Finite Group Databasebeta

← #13144490

Subgroups — #13144490

Order 1728.

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

normal subgroups only

This group has 800 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×#123C211yesmaximal, characteristic, complemented
3#11721818C212yesmaximal, complemented
4#11720577C211yesmaximal, characteristic, complemented
5#11720523C212yesmaximal, complemented
6#10592752—31nomaximal
7#10588733—33nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C22×C3×C4×C9C2211yescharacteristic, complemented
10C2×#6233—21no—
11C2×#6233C2212yes—
12#88742—22no—
13D4×C3⋊D9—22no—
14#88736—21no—
15C3×C9×C4⋊C4—21no—
16C3×C9×C22⋊C4—21no—
17C3×C9×C22⋊C4C2211yescharacteristic, complemented
18#88551C2212yescomplemented
19#88536—22no—
20#88534—22no—
21#64361—31no—
22C32×#123D311yescharacteristic
23#64138—32no—
24#63710—33no—
25#63661—31no—
26#63633—32no—
27C9×#123D313yescomplemented
28#63491—36no—
29#63393—33no—
30#63365—36no—
31C23×C3×C9D411yescharacteristic
32C23×C3×C9C2311yescharacteristic
33C22×C3⋊D9—21no—
34C22×C3⋊D9D412yes—
35C3×D4×C9—22no—
36C2×C3×C4×C9—22no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9D411yescharacteristic
40#6233—22no—
41#6233—22no—
42#6233—22no—
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9—22no—
45C2×C3⋊Dic9D412yes—
46C4⋊(C3⋊D9)—21no—
47C4×C3⋊D9—22no—
48#6226—21no—
49#5053—93no—
50#5053—91no—
51C2×C32×D4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C2×#638—32no—
54C2×#638—61no—
55#3830—62no—
56D4×C3⋊D3—62no—
57#3828—61no—
58C32×C4⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C22⋊C4C2×D311yescharacteristic
61#3757—32no—
62#3752—62no—
63#3750—62no—
64C2×D4×C9C2×D313yes—
65C22×C4×C9C2×D313yescomplemented
66C2×#611—36no—
67C2×#611—63no—
68#3699—66no—
69D4×D9—66no—
70#3697—63no—
71C9×C4⋊C4—23no—
72C9×C22⋊C4—23no—
73C9×C22⋊C4C2×D313yescomplemented
74#3676—36no—
75(C2×D9)⋊C4—66no—
76Dic9⋊C4—66no—
77C22×C3×C9—21no—
78C22×C3×C9—41no—
79C22×C3×C9—41no—
80C22×C3×C9C2×D412yes—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—42no—
84C2×C3⋊D9—81no—
85C3×C4×C9—21no—
86C3×C4×C9—41no—
87C3×C4×C9—41no—
88C3⋊Dic9—22no—
89C3⋊Dic9—42no—
90#1004—93no—
91#1004—91no—
92C3×#123D913yescomplemented
93C3×#123C3⋊D311yescharacteristic
94#929—96no—
95#929—92no—
96#834—93no—
97#834—91no—
98#806—96no—
99#806—92no—
100C23×C32#6711yescharacteristic
101C23×C32C22×D311yescharacteristic
102C22×C3⋊D3—32no—
103C22×C3⋊D3—61no—
104C32×D4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4#6711yescharacteristic
109#638—62no—
110#638—62no—
111#638—62no—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—62no—
115C4⋊(C3⋊D3)—61no—
116C4×C3⋊D3—62no—
117#634—61no—
118C23×C9#6713yes—
119C23×C9C22×D313yes—
120C22×D9—36no—
121C22×D9—63no—
122D4×C9—26no—
123C2×C4×C9—26no—
124C2×C4×C9—23no—
125C2×C4×C9—23no—
126C2×C4×C9#6713yes—
127#611—66no—
128#611—66no—
129#611—66no—
130C2×Dic9—36no—
131C2×Dic9—63no—
132C2×Dic9—66no—
133D36—63no—
134C4×D9—66no—
135Dic18—63no—
136#458—271noSylow
137C2×C3×C9—22no—
138C2×C3×C9—21no—
139C2×C3×C9—41no—
140C2×C3×C9#12011yescharacteristic
141C3⋊D9—42no—
142C2×C3×D4C2×D913yes—
143C2×C3×D4C2×C3⋊D311yescharacteristic
144C22×C3×C4C2×D913yescomplemented
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C2×#67—96no—
147C2×#67—92no—
148C2×#67—183no—
149C2×#67—181no—
150#237—182no—
151#237—186no—
152D3×D4—182no—
153D3×D4—186no—
154#235—183no—
155#235—181no—
156C3×C4⋊C4—23no—
157C3×C4⋊C4—21no—
158C3×C22⋊C4—23no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4C2×D913yescomplemented
161C3×C22⋊C4C2×C3⋊D311yescharacteristic
162#217—96no—
163#217—92no—
164(C2×D3)⋊C4—182no—
165(C2×D3)⋊C4—186no—
166Dic3⋊C4—182no—
167Dic3⋊C4—186no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32D3×D412yes—
172C22×C32C2×#6711yescharacteristic
173C2×C3⋊D3—62no—
174C2×C3⋊D3—122no—
175C2×C3⋊D3—241no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—41no—
179C3⋊Dic3—62no—
180C3⋊Dic3—122no—
181C22×C9—23no—
182C22×C9—43no—
183C22×C9—43no—
184C22×C9D3×D416yes—
185C22×C9C2×#6713yes—
186C2×D9—66no—
187C2×D9—126no—
188C2×D9—243no—
189C4×C9—23no—
190C4×C9—43no—
191C4×C9—43no—
192Dic9—66no—
193Dic9—126no—
194H(5,𝔽2)—271no—
195#123—272no—
196#123C3⋊D911yescharacteristic, complemented
197#99—271no—
198#99—272no—
199C3×C9#45811yescharacteristic, complemented, Sylow
200C23×C3#61113yes—
201C23×C3C22×D913yes—
202C23×C3#63811yescharacteristic
203C23×C3C22×C3⋊D311yescharacteristic
204C22×D3—96no—
205C22×D3—92no—
206C22×D3—181no—
207C22×D3—183no—
208C3×D4—26no—
209C3×D4—22no—
210C2×C3×C4—26no—
211C2×C3×C4—23no—
212C2×C3×C4—23no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4#61113yes—
217C2×C3×C4#63811yescharacteristic
218#67—182no—
219#67—186no—
220#67—186no—
221#67—186no—
222#67—182no—
223#67—182no—
224C2×Dic3—96no—
225C2×Dic3—92no—
226C2×Dic3—181no—
227C2×Dic3—183no—
228C2×Dic3—186no—
229C2×Dic3—182no—
230D12—181no—
231D12—183no—
232C4×D3—182no—
233C4×D3—186no—
234Dic6—181no—
235Dic6—183no—
236C2×C32—22no—
237C2×C32—21no—
238C2×C32—41no—
239C2×C32#93711yescharacteristic
240C3⋊D3—122no—
241C2×C9—26no—
242C2×C9—23no—
243C2×C9—43no—
244C2×C9#93713yes—
245D9—126no—
246#41—542no—
247#41—541no—
248C2×D4—272no—
249C2×D4—542no—
250C2×D4—541no—
251C2×D4C2×C3⋊D911yescharacteristic
252C22×C4C2×C3⋊D911yescharacteristic, complemented
253C4⋊C4—21no—
254C4⋊C4—542no—
255C22⋊C4—21no—
256C22⋊C4—272no—
257C22⋊C4—542no—
258C22⋊C4C2×C3⋊D911yescharacteristic, complemented
259C22×C3—23no—
260C22×C3—21no—
261C22×C3—43no—
262C22×C3—41no—
263C22×C3—43no—
264C22×C3—41no—
265C22×C3D4×D916yes—
266C22×C3C2×#61113yes—
267C22×C3D4×C3⋊D312yes—
268C22×C3C2×#63811yescharacteristic
269C2×D3—186no—
270C2×D3—182no—
271C2×D3—362no—
272C2×D3—366no—
273C2×D3—721no—
274C2×D3—723no—
275C3×C4—23no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—43no—
279C3×C4—43no—
280C3×C4—41no—
281Dic3—186no—
282Dic3—182no—
283Dic3—362no—
284Dic3—366no—
285C32#505311yescharacteristic
286C9#505313yescomplemented
287C23—272no—
288C23—541no—
289C23#623311yescharacteristic
290C23C22×C3⋊D911yescharacteristic
291Q8—541no—
292D4—22no—
293D4—541no—
294D4—542no—
295D4—542no—
296D4—542no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—272no—
301C2×C4—542no—
302C2×C4—541no—
303C2×C4—542no—
304C2×C4#623311yescharacteristic
305C2×C3—26no—
306C2×C3—23no—
307C2×C3—22no—
308C2×C3—21no—
309C2×C3—43no—
310C2×C3—41no—
311C2×C3#6349913yes—
312C2×C3#6414611yescharacteristic
313D3—366no—
314D3—362no—
315C22—21no—
316C22—41no—
317C22—41no—
318C22—542no—
319C22—1082no—
320C22—2161no—
321C22D4×C3⋊D912yes—
322C22C2×#623311yescharacteristic
323C4—21no—
324C4—41no—
325C4—41no—
326C4—542no—
327C4—1082no—
328C3#1058873313yescomplemented
329C3#1059275211yescharacteristic
330C2—21no—
331C2—22no—
332C2—41no—
333C2—1082no—
334C2#1172183711yescharacteristic
subgroupG/H
centerC2#11721837
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#123C2
socleC2×C32#937
prime radical#11722661C2
hypercenter#123C3⋊D9
hypocenterC3×C9#458
LCSG ▷ C23×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C3×C9×#123C2semidirect
3. #11721818C2semidirect
4. #11720577C2semidirect
5. #11720523C2semidirect
9. C22×C3×C4×C9C22semidirect
17. C3×C9×C22⋊C4C22semidirect
18. #88551C22semidirect
27. C9×#123D3semidirect
65. C22×C4×C9C2×D3semidirect
73. C9×C22⋊C4C2×D3semidirect
92. C3×#123D9semidirect
144. C22×C3×C4C2×D9semidirect
160. C3×C22⋊C4C2×D9semidirect
196. #123C3⋊D9semidirect
199. C3×C9#458semidirect
252. C22×C4C2×C3⋊D9semidirect
258. C22⋊C4C2×C3⋊D9semidirect
286. C9#5053semidirect
328. C3#10588733semidirect

Maximal subgroups

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

subgroupindex
#117226612
C3×C9×#1232
#117218182
#117205772
#117205232
#105927523
#105887333