no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63#3750—61no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C3×C4⋊Dic3—34no—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—34no—
72C2×C3×C3⋊C8C2×D314yes—
73Dic3⋊Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3SD1612yes—
81C3×C3⋊Dic3SD1612yes—
82C32×Dic3—41no—
83C3×C4⋊Q8—91no—
84#888—34no—
85#888—94no—
86#869—91no—
87C3×#106—34no—
88#817—94no—
89C8×Dic3—94no—
90#808—94no—
91#808—94no—
92#808—91no—
93#808—91no—
94#808—91no—
95#808—184no—
96#808C3⋊D311yescharacteristic
97C4×C3⋊C8—91no—
98C2×C32×C4—21no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—44no—
102C2×C32×C4—61no—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—64no—
108C2×C3⋊Dic3—124no—
109C2×C3⋊Dic3#6711yescharacteristic
110C2×C3⋊Dic3#6711yescharacteristic
111#634—42no—
112#634—122no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—64no—
117C3×Dic6—128no—
118#627—128no—
119C32×C8—32no—
120C3⋊(C3⋊C8)—32no—
121C3⋊(C3⋊C8)—38no—
122C3⋊(C3⋊C8)—68no—
123C3×C3⋊C8Dic618yescomplemented
124#498—271noSylow
125C2×C33C4⋊Q811yescharacteristic
126C2×C33C2×SD1611yescharacteristic
127C2×C33C2×SD1611yescharacteristic
128C2×C3×Q8—181no—
129C2×Dic6—64no—
130C2×Dic6—181no—
131C2×Dic6—184no—
132C2×C3×C8—34no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—34no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C4⋊C4D3214yes—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C3×C42—91no—
140C4⋊Dic3—34no—
141C4⋊Dic3—94no—
142C4⋊Dic3—91no—
143C4⋊Dic3—184no—
144C4⋊Dic3—181no—
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—64no—
147Dic3⋊C4—184no—
148C4×Dic3—34no—
149C2×C3⋊C8—94no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—184no—
152C2×C3⋊C8C2×C3⋊D311yescharacteristic
153C22×C32—24no—
154C22×C32D3×D414yes—
155C22×C32C2×#6711yescharacteristic
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—41no—
159C32×C4—44no—
160C32×C4—84no—
161C32×C4—121no—
162C32×C4C2×Dic614yes—
163C32×C4D3×Q814yes—
164C32×C4D3×Q811yescharacteristic
165C32×C4D3×Q811yescharacteristic
166C3⋊Dic3—41no—
167C3⋊Dic3—121no—
168C3⋊Dic3—124no—
169C3⋊Dic3—244no—
170C3⋊Dic3#21412yes—
171C3⋊Dic3#21512yes—
172C3×Dic3—38no—
173C3×Dic3—38no—
174C3×Dic3—44no—
175C3×Dic3—124no—
176C4⋊Q8—91no—
177C4⋊Q8—271no—
178#106—31no—
179#106—271no—
180#106—271no—
181#106—271no—
182C4×C8—271no—
183C33#49811yescharacteristic, complemented, Sylow
184C3×Q8—362no—
185C2×C3×C4—24no—
186C2×C3×C4—21no—
187C2×C3×C4—24no—
188C2×C3×C4—44no—
189C2×C3×C4—64no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—181no—
193C2×C3×C4C2×D3214yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195C2×Dic3—21no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—64no—
199C2×Dic3—184no—
200C2×Dic3—181no—
201C2×Dic3—364no—
202Dic6—128no—
203Dic6—368no—
204Dic6—362no—
205C3×C8—38no—
206C3⋊C8—98no—
207C3⋊C8—92no—
208C3⋊C8—188no—
209C3⋊C8#63412yescomplemented
210C2×C32—24no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32#88814yes—
214C2×C32D3×SD1614yes—
215C2×C32D3×SD1614yes—
216C2×C32C2×#21411yescharacteristic
217C2×C32C2×#21511yescharacteristic
218C2×C32#94411yescharacteristic
219C2×Q8—181no—
220C2×Q8—541no—
221C2×C8—31no—
222C2×C8—271no—
223C4⋊C4—31no—
224C4⋊C4—91no—
225C4⋊C4—181no—
226C4⋊C4—271no—
227C4⋊C4—541no—
228C4⋊C4D3×C3⋊D311yescharacteristic
229C42—91no—
230C22×C3—24no—
231C22×C3D3×#6714yes—
232C22×C3D4×C3⋊D311yescharacteristic
233C3×C4—24no—
234C3×C4—24no—
235C3×C4—41no—
236C3×C4—44no—
237C3×C4—84no—
238C3×C4—92no—
239C3×C4—92no—
240C3×C4—124no—
241C3×C4—361no—
242C3×C4D3×Dic614yes—
243C3×C4#379714yes—
244C3×C4C2×#63411yescharacteristic
245C3×C4Q8×C3⋊D311yescharacteristic
246Dic3—38no—
247Dic3—38no—
248Dic3—41no—
249Dic3—124no—
250Dic3—361no—
251Dic3—364no—
252Dic3—724no—
253C32—24no—
254C32#485514yescomplemented
255C32#508011yescharacteristic, complemented
256Q8—362no—
257Q8—1082no—
258C2×C4—21no—
259C2×C4—61no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—181no—
263C2×C4—541no—
264C2×C4C2×D3×C3⋊D311yescharacteristic
265C8—32no—
266C8—272no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3#6383914yes—
271C2×C3D3×#21414yes—
272C2×C3D3×#21514yes—
273C2×C3#6409711yescharacteristic
274C2×C3SD16×C3⋊D311yescharacteristic
275C2×C3SD16×C3⋊D311yescharacteristic
276C22C3⋊D3×#6711yescharacteristic
277C4—41no—
278C4—92no—
279C4—92no—
280C4—121no—
281C4—361no—
282C4—1081no—
283C4D3×#63411yescharacteristic
284C4#8901911yescharacteristic
285C3—24no—
286C3#1059041214yescomplemented
287C3#1059255411yescharacteristic, complemented
288C2#1172339411yescharacteristic
289C2C3⋊D3×#21411yescharacteristic
290C2C3⋊D3×#21511yescharacteristic
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#498
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
45. C3⋊(C3⋊(C3⋊C8))Q8semidirect
46. C32×C3⋊C8Q8semidirect
123. C3×C3⋊C8Dic6semidirect
183. C33#498semidirect
209. C3⋊C8#634semidirect
254. C32#4855semidirect
255. C32#5080semidirect
286. C3#10590412semidirect
287. C3#10592554semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
#117235352
#117221782
C32×#8082
#117220532
#117220522
C3⋊C8×C3⋊Dic32
#105925543
#105904123