Finite Group Databasebeta

← #13138129

Subgroups — #13138129

Order 1728.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88521C211yesmaximal, characteristic
2#11720932C211yesmaximal, characteristic
3#11720912C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic
5C9×Dic3⋊C8C211yesmaximal, characteristic
6C3×#63376C211yesmaximal, characteristic
7#11720277C211yesmaximal, characteristic
8#10590808—31nomaximal
9#10588513—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14#88521—22no—
15C4×C9×Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18Dic3⋊Dic9—21no—
19Dic3⋊Dic9—21no—
20#88442—21no—
21Dic3×Dic9—21no—
22#88418—22no—
23C2×#3746—31no—
24#63866—31no—
25#63844—31no—
26C2×#3719—31no—
27C3×#816—31no—
28C3×Dic3⋊C8D311yescharacteristic
29#63567—31no—
30C2×Dic36—31no—
31C2×Dic36—31no—
32C2×Dic36—61no—
33#63431—31no—
34C4×Dic18—31no—
35C9×C4⋊C8—31no—
36#63376—31no—
37#63376D311yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9—21no—
41#6226—21no—
42#6226—21no—
43#6226—21no—
44#6226D412yes—
45C2×C9×Dic3D411yescharacteristic
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53#4849—91no—
54#4795—91no—
55C2×#634—31no—
56C2×#634—31no—
57C2×C3×Dic6—31no—
58C2×C32×C8C2×D311yescharacteristic
59#3746—62no—
60C3×C4⋊Dic3—31no—
61C3×C4×Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8C2×D311yescharacteristic
63#3724—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic32—61no—
67#3719—62no—
68C2×Dic18—31no—
69C2×Dic18—31no—
70C2×Dic18—61no—
71C2×Dic18—61no—
72C2×Dic18C2×D311yescharacteristic
73C2×C8×C9—21no—
74C2×C8×C9—31no—
75C2×C8×C9C2×D311yescharacteristic
76C42×C9—31no—
77C4⋊Dic9—61no—
78C4⋊Dic9—61no—
79C4⋊Dic9C2×D311yescharacteristic
80Dic9⋊C4—61no—
81C4×Dic9—61no—
82Dic36—62no—
83Dic36—64no—
84Dic36—62no—
85C22×C3×C9C2×D411yescharacteristic
86C3×C4×C9C2×D411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—22no—
89C3⋊Dic9—41no—
90C9×Dic3—21no—
91C9×Dic3Q1612yescomplemented
92C3×Dic9—41no—
93C3×Dic9—41no—
94#944—91no—
95C2×#216—91no—
96C2×Dic12—91no—
97C2×Dic12—91no—
98C2×Dic12—91no—
99C2×Dic12—181no—
100#887—91no—
101#869—91no—
102C4×Dic6—91no—
103C3×C4⋊C8—31no—
104C3×#103—91no—
105#816—31no—
106#816—91no—
107Dic3⋊C8D911yescharacteristic
108#810—91no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112#634—32no—
113#634—61no—
114#634—61no—
115#634—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3D1211yescharacteristic
119C2×C3×Dic3D1211yescharacteristic
120C3×Dic6—61no—
121C3×Dic6—61no—
122C32×C8—21no—
123C3×C3⋊C8—21no—
124C2×C4×C9—21no—
125C2×C4×C9—31no—
126C2×C4×C9—31no—
127C2×C4×C9C22×D311yescharacteristic
128C2×Dic9—21no—
129C2×Dic9—21no—
130C2×Dic9—61no—
131C2×Dic9—61no—
132C2×Dic9—121no—
133C2×Dic9—121no—
134Dic18—21no—
135Dic18—21no—
136Dic18—32no—
137Dic18—62no—
138Dic18—61no—
139Dic18—62no—
140Dic18—61no—
141Dic18—61no—
142Dic18—121no—
143Dic18—121no—
144C8×C9—22no—
145C8×C9—21no—
146C8×C9—61no—
147#462—271noSylow
148C2×C3×C9#12111yescharacteristic
149C2×C3×C9C2×Q1611yescharacteristic
150C2×C3×C9#13711yescharacteristic
151C2×C3×Q8—91no—
152C2×Dic6—31no—
153C2×Dic6—91no—
154C2×Dic6—91no—
155C2×Dic6—91no—
156C2×Dic6—91no—
157C2×Dic6—181no—
158C2×Dic6—181no—
159C2×C3×C8—21no—
160C2×C3×C8—31no—
161C2×C3×C8C2×D911yescharacteristic
162C2×C3×C8D3211yescharacteristic
163C3×C4⋊C4—91no—
164C3×C42—31no—
165#216—182no—
166C4⋊Dic3—31no—
167C4⋊Dic3—181no—
168C4⋊Dic3—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173C4×Dic3—181no—
174C4×Dic3—181no—
175C4×Dic3C2×D911yescharacteristic
176C2×C3⋊C8C2×D911yescharacteristic
177Dic12—182no—
178Dic12—182no—
179Dic12—184no—
180Dic12—182no—
181C22×C32C2×D1211yescharacteristic
182C32×C4D3×D411yescharacteristic
183C32×C4#23911yescharacteristic
184C3⋊Dic3—62no—
185C3⋊Dic3—121no—
186C3×Dic3—21no—
187C3×Dic3—121no—
188C3×Dic3—121no—
189C3×Dic3Dic1212yes—
190C22×C9—21no—
191C22×C9D3×D411yescharacteristic
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—32no—
195C4×C9—61no—
196C4×C9#23511yescharacteristic
197C4×C9D3×D411yescharacteristic
198Dic9—41no—
199Dic9—41no—
200Dic9—62no—
201Dic9—122no—
202Dic9—122no—
203Dic9—121no—
204C2×Q16—271no—
205C2×Q16—271no—
206C4⋊Q8—271no—
207C4×Q8—271no—
208C4⋊C8—31no—
209#103—91no—
210#103—271no—
211C3×C9#46211yescharacteristic, complemented, Sylow
212C3×Q8—181no—
213C3×Q8—181no—
214C2×C3×C4—21no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C22×D911yescharacteristic
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C2×Dic3—181no—
227C2×Dic3—361no—
228C2×Dic3—361no—
229C2×Dic3D3611yescharacteristic
230C2×Dic3D3611yescharacteristic
231Dic6—61no—
232Dic6—61no—
233Dic6—92no—
234Dic6—92no—
235Dic6—182no—
236Dic6—181no—
237Dic6—181no—
238Dic6—182no—
239Dic6—181no—
240Dic6—181no—
241Dic6—181no—
242Dic6—181no—
243Dic6—361no—
244Dic6—361no—
245C3×C8—22no—
246C3×C8—21no—
247C3×C8—21no—
248C3×C8—61no—
249C3⋊C8—21no—
250C2×C32#89511yescharacteristic
251C2×C32C2×Dic1211yescharacteristic
252C2×C32#90911yescharacteristic
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9#88411yescharacteristic
257C2×C9#91511yescharacteristic
258C2×C9D3×Q1611yescharacteristic
259C2×Q8—91no—
260C2×Q8—271no—
261C2×Q8—271no—
262Q16—542no—
263Q16—542no—
264C2×C8—31no—
265C2×C8D3×D911yescharacteristic
266C4⋊C4—91no—
267C4⋊C4—541no—
268C4⋊C4—541no—
269C4⋊C4—541no—
270C42—31no—
271C42—541no—
272C22×C3—21no—
273C22×C3C2×D3611yescharacteristic
274C22×C3D3×D1211yescharacteristic
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—32no—
278C3×C4—61no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4D4×D911yescharacteristic
282C3×C4#370111yescharacteristic
283C3×C4#379911yescharacteristic
284C3×C4#381111yescharacteristic
285Dic3—21no—
286Dic3—121no—
287Dic3—121no—
288Dic3—182no—
289Dic3—182no—
290Dic3—362no—
291Dic3—362no—
292Dic3—361no—
293Dic3—361no—
294Dic3Dic3612yescomplemented
295C32#484911yescharacteristic
296C9—21no—
297C9#479511yescharacteristic, complemented
298Q8—181no—
299Q8—181no—
300Q8—272no—
301Q8—541no—
302Q8—541no—
303Q8—541no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4—541no—
309C2×C4—541no—
310C2×C4C2×D3×D911yescharacteristic
311C8—21no—
312C8—61no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#6345711yescharacteristic
317C2×C3C2×Dic3611yescharacteristic
318C2×C3#6347111yescharacteristic
319C2×C3D3×Dic1211yescharacteristic
320C2×C3#6380011yescharacteristic
321C2×C3#6391011yescharacteristic
322C22D3×D3611yescharacteristic
323C4—32no—
324C4—61no—
325C4—361no—
326C4—361no—
327C4—542no—
328C4—1081no—
329C4#8863511yescharacteristic
330C4#8866811yescharacteristic
331C3—21no—
332C3#1058851311yescharacteristic, complemented
333C3#1059080811yescharacteristic
334C2D3×Dic3611yescharacteristic
335C2#1172079611yescharacteristic
336C2#1172100511yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32C2×D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#462
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
91. C9×Dic3Q16semidirect
211. C3×C9#462semidirect
294. Dic3Dic36semidirect
297. C9#4795semidirect
332. C3#10588513semidirect

Maximal subgroups

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

subgroupindex
C2×#885212
#117209322
#117209122
C2×#884182
C9×Dic3⋊C82
C3×#633762
#117202772
#105908083
#105885133