Finite Group Databasebeta

← #13143866

Subgroups — #13143866

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
2#11721835C211yesmaximal, characteristic, complemented
3#11721732C212yesmaximal, complemented
4C3×C9×#99C211yesmaximal, characteristic, complemented
5#11720523C212yesmaximal, complemented
6#10592441—31nomaximal
7#10588422—33nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233—21no—
10C2×#6233C2212yes—
11C22×C3⋊Dic9C2211yescharacteristic, complemented
12#88742—22no—
13D4×C3⋊D9—22no—
14#88736—21no—
15C3×C9×C22⋊C4C2212yescomplemented
16#88551—21no—
17#88551C2211yescharacteristic, complemented
18#88536—22no—
19C4⋊(C3⋊Dic9)—22no—
20#88534—21no—
21#64361—31no—
22#64144—31no—
23#64095—32no—
24#63710—33no—
25C32×#99D311yescharacteristic
26#63633—32no—
27#63497—33no—
28#63448—36no—
29C9×#99D313yescomplemented
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×C9D411yescharacteristic
38#6233—22no—
39#6233—22no—
40#6233—22no—
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—22no—
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D412yes—
46C4⋊(C3⋊D9)—21no—
47C4×C3⋊D9—22no—
48#6226—21no—
49#4742—93no—
50#4742—91no—
51C2×C32×D4C2×D311yescharacteristic
52C2×#638—32no—
53C2×#638—61no—
54C22×C3⋊Dic3—31no—
55#3830—62no—
56D4×C3⋊D3—62no—
57#3828—61no—
58C32×C22⋊C4C2×D312yes—
59#3757—31no—
60#3757—61no—
61#3752—62no—
62C4⋊(C3⋊Dic3)—62no—
63#3750—61no—
64C2×D4×C9C2×D313yes—
65C2×#611—36no—
66C2×#611—63no—
67C22×Dic9—33no—
68#3699—66no—
69D4×D9—66no—
70#3697—63no—
71C9×C22⋊C4C2×D316yescomplemented
72#3676—33no—
73#3676—63no—
74(C2×D9)⋊C4—66no—
75C4⋊Dic9—66no—
76Dic9⋊C4—63no—
77C22×C3×C9—21no—
78C22×C3×C9—41no—
79C22×C3×C9—41no—
80C22×C3×C9C2×D411yescharacteristic
81C22×C3×C9C2×D412yes—
82C2×C3⋊D9—22no—
83C2×C3⋊D9—42no—
84C2×C3⋊D9—81no—
85C3×C4×C9—21no—
86C3×C4×C9—42no—
87C3⋊Dic9—22no—
88C3⋊Dic9—41no—
89C3⋊Dic9—41no—
90#1004—93no—
91#1004—91no—
92#935—93no—
93#935—91no—
94#886—96no—
95#886—92no—
96C3×#99D913yescomplemented
97C3×#99C3⋊D311yescharacteristic
98#806—96no—
99#806—92no—
100C23×C32D1211yescharacteristic
101C23×C32C22×D311yescharacteristic
102C22×C3⋊D3—32no—
103C22×C3⋊D3—61no—
104C32×D4—22no—
105C2×C32×C4—22no—
106C2×C32×C4D1211yescharacteristic
107#638—62no—
108#638—62no—
109#638—62no—
110C2×C3⋊Dic3—32no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—62no—
115C4⋊(C3⋊D3)—61no—
116C4×C3⋊D3—62no—
117#634—61no—
118C23×C9D1213yes—
119C23×C9C22×D313yes—
120C22×D9—36no—
121C22×D9—63no—
122D4×C9—26no—
123C2×C4×C9—26no—
124C2×C4×C9D1213yes—
125#611—66no—
126#611—66no—
127#611—66no—
128C2×Dic9—36no—
129C2×Dic9—63no—
130C2×Dic9—63no—
131C2×Dic9—63no—
132C2×Dic9—66no—
133D36—63no—
134C4×D9—66no—
135Dic18—63no—
136#458—271noSylow
137C2×C3×C9—21no—
138C2×C3×C9—22no—
139C2×C3×C9—41no—
140C2×C3×C9#12011yescharacteristic
141C3⋊D9—42no—
142C2×C3×D4C2×D913yes—
143C2×C3×D4C2×C3⋊D311yescharacteristic
144C2×#67—96no—
145C2×#67—92no—
146C2×#67—183no—
147C2×#67—181no—
148C22×Dic3—93no—
149C22×Dic3—91no—
150#237—182no—
151#237—186no—
152D3×D4—182no—
153D3×D4—186no—
154#235—183no—
155#235—181no—
156C3×C22⋊C4C2×D916yescomplemented
157C3×C22⋊C4C2×C3⋊D312yes—
158#217—93no—
159#217—91no—
160#217—183no—
161#217—181no—
162(C2×D3)⋊C4—182no—
163(C2×D3)⋊C4—186no—
164C4⋊Dic3—182no—
165C4⋊Dic3—186no—
166Dic3⋊C4—183no—
167Dic3⋊C4—181no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32C2×D1211yescharacteristic
172C22×C32D3×D412yes—
173C2×C3⋊D3—62no—
174C2×C3⋊D3—122no—
175C2×C3⋊D3—241no—
176C32×C4—21no—
177C32×C4—42no—
178C3⋊Dic3—62no—
179C3⋊Dic3—121no—
180C3⋊Dic3—121no—
181C22×C9—23no—
182C22×C9—43no—
183C22×C9—43no—
184C22×C9C2×D1213yes—
185C22×C9D3×D416yes—
186C2×D9—66no—
187C2×D9—126no—
188C2×D9—243no—
189C4×C9—23no—
190C4×C9—46no—
191Dic9—66no—
192Dic9—123no—
193Dic9—123no—
194H(5,𝔽2)—271no—
195#123—271no—
196#123—272no—
197#99—272no—
198#99C3⋊D911yescharacteristic, complemented
199C3×C9#45811yescharacteristic, complemented, Sylow
200C23×C3D3613yes—
201C23×C3C22×D913yes—
202C23×C3C4⋊(C3⋊D3)11yescharacteristic
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—22no—
212C2×C3×C4D3613yes—
213C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
214#67—182no—
215#67—186no—
216#67—186no—
217#67—186no—
218#67—182no—
219#67—182no—
220C2×Dic3—96no—
221C2×Dic3—92no—
222C2×Dic3—181no—
223C2×Dic3—183no—
224C2×Dic3—183no—
225C2×Dic3—183no—
226C2×Dic3—186no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—182no—
230D12—181no—
231D12—183no—
232C4×D3—182no—
233C4×D3—186no—
234Dic6—181no—
235Dic6—183no—
236C2×C32—21no—
237C2×C32—22no—
238C2×C32—41no—
239C2×C32#88211yescharacteristic
240C3⋊D3—122no—
241C2×C9—23no—
242C2×C9—26no—
243C2×C9—43no—
244C2×C9#88213yes—
245D9—126no—
246#41—542no—
247#41—541no—
248C2×D4—272no—
249C2×D4—542no—
250C2×D4—541no—
251C2×D4C2×C3⋊D911yescharacteristic
252C22×C4—271no—
253C4⋊C4—542no—
254C4⋊C4—541no—
255C22⋊C4—271no—
256C22⋊C4—542no—
257C22⋊C4—541no—
258C22⋊C4C2×C3⋊D912yescomplemented
259C22×C3—21no—
260C22×C3—23no—
261C22×C3—43no—
262C22×C3—41no—
263C22×C3—43no—
264C22×C3—41no—
265C22×C3C2×D3613yes—
266C22×C3D4×D916yes—
267C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
268C22×C3D4×C3⋊D312yes—
269C2×D3—182no—
270C2×D3—186no—
271C2×D3—362no—
272C2×D3—366no—
273C2×D3—721no—
274C2×D3—723no—
275C3×C4—21no—
276C3×C4—23no—
277C3×C4—42no—
278C3×C4—46no—
279Dic3—182no—
280Dic3—186no—
281Dic3—361no—
282Dic3—363no—
283Dic3—361no—
284Dic3—363no—
285C32#474211yescharacteristic
286C9#474213yescomplemented
287C23—272no—
288C23—541no—
289C23C4⋊(C3⋊D9)11yescharacteristic
290C23C22×C3⋊D911yescharacteristic
291Q8—541no—
292D4—22no—
293D4—541no—
294D4—542no—
295D4—542no—
296D4—542no—
297C2×C4—22no—
298C2×C4—272no—
299C2×C4—542no—
300C2×C4—541no—
301C2×C4—541no—
302C2×C4—541no—
303C2×C4—542no—
304C2×C4C4⋊(C3⋊D9)11yescharacteristic
305C2×C3—23no—
306C2×C3—26no—
307C2×C3—21no—
308C2×C3—22no—
309C2×C3—43no—
310C2×C3—41no—
311C2×C3#6344413yes—
312C2×C3#6409111yescharacteristic
313D3—366no—
314D3—362no—
315C22—21no—
316C22—41no—
317C22—41no—
318C22—542no—
319C22—1082no—
320C22—2161no—
321C22C2×C4⋊(C3⋊D9)11yescharacteristic
322C22D4×C3⋊D912yes—
323C4—21no—
324C4—42no—
325C4—542no—
326C4—1081no—
327C4—1081no—
328C3#1058842213yescomplemented
329C3#1059244111yescharacteristic
330C2—22no—
331C2—21no—
332C2—41no—
333C2—1082no—
334C2#1172172811yescharacteristic
subgroupG/H
centerC2#11721728
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#99C2
socleC2×C32#882
prime radical#11722661C2
hypercenter#99C3⋊D9
hypocenterC3×C9#458
LCSG ▷ C23×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #11721835C2semidirect
3. #11721732C2semidirect
4. C3×C9×#99C2semidirect
5. #11720523C2semidirect
11. C22×C3⋊Dic9C22semidirect
15. C3×C9×C22⋊C4C22semidirect
17. #88551C22semidirect
29. C9×#99D3semidirect
71. C9×C22⋊C4C2×D3semidirect
96. C3×#99D9semidirect
156. C3×C22⋊C4C2×D9semidirect
198. #99C3⋊D9semidirect
199. C3×C9#458semidirect
258. C22⋊C4C2×C3⋊D9semidirect
286. C9#4742semidirect
328. C3#10588422semidirect

Maximal subgroups

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

subgroupindex
#117226612
#117218352
#117217322
C3×C9×#992
#117205232
#105924413
#105884223