Finite Group Databasebeta

← #13140765

Subgroups — #13140765

Order 1728.

514 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 85 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×#63713C211yesmaximal, characteristic, complemented
2#11721857C211yesmaximal, characteristic, complemented
3#11720873C212yesmaximal, complemented
4C9×#836C211yesmaximal, characteristic, complemented
5#11720291C212yesmaximal, complemented
6#10591646—31nomaximal
7#10588433—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9C2×C4⋊(C3⋊D9)C2211yescharacteristic
10C3×#3701—22no—
11C3×Q8×D9—22no—
12C3×#3697—22no—
13C3×#3697C2212yes—
14#88536—22no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×#208C2212yes—
17#88417—22no—
18#88416—22no—
19C3×#1007—31no—
20#64156—31no—
21#63825—32no—
22#63713D311yescharacteristic, complemented
23C3×#836D311yescharacteristic
24#63568—32no—
25#63509—31no—
26#63509—61no—
27#63470—32no—
28C9×#101—31no—
29#63384—32no—
30C22×C3⋊D9—21no—
31C3×Q8×C9—22no—
32C2×C3×C4×C9D412yes—
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3⋊Dic9—21no—
35C4⋊(C3⋊D9)—22no—
36C3×#611—22no—
37C3×D36—22no—
38C3×D36D412yes—
39C3×C4×D9—24no—
40C3×C4×D9—22no—
41C3×Dic18—22no—
42C3×Dic18D412yes—
43C9×C3⋊C8—22no—
44#5245—91no—
45#4753—91no—
46C2×C32×Q8C2×D311yescharacteristic
47C2×C4⋊(C3⋊D3)—31no—
48C3×#239—62no—
49C3×D3×Q8—62no—
50C3×#235—32no—
51C3×#235—62no—
52#3752—62no—
53C4×C3⋊Dic3—31no—
54C3×#208C2×D312yes—
55#3717—62no—
56#3714—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C2×D311yescharacteristic
59#3701—22no—
60Q8×D9—22no—
61#3697—22no—
62#3697C2×D312yes—
63C2×D36—31no—
64C2×D36—61no—
65C9×M16—32no—
66(C2×D9)⋊C4—64no—
67(C2×D9)⋊C4—62no—
68C4×Dic9—31no—
69C4×Dic9—61no—
70D72—62no—
71#3664—62no—
72C22×C3×C9C2×D411yescharacteristic
73C2×C3⋊D9—42no—
74C2×C3⋊D9—41no—
75C2×C3×D9—22no—
76C3×C4×C9—22no—
77C3×C4×C9C2×D412yes—
78C3⋊Dic9—41no—
79C3×Dic9—22no—
80C3×2−1+4—91no—
81#1007—31no—
82#949—92no—
83#947—91no—
84#947—91no—
85#947—181no—
86#908—92no—
87C3×#101—31no—
88#837—92no—
89#836D911yescharacteristic, complemented
90#825—92no—
91C22×C3⋊D3—61no—
92C32×Q8—22no—
93C2×C32×C4D1212yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—61no—
96C4⋊(C3⋊D3)—62no—
97C3×#67—62no—
98C3×D12—32no—
99C3×D12—62no—
100C3×C4×D3—64no—
101C3×C4×D3—62no—
102C3×Dic6—32no—
103C3×Dic6—62no—
104C22×D9—62no—
105C22×D9—61no—
106C3×C3⋊C8—22no—
107Q8×C9—22no—
108Q8×C9—42no—
109C2×C4×C9—22no—
110C2×C4×C9—21no—
111C2×C4×C9#6712yes—
112C2×C4×C9C22×D311yescharacteristic
113#611—22no—
114C2×Dic9—62no—
115C2×Dic9—61no—
116D36—22no—
117D36—62no—
118D36—122no—
119D36#6712yes—
120C4×D9—24no—
121C4×D9—22no—
122Dic18—22no—
123Dic18#6712yes—
124C8×C9—62no—
125#454—271noSylow
126C2×C3×C9—21no—
127C2×C3×C9#12011yescharacteristic
128C3⋊D9—81no—
129C3×D9—42no—
130C3×#41—92no—
131C3×#41—182no—
132C3×#41—182no—
133C2×C3×Q8—21no—
134C2×C3×Q8—182no—
135C2×C3×Q8C2×D911yescharacteristic
136C2×C3×Q8D3211yescharacteristic
137#239—62no—
138D3×Q8—62no—
139#235—32no—
140#235—62no—
141C2×D12—91no—
142C2×D12—91no—
143C2×D12—181no—
144C3×M16—32no—
145#215—182no—
146#213—182no—
147(C2×D3)⋊C4—184no—
148(C2×D3)⋊C4—182no—
149(C2×D3)⋊C4—182no—
150C4×Dic3—91no—
151C4×Dic3—91no—
152C4×Dic3—181no—
153#208C2×D912yes—
154D24—182no—
155#204—182no—
156C22×C32C2×D1211yescharacteristic
157C2×C3⋊D3—122no—
158C2×C3⋊D3—121no—
159C2×C3×D3—62no—
160C32×C4—22no—
161C32×C4D3×D412yes—
162C3⋊Dic3—121no—
163C3×Dic3—62no—
164C22×C9—21no—
165C22×C9C2×#6711yescharacteristic
166C2×D9—22no—
167C2×D9—122no—
168C2×D9—124no—
169C2×D9—122no—
170C2×D9—121no—
171C4×C9—22no—
172C4×C9—22no—
173C4×C9—42no—
174C4×C9C2×#6712yes—
175Dic9—22no—
176Dic9—122no—
177Dic9—121no—
1782−1+4—91no—
179AGL(1,ℤ/8)—272no—
180#124—271no—
181#104—272no—
182#101—31no—
183C3×C9#45411yescharacteristic, complemented, Sylow
184C22×D3—182no—
185C22×D3—181no—
186C22×D3—181no—
187C3×Q8—22no—
188C3×Q8—22no—
189C3×Q8—42no—
190C3×Q8—92no—
191C3×Q8—182no—
192C3×D4—92no—
193C3×D4—182no—
194C3×D4—182no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—184no—
198C2×C3×C4—182no—
199C2×C3×C4D3612yes—
200C2×C3×C4C22×D911yescharacteristic
201C2×C3×C4#62612yes—
202C2×C3×C4C2×D3211yescharacteristic
203#67—62no—
204C2×Dic3—182no—
205C2×Dic3—181no—
206C2×Dic3—181no—
207D12—32no—
208D12—62no—
209D12—182no—
210D12—182no—
211D12—362no—
212C4×D3—64no—
213C4×D3—62no—
214Dic6—32no—
215Dic6—62no—
216C3×C8—62no—
217C3⋊C8—22no—
218C2×C32—21no—
219C2×C32#88211yescharacteristic
220C3⋊D3—241no—
221C3×D3—122no—
222C2×C9—22no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9#95311yescharacteristic
226D9—42no—
227D9—242no—
228D9—241no—
229#41—92no—
230#41—182no—
231#41—182no—
232C2×Q8—182no—
233C2×Q8D3×D911yescharacteristic
234C2×D4—271no—
235SD16—542no—
236D8—542no—
237M16—32no—
238C22⋊C4—542no—
239C42—271no—
240C22×C3—21no—
241C22×C3—182no—
242C22×C3C2×D3611yescharacteristic
243C22×C3C2×#62611yescharacteristic
244C2×D3—62no—
245C2×D3—362no—
246C2×D3—362no—
247C2×D3—364no—
248C2×D3—362no—
249C2×D3—361no—
250C2×D3—361no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—42no—
255C3×C4—182no—
256C3×C4D4×D912yes—
257C3×C4D3×#6712yes—
258Dic3—62no—
259Dic3—361no—
260Dic3—362no—
261Dic3—361no—
262C32#475311yescharacteristic
263C9—21no—
264C9#524511yescharacteristic, complemented
265C23—541no—
266Q8—22no—
267Q8—92no—
268Q8—182no—
269D4—92no—
270D4—182no—
271D4—182no—
272D4—542no—
273C2×C4—184no—
274C2×C4—182no—
275C2×C4—541no—
276C2×C4#619212yes—
277C2×C4C2×D3×D911yescharacteristic
278C8—62no—
279C2×C3—22no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—362no—
284C2×C3#6344411yescharacteristic
285C2×C3#6397711yescharacteristic
286D3—122no—
287D3—722no—
288D3—721no—
289D3—721no—
290C22—182no—
291C22—1082no—
292C22—1081no—
293C22C2×#619211yescharacteristic
294C4—22no—
295C4—182no—
296C4—1081no—
297C4D9×#6712yes—
298C3—21no—
299C3#1058843311yescharacteristic, complemented
300C3#1059164611yescharacteristic
301C2—21no—
302C2—362no—
303C2—2161no—
304C2#1172127611yescharacteristic
subgroupG/H
centerC2#11721276
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32#882
hypercenterC2×Q8D3×D9
hypocenterC3×C9#454
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#63713C2semidirect
2. #11721857C2semidirect
3. #11720873C2semidirect
4. C9×#836C2semidirect
5. #11720291C2semidirect
22. #63713D3semidirect
89. #836D9semidirect
183. C3×C9#454semidirect
264. C9#5245semidirect
299. C3#10588433semidirect

Maximal subgroups

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

subgroupindex
C3×#637132
#117218572
#117208732
C9×#8362
#117202912
#105916463
#105884333