Finite Group Databasebeta

← #13143873

Subgroups — #13143873

Order 1728.

750 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#88742C211yesmaximal, characteristic
2M16×C3⋊D9C212yesmaximal, complemented
3C3×C9×#100C211yesmaximal, characteristic, complemented
4#11720575C211yesmaximal, characteristic, complemented
5#11720556C212yesmaximal, complemented
6#10592444—31nomaximal
7#10588425—33nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233C412yes—
10C22×C3⋊Dic9C412yes—
11#88742—22no—
12#88742—22no—
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×#6226C2211yescharacteristic
15C3×C9×M16C2212yes—
16#88532C2212yes—
17#88523—22no—
18C8×C3⋊D9—22no—
19C2×#3830—31no—
20M16×C3⋊D3—32no—
21C2×#3699—33no—
22C32×#100D311yescharacteristic
23#63660—31no—
24#63651—32no—
25M16×D9—36no—
26C9×#100D313yescomplemented
27#63391—33no—
28#63382—36no—
29C23×C3×C9C2×C412yes—
30C22×C3⋊D9C2×C411yescharacteristic
31C3×D4×C9—22no—
32C2×C3×C4×C9C2311yescharacteristic
33#6233—24no—
34C2×C3⋊Dic9—22no—
35C2×C3⋊Dic9—22no—
36C2×C3⋊Dic9C2×C412yes—
37C2×C3⋊Dic9C2×C411yescharacteristic
38C4×C3⋊D9D412yes—
39C4×C3⋊D9D412yes—
40#6226—22no—
41C3×C8×C9—22no—
42C3⋊(C9⋊C8)—22no—
43#4745—93no—
44#4745—91no—
45C2×C32×D4C2×D311yescharacteristic
46C2×#638—32no—
47C22×C3⋊Dic3—32no—
48#3830—62no—
49#3830—62no—
50C2×C4×C3⋊D3—31no—
51C2×#634—31no—
52C32×M16C2×D312yes—
53#3748—32no—
54#3743—62no—
55C8×C3⋊D3—62no—
56C2×D4×C9C2×D313yes—
57C2×#611—36no—
58C22×Dic9—36no—
59#3699—66no—
60#3699—66no—
61C2×C4×D9—33no—
62C2×Dic18—33no—
63C9×M16C2×D316yes—
64#3667—36no—
65#3663—66no—
66C8×D9—66no—
67C22×C3×C9—22no—
68C22×C3×C9—42no—
69C22×C3×C9C22×C411yescharacteristic
70C2×C3⋊D9—41no—
71C2×C3⋊D9C22⋊C412yes—
72C3×C4×C9C2×D412yes—
73C3⋊Dic9—22no—
74C3⋊Dic9C22⋊C411yescharacteristic
75C3⋊Dic9C22⋊C411yescharacteristic
76C2×#237—93no—
77C2×#237—91no—
78D3×M16—96no—
79D3×M16—92no—
80C3×#100D913yescomplemented
81C3×#100C3⋊D311yescharacteristic
82#833—93no—
83#833—91no—
84#824—96no—
85#824—92no—
86C23×C32C4×D312yes—
87C22×C3⋊D3—31no—
88C32×D4—22no—
89C2×C32×C4C22×D311yescharacteristic
90#638—64no—
91C2×C3⋊Dic3—31no—
92C2×C3⋊Dic3—32no—
93C2×C3⋊Dic3—62no—
94C2×C3⋊Dic3—62no—
95C4×C3⋊D3—32no—
96C4×C3⋊D3—32no—
97#634—62no—
98C23×C9C4×D316yes—
99C22×D9—33no—
100C32×C8—22no—
101C3⋊(C3⋊C8)—62no—
102D4×C9—26no—
103C2×C4×C9C22×D313yes—
104#611—612no—
105C2×Dic9—33no—
106C2×Dic9—36no—
107C2×Dic9—66no—
108C2×Dic9—66no—
109C4×D9—36no—
110C4×D9—36no—
111Dic18—66no—
112C8×C9—26no—
113C9⋊C8—66no—
114#413—271noSylow
115C2×C3×C9—21no—
116C2×C3×C9—42no—
117C2×C3×C9C2×C22⋊C411yescharacteristic
118C3⋊D9—22no—
119C2×C3×D4C2×D913yes—
120C2×C3×D4C2×C3⋊D311yescharacteristic
121C2×#67—96no—
122C2×#67—92no—
123C22×Dic3—96no—
124C22×Dic3—92no—
125#237—182no—
126#237—182no—
127#237—186no—
128#237—186no—
129C2×C4×D3—93no—
130C2×C4×D3—91no—
131C2×Dic6—93no—
132C2×Dic6—91no—
133C3×M16C2×D916yes—
134C3×M16C2×C3⋊D312yes—
135#208—96no—
136#208—92no—
137ΓL(1,𝔽25)—182no—
138ΓL(1,𝔽25)—186no—
139D3×C8—182no—
140D3×C8—186no—
141C22×C32—22no—
142C22×C32—42no—
143C22×C32C2×C4×D311yescharacteristic
144C2×C3⋊D3—32no—
145C2×C3⋊D3—121no—
146C32×C4D3×D412yes—
147C3⋊Dic3—31no—
148C3⋊Dic3—31no—
149C3⋊Dic3—62no—
150C22×C9—26no—
151C22×C9—46no—
152C22×C9C2×C4×D313yes—
153C2×D9—36no—
154C2×D9—123no—
155C4×C9D3×D416yes—
156Dic9—33no—
157Dic9—33no—
158Dic9—66no—
159C2×#41—271no—
160C2×M16—272no—
161#101—272no—
162#100—271no—
163#100C3⋊D911yescharacteristic, complemented
164C3×C9#41311yescharacteristic, complemented, Sylow
165C23×C3C4×D916yes—
166C23×C3C4×C3⋊D312yes—
167C22×D3—93no—
168C22×D3—91no—
169C3×D4—26no—
170C3×D4—22no—
171C2×C3×C4C22×D913yes—
172C2×C3×C4C22×C3⋊D311yescharacteristic
173#67—184no—
174#67—1812no—
175C2×Dic3—93no—
176C2×Dic3—91no—
177C2×Dic3—96no—
178C2×Dic3—92no—
179C2×Dic3—182no—
180C2×Dic3—182no—
181C2×Dic3—186no—
182C2×Dic3—186no—
183C4×D3—96no—
184C4×D3—96no—
185C4×D3—92no—
186C4×D3—92no—
187Dic6—186no—
188Dic6—182no—
189C3×C8—26no—
190C3×C8—22no—
191C3⋊C8—186no—
192C3⋊C8—182no—
193C2×C32—21no—
194C2×C32—42no—
195C2×C32D3×C22⋊C411yescharacteristic
196C3⋊D3—62no—
197C2×C9—23no—
198C2×C9—46no—
199C2×C9D3×C22⋊C413yes—
200D9—66no—
201#41—542no—
202#41—542no—
203C2×Q8—271no—
204C2×D4—272no—
205C2×D4C2×C3⋊D911yescharacteristic
206C22×C4—271no—
207C22×C4—272no—
208M16—272no—
209M16—542no—
210M16C2×C3⋊D912yes—
211C2×C8—542no—
212C22×C3—22no—
213C22×C3—26no—
214C22×C3—46no—
215C22×C3—42no—
216C22×C3C2×C4×D913yes—
217C22×C3C2×C4×C3⋊D311yescharacteristic
218C2×D3—92no—
219C2×D3—96no—
220C2×D3—361no—
221C2×D3—363no—
222C3×C4D4×D916yes—
223C3×C4D4×C3⋊D312yes—
224Dic3—91no—
225Dic3—91no—
226Dic3—93no—
227Dic3—93no—
228Dic3—182no—
229Dic3—186no—
230C32#474511yescharacteristic
231C9#474513yescomplemented
232C23—271no—
233C23C4×C3⋊D912yes—
234Q8—542no—
235D4—22no—
236D4—544no—
237C2×C4—271no—
238C2×C4—272no—
239C2×C4—272no—
240C2×C4—272no—
241C2×C4—542no—
242C2×C4—542no—
243C2×C4C22×C3⋊D911yescharacteristic
244C8—22no—
245C8—542no—
246C2×C3—23no—
247C2×C3—21no—
248C2×C3—46no—
249C2×C3—42no—
250C2×C3C22⋊C4×D913yes—
251C2×C3C22⋊C4×C3⋊D311yescharacteristic
252D3—186no—
253D3—182no—
254C22—22no—
255C22—42no—
256C22—272no—
257C22—1081no—
258C22C2×C4×C3⋊D911yescharacteristic
259C4—271no—
260C4—271no—
261C4—542no—
262C4D4×C3⋊D912yes—
263C3#1058842513yescomplemented
264C3#1059244411yescharacteristic
265C2—21no—
266C2—42no—
267C2—542no—
268C2C22⋊C4×C3⋊D911yescharacteristic
subgroupG/H
centerC2C22⋊C4×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#100C2
socleC2×C32D3×C22⋊C4
prime radicalC2×#88742C2
cosocleC2×C3×C4×C9C23
hypercenter#100C3⋊D9
hypocenterC3×C9#413
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D9C2semidirect
3. C3×C9×#100C2semidirect
4. #11720575C2semidirect
5. #11720556C2semidirect
26. C9×#100D3semidirect
80. C3×#100D9semidirect
163. #100C3⋊D9semidirect
164. C3×C9#413semidirect
231. C9#4745semidirect
263. C3#10588425semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
M16×C3⋊D92
C3×C9×#1002
#117205752
#117205562
#105924443
#105884253