Finite Group Databasebeta

← #101×C3⋊D9 #13143884

Subgroups — #101×C3⋊D9 #13143884

Order 1728.

730 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 149 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×Q8×C3⋊D9C211yesmaximal, characteristic
2M16×C3⋊D9C212yesmaximal
3C3×C9×#101C211yesmaximal, characteristic, complemented
4#11720581C211yesmaximal, characteristic, complemented
5#11720556C212yesmaximal, complemented
6C3⋊D3×#101—31nomaximal
7D9×#101—33nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9Q8×C3⋊D9—22no—
10Q8×C3⋊D9—22no—
11C2×C4×C3⋊D9C412yes—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×#6226C412yes—
14C2×#6226C2211yescharacteristic
15C3×C9×M16C2212yes—
16#88532C2212yes—
17#88523—22no—
18C8×C3⋊D9—22no—
19C2×Q8×C3⋊D3—31no—
20M16×C3⋊D3—32no—
21C2×Q8×D9—33no—
22C32×#101D311yescharacteristic
23#63663—31no—
24#63651—32no—
25M16×D9—36no—
26C9×#101D313yescomplemented
27#63394—33no—
28#63382—36no—
29C22×C3⋊D9C2×C411yescharacteristic
30C3×Q8×C9—22no—
31C2×C3×C4×C9C2×C412yes—
32C2×C3×C4×C9C2311yescharacteristic
33C2×C3⋊Dic9C2×C412yes—
34C2×C3⋊Dic9C2×C411yescharacteristic
35C4×C3⋊D9—22no—
36C4×C3⋊D9—22no—
37C4×C3⋊D9D412yes—
38C4×C3⋊D9D412yes—
39#6226—22no—
40#6226—24no—
41C3×C8×C9—22no—
42C3⋊(C9⋊C8)—22no—
43D3×#101—93no—
44D3×#101—91no—
45C2×C32×Q8C2×D311yescharacteristic
46Q8×C3⋊D3—62no—
47Q8×C3⋊D3—62no—
48C2×C4×C3⋊D3—31no—
49C2×C4×C3⋊D3—32no—
50C2×#634—32no—
51C2×#634—31no—
52C32×M16C2×D312yes—
53#3748—32no—
54#3743—62no—
55C8×C3⋊D3—62no—
56C2×Q8×C9C2×D313yes—
57Q8×D9—66no—
58Q8×D9—66no—
59C2×C4×D9—33no—
60C2×C4×D9—36no—
61C2×Dic18—36no—
62C2×Dic18—33no—
63C9×M16C2×D316yes—
64#3667—36no—
65#3663—66no—
66C8×D9—66no—
67C22×C3×C9C22×C411yescharacteristic
68C2×C3⋊D9—22no—
69C2×C3⋊D9C22⋊C411yescharacteristic
70C2×C3⋊D9C22⋊C411yescharacteristic
71C3×C4×C9—22no—
72C3×C4×C9C2×D412yes—
73C3⋊Dic9—22no—
74C3⋊Dic9C22⋊C412yes—
75C2×D3×Q8—93no—
76C2×D3×Q8—91no—
77D3×M16—96no—
78D3×M16—92no—
79C3×#101D913yescomplemented
80C3×#101C3⋊D311yescharacteristic
81#836—93no—
82#836—91no—
83#824—96no—
84#824—92no—
85C22×C3⋊D3—31no—
86C32×Q8—22no—
87C2×C32×C4C4×D312yes—
88C2×C32×C4C22×D311yescharacteristic
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—32no—
91C4×C3⋊D3—32no—
92C4×C3⋊D3—32no—
93C4×C3⋊D3—62no—
94C4×C3⋊D3—62no—
95#634—64no—
96#634—62no—
97C22×D9—33no—
98C32×C8—22no—
99C3⋊(C3⋊C8)—62no—
100Q8×C9—26no—
101C2×C4×C9C4×D316yes—
102C2×C4×C9C22×D313yes—
103C2×Dic9—33no—
104C2×Dic9—36no—
105C4×D9—36no—
106C4×D9—36no—
107C4×D9—66no—
108C4×D9—66no—
109Dic18—612no—
110Dic18—66no—
111C8×C9—26no—
112C9⋊C8—66no—
113C2×#101—271noSylow
114C2×C3×C9—21no—
115C2×C3×C9C2×C22⋊C411yescharacteristic
116C3⋊D9—21no—
117C3⋊D9#10112yesdirect product factor
118C2×C3×Q8C2×D913yes—
119C2×C3×Q8C2×C3⋊D311yescharacteristic
120D3×Q8—182no—
121D3×Q8—182no—
122D3×Q8—186no—
123D3×Q8—186no—
124C2×C4×D3—93no—
125C2×C4×D3—91no—
126C2×C4×D3—96no—
127C2×C4×D3—92no—
128C2×Dic6—96no—
129C2×Dic6—93no—
130C2×Dic6—92no—
131C2×Dic6—91no—
132C3×M16C2×D916yes—
133C3×M16C2×C3⋊D312yes—
134#208—96no—
135#208—92no—
136ΓL(1,𝔽25)—182no—
137ΓL(1,𝔽25)—186no—
138D3×C8—182no—
139D3×C8—186no—
140C22×C32C2×C4×D311yescharacteristic
141C2×C3⋊D3—31no—
142C2×C3⋊D3—31no—
143C2×C3⋊D3—62no—
144C32×C4—22no—
145C32×C4D3×D412yes—
146C3⋊Dic3—32no—
147C3⋊Dic3—62no—
148C22×C9C2×C4×D313yes—
149C2×D9—33no—
150C2×D9—33no—
151C2×D9—66no—
152C4×C9—26no—
153C4×C9D3×D416yes—
154Dic9—36no—
155Dic9—66no—
156C22×Q8—271no—
157C2×M16—272no—
158#101—271no—
159#101—272no—
160#101C3⋊D911yescharacteristic, direct product factor
161C3×C9C2×#10111yescharacteristic, complemented, Sylow
162C22×D3—93no—
163C22×D3—91no—
164C3×Q8—26no—
165C3×Q8—22no—
166C2×C3×C4C4×D916yes—
167C2×C3×C4C22×D913yes—
168C2×C3×C4C4×C3⋊D312yes—
169C2×C3×C4C22×C3⋊D311yescharacteristic
170C2×Dic3—93no—
171C2×Dic3—91no—
172C2×Dic3—96no—
173C2×Dic3—92no—
174C4×D3—96no—
175C4×D3—96no—
176C4×D3—92no—
177C4×D3—92no—
178C4×D3—182no—
179C4×D3—182no—
180C4×D3—186no—
181C4×D3—186no—
182Dic6—184no—
183Dic6—1812no—
184Dic6—186no—
185Dic6—182no—
186C3×C8—26no—
187C3×C8—22no—
188C3⋊C8—186no—
189C3⋊C8—182no—
190C2×C32—21no—
191C2×C32D3×C22⋊C411yescharacteristic
192C3⋊D3—32no—
193C3⋊D3—61no—
194C2×C9—23no—
195C2×C9D3×C22⋊C413yes—
196D9—36no—
197D9—63no—
198C2×Q8—272no—
199C2×Q8—271no—
200C2×Q8—542no—
201C2×Q8—542no—
202C2×Q8C2×C3⋊D911yescharacteristic
203C22×C4—271no—
204C22×C4—272no—
205M16—272no—
206M16—542no—
207M16C2×C3⋊D912yes—
208C2×C8—542no—
209C22×C3C2×C4×D913yes—
210C22×C3C2×C4×C3⋊D311yescharacteristic
211C2×D3—91no—
212C2×D3—93no—
213C2×D3—91no—
214C2×D3—93no—
215C2×D3—182no—
216C2×D3—186no—
217C3×C4—22no—
218C3×C4—26no—
219C3×C4D4×D916yes—
220C3×C4D4×C3⋊D312yes—
221Dic3—92no—
222Dic3—96no—
223Dic3—182no—
224Dic3—186no—
225C32D3×#10111yescharacteristic
226C9D3×#10113yescomplemented
227C23—271no—
228Q8—22no—
229Q8—544no—
230Q8—542no—
231C2×C4—271no—
232C2×C4—272no—
233C2×C4—272no—
234C2×C4—272no—
235C2×C4—542no—
236C2×C4—542no—
237C2×C4C4×C3⋊D912yes—
238C2×C4C22×C3⋊D911yescharacteristic
239C8—22no—
240C8—542no—
241C2×C3—23no—
242C2×C3—21no—
243C2×C3C22⋊C4×D913yes—
244C2×C3C22⋊C4×C3⋊D311yescharacteristic
245D3—96no—
246D3—92no—
247D3—183no—
248D3—181no—
249C22—271no—
250C22—271no—
251C22—542no—
252C22C2×C4×C3⋊D911yescharacteristic
253C4—22no—
254C4—272no—
255C4—542no—
256C4D4×C3⋊D912yes—
257C3D9×#10113yescomplemented
258C3C3⋊D3×#10111yescharacteristic
259C2—21no—
260C2—272no—
261C2—541no—
262C2C22⋊C4×C3⋊D911yescharacteristic
subgroupG/H
centerC2C22⋊C4×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#101C2
socleC2×C32D3×C22⋊C4
prime radicalC22×C3⋊D9C2×C4
cosocleC2×C3×C4×C9C23
hypercenter#101C3⋊D9
hypocenterC3×C9C2×#101
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
3. C3×C9×#101C2semidirect
4. #11720581C2semidirect
5. #11720556C2semidirect
26. C9×#101D3semidirect
79. C3×#101D9semidirect
117. C3⋊D9#101direct (N × H)
160. #101C3⋊D9direct (N × H)
161. C3×C9C2×#101semidirect
226. C9D3×#101semidirect
257. C3D9×#101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
M16×C3⋊D92
C3×C9×#1012
#117205812
#117205562
C3⋊D3×#1013
D9×#1013