Finite Group Databasebeta

← #13144499

Subgroups — #13144499

Order 1728.

760 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#11722661C211yesmaximal, characteristic
2C3×C9×#124C211yesmaximal, characteristic, complemented
3#11721853C212yesmaximal, complemented
4#11720575C211yesmaximal, characteristic, complemented
5#11720518C212yesmaximal, complemented
6#10592760—31nomaximal
7#10588741—33nomaximal
8C2×C3×Q8×C9C2211yescharacteristic, complemented
9C2×C3×D4×C9C2211yescharacteristic
10C2×#6233—22no—
11#88742—22no—
12D4×C3⋊D9—22no—
13#88736C2212yes—
14C3×C9×C22⋊C4—22no—
15C3×C42×C9C2211yescharacteristic, complemented
16#88548—22no—
17#88547—22no—
18#88532C2212yescomplemented
19#64361—31no—
20C32×#124D311yescharacteristic
21#64151—32no—
22#63710—33no—
23#63660—31no—
24#63632—32no—
25C9×#124D313yescomplemented
26#63505—36no—
27#63391—33no—
28#63364—36no—
29C23×C3×C9D412yes—
30C22×C3⋊D9—22no—
31C3×Q8×C9—22no—
32C3×D4×C9—22no—
33C2×C3×C4×C9—21no—
34C2×C3×C4×C9—21no—
35C2×C3×C4×C9C2311yescharacteristic
36#6233—24no—
37#6233—22no—
38C2×C3⋊Dic9—22no—
39C4⋊(C3⋊D9)D412yes—
40C4×C3⋊D9—22no—
41#6226D412yes—
42C3⋊(C9⋊C8)—22no—
43#5061—93no—
44#5061—91no—
45C2×C32×Q8C2×D311yescharacteristic
46C2×C32×D4C2×D311yescharacteristic
47C2×#638—62no—
48#3830—62no—
49D4×C3⋊D3—62no—
50#3828—32no—
51C32×C22⋊C4—22no—
52C32×C42C2×D311yescharacteristic
53#3756—62no—
54#3755—62no—
55#3748—32no—
56C2×Q8×C9C2×D313yescomplemented
57C2×D4×C9C2×D313yes—
58C2×#611—66no—
59#3699—66no—
60D4×D9—66no—
61#3697—36no—
62C9×C22⋊C4—26no—
63C42×C9C2×D313yescomplemented
64#3675—66no—
65#3674—66no—
66#3667—36no—
67C22×C3×C9—22no—
68C22×C3×C9—42no—
69C22×C3×C9C2×D411yescharacteristic
70C2×C3⋊D9—22no—
71C2×C3⋊D9—82no—
72C3×C4×C9—41no—
73C3×C4×C9—41no—
74C3×C4×C9C2×D412yes—
75C3⋊Dic9—22no—
76#1004—93no—
77#1004—91no—
78C3×#124D913yescomplemented
79C3×#124C3⋊D311yescharacteristic
80#942—96no—
81#942—92no—
82#833—93no—
83#833—91no—
84#805—96no—
85#805—92no—
86C23×C32#6712yes—
87C22×C3⋊D3—62no—
88C32×Q8—22no—
89C32×D4—22no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4C22×D311yescharacteristic
93#638—64no—
94#638—62no—
95C2×C3⋊Dic3—62no—
96C4⋊(C3⋊D3)—32no—
97C4×C3⋊D3—62no—
98#634—32no—
99C23×C9#6716yes—
100C22×D9—66no—
101C3⋊(C3⋊C8)—62no—
102Q8×C9—26no—
103D4×C9—26no—
104C2×C4×C9—23no—
105C2×C4×C9—23no—
106C2×C4×C9C22×D313yes—
107#611—612no—
108#611—66no—
109C2×Dic9—66no—
110D36—36no—
111C4×D9—66no—
112Dic18—36no—
113C9⋊C8—66no—
114#455—271noSylow
115C2×C3×C9—21no—
116C2×C3×C9—42no—
117C2×C3×C9#12011yescharacteristic
118C3⋊D9—42no—
119C2×C3×Q8C2×D913yescomplemented
120C2×C3×Q8C2×C3⋊D311yescharacteristic
121C2×C3×D4C2×D913yes—
122C2×C3×D4C2×C3⋊D311yescharacteristic
123C2×#67—186no—
124C2×#67—182no—
125#237—182no—
126#237—186no—
127D3×D4—182no—
128D3×D4—186no—
129#235—96no—
130#235—92no—
131C3×C22⋊C4—26no—
132C3×C22⋊C4—22no—
133C3×C42C2×D913yescomplemented
134C3×C42C2×C3⋊D311yescharacteristic
135#216—182no—
136#216—186no—
137#215—182no—
138#215—186no—
139#208—96no—
140#208—92no—
141C22×C32—22no—
142C22×C32—42no—
143C22×C32C2×#6711yescharacteristic
144C2×C3⋊D3—62no—
145C2×C3⋊D3—242no—
146C32×C4—41no—
147C32×C4—41no—
148C32×C4D3×D412yes—
149C3⋊Dic3—62no—
150C22×C9—26no—
151C22×C9—46no—
152C22×C9C2×#6713yes—
153C2×D9—66no—
154C2×D9—246no—
155C4×C9—43no—
156C4×C9—43no—
157C4×C9D3×D416yes—
158Dic9—66no—
159H(5,𝔽2)—271no—
160#137—272no—
161#124C3⋊D911yescharacteristic, complemented
162#104—272no—
163#100—271no—
164C3×C9#45511yescharacteristic, complemented, Sylow
165C23×C3#61116yes—
166C23×C3#63812yes—
167C22×D3—182no—
168C22×D3—186no—
169C3×Q8—26no—
170C3×Q8—22no—
171C3×D4—26no—
172C3×D4—22no—
173C2×C3×C4—23no—
174C2×C3×C4—23no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4C22×D913yes—
178C2×C3×C4C22×C3⋊D311yescharacteristic
179#67—184no—
180#67—1812no—
181#67—186no—
182#67—182no—
183C2×Dic3—182no—
184C2×Dic3—186no—
185D12—96no—
186D12—92no—
187C4×D3—186no—
188C4×D3—182no—
189Dic6—96no—
190Dic6—92no—
191C3⋊C8—186no—
192C3⋊C8—182no—
193C2×C32—21no—
194C2×C32—42no—
195C2×C32#93711yescharacteristic
196C3⋊D3—122no—
197C2×C9—23no—
198C2×C9—46no—
199C2×C9#93713yes—
200D9—126no—
201#41—272no—
202#41—542no—
203C2×Q8C2×C3⋊D911yescharacteristic, complemented
204C2×D4—542no—
205C2×D4—542no—
206C2×D4C2×C3⋊D911yescharacteristic
207Q16—542no—
208SD16—542no—
209M16—272no—
210C22⋊C4—22no—
211C42C2×C3⋊D911yescharacteristic, complemented
212C22×C3—26no—
213C22×C3—22no—
214C22×C3—46no—
215C22×C3—42no—
216C22×C3C2×#61113yes—
217C22×C3C2×#63811yescharacteristic
218C2×D3—186no—
219C2×D3—182no—
220C2×D3—722no—
221C2×D3—726no—
222C3×C4—41no—
223C3×C4—43no—
224C3×C4—43no—
225C3×C4—41no—
226C3×C4D4×D916yes—
227C3×C4D4×C3⋊D312yes—
228Dic3—186no—
229Dic3—182no—
230C32#506111yescharacteristic
231C9#506113yescomplemented
232C23—542no—
233C23#623312yes—
234Q8—22no—
235Q8—272no—
236D4—22no—
237D4—272no—
238D4—544no—
239D4—542no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—542no—
243C2×C4—542no—
244C2×C4C22×C3⋊D911yescharacteristic
245C8—542no—
246C2×C3—23no—
247C2×C3—21no—
248C2×C3—46no—
249C2×C3—42no—
250C2×C3#6349913yes—
251C2×C3#6414611yescharacteristic
252D3—366no—
253D3—362no—
254C22—22no—
255C22—42no—
256C22—542no—
257C22—2162no—
258C22C2×#623311yescharacteristic
259C4—41no—
260C4—41no—
261C4—542no—
262C4D4×C3⋊D912yes—
263C3#1058874113yescomplemented
264C3#1059276011yescharacteristic
265C2—21no—
266C2—42no—
267C2—1082no—
268C2#1172183711yescharacteristic
subgroupG/H
centerC2#11721837
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#124C2
socleC2×C32#937
prime radical#11722661C2
hypercenter#124C3⋊D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. C3×C9×#124C2semidirect
3. #11721853C2semidirect
4. #11720575C2semidirect
5. #11720518C2semidirect
8. C2×C3×Q8×C9C22semidirect
15. C3×C42×C9C22semidirect
18. #88532C22semidirect
25. C9×#124D3semidirect
56. C2×Q8×C9C2×D3semidirect
63. C42×C9C2×D3semidirect
78. C3×#124D9semidirect
119. C2×C3×Q8C2×D9semidirect
133. C3×C42C2×D9semidirect
161. #124C3⋊D9semidirect
164. C3×C9#455semidirect
203. C2×Q8C2×C3⋊D9semidirect
211. C42C2×C3⋊D9semidirect
231. C9#5061semidirect
263. C3#10588741semidirect

Maximal subgroups

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

subgroupindex
#117226612
C3×C9×#1242
#117218532
#117205752
#117205182
#105927603
#105887413