Finite Group Databasebeta

← #13140460

Subgroups — #13140460

Order 1728.

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

normal subgroups only

This group has 502 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×#63710C211yesmaximal, characteristic
2C9×#936C211yesmaximal, characteristic, complemented
3#11720893C212yesmaximal, complemented
4#11720575C211yesmaximal, characteristic, complemented
5#11720260C212yesmaximal, complemented
6#10591512—31nomaximal
7#10588741—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×C3×#611—22no—
10C3×#3699—22no—
11C3×D4×D9—22no—
12C3×#3697C2212yes—
13C2×C9×Dic6C2211yescharacteristic, complemented
14#88532C2212yescomplemented
15C9×#217—22no—
16C4×C9×Dic3C2211yescharacteristic, complemented
17#88427—22no—
18#88425—22no—
19C3×#1004—31no—
20C3×#936D311yescharacteristic
21#63827—32no—
22#63710D311yescharacteristic
23#63660—31no—
24#63569—32no—
25C9×#124—31no—
26#63505—32no—
27#63391—31no—
28#63391—61no—
29#63364—32no—
30C23×C3×C9D412yes—
31C22×C3×D9—22no—
32C3×D4×C9—22no—
33C2×C3×C4×C9C2311yescharacteristic
34C3×#611—24no—
35C3×#611—22no—
36C2×C9×Dic3—21no—
37C2×C9×Dic3—21no—
38C2×C3×Dic9—22no—
39C3×D36D412yes—
40C3×C4×D9—22no—
41C9×Dic6—22no—
42C3×Dic18D412yes—
43C3⋊(C9⋊C8)—22no—
44#5242—91no—
45#5061—91no—
46C2×C32×D4C2×D311yescharacteristic
47C2×C3×#67—62no—
48C3×#237—62no—
49C3×D3×D4—62no—
50C3×#235—32no—
51C2×C3×Dic6C2×D311yescharacteristic
52#3748—32no—
53C3×#217—22no—
54C3×C4×Dic3C2×D311yescharacteristic
55#3718—62no—
56#3715—62no—
57C2×Q8×C9—31no—
58C2×D4×C9—21no—
59C2×D4×C9C2×D311yescharacteristic
60C2×#611—22no—
61#3699—22no—
62D4×D9—22no—
63#3697C2×D312yes—
64C9×C22⋊C4—62no—
65C42×C9—31no—
66#3675—62no—
67#3674—62no—
68#3667—32no—
69#3667—62no—
70C22×C3×C9—22no—
71C22×C3×C9—42no—
72C22×C3×C9C2×D411yescharacteristic
73C2×C3×D9—22no—
74C2×C3×D9—82no—
75C3×C4×C9C2×D412yes—
76C9×Dic3—41no—
77C9×Dic3—41no—
78C3×Dic9—22no—
79C3×H(5,𝔽2)—91no—
80#1004—31no—
81C3×#124—31no—
82#951—92no—
83#942—92no—
84#936D911yescharacteristic, complemented
85#837—92no—
86#833—91no—
87#833—91no—
88#833—181no—
89#805—92no—
90C23×C32#6712yes—
91C22×C3×D3—62no—
92C32×D4—22no—
93C2×C32×C4C22×D311yescharacteristic
94C3×#67—64no—
95C3×#67—62no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—62no—
99C3×D12—32no—
100C3×C4×D3—62no—
101C3×Dic6—22no—
102C3×Dic6—32no—
103C23×C9—22no—
104C23×C9#6712yes—
105C22×D9—22no—
106C3⋊(C3⋊C8)—62no—
107Q8×C9—62no—
108D4×C9—22no—
109D4×C9—42no—
110C2×C4×C9—21no—
111C2×C4×C9—61no—
112C2×C4×C9—61no—
113C2×C4×C9C22×D311yescharacteristic
114#611—24no—
115#611—22no—
116C2×Dic9—22no—
117D36#6712yes—
118C4×D9—22no—
119Dic18#6712yes—
120C9⋊C8—62no—
121C9⋊C8—122no—
122#455—271noSylow
123C2×C3×C9—21no—
124C2×C3×C9—42no—
125C2×C3×C9#12011yescharacteristic
126C3×D9—42no—
127C3×#41—92no—
128C3×#41—182no—
129C2×C3×Q8—31no—
130C2×C3×D4—21no—
131C2×C3×D4—182no—
132C2×C3×D4—182no—
133C2×C3×D4C2×D911yescharacteristic
134C2×C3×D4D3211yescharacteristic
135C2×#67—62no—
136#237—62no—
137D3×D4—62no—
138#235—32no—
139C2×Dic6C2×D911yescharacteristic, complemented
140C3×C22⋊C4—62no—
141C3×C42—31no—
142#217—22no—
143#216—182no—
144#216—182no—
145#215—182no—
146#214—182no—
147C4×Dic3C2×D911yescharacteristic, complemented
148#208—92no—
149#208—92no—
150#208—182no—
151C22×C32—22no—
152C22×C32—42no—
153C22×C32C2×#6711yescharacteristic
154C2×C3×D3—62no—
155C2×C3×D3—242no—
156C32×C4D3×D412yes—
157C3×Dic3—41no—
158C3×Dic3—41no—
159C3×Dic3—62no—
160C22×C9—22no—
161C22×C9—21no—
162C22×C9—42no—
163C22×C9—42no—
164C22×C9—82no—
165C22×C9C2×#6711yescharacteristic
166C2×D9—22no—
167C2×D9—82no—
168C4×C9—22no—
169C4×C9—121no—
170C4×C9—121no—
171C4×C9C2×#6712yes—
172Dic9—22no—
173H(5,𝔽2)—91no—
174#137—272no—
175#124—31no—
176#104—272no—
177#100—271no—
178C3×C9#45511yescharacteristic, complemented, Sylow
179C23×C3—22no—
180C23×C3—182no—
181C23×C3#61112yes—
182C23×C3#62512yes—
183C22×D3—62no—
184C3×Q8—62no—
185C3×Q8—92no—
186C3×D4—22no—
187C3×D4—22no—
188C3×D4—42no—
189C3×D4—92no—
190C3×D4—184no—
191C3×D4—182no—
192C2×C3×C4—21no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—182no—
196C2×C3×C4—182no—
197C2×C3×C4C22×D911yescharacteristic
198C2×C3×C4C2×D3211yescharacteristic
199#67—64no—
200#67—62no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—62no—
204D12—32no—
205C4×D3—62no—
206Dic6—22no—
207Dic6—32no—
208C3⋊C8—182no—
209C3⋊C8—182no—
210C3⋊C8—362no—
211C2×C32—21no—
212C2×C32—42no—
213C2×C32#93711yescharacteristic
214C3×D3—122no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9—41no—
218C2×C9—42no—
219C2×C9—82no—
220C2×C9#95311yescharacteristic
221D9—42no—
222#41—92no—
223#41—182no—
224C2×Q8—31no—
225C2×D4—182no—
226C2×D4—182no—
227C2×D4D3×D911yescharacteristic
228Q16—542no—
229SD16—542no—
230M16—272no—
231C22⋊C4—62no—
232C42—31no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—82no—
240C22×C3—182no—
241C22×C3—722no—
242C22×C3C2×#61111yescharacteristic
243C22×C3C2×#62511yescharacteristic
244C2×D3—62no—
245C2×D3—242no—
246C3×C4—22no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—182no—
250C3×C4D4×D912yes—
251C3×C4D3×#6712yes—
252Dic3—41no—
253Dic3—41no—
254Dic3—62no—
255C32#506111yescharacteristic
256C9—21no—
257C9#524211yescharacteristic, complemented
258C23—182no—
259C23#619412yes—
260Q8—62no—
261Q8—92no—
262D4—22no—
263D4—92no—
264D4—184no—
265D4—182no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4C2×D3×D911yescharacteristic
271C8—542no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—41no—
276C2×C3—42no—
277C2×C3—42no—
278C2×C3—82no—
279C2×C3—362no—
280C2×C3#6349911yescharacteristic
281C2×C3#6397611yescharacteristic
282D3—122no—
283C22—22no—
284C22—42no—
285C22—182no—
286C22—722no—
287C22C2×#619411yescharacteristic
288C4—121no—
289C4—121no—
290C4—182no—
291C4D9×#6712yes—
292C3—21no—
293C3#1058874111yescharacteristic, complemented
294C3#1059151211yescharacteristic
295C2—21no—
296C2—42no—
297C2—362no—
298C2#1172128511yescharacteristic
subgroupG/H
centerC2#11721285
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#937
prime radicalC3×#63710C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C9×#936C2semidirect
3. #11720893C2semidirect
4. #11720575C2semidirect
5. #11720260C2semidirect
13. C2×C9×Dic6C22semidirect
14. #88532C22semidirect
16. C4×C9×Dic3C22semidirect
84. #936D9semidirect
139. C2×Dic6C2×D9semidirect
147. C4×Dic3C2×D9semidirect
178. C3×C9#455semidirect
257. C9#5242semidirect
293. C3#10588741semidirect

Maximal subgroups

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

subgroupindex
C3×#637102
C9×#9362
#117208932
#117205752
#117202602
#105915123
#105887413