Finite Group Databasebeta

← #13140688

Subgroups — #13140688

Order 1728.

384 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88547C211yesmaximal, characteristic
2C3×#63506C211yesmaximal, characteristic
3#11720913C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10591575—31nomaximal
9#10588475—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13#88547—22no—
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×Dic9⋊C4—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20#88442—21no—
21Dic3×Dic9—21no—
22#88418—22no—
23C2×#3756—31no—
24C3×#944—31no—
25#63844—31no—
26C2×#3719—31no—
27C3×#835D311yescharacteristic
28Dic3⋊(C3⋊C8)—31no—
29#63565—31no—
30#63506D311yescharacteristic
31C2×#3674—31no—
32C2×#3674—61no—
33C2×Dic36—31no—
34#63449—31no—
35C9×#103—31no—
36Dic9⋊C8—31no—
37#63368—31no—
38C3×Q8×C9—21no—
39C3×Q8×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226D412yes—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C2×C3×Dic9D411yescharacteristic
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8—21no—
53#5090—91no—
54#4795—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×#634—31no—
57C2×C3×Dic6—31no—
58#3756—62no—
59C2×C3⋊(C3⋊C8)—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×C4×Dic3—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65#3724—61no—
66Dic32—61no—
67#3719—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C2×D311yescharacteristic
70C2×Dic18—31no—
71C2×Dic18—61no—
72C2×Dic18C2×D311yescharacteristic
73C2×C8×C9—31no—
74C9×C4⋊C4—31no—
75#3674—62no—
76#3674—64no—
77Dic9⋊C4—21no—
78Dic9⋊C4—21no—
79Dic9⋊C4—61no—
80C4×Dic9—61no—
81C4×Dic9C2×D311yescharacteristic
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84Dic36—62no—
85C22×C3×C9C2×D411yescharacteristic
86C3×C4×C9—41no—
87C3×C4×C9C2×D411yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—22no—
90C9×Dic3—41no—
91C3×Dic9—21no—
92C3×Dic9—41no—
93C3×Dic9Q1612yescomplemented
94C3×C4⋊Q8—91no—
95#944—31no—
96C2×#216—91no—
97C2×#216—91no—
98C2×#216—91no—
99C2×#216—181no—
100C2×Dic12—91no—
101#887—91no—
102C4×Dic6—91no—
103C3×#103—31no—
104#835—91no—
105#835D911yescharacteristic
106Dic3⋊C8—91no—
107#810—91no—
108C4⋊(C3⋊C8)—91no—
109C32×Q8—21no—
110C32×Q8—21no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—61no—
114#634—32no—
115#634—61no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C3×Dic6—61no—
121C3×Dic6—61no—
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—21no—
124Q8×C9—22no—
125Q8×C9—21no—
126Q8×C9—21no—
127Q8×C9—41no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—41no—
131C2×C4×C9—61no—
132C2×C4×C9C22×D311yescharacteristic
133C2×Dic9—21no—
134C2×Dic9—61no—
135C2×Dic9—121no—
136C2×Dic9#6711yescharacteristic
137C2×Dic9#6711yescharacteristic
138Dic18—21no—
139Dic18—21no—
140Dic18—32no—
141Dic18—62no—
142Dic18—61no—
143Dic18—121no—
144C8×C9—61no—
145C9⋊C8—62no—
146C9⋊C8—61no—
147#462—271noSylow
148C2×C3×C9#12111yescharacteristic
149C2×C3×C9C2×Q1611yescharacteristic
150C2×C3×C9#13711yescharacteristic
151C2×C3×Q8—21no—
152C2×C3×Q8—91no—
153C2×C3×Q8C2×D911yescharacteristic
154C2×C3×Q8D3211yescharacteristic
155C2×Dic6—31no—
156C2×Dic6—91no—
157C2×Dic6—91no—
158C2×Dic6—181no—
159C2×C3×C8—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4—181no—
163C3×C42—91no—
164#216—182no—
165#216—182no—
166#216—182no—
167#216—184no—
168C4⋊Dic3C2×D911yescharacteristic
169Dic3⋊C4—61no—
170Dic3⋊C4—61no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173C4×Dic3—31no—
174C4×Dic3—181no—
175C4×Dic3—181no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—181no—
179C2×C3⋊C8C2×D911yescharacteristic
180Dic12—182no—
181C22×C32D3×D411yescharacteristic
182C32×C4—41no—
183C32×C4#23511yescharacteristic
184C32×C4D3×D411yescharacteristic
185C3⋊Dic3—62no—
186C3×Dic3—32no—
187C3×Dic3—41no—
188C3×Dic3—61no—
189C3×Dic3—121no—
190C22×C9—21no—
191C22×C9C2×#6711yescharacteristic
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—42no—
195C4×C9—41no—
196C4×C9—121no—
197C4×C9D3×D411yescharacteristic
198C4×C9#23711yescharacteristic
199Dic9—21no—
200Dic9—41no—
201Dic9—62no—
202Dic9—122no—
203Dic9#21612yescomplemented
204C2×Q16—271no—
205C2×Q16—271no—
206C4⋊Q8—91no—
207C4×Q8—271no—
208C4⋊C8—271no—
209#103—31no—
210#103—271no—
211C3×C9#46211yescharacteristic, complemented, Sylow
212C3×Q8—22no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—41no—
218C3×Q8—181no—
219C3×Q8—181no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—61no—
234C2×Dic3—181no—
235C2×Dic3—181no—
236C2×Dic3—361no—
237Dic6—61no—
238Dic6—61no—
239Dic6—92no—
240Dic6—92no—
241Dic6—182no—
242Dic6—181no—
243Dic6—181no—
244Dic6—361no—
245C3×C8—61no—
246C3⋊C8—21no—
247C3⋊C8—182no—
248C3⋊C8—181no—
249C3⋊C8—181no—
250C2×C32#88411yescharacteristic
251C2×C32#91511yescharacteristic
252C2×C32D3×Q1611yescharacteristic
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9#93811yescharacteristic
257C2×C9#94211yescharacteristic
258C2×C9C2×#21611yescharacteristic
259C2×Q8—91no—
260C2×Q8—271no—
261C2×Q8D3×D911yescharacteristic
262Q16—542no—
263Q16—542no—
264C2×C8—31no—
265C2×C8—271no—
266C4⋊C4—31no—
267C4⋊C4—181no—
268C4⋊C4—181no—
269C4⋊C4—541no—
270C42—91no—
271C42—541no—
272C22×C3—21no—
273C22×C3D4×D911yescharacteristic
274C22×C3D3×#6711yescharacteristic
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—42no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—92no—
281C3×C4—121no—
282C3×C4—181no—
283C3×C4—361no—
284C3×C4#369711yescharacteristic
285C3×C4D4×D911yescharacteristic
286C3×C4#380411yescharacteristic
287C3×C4#381111yescharacteristic
288Dic3—32no—
289Dic3—41no—
290Dic3—61no—
291Dic3—121no—
292Dic3—182no—
293Dic3—182no—
294Dic3—362no—
295C32#479511yescharacteristic
296C9—21no—
297C9#509011yescharacteristic, complemented
298Q8—21no—
299Q8—21no—
300Q8—181no—
301Q8—181no—
302Q8—272no—
303Q8—541no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—541no—
310C2×C4C2×D3×D911yescharacteristic
311C8—61no—
312C8—541no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#6344611yescharacteristic
317C2×C3#6347711yescharacteristic
318C2×C3Q16×D911yescharacteristic
319C2×C3D3×#21611yescharacteristic
320C2×C3#6394611yescharacteristic
321C2×C3#6395811yescharacteristic
322C22D9×#6711yescharacteristic
323C4—41no—
324C4—92no—
325C4—121no—
326C4—181no—
327C4—361no—
328C4—542no—
329C4#8865811yescharacteristic
330C4#8866811yescharacteristic
331C3—21no—
332C3#1058847511yescharacteristic, complemented
333C3#1059157511yescharacteristic
334C2#1172121411yescharacteristic
335C2D9×#21611yescharacteristic
336C2#1172125611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#462
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
93. C3×Dic9Q16semidirect
203. Dic9#216semidirect
211. C3×C9#462semidirect
297. C9#5090semidirect
332. C3#10588475semidirect

Maximal subgroups

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

subgroupindex
C2×#885472
C3×#635062
#117209132
C2×#884182
C9×#8352
Dic9⋊(C3⋊C8)2
#117202682
#105915753
#105884753