Finite Group Databasebeta

← #13154265

Subgroups — #13154265

Order 1728.

1200 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 271 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×C9×#127C211yesmaximal, characteristic, complemented
2#11721839C214yesmaximal, complemented
3#11721838C212yesmaximal, complemented
4#11721835C214yesmaximal, complemented
5D4×C3⋊Dic9C212yesmaximal, complemented
6#11721709C211yesmaximal, characteristic, complemented
7#11721707C211yesmaximal, characteristic, complemented
8#10596190—31nomaximal
9#10589821—33nomaximal
10C2×C3×D4×C9C2212yes—
11C2×C3×D4×C9C2214yes—
12C2×#6233C2214yes—
13C22×C3⋊Dic9C2214yes—
14C2×C4×C3⋊D9C2211yescharacteristic
15C3×C42×C9C2211yescharacteristic
16#88551C2214yes—
17#88551C2214yes—
18#88536C2212yes—
19C4⋊(C3⋊Dic9)C2212yes—
20#88534C2212yes—
21#88534C2214yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C32×#127D311yescharacteristic
24#64148—34no—
25#64147—32no—
26#64144—34no—
27D4×C3⋊Dic3—32no—
28#64081—31no—
29#64079—31no—
30C9×#127D313yescomplemented
31#63501—312no—
32#63500—36no—
33#63497—312no—
34D4×Dic9—36no—
35#63434—33no—
36#63432—33no—
37C23×C3×C9C2314yes—
38C22×C3⋊D9C2311yescharacteristic
39C3×D4×C9—24no—
40C3×D4×C9—44no—
41C2×C3×C4×C9C2312yes—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—44no—
44C2×C3⋊Dic9—44no—
45C2×C3⋊Dic9C2311yescharacteristic
46C2×C3⋊Dic9C2312yes—
47C2×C3⋊Dic9C2314yes—
48C4×C3⋊D9—22no—
49#5719—93no—
50#5719—91no—
51C2×C32×D4C2×D312yes—
52C2×C32×D4C2×D314yes—
53C2×#638—34no—
54C22×C3⋊Dic3—34no—
55C2×C4×C3⋊D3—31no—
56C32×C42C2×D311yescharacteristic
57#3757—34no—
58#3757—34no—
59#3752—32no—
60C4⋊(C3⋊Dic3)—32no—
61#3750—34no—
62#3750—32no—
63C4×C3⋊Dic3—31no—
64C2×D4×C9C2×D316yes—
65C2×D4×C9C2×D3112yes—
66C2×#611—312no—
67C22×Dic9—312no—
68C2×C4×D9—33no—
69C42×C9C2×D313yes—
70#3676—312no—
71#3676—312no—
72(C2×D9)⋊C4—36no—
73C4⋊Dic9—36no—
74Dic9⋊C4—36no—
75Dic9⋊C4—312no—
76C4×Dic9—33no—
77C22×C3×C9—24no—
78C22×C3×C9—24no—
79C22×C3×C9—24no—
80C22×C3×C9C2411yescharacteristic
81C2×C3⋊D9—21no—
82C2×C3⋊D9—22no—
83C3×C4×C9—22no—
84C3×C4×C9#4112yes—
85C3⋊Dic9—21no—
86C3⋊Dic9—22no—
87C3⋊Dic9—24no—
88C3×#127D913yescomplemented
89C3×#127C3⋊D311yescharacteristic
90#939—912no—
91#939—94no—
92#938—96no—
93#938—92no—
94#935—912no—
95#935—94no—
96D4×Dic3—96no—
97D4×Dic3—92no—
98#872—93no—
99#872—91no—
100#870—93no—
101#870—91no—
102C23×C32C22×D314yes—
103C22×C3⋊D3—31no—
104C32×D4—24no—
105C32×D4—44no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108#638—124no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—34no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—124no—
113C4×C3⋊D3—62no—
114C23×C9C22×D3112yes—
115C22×D9—33no—
116D4×C9—212no—
117D4×C9—412no—
118C2×C4×C9C22×D316yes—
119C2×C4×C9C22×D313yes—
120#611—1212no—
121C2×Dic9—36no—
122C2×Dic9—312no—
123C2×Dic9—33no—
124C2×Dic9—1212no—
125C4×D9—66no—
126#540—271noSylow
127C2×C3×C9—44no—
128C2×C3×C9C2×#4111yescharacteristic
129C2×C3×C9H(5,𝔽2)12yes—
130C3⋊D9—41no—
131C2×C3×D4C2×D916yes—
132C2×C3×D4C2×D9112yes—
133C2×C3×D4C2×C3⋊D312yes—
134C2×C3×D4C2×C3⋊D314yes—
135C2×#67—912no—
136C2×#67—94no—
137C22×Dic3—912no—
138C22×Dic3—94no—
139C2×C4×D3—93no—
140C2×C4×D3—91no—
141C3×C42C2×D913yes—
142C3×C42C2×C3⋊D311yescharacteristic
143#217—912no—
144#217—912no—
145#217—94no—
146#217—94no—
147(C2×D3)⋊C4—96no—
148(C2×D3)⋊C4—92no—
149C4⋊Dic3—96no—
150C4⋊Dic3—92no—
151Dic3⋊C4—912no—
152Dic3⋊C4—96no—
153Dic3⋊C4—94no—
154Dic3⋊C4—92no—
155C4×Dic3—93no—
156C4×Dic3—91no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—24no—
160C22×C32C23×D311yescharacteristic
161C2×C3⋊D3—61no—
162C2×C3⋊D3—62no—
163C32×C4—22no—
164C32×C4#23712yes—
165C3⋊Dic3—61no—
166C3⋊Dic3—62no—
167C3⋊Dic3—64no—
168C22×C9—212no—
169C22×C9—212no—
170C22×C9—212no—
171C22×C9C23×D313yes—
172C2×D9—63no—
173C2×D9—66no—
174C4×C9—26no—
175C4×C9#23716yes—
176Dic9—63no—
177Dic9—66no—
178Dic9—612no—
179#127C3⋊D911yescharacteristic, complemented
180#125—271no—
181#123—274no—
182#121—272no—
183#121—274no—
184C4×D4—272no—
185#117—271no—
186C3×C9#54011yescharacteristic, complemented, Sylow
187C23×C3C22×D9112yes—
188C23×C3C22×C3⋊D314yes—
189C22×D3—91no—
190C22×D3—93no—
191C3×D4—212no—
192C3×D4—24no—
193C3×D4—44no—
194C3×D4—412no—
195C2×C3×C4C22×D916yes—
196C2×C3×C4C22×D913yes—
197C2×C3×C4C22×C3⋊D312yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199#67—364no—
200#67—3612no—
201C2×Dic3—92no—
202C2×Dic3—94no—
203C2×Dic3—91no—
204C2×Dic3—96no—
205C2×Dic3—912no—
206C2×Dic3—93no—
207C2×Dic3—364no—
208C2×Dic3—3612no—
209C4×D3—186no—
210C4×D3—182no—
211C2×C32—44no—
212C2×C32C2×#23711yescharacteristic
213C2×C32#100412yes—
214C3⋊D3—121no—
215C2×C9—412no—
216C2×C9C2×#23713yes—
217C2×C9#100416yes—
218D9—123no—
219C2×D4—274no—
220C2×D4C2×C3⋊D912yes—
221C2×D4C2×C3⋊D914yes—
222C22×C4—274no—
223C22×C4—271no—
224C4⋊C4—272no—
225C4⋊C4—272no—
226C4⋊C4—274no—
227C22⋊C4—274no—
228C22⋊C4—274no—
229C22⋊C4—272no—
230C42—271no—
231C42C2×C3⋊D911yescharacteristic
232C22×C3—24no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—212no—
236C22×C3—212no—
237C22×C3—212no—
238C22×C3C23×D913yes—
239C22×C3C23×C3⋊D311yescharacteristic
240C2×D3—181no—
241C2×D3—182no—
242C2×D3—183no—
243C2×D3—186no—
244C3×C4—22no—
245C3×C4—26no—
246C3×C4#369916yes—
247C3×C4#383012yes—
248Dic3—181no—
249Dic3—182no—
250Dic3—184no—
251Dic3—183no—
252Dic3—186no—
253Dic3—1812no—
254C32#571911yescharacteristic
255C9#571913yescomplemented
256C23—271no—
257C23C22×C3⋊D914yes—
258D4—24no—
259D4—44no—
260D4—1084no—
261C2×C4—272no—
262C2×C4—271no—
263C2×C4—274no—
264C2×C4—542no—
265C2×C4—1084no—
266C2×C4C22×C3⋊D912yes—
267C2×C4C22×C3⋊D911yescharacteristic
268C2×C3—44no—
269C2×C3—412no—
270C2×C3C2×#369913yes—
271C2×C3#6371016yes—
272C2×C3C2×#383011yescharacteristic
273C2×C3#6436112yes—
274D3—361no—
275D3—363no—
276C22—24no—
277C22—24no—
278C22—24no—
279C22—541no—
280C22—542no—
281C22C23×C3⋊D911yescharacteristic
282C4—22no—
283C4—541no—
284C4—542no—
285C4—544no—
286C4#8874212yes—
287C3#1058982113yescomplemented
288C3#1059619011yescharacteristic
289C2—44no—
290C2—1081no—
291C2C2×#8874211yescharacteristic
292C2#1172266112yes—
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#127C2
socleC22×C32C23×D3
hypercenter#127C3⋊D9
hypocenterC3×C9#540
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×C9×#127C2semidirect
2. #11721839C2semidirect
3. #11721838C2semidirect
4. #11721835C2semidirect
5. D4×C3⋊Dic9C2semidirect
6. #11721709C2semidirect
7. #11721707C2semidirect
30. C9×#127D3semidirect
88. C3×#127D9semidirect
179. #127C3⋊D9semidirect
186. C3×C9#540semidirect
255. C9#5719semidirect
287. C3#10589821semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1272
#117218392
#117218382
#117218352
D4×C3⋊Dic92
#117217092
#117217072
#105961903
#105898213