Finite Group Databasebeta

← #13137648

Subgroups — #13137648

Order 1728.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#63449C211yesmaximal, characteristic, complemented
2#11720992C211yesmaximal, characteristic
3C2×#88416C211yesmaximal, characteristic
4#11720528C211yesmaximal, characteristic
5C9×#808C211yesmaximal, characteristic, complemented
6C3⋊C8×Dic9C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590458—31nomaximal
9#10588532—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C2×(C3⋊D9)⋊C4—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20#88445—21no—
21#88416C414yescomplemented
22C3×#887—31no—
23#63894—31no—
24C2×#3717—31no—
25#63636—31no—
26C3×#808D311yescharacteristic
27#63565—31no—
28Dic3×C3⋊C8—31no—
29C2×#3664—31no—
30#63455—31no—
31#63449D311yescharacteristic, complemented
32C9×#106—31no—
33C8×Dic9—31no—
34#63369—31no—
35#63369—61no—
36#63368—31no—
37C22×C3⋊D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)C2×C412yes—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C3×Dic18—21no—
47C3×Dic18C2×C412yescomplemented
48(C3⋊D9)⋊C4—42no—
49C3⋊(C9⋊C8)—21no—
50C9×C3⋊C8C2×C412yescomplemented
51C4×#215—91no—
52#4852—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C3×Dic6—31no—
55C2×#624—61no—
56C32×C4⋊C4C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63#3722—61no—
64#3717—34no—
65C2×D36—31no—
66C2×D36—61no—
67C2×C4×D9—61no—
68C2×Dic18C2×D311yescharacteristic
69C2×C8×C9—31no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4—31no—
72C9×C4⋊C4C2×D311yescharacteristic
73(C2×D9)⋊C4—61no—
74Dic9⋊C4—21no—
75C4×Dic9—21no—
76C4×Dic9C2×D311yescharacteristic
77C2×C9⋊C8—31no—
78C2×C9⋊C8—61no—
79#3664—34no—
80C22×C3×C9C2×D411yescharacteristic
81C2×C3⋊D9—22no—
82C2×C3⋊D9—41no—
83C2×C3⋊D9—41no—
84C3×C4×C9—41no—
85C3×C4×C9C22×C411yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C9×Dic3—41no—
88C3×Dic9—22no—
89C3×Dic9—21no—
90C3×Dic9SD1612yescomplemented
91C3×C4×Q8—91no—
92C2×#215—91no—
93C2×#204—91no—
94#893—91no—
95#887—31no—
96C4×D12—91no—
97C3×#106—31no—
98#835—91no—
99C8×Dic3—91no—
100#810—91no—
101#809—91no—
102#809—91no—
103#809—181no—
104#808D911yescharacteristic, complemented
105C4×C3⋊C8—91no—
106C22×C3⋊D3—61no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C4⋊(C3⋊D3)—32no—
110C4⋊(C3⋊D3)—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—61no—
115C3×Dic6—32no—
116C3×Dic6—61no—
117#624—122no—
118C22×D9—61no—
119C22×D9—121no—
120C3⋊(C3⋊C8)—61no—
121C3×C3⋊C8C4×D312yes—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—41no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—21no—
128C2×Dic9#6711yescharacteristic
129C2×Dic9#6711yescharacteristic
130D36—32no—
131D36—62no—
132D36—61no—
133D36—121no—
134C4×D9—122no—
135Dic18—21no—
136Dic18C4×D312yescomplemented
137C8×C9—32no—
138C9⋊C8—61no—
139C9⋊C8—121no—
140C4×SD16—271noSylow
141C2×C3×C9C4×D411yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9#13511yescharacteristic
144C3⋊D9—42no—
145C2×C3×Q8—91no—
146C2×D12—91no—
147C2×D12—91no—
148C2×D12—181no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C2×Dic6—31no—
152C2×C3×C8—31no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—181no—
156C3×C4⋊C4C2×D911yescharacteristic
157C3×C4⋊C4D3211yescharacteristic
158C3×C42—91no—
159C3×C42—181no—
160#215—94no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3C2×D911yescharacteristic
164Dic3⋊C4—61no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—181no—
170C2×C3⋊C8C2×D911yescharacteristic
171#204—94no—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—62no—
174C2×C3⋊D3—121no—
175C2×C3⋊D3—121no—
176C32×C4—41no—
177C32×C4C2×C4×D311yescharacteristic
178C32×C4#23911yescharacteristic
179C3×Dic3—32no—
180C3×Dic3—41no—
181C3×Dic3—62no—
182C3×Dic3—61no—
183C22×C9—21no—
184C22×C9C2×#6711yescharacteristic
185C2×D9—62no—
186C2×D9—122no—
187C2×D9—121no—
188C2×D9—121no—
189C2×D9—241no—
190C2×D9—241no—
191C4×C9—21no—
192C4×C9—21no—
193C4×C9—41no—
194C4×C9—81no—
195C4×C9—121no—
196C4×C9C2×C4×D311yescharacteristic
197C4×C9#23511yescharacteristic
198Dic9—22no—
199Dic9—21no—
200Dic9#21512yescomplemented
201C2×SD16—271no—
202C4×Q8—91no—
203C4×D4—271no—
204#106—31no—
205#103—271no—
206#102—271no—
207C4×C8—271no—
208C3×C9C4×SD1611yescharacteristic, complemented, Sylow
209C22×D3—181no—
210C22×D3—181no—
211C22×D3—361no—
212C3×Q8—92no—
213C3×Q8—181no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—61no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—181no—
222C2×C3×C4C22×D911yescharacteristic
223C2×C3×C4C2×D3211yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—31no—
226C2×Dic3—31no—
227C2×Dic3—61no—
228D12—92no—
229D12—92no—
230D12—182no—
231D12—181no—
232D12—181no—
233D12—361no—
234C4×D3—362no—
235C4×D3—362no—
236Dic6—32no—
237Dic6—61no—
238C3×C8—32no—
239C3⋊C8—181no—
240C3⋊C8—181no—
241C3⋊C8—361no—
242C3⋊C8C4×D912yescomplemented
243C2×C32#89311yescharacteristic
244C2×C32D3×SD1611yescharacteristic
245C2×C32#91611yescharacteristic
246C3⋊D3—122no—
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—21no—
250C2×C9C4×#6711yescharacteristic
251C2×C9C2×#21511yescharacteristic
252C2×C9#95011yescharacteristic
253D9—122no—
254D9—242no—
255C2×Q8—91no—
256C2×D4—271no—
257C22×C4—541no—
258SD16—274no—
259C2×C8—31no—
260C2×C8—271no—
261C4⋊C4—31no—
262C4⋊C4—181no—
263C4⋊C4D3×D911yescharacteristic
264C22⋊C4—541no—
265C42—91no—
266C42—181no—
267C22×C3—21no—
268C22×C3D4×D911yescharacteristic
269C22×C3D3×#6711yescharacteristic
270C2×D3—182no—
271C2×D3—182no—
272C2×D3—362no—
273C2×D3—361no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—721no—
278C2×D3—721no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—81no—
284C3×C4—92no—
285C3×C4—121no—
286C3×C4—182no—
287C3×C4—181no—
288C3×C4C2×C4×D911yescharacteristic
289C3×C4#370111yescharacteristic
290C3×C4#379911yescharacteristic
291C3×C4C4×D3211yescharacteristic
292Dic3—32no—
293Dic3—41no—
294Dic3—62no—
295Dic3—61no—
296C32#485211yescharacteristic
297C9—21no—
298C9C4×#21511yescharacteristic, complemented
299C23—541no—
300Q8—92no—
301Q8—181no—
302D4—272no—
303D4—541no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—1082no—
310C2×C4C2×D3×D911yescharacteristic
311C8—32no—
312C8—541no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#6345511yescharacteristic
317C2×C3SD16×D911yescharacteristic
318C2×C3#6347811yescharacteristic
319C2×C3#6390411yescharacteristic
320C2×C3#6393511yescharacteristic
321C2×C3D3×#21511yescharacteristic
322D3—362no—
323D3—362no—
324D3—722no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22D9×#6711yescharacteristic
329C4—41no—
330C4—92no—
331C4—121no—
332C4—182no—
333C4—181no—
334C4#8863511yescharacteristic
335C4C4×D3×D911yescharacteristic
336C3—21no—
337C3#1058853211yescharacteristic, complemented
338C3#1059045811yescharacteristic
339C2—1082no—
340C2#1172100211yescharacteristic
341C2#1172118511yescharacteristic
342C2D9×#21511yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×SD16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#63449C2semidirect
5. C9×#808C2semidirect
6. C3⋊C8×Dic9C2semidirect
7. #11720268C2semidirect
21. #88416C4semidirect
31. #63449D3semidirect
47. C3×Dic18C2×C4semidirect
50. C9×C3⋊C8C2×C4semidirect
90. C3×Dic9SD16semidirect
104. #808D9semidirect
136. Dic18C4×D3semidirect
200. Dic9#215semidirect
208. C3×C9C4×SD16semidirect
242. C3⋊C8C4×D9semidirect
298. C9C4×#215semidirect
337. C3#10588532semidirect

Maximal subgroups

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

subgroupindex
C3×#634492
#117209922
C2×#884162
#117205282
C9×#8082
C3⋊C8×Dic92
#117202682
#105904583
#105885323