Finite Group Databasebeta

← #13140367

Subgroups — #13140367

Order 1728.

524 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 121 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×C3×#3699C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic, complemented
3C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6#11720276C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10591420—31nomaximal
9#10588452—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C22×C3×Dic9—21no—
13C3×#3699—22no—
14C3×#3699C414yescomplemented
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21(C2×D9)⋊Dic3—21no—
22Dic3×Dic9—21no—
23D9×C3⋊C8—22no—
24C2×C3×#237—31no—
25#63841—31no—
26C2×D3×C3⋊C8—31no—
27C2×#3699D311yescharacteristic
28#63659—31no—
29C3×#832D311yescharacteristic
30#63566—31no—
31#63565—31no—
32C2×C8×D9—31no—
33#63454—31no—
34C9×#102—31no—
35#63392—31no—
36#63392—61no—
37#63376—31no—
38#63368—31no—
39C23×C3×C9—21no—
40C22×C3×D9D411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C412yes—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C3×#611—24no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9—24no—
49C2×C3×Dic9D411yescharacteristic
50C3×C4×D9C2×C412yes—
51C3×C4×D9D412yes—
52C3×Dic18—21no—
53C3×Dic18C2×C412yes—
54C3⋊(C9⋊C8)—21no—
55C9×C3⋊C8—21no—
56#5233—91no—
57#4772—91no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C3×#67—61no—
60C22×C3×Dic3—61no—
61C3×#237—34no—
62C3×#237—62no—
63C2×C3×C4×D3—31no—
64C2×C3×Dic6—31no—
65C4⋊(C3⋊Dic3)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C3×C4⋊Dic3C2×D311yescharacteristic
68C2×C3×C3⋊C8C2×D311yescharacteristic
69(C2×D3)⋊Dic3—61no—
70Dic32—61no—
71D3×C3⋊C8—62no—
72C2×D4×C9—21no—
73C2×D4×C9C2×D311yescharacteristic
74C2×#611—21no—
75C22×Dic9—21no—
76#3699—22no—
77#3699Dic314yescomplemented
78C2×C4×D9C2×D311yescharacteristic
79C2×Dic18C2×D311yescharacteristic
80C2×C8×C9—31no—
81C9×C4⋊C4—31no—
82(C2×D9)⋊C4—61no—
83C4⋊Dic9—31no—
84C4⋊Dic9—61no—
85C4×Dic9—61no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88C8×D9—62no—
89C22×C3×C9—22no—
90C22×C3×C9—41no—
91C22×C3×C9—41no—
92C22×C3×C9C2×D411yescharacteristic
93C2×C3×D9—21no—
94C2×C3×D9—21no—
95C2×C3×D9C22⋊C412yes—
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—41no—
99C9×Dic3—41no—
100C3×Dic9—22no—
101C3×Dic9C22⋊C412yes—
102C2×C3×#41—91no—
103C2×#237—31no—
104#926—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—91no—
107#892—91no—
108C3×#102—31no—
109#835—91no—
110#835—91no—
111#832—91no—
112#832—91no—
113#832—181no—
114#832D911yescharacteristic, complemented
115#816—91no—
116#810—91no—
117C23×C32—21no—
118C22×C3×D3—31no—
119C32×D4—21no—
120C32×D4C4×D312yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—61no—
123C3×#67—64no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—61no—
128C3×C4×D3—32no—
129C3×C4×D3—32no—
130C3×Dic6—32no—
131C3×Dic6—61no—
132C23×C9—21no—
133C23×C9—41no—
134C22×D9#6711yescharacteristic
135C3⋊(C3⋊C8)—61no—
136C3×C3⋊C8—21no—
137D4×C9—21no—
138D4×C9—22no—
139D4×C9—41no—
140D4×C9C2×Dic312yes—
141C2×C4×C9—21no—
142C2×C4×C9—61no—
143C2×C4×C9C22×D311yescharacteristic
144#611—24no—
145C2×Dic9—21no—
146C2×Dic9—24no—
147C2×Dic9—61no—
148C2×Dic9—121no—
149C2×Dic9#6711yescharacteristic
150C4×D9C2×Dic312yes—
151C4×D9#6712yes—
152Dic18—21no—
153Dic18C2×Dic312yes—
154C8×C9—61no—
155C9⋊C8—61no—
156C9⋊C8—121no—
157#416—271noSylow
158C2×C3×C9—42no—
159C2×C3×C9C2×C22⋊C411yescharacteristic
160C2×C3×C9#13511yescharacteristic
161C2×C3×C9#13511yescharacteristic
162C3×D9—22no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×Q8—91no—
166C2×C3×D4—21no—
167C2×C3×D4—181no—
168C2×C3×D4C2×D911yescharacteristic
169C2×C3×D4D3211yescharacteristic
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172C2×#67—61no—
173C22×Dic3—61no—
174#237—34no—
175#237—62no—
176C2×C4×D3—31no—
177C2×Dic6—31no—
178C2×C3×C8—31no—
179C3×C4⋊C4—31no—
180#217—181no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—91no—
183C4⋊Dic3—91no—
184C4⋊Dic3—181no—
185C4⋊Dic3C2×D911yescharacteristic
186C4×Dic3—181no—
187C4×Dic3—181no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—181no—
191C2×C3⋊C8—182no—
192C2×C3⋊C8C2×D911yescharacteristic
193D3×C8—182no—
194C22×C32—22no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32D3×D411yescharacteristic
198C2×C3×D3—32no—
199C2×C3×D3—61no—
200C2×C3×D3—61no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—121no—
204C3×Dic3—32no—
205C3×Dic3—41no—
206C3×Dic3—62no—
207C22×C9—22no—
208C22×C9—21no—
209C22×C9—42no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9—81no—
213C22×C9—81no—
214C22×C9C2×#6711yescharacteristic
215C2×D9—21no—
216C2×D9—21no—
217C2×D9#21712yes—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—121no—
221C4×C9C22×Dic311yescharacteristic
222C4×C9C2×#6711yescharacteristic
223Dic9—22no—
224Dic9—121no—
225Dic9—241no—
226Dic9#21712yes—
227C2×#41—91no—
228C22×C8—271no—
229#117—271no—
230#103—271no—
231#103—271no—
232#102—31no—
233#102—271no—
234C3×C9#41611yescharacteristic, complemented, Sylow
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—41no—
238C23×C3—91no—
239C22×D3—31no—
240C3×Q8—92no—
241C3×Q8—181no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—22no—
245C3×D4—41no—
246C3×D4—184no—
247C3×D4C4×D912yes—
248C3×D4D3×Dic312yes—
249C2×C3×C4—21no—
250C2×C3×C4—61no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—92no—
254C2×C3×C4—184no—
255C2×C3×C4—181no—
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4C2×D3211yescharacteristic
258#67—64no—
259C2×Dic3—21no—
260C2×Dic3—31no—
261C2×Dic3—64no—
262C2×Dic3—61no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265C2×Dic3—361no—
266C4×D3—32no—
267C4×D3—32no—
268Dic6—32no—
269Dic6—61no—
270C3×C8—61no—
271C3⋊C8—21no—
272C3⋊C8—181no—
273C3⋊C8—181no—
274C3⋊C8—361no—
275C2×C32—42no—
276C2×C32D3×C22⋊C411yescharacteristic
277C2×C32#91211yescharacteristic
278C2×C32#91611yescharacteristic
279C3×D3—62no—
280C2×C9—21no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9—42no—
284C2×C9—82no—
285C2×C9#95011yescharacteristic
286C2×C9#95011yescharacteristic
287C2×C9C2×#21711yescharacteristic
288D9—22no—
289#41—94no—
290#41—182no—
291C2×Q8—91no—
292C2×D4—181no—
293C2×D4D3×D911yescharacteristic
294C22×C4—91no—
295C22×C4—181no—
296C2×C8—31no—
297C2×C8—271no—
298C2×C8—542no—
299C4⋊C4—31no—
300C4⋊C4—271no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—92no—
314C22×C3—181no—
315C22×C3—181no—
316C22×C3D4×D911yescharacteristic
317C22×C3D3×#6711yescharacteristic
318C2×D3—32no—
319C2×D3—61no—
320C2×D3—61no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—92no—
324C3×C4—121no—
325C3×C4—182no—
326C3×C4C2×C4×D911yescharacteristic
327C3×C4D4×D911yescharacteristic
328C3×C4C2×D3×Dic311yescharacteristic
329C3×C4D3×#6711yescharacteristic
330Dic3—32no—
331Dic3—41no—
332Dic3—62no—
333Dic3—361no—
334Dic3—361no—
335Dic3—721no—
336C32#477211yescharacteristic
337C9—21no—
338C9#523311yescharacteristic, complemented
339C23—21no—
340C23—91no—
341Q8—92no—
342Q8—181no—
343D4—21no—
344D4—184no—
345D4Dic3×D912yes—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—184no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4C2×D3×D911yescharacteristic
354C8—61no—
355C8—541no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—42no—
360C2×C3—42no—
361C2×C3—82no—
362C2×C3—182no—
363C2×C3C22⋊C4×D911yescharacteristic
364C2×C3#6347411yescharacteristic
365C2×C3#6347811yescharacteristic
366C2×C3#6393311yescharacteristic
367C2×C3#6393511yescharacteristic
368C2×C3D3×#21711yescharacteristic
369D3—62no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22D9×#6711yescharacteristic
377C4—92no—
378C4—121no—
379C4—182no—
380C4—1081no—
381C4C2×Dic3×D911yescharacteristic
382C4D9×#6711yescharacteristic
383C3—21no—
384C3#1058845211yescharacteristic, complemented
385C3#1059142011yescharacteristic
386C2—42no—
387C2—182no—
388C2#1172117711yescharacteristic
389C2#1172118511yescharacteristic
390C2D9×#21711yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#3699C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#416
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720907C2semidirect
3. C2×D9×C3⋊C8C2semidirect
4. #11720576C2semidirect
5. C9×#832C2semidirect
6. #11720276C2semidirect
7. #11720268C2semidirect
14. C3×#3699C4semidirect
77. #3699Dic3semidirect
114. #832D9semidirect
234. C3×C9#416semidirect
338. C9#5233semidirect
384. C3#10588452semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
#117209072
C2×D9×C3⋊C82
#117205762
C9×#8322
#117202762
#117202682
#105914203
#105884523