Finite Group Databasebeta

← #13140317

Subgroups — #13140317

Order 1728.

460 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 109 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
1D4×C3⋊Dic9C211yesmaximal, characteristic
2C2×C3×#3672C211yesmaximal, characteristic
3#11720910C211yesmaximal, characteristic, complemented
4C9×#832C211yesmaximal, characteristic
5#11720276C211yesmaximal, characteristic, complemented
6#11720271C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10591445—31nomaximal
9#10588438—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×C3×Dic18C2211yescharacteristic
13#88551—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C4×C3⋊Dic9C2211yescharacteristic
16C3×#3672C414yescomplemented
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21Dic3×Dic9—21no—
22D4×C3⋊Dic3—31no—
23C2×C3×#214—31no—
24#63843—31no—
25C3×#832D311yescharacteristic
26#63573—31no—
27#63566—31no—
28#63553—31no—
29D4×Dic9—31no—
30D4×Dic9—61no—
31C2×#3672D311yescharacteristic
32#63449—31no—
33C9×#102—31no—
34#63376—31no—
35C8×Dic9—31no—
36#63367—31no—
37C23×C3×C9—21no—
38C3×D4×C9—21no—
39C3×D4×C9C2×C412yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—42no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C3×Dic18—21no—
48C3×Dic18C2×C412yescomplemented
49C9×C3⋊C8—21no—
50C3×C9⋊C8C2×C412yescomplemented
51Dic3×SD16—91no—
52#4758—91no—
53C2×C32×D4C2×D311yescharacteristic
54C22×C3⋊Dic3—61no—
55C2×C3×Dic6—31no—
56#3757—61no—
57C4⋊(C3⋊Dic3)—31no—
58C4×C3⋊Dic3—31no—
59C3×#214—34no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63Dic3⋊Dic3—61no—
64Dic32—61no—
65C2×D4×C9—21no—
66C2×D4×C9C2×D311yescharacteristic
67C22×Dic9—61no—
68C22×Dic9—62no—
69C2×Dic18C2×D311yescharacteristic
70C2×C8×C9—31no—
71C9×C4⋊C4—31no—
72#3676—61no—
73#3676—62no—
74#3672Dic314yescomplemented
75C4⋊Dic9—31no—
76C4⋊Dic9—61no—
77Dic9⋊C4—61no—
78C4×Dic9—31no—
79C4×Dic9—61no—
80C4×Dic9—61no—
81C2×C9⋊C8C2×D311yescharacteristic
82C22×C3×C9—22no—
83C22×C3×C9—41no—
84C22×C3×C9—41no—
85C22×C3×C9C2×D411yescharacteristic
86C3×C4×C9C22×C411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—21no—
89C3⋊Dic9—41no—
90C3⋊Dic9SD1612yescomplemented
91C9×Dic3—41no—
92C3×Dic9—22no—
93C2×C3×SD16—91no—
94Q8×Dic3—91no—
95D4×Dic3—91no—
96D4×Dic3—91no—
97D4×Dic3—181no—
98C2×#214—31no—
99#887—91no—
100C3×#102—31no—
101#835—91no—
102#832D911yescharacteristic
103#817—91no—
104#816—91no—
105C8×Dic3—91no—
106C8×Dic3—91no—
107#808—91no—
108C23×C32—21no—
109C32×D4—21no—
110C32×D4C4×D312yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—122no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—61no—
118C3×Dic6—32no—
119C3×Dic6—61no—
120C23×C9—22no—
121C23×C9—21no—
122C3×C3⋊C8—21no—
123C3×C3⋊C8—32no—
124D4×C9—22no—
125D4×C9—21no—
126D4×C9—22no—
127D4×C9C2×Dic312yes—
128C2×C4×C9—21no—
129C2×C4×C9—61no—
130C2×C4×C9C22×D311yescharacteristic
131C2×Dic9—21no—
132C2×Dic9—31no—
133C2×Dic9—31no—
134C2×Dic9—61no—
135C2×Dic9—61no—
136C2×Dic9—62no—
137C2×Dic9—61no—
138C2×Dic9—122no—
139C2×Dic9—124no—
140Dic18—21no—
141Dic18C2×Dic312yescomplemented
142C8×C9—61no—
143C9⋊C8C2×Dic312yescomplemented
144C4×SD16—271noSylow
145C2×C3×C9—42no—
146C2×C3×C9C4×D411yescharacteristic
147C2×C3×C9C2×SD1611yescharacteristic
148C2×C3×C9#13511yescharacteristic
149C2×C3×Q8—91no—
150C2×C3×D4—21no—
151C2×C3×D4C2×D911yescharacteristic
152C2×C3×D4D3211yescharacteristic
153C22×Dic3—181no—
154C22×Dic3—181no—
155C22×Dic3—182no—
156C2×Dic6—31no—
157C3×SD16—94no—
158C2×C3×C8—31no—
159C2×C3×C8—91no—
160C3×C4⋊C4—31no—
161#217—181no—
162#217—181no—
163#217—182no—
164#214—34no—
165C4⋊Dic3—91no—
166C4⋊Dic3—91no—
167C4⋊Dic3—181no—
168C4⋊Dic3—181no—
169C4⋊Dic3C2×D911yescharacteristic
170Dic3⋊C4—181no—
171C4×Dic3—91no—
172C4×Dic3—91no—
173C4×Dic3—181no—
174C4×Dic3—181no—
175C4×Dic3—181no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8C2×D911yescharacteristic
178C22×C32—22no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32D3×D411yescharacteristic
182C32×C4C2×C4×D311yescharacteristic
183C32×C4#23711yescharacteristic
184C3⋊Dic3—32no—
185C3⋊Dic3—61no—
186C3⋊Dic3—121no—
187C3×Dic3—41no—
188C3×Dic3—62no—
189C22×C9—24no—
190C22×C9—22no—
191C22×C9—21no—
192C22×C9—42no—
193C22×C9—42no—
194C22×C9—41no—
195C22×C9—41no—
196C22×C9D3×D411yescharacteristic
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—121no—
200C4×C9#23711yescharacteristic
201C4×C9C22×Dic311yescharacteristic
202Dic9—22no—
203Dic9—32no—
204Dic9—62no—
205Dic9—61no—
206Dic9—121no—
207Dic9—122no—
208Dic9—121no—
209C2×SD16—91no—
210C4×Q8—271no—
211C4×D4—271no—
212#106—271no—
213#103—271no—
214#102—31no—
215C4×C8—271no—
216C3×C9C4×SD1611yescharacteristic, complemented, Sylow
217C23×C3—22no—
218C23×C3—21no—
219C23×C3—21no—
220C3×Q8—92no—
221C3×Q8—181no—
222C3×D4—22no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—22no—
226C3×D4C4×D912yes—
227C3×D4D3×Dic312yes—
228C2×C3×C4—21no—
229C2×C3×C4—61no—
230C2×C3×C4—181no—
231C2×C3×C4C22×D911yescharacteristic
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—61no—
235C2×Dic3—91no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241C2×Dic3—182no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C2×Dic3—362no—
245C2×Dic3—362no—
246C2×Dic3—364no—
247Dic6—32no—
248Dic6—61no—
249C3×C8—61no—
250C3×C8—92no—
251C3⋊C8—21no—
252C3⋊C8—32no—
253C2×C32—42no—
254C2×C32#67⋊C411yescharacteristic
255C2×C32#91211yescharacteristic
256C2×C32D3×SD1611yescharacteristic
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—44no—
261C2×C9—42no—
262C2×C9D3×SD1611yescharacteristic
263C2×C9#91611yescharacteristic
264C2×C9D4×Dic311yescharacteristic
265C2×Q8—91no—
266C2×D4D3×D911yescharacteristic
267C22×C4—541no—
268SD16—94no—
269C2×C8—31no—
270C2×C8—91no—
271C4⋊C4—31no—
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—541no—
275C42—271no—
276C42—541no—
277C22×C3—24no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—21no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3D4×D911yescharacteristic
288C22×C3#381111yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—121no—
292C3×C4—182no—
293C3×C4C2×C4×D911yescharacteristic
294C3×C4#369911yescharacteristic
295C3×C4C2×D3×Dic311yescharacteristic
296C3×C4#380511yescharacteristic
297Dic3—41no—
298Dic3—62no—
299Dic3—92no—
300Dic3—92no—
301Dic3—182no—
302Dic3—181no—
303Dic3—181no—
304Dic3—361no—
305Dic3—362no—
306Dic3—361no—
307Dic3—361no—
308C32#475811yescharacteristic
309C9—21no—
310C9Dic3×SD1611yescharacteristic, complemented
311C23—21no—
312Q8—92no—
313Q8—181no—
314D4—21no—
315D4Dic3×D912yes—
316C2×C4—61no—
317C2×C4—181no—
318C2×C4—271no—
319C2×C4—271no—
320C2×C4—541no—
321C2×C4—1082no—
322C2×C4C2×D3×D911yescharacteristic
323C8—61no—
324C8—92no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—44no—
329C2×C3—42no—
330C2×C3—42no—
331C2×C3#611⋊C411yescharacteristic
332C2×C3#6347411yescharacteristic
333C2×C3SD16×D911yescharacteristic
334C2×C3#6393011yescharacteristic
335C2×C3#6393611yescharacteristic
336C2×C3#67⋊Dic311yescharacteristic
337C22—22no—
338C22—41no—
339C22—41no—
340C22#8866811yescharacteristic
341C4—121no—
342C4—182no—
343C4—272no—
344C4—541no—
345C4—1081no—
346C4C2×Dic3×D911yescharacteristic
347C4#8866311yescharacteristic
348C3—21no—
349C3#1058843811yescharacteristic, complemented
350C3#1059144511yescharacteristic
351C2—42no—
352C2#1172117811yescharacteristic
353C2#1172118211yescharacteristic
354C2#611⋊Dic311yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9C4×SD16
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
3. #11720910C2semidirect
5. #11720276C2semidirect
6. #11720271C2semidirect
7. #11720264C2semidirect
16. C3×#3672C4semidirect
48. C3×Dic18C2×C4semidirect
50. C3×C9⋊C8C2×C4semidirect
74. #3672Dic3semidirect
90. C3⋊Dic9SD16semidirect
141. Dic18C2×Dic3semidirect
143. C9⋊C8C2×Dic3semidirect
216. C3×C9C4×SD16semidirect
310. C9Dic3×SD16semidirect
349. C3#10588438semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C2×C3×#36722
#117209102
C9×#8322
#117202762
#117202712
#117202642
#105914453
#105884383