Finite Group Databasebeta

← #13143975

Subgroups — #13143975

Order 1728.

750 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×Q8×C3⋊D9C211yesmaximal, characteristic
2C2×#88523C211yesmaximal, characteristic
3#11721748C211yesmaximal, characteristic
4C3×C9×#103C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720539C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592501—31nomaximal
9#10588482—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11Q8×C3⋊D9—22no—
12Q8×C3⋊D9C414yescomplemented
13C2×C4×C3⋊D9—21no—
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226—21no—
16C2×#6226C2211yescharacteristic
17C2×C3×C8×C9C2211yescharacteristic
18C3×C9×C4⋊C4C2211yescharacteristic
19#88536—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4×C3⋊Dic9—21no—
22C2×C3⋊(C9⋊C8)C2211yescharacteristic
23#88523—22no—
24C2×Q8×C3⋊D3—31no—
25C2×#3743—31no—
26#64101—31no—
27C2×Q8×D9—33no—
28C32×#103D311yescharacteristic
29#63662—31no—
30#63643—31no—
31#63637—31no—
32C2×#3663—33no—
33#63454—33no—
34C9×#103D313yescomplemented
35#63395—33no—
36#63376—33no—
37#63368—33no—
38C22×C3⋊D9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9C2×C412yes—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C4×C3⋊D9—24no—
48C4×C3⋊D9C2×C412yes—
49C4×C3⋊D9D412yes—
50#6226—21no—
51#6226—24no—
52#6226C2×C412yes—
53C3×C8×C9—21no—
54C3⋊(C9⋊C8)—21no—
55#4802—93no—
56#4802—91no—
57C2×C32×Q8C2×D311yescharacteristic
58Q8×C3⋊D3—34no—
59Q8×C3⋊D3—62no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—61no—
62C2×#634—31no—
63C2×#634—61no—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)—31no—
70#3743—62no—
71C2×Q8×C9C2×D313yes—
72Q8×D9—312no—
73Q8×D9—66no—
74C2×C4×D9—33no—
75C2×C4×D9—63no—
76C2×Dic18—33no—
77C2×Dic18—63no—
78C2×C8×C9C2×D313yes—
79C9×C4⋊C4C2×D313yes—
80(C2×D9)⋊C4—63no—
81C4⋊Dic9—33no—
82C4×Dic9—63no—
83C2×C9⋊C8—33no—
84#3663—66no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3⋊D9—21no—
87C2×C3⋊D9—21no—
88C2×C3⋊D9C22⋊C412yes—
89C3×C4×C9—22no—
90C3×C4×C9—41no—
91C3×C4×C9C22×C411yescharacteristic
92C3×C4×C9C2×D411yescharacteristic
93C3⋊Dic9—22no—
94C3⋊Dic9—41no—
95C3⋊Dic9C22⋊C412yes—
96C2×D3×Q8—93no—
97C2×D3×Q8—91no—
98C2×ΓL(1,𝔽25)—93no—
99C2×ΓL(1,𝔽25)—91no—
100#892—93no—
101#892—91no—
102C3×#103D913yescomplemented
103C3×#103C3⋊D311yescharacteristic
104#835—93no—
105#835—91no—
106#816—93no—
107#816—91no—
108#810—93no—
109#810—91no—
110C22×C3⋊D3—31no—
111C32×Q8—21no—
112C32×Q8C4×D312yes—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—61no—
119C4×C3⋊D3—32no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—64no—
122#634—32no—
123#634—64no—
124#634—61no—
125C22×D9—33no—
126C32×C8—21no—
127C3⋊(C3⋊C8)—61no—
128Q8×C9—23no—
129Q8×C9C4×D316yes—
130C2×C4×C9—23no—
131C2×C4×C9—23no—
132C2×C4×C9C22×D313yes—
133C2×Dic9—33no—
134C2×Dic9—63no—
135C2×Dic9—63no—
136C4×D9—36no—
137C4×D9—36no—
138C4×D9—612no—
139Dic18—36no—
140Dic18—612no—
141Dic18—63no—
142C8×C9—23no—
143C9⋊C8—63no—
144#419—271noSylow
145C2×C3×C9C2×C22⋊C411yescharacteristic
146C2×C3×C9#13711yescharacteristic
147C2×C3×C9#13711yescharacteristic
148C3⋊D9—22no—
149C2×C3×Q8C2×D913yes—
150C2×C3×Q8C2×C3⋊D311yescharacteristic
151D3×Q8—912no—
152D3×Q8—94no—
153D3×Q8—182no—
154D3×Q8—186no—
155C2×C4×D3—93no—
156C2×C4×D3—91no—
157C2×C4×D3—181no—
158C2×C4×D3—183no—
159C2×Dic6—93no—
160C2×Dic6—91no—
161C2×Dic6—181no—
162C2×Dic6—183no—
163C2×C3×C8C2×D913yes—
164C2×C3×C8C2×C3⋊D311yescharacteristic
165C3×C4⋊C4C2×D913yes—
166C3×C4⋊C4C2×C3⋊D311yescharacteristic
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—183no—
169C4⋊Dic3—93no—
170C4⋊Dic3—91no—
171C4×Dic3—181no—
172C4×Dic3—183no—
173C2×C3⋊C8—93no—
174C2×C3⋊C8—91no—
175ΓL(1,𝔽25)—182no—
176ΓL(1,𝔽25)—186no—
177C22×C32D3×D411yescharacteristic
178C2×C3⋊D3—32no—
179C2×C3⋊D3—61no—
180C2×C3⋊D3—61no—
181C32×C4—22no—
182C32×C4—41no—
183C32×C4C2×C4×D311yescharacteristic
184C32×C4D3×D411yescharacteristic
185C3⋊Dic3—32no—
186C3⋊Dic3—62no—
187C3⋊Dic3—121no—
188C22×C9D3×D413yes—
189C2×D9—36no—
190C2×D9—63no—
191C2×D9—63no—
192C4×C9—26no—
193C4×C9—43no—
194C4×C9C2×C4×D313yes—
195C4×C9D3×D413yes—
196Dic9—36no—
197Dic9—66no—
198Dic9—123no—
199C22×Q8—271no—
200C2×M16—271no—
201#117—271no—
202#103—271no—
203#103—271no—
204#103—271no—
205#103C3⋊D911yescharacteristic, complemented
206C3×C9#41911yescharacteristic, complemented, Sylow
207C22×D3—93no—
208C22×D3—91no—
209C3×Q8—23no—
210C3×Q8—21no—
211C3×Q8C4×D916yes—
212C3×Q8C4×C3⋊D312yes—
213C2×C3×C4—23no—
214C2×C3×C4—21no—
215C2×C3×C4—23no—
216C2×C3×C4—21no—
217C2×C3×C4C22×D913yes—
218C2×C3×C4C22×C3⋊D311yescharacteristic
219C2×Dic3—93no—
220C2×Dic3—91no—
221C2×Dic3—183no—
222C2×Dic3—181no—
223C2×Dic3—183no—
224C2×Dic3—181no—
225C4×D3—96no—
226C4×D3—96no—
227C4×D3—92no—
228C4×D3—92no—
229C4×D3—184no—
230C4×D3—1812no—
231Dic6—96no—
232Dic6—92no—
233Dic6—184no—
234Dic6—1812no—
235Dic6—183no—
236Dic6—181no—
237C3×C8—23no—
238C3×C8—21no—
239C3⋊C8—183no—
240C3⋊C8—181no—
241C2×C32D3×C22⋊C411yescharacteristic
242C2×C32#91511yescharacteristic
243C2×C32#91811yescharacteristic
244C3⋊D3—62no—
245C2×C9D3×C22⋊C413yes—
246C2×C9#91513yes—
247C2×C9#91813yes—
248D9—66no—
249C2×Q8—274no—
250C2×Q8—271no—
251C2×Q8—541no—
252C2×Q8—542no—
253C2×Q8C2×C3⋊D911yescharacteristic
254C22×C4—271no—
255C22×C4—541no—
256M16—542no—
257C2×C8—271no—
258C2×C8C2×C3⋊D911yescharacteristic
259C4⋊C4—271no—
260C4⋊C4C2×C3⋊D911yescharacteristic
261C22⋊C4—541no—
262C42—541no—
263C22×C3D4×D913yes—
264C22×C3D4×C3⋊D311yescharacteristic
265C2×D3—92no—
266C2×D3—96no—
267C2×D3—181no—
268C2×D3—181no—
269C2×D3—183no—
270C2×D3—183no—
271C3×C4—22no—
272C3×C4—26no—
273C3×C4—41no—
274C3×C4—43no—
275C3×C4C2×C4×D913yes—
276C3×C4D4×D913yes—
277C3×C4C2×C4×C3⋊D311yescharacteristic
278C3×C4D4×C3⋊D311yescharacteristic
279Dic3—92no—
280Dic3—96no—
281Dic3—182no—
282Dic3—186no—
283Dic3—361no—
284Dic3—363no—
285C32#480211yescharacteristic
286C9#480213yescomplemented
287C23—271no—
288Q8—21no—
289Q8—272no—
290Q8—544no—
291Q8—541no—
292Q8C4×C3⋊D912yes—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—271no—
296C2×C4—272no—
297C2×C4—272no—
298C2×C4—544no—
299C2×C4—541no—
300C2×C4—541no—
301C2×C4C22×C3⋊D911yescharacteristic
302C8—21no—
303C8—541no—
304C2×C3C22⋊C4×D913yes—
305C2×C3#6347713yes—
306C2×C3#6348013yes—
307C2×C3C22⋊C4×C3⋊D311yescharacteristic
308C2×C3#6412411yescharacteristic
309C2×C3#6412711yescharacteristic
310D3—186no—
311D3—182no—
312C22—272no—
313C22—541no—
314C22—541no—
315C22D4×C3⋊D911yescharacteristic
316C4—22no—
317C4—41no—
318C4—272no—
319C4—542no—
320C4—1081no—
321C4C2×C4×C3⋊D911yescharacteristic
322C4D4×C3⋊D911yescharacteristic
323C3#1058848213yescomplemented
324C3#1059250111yescharacteristic
325C2—542no—
326C2C22⋊C4×C3⋊D911yescharacteristic
327C2#1172179711yescharacteristic
328C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenter#103C3⋊D9
hypocenterC3×C9#419
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
4. C3×C9×#103C2semidirect
5. #11720582C2semidirect
6. #11720539C2semidirect
7. #11720527C2semidirect
12. Q8×C3⋊D9C4semidirect
34. C9×#103D3semidirect
102. C3×#103D9semidirect
205. #103C3⋊D9semidirect
206. C3×C9#419semidirect
286. C9#4802semidirect
323. C3#10588482semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
C2×#885232
#117217482
C3×C9×#1032
#117205822
#117205392
#117205272
#105925013
#105884823