Finite Group Databasebeta

← #13143913

Subgroups — #13143913

Order 1728.

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

normal subgroups only

This group has 790 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×#88742C211yesmaximal, characteristic
2C2×C8×C3⋊D9C211yesmaximal, characteristic, complemented
3#11721748C211yesmaximal, characteristic, complemented
4C3×C9×#102C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6#11720539C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592471—31nomaximal
9#10588452—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C22×C3⋊Dic9—21no—
13#88742—22no—
14#88742C414yescomplemented
15C2×C4×C3⋊D9C2211yescharacteristic
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
23C8×C3⋊D9—22no—
24C2×#3830—31no—
25C2×C8×C3⋊D3—31no—
26#64101—31no—
27C2×#3699—33no—
28C32×#102D311yescharacteristic
29#63659—31no—
30#63643—31no—
31#63637—31no—
32C2×C8×D9—33no—
33#63454—33no—
34C9×#102D313yescomplemented
35#63392—33no—
36#63376—33no—
37#63368—33no—
38C23×C3×C9—21no—
39C22×C3⋊D9D411yescharacteristic
40C3×D4×C9—21no—
41C3×D4×C9C2×C412yes—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44#6233—24no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9—24no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9D411yescharacteristic
49C4×C3⋊D9C2×C412yes—
50C4×C3⋊D9D412yes—
51#6226—21no—
52#6226C2×C412yes—
53C3×C8×C9—21no—
54C3⋊(C9⋊C8)—21no—
55#4772—93no—
56#4772—91no—
57C2×C32×D4C2×D311yescharacteristic
58C2×#638—61no—
59C22×C3⋊Dic3—61no—
60#3830—34no—
61#3830—62no—
62C2×C4×C3⋊D3—31no—
63C2×#634—31no—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)—31no—
70C8×C3⋊D3—62no—
71C2×D4×C9C2×D313yes—
72C2×#611—63no—
73C22×Dic9—63no—
74#3699—312no—
75#3699—66no—
76C2×C4×D9—33no—
77C2×Dic18—33no—
78C2×C8×C9C2×D313yes—
79C9×C4⋊C4C2×D313yes—
80(C2×D9)⋊C4—63no—
81C4⋊Dic9—33no—
82C4×Dic9—63no—
83C2×C9⋊C8—33no—
84C8×D9—66no—
85C22×C3×C9—22no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9C2×D411yescharacteristic
89C2×C3⋊D9—21no—
90C2×C3⋊D9—21no—
91C2×C3⋊D9C22⋊C412yes—
92C3×C4×C9—41no—
93C3×C4×C9C22×C411yescharacteristic
94C3×C4×C9C2×D411yescharacteristic
95C3⋊Dic9—22no—
96C3⋊Dic9—41no—
97C3⋊Dic9C22⋊C412yes—
98C2×#237—93no—
99C2×#237—91no—
100C2×D3×C8—93no—
101C2×D3×C8—91no—
102#892—93no—
103#892—91no—
104C3×#102D913yescomplemented
105C3×#102C3⋊D311yescharacteristic
106#832—93no—
107#832—91no—
108#816—93no—
109#816—91no—
110#810—93no—
111#810—91no—
112C23×C32—21no—
113C22×C3⋊D3—31no—
114C32×D4—21no—
115C32×D4C4×D312yes—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118#638—64no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—61no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—32no—
125#634—32no—
126#634—61no—
127C23×C9—23no—
128C22×D9—33no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—61no—
131D4×C9—23no—
132D4×C9C4×D316yes—
133C2×C4×C9—23no—
134C2×C4×C9C22×D313yes—
135#611—612no—
136C2×Dic9—33no—
137C2×Dic9—612no—
138C2×Dic9—63no—
139C2×Dic9—63no—
140C4×D9—36no—
141C4×D9—36no—
142Dic18—36no—
143Dic18—63no—
144C8×C9—23no—
145C9⋊C8—63no—
146#416—271noSylow
147C2×C3×C9—42no—
148C2×C3×C9C2×C22⋊C411yescharacteristic
149C2×C3×C9#13511yescharacteristic
150C2×C3×C9#13511yescharacteristic
151C3⋊D9—22no—
152C2×C3×D4C2×D913yes—
153C2×C3×D4C2×C3⋊D311yescharacteristic
154C2×#67—181no—
155C2×#67—183no—
156C22×Dic3—181no—
157C22×Dic3—183no—
158#237—912no—
159#237—94no—
160#237—182no—
161#237—186no—
162C2×C4×D3—93no—
163C2×C4×D3—91no—
164C2×Dic6—93no—
165C2×Dic6—91no—
166C2×C3×C8C2×D913yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168C3×C4⋊C4C2×D913yes—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—183no—
172C4⋊Dic3—93no—
173C4⋊Dic3—91no—
174C4×Dic3—181no—
175C4×Dic3—183no—
176C2×C3⋊C8—93no—
177C2×C3⋊C8—91no—
178D3×C8—182no—
179D3×C8—186no—
180C22×C32—22no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32D3×D411yescharacteristic
184C2×C3⋊D3—32no—
185C2×C3⋊D3—61no—
186C2×C3⋊D3—61no—
187C32×C4—41no—
188C32×C4C2×C4×D311yescharacteristic
189C32×C4D3×D411yescharacteristic
190C3⋊Dic3—32no—
191C3⋊Dic3—62no—
192C3⋊Dic3—121no—
193C22×C9—26no—
194C22×C9—43no—
195C22×C9—43no—
196C22×C9D3×D413yes—
197C2×D9—36no—
198C2×D9—63no—
199C2×D9—63no—
200C4×C9—43no—
201C4×C9C2×C4×D313yes—
202C4×C9D3×D413yes—
203Dic9—36no—
204Dic9—66no—
205Dic9—123no—
206C2×#41—271no—
207C22×C8—271no—
208#117—271no—
209#103—271no—
210#103—271no—
211#102—271no—
212#102C3⋊D911yescharacteristic, complemented
213C3×C9#41611yescharacteristic, complemented, Sylow
214C23×C3—23no—
215C23×C3—21no—
216C22×D3—93no—
217C22×D3—91no—
218C3×D4—23no—
219C3×D4—21no—
220C3×D4C4×D916yes—
221C3×D4C4×C3⋊D312yes—
222C2×C3×C4—23no—
223C2×C3×C4—21no—
224C2×C3×C4C22×D913yes—
225C2×C3×C4C22×C3⋊D311yescharacteristic
226#67—184no—
227#67—1812no—
228C2×Dic3—93no—
229C2×Dic3—91no—
230C2×Dic3—184no—
231C2×Dic3—1812no—
232C2×Dic3—183no—
233C2×Dic3—181no—
234C2×Dic3—183no—
235C2×Dic3—181no—
236C4×D3—96no—
237C4×D3—96no—
238C4×D3—92no—
239C4×D3—92no—
240Dic6—96no—
241Dic6—92no—
242Dic6—183no—
243Dic6—181no—
244C3×C8—23no—
245C3×C8—21no—
246C3⋊C8—183no—
247C3⋊C8—181no—
248C2×C32—42no—
249C2×C32D3×C22⋊C411yescharacteristic
250C2×C32#91211yescharacteristic
251C2×C32#91611yescharacteristic
252C3⋊D3—62no—
253C2×C9—46no—
254C2×C9D3×C22⋊C413yes—
255C2×C9#91213yes—
256C2×C9#91613yes—
257D9—66no—
258#41—274no—
259#41—542no—
260C2×Q8—271no—
261C2×D4—541no—
262C2×D4C2×C3⋊D911yescharacteristic
263C22×C4—271no—
264C22×C4—541no—
265C2×C8—271no—
266C2×C8—542no—
267C2×C8C2×C3⋊D911yescharacteristic
268C4⋊C4—271no—
269C4⋊C4C2×C3⋊D911yescharacteristic
270C22⋊C4—541no—
271C42—541no—
272C22×C3—22no—
273C22×C3—26no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—43no—
277C22×C3—43no—
278C22×C3D4×D913yes—
279C22×C3D4×C3⋊D311yescharacteristic
280C2×D3—92no—
281C2×D3—96no—
282C2×D3—181no—
283C2×D3—181no—
284C2×D3—183no—
285C2×D3—183no—
286C3×C4—41no—
287C3×C4—43no—
288C3×C4C2×C4×D913yes—
289C3×C4D4×D913yes—
290C3×C4C2×C4×C3⋊D311yescharacteristic
291C3×C4D4×C3⋊D311yescharacteristic
292Dic3—92no—
293Dic3—96no—
294Dic3—182no—
295Dic3—186no—
296Dic3—361no—
297Dic3—363no—
298C32#477211yescharacteristic
299C9#477213yescomplemented
300C23—21no—
301C23—271no—
302Q8—272no—
303Q8—541no—
304D4—21no—
305D4—544no—
306D4C4×C3⋊D912yes—
307C2×C4—21no—
308C2×C4—271no—
309C2×C4—272no—
310C2×C4—272no—
311C2×C4—544no—
312C2×C4—541no—
313C2×C4—541no—
314C2×C4C22×C3⋊D911yescharacteristic
315C8—21no—
316C8—541no—
317C2×C3—46no—
318C2×C3—42no—
319C2×C3C22⋊C4×D913yes—
320C2×C3#6347413yes—
321C2×C3#6347813yes—
322C2×C3C22⋊C4×C3⋊D311yescharacteristic
323C2×C3#6412111yescharacteristic
324C2×C3#6412511yescharacteristic
325D3—186no—
326D3—182no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—272no—
331C22—541no—
332C22—541no—
333C22D4×C3⋊D911yescharacteristic
334C4—41no—
335C4—272no—
336C4—542no—
337C4—1081no—
338C4C2×C4×C3⋊D911yescharacteristic
339C4D4×C3⋊D911yescharacteristic
340C3#1058845213yescomplemented
341C3#1059247111yescharacteristic
342C2—42no—
343C2—542no—
344C2C22⋊C4×C3⋊D911yescharacteristic
345C2#1172179111yescharacteristic
346C2#1172179811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radicalC2×#88742C2
cosocleC2×C3×C4×C9C23
hypercenter#102C3⋊D9
hypocenterC3×C9#416
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×C8×C3⋊D9C2semidirect
3. #11721748C2semidirect
4. C3×C9×#102C2semidirect
5. #11720576C2semidirect
6. #11720539C2semidirect
7. #11720527C2semidirect
14. #88742C4semidirect
34. C9×#102D3semidirect
104. C3×#102D9semidirect
212. #102C3⋊D9semidirect
213. C3×C9#416semidirect
299. C9#4772semidirect
340. C3#10588452semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C2×C8×C3⋊D92
#117217482
C3×C9×#1022
#117205762
#117205392
#117205272
#105924713
#105884523