Finite Group Databasebeta

← #13140834

Subgroups — #13140834

Order 1728.

426 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 97 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
3C2×C9×#216C211yesmaximal, characteristic
4#11720924C211yesmaximal, characteristic
5C2×#88427C211yesmaximal, characteristic
6C2×#88418C211yesmaximal, characteristic
7C3⋊C8×Dic9C211yesmaximal, characteristic
8#10591626—31nomaximal
9#10588863—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14#88547—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×#216—22no—
17C3×Dic9⋊C4—22no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic9⋊Dic3—22no—
21#88427—22no—
22#88418—22no—
23C2×#3756—31no—
24C3×#944—31no—
25C2×C3×#216D311yescharacteristic
26#63866—31no—
27C2×#3719—31no—
28C2×#3718—31no—
29Dic3×C3⋊C8—31no—
30C2×C9×Q16—31no—
31#63506—31no—
32#63506D311yescharacteristic
33C2×#3674—31no—
34C2×#3674—31no—
35C2×#3674—61no—
36C2×Dic36—31no—
37C8×Dic9—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#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D412yes—
48C9×Dic6—21no—
49C9×Dic6—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C3⋊(C9⋊C8)D412yes—
53C9×C3⋊C8D412yes—
54#5183—91no—
55#5092—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×#634—31no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6C2×D311yescharacteristic
60#3756—62no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×#216—22no—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66Dic3⋊Dic3—62no—
67#3719—62no—
68#3718—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9—31no—
71C2×Q8×C9C2×D311yescharacteristic
72C2×Dic18—31no—
73C2×Dic18—61no—
74C2×Dic18C2×D311yescharacteristic
75C9×Q16—62no—
76C2×C8×C9—31no—
77#3674—62no—
78#3674—62no—
79#3674—64no—
80Dic9⋊C4—22no—
81Dic9⋊C4—62no—
82C4×Dic9C2×D311yescharacteristic
83C2×C9⋊C8—31no—
84C2×C9⋊C8—61no—
85Dic36—62no—
86C22×C3×C9C2×D411yescharacteristic
87C3×C4×C9—41no—
88C3×C4×C9C2×D411yescharacteristic
89C3×C4×C9C2×D411yescharacteristic
90C3⋊Dic9—41no—
91C9×Dic3—41no—
92C3×Dic9—41no—
93C3×Dic9Q1614yescomplemented
94C2×C3×Q16—31no—
95C3×C4⋊Q8—91no—
96#944—31no—
97#944—91no—
98C2×#216—91no—
99C2×#216—91no—
100C2×#216—91no—
101C2×#216—91no—
102C2×#216—91no—
103C2×#216—181no—
104C2×#216D911yescharacteristic
105C2×Dic12—91no—
106#869—91no—
107C8×Dic3—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—61no—
115#634—61no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—61no—
119C3×Dic6—21no—
120C3×Dic6—21no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C3⋊(C3⋊C8)—32no—
124C3×C3⋊C8#6712yes—
125Q8×C9—22no—
126Q8×C9—21no—
127Q8×C9—21no—
128Q8×C9—41no—
129Q8×C9—61no—
130Q8×C9—61no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—41no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136C2×Dic9—21no—
137C2×Dic9—61no—
138C2×Dic9—121no—
139C2×Dic9#6712yes—
140Dic18—21no—
141Dic18—21no—
142Dic18—62no—
143Dic18—61no—
144Dic18—61no—
145Dic18—121no—
146C8×C9—32no—
147C9⋊C8—32no—
148C9⋊C8—62no—
149#494—271noSylow
150C2×C3×C9#12711yescharacteristic
151C2×C3×C9C2×Q1612yes—
152C2×C3×Q8—21no—
153C2×C3×Q8—31no—
154C2×C3×Q8—91no—
155C2×C3×Q8C2×D911yescharacteristic
156C2×C3×Q8D3211yescharacteristic
157C2×Dic6—31no—
158C2×Dic6—91no—
159C2×Dic6—91no—
160C2×Dic6—181no—
161C2×Dic6C2×D911yescharacteristic
162C3×Q16—62no—
163C2×C3×C8—31no—
164C3×C4⋊C4—182no—
165C3×C42—91no—
166#216—22no—
167#216—182no—
168#216—182no—
169#216—182no—
170#216—182no—
171#216—184no—
172#216—182no—
173C4⋊Dic3—182no—
174Dic3⋊C4—62no—
175Dic3⋊C4—182no—
176C4×Dic3—31no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—181no—
180C2×C3⋊C8C2×D911yescharacteristic
181Dic12—182no—
182C22×C32D3×D411yescharacteristic
183C32×C4—41no—
184C32×C4D3×D411yescharacteristic
185C32×C4C2×#6711yescharacteristic
186C3⋊Dic3—121no—
187C3×Dic3—34no—
188C3×Dic3—41no—
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×C9D3×D411yescharacteristic
199Dic9—41no—
200Dic9—122no—
201Dic9—121no—
202Dic9#21614yescomplemented
203C2×Q16—31no—
204C2×Q16—271no—
205C2×Q16—271no—
206C2×Q16—271no—
207C4⋊Q8—91no—
208C4⋊Q8—271no—
209C4×C8—271no—
210C3×C9#49411yescharacteristic, complemented, Sylow
211C3×Q8—22no—
212C3×Q8—21no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—41no—
217C3×Q8—61no—
218C3×Q8—61no—
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—61no—
226C2×C3×C4—92no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—32no—
232C2×Dic3—61no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—361no—
236Dic6—21no—
237Dic6—21no—
238Dic6—61no—
239Dic6—61no—
240Dic6—182no—
241Dic6—181no—
242Dic6—181no—
243Dic6—181no—
244Dic6—181no—
245Dic6—361no—
246C3×C8—32no—
247C3⋊C8—92no—
248C3⋊C8—92no—
249C3⋊C8—182no—
250C3⋊C8#61112yes—
251C2×C32D3×Q1612yes—
252C2×C32#94011yescharacteristic
253C2×C9—22no—
254C2×C9—21no—
255C2×C9#94011yescharacteristic
256C2×C9C2×#21612yes—
257C2×Q8—31no—
258C2×Q8—91no—
259C2×Q8—271no—
260C2×Q8D3×D911yescharacteristic
261Q16—62no—
262Q16—542no—
263Q16—542no—
264Q16—542no—
265C2×C8—31no—
266C2×C8—271no—
267C4⋊C4—182no—
268C4⋊C4—542no—
269C42—91no—
270C22×C3—21no—
271C22×C3D4×D911yescharacteristic
272C22×C3D3×#6711yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—42no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—94no—
279C3×C4—121no—
280C3×C4—361no—
281C3×C4D4×D911yescharacteristic
282C3×C4C2×#61111yescharacteristic
283C3×C4D3×#6711yescharacteristic
284C3×C4#381111yescharacteristic
285Dic3—34no—
286Dic3—41no—
287Dic3—121no—
288Dic3—362no—
289Dic3—361no—
290Dic3—361no—
291C32#518311yescharacteristic
292C9—21no—
293C9#509211yescharacteristic, complemented
294Q8—21no—
295Q8—21no—
296Q8—61no—
297Q8—61no—
298Q8—181no—
299Q8—181no—
300Q8—541no—
301Q8—541no—
302C2×C4—21no—
303C2×C4—61no—
304C2×C4—92no—
305C2×C4—181no—
306C2×C4—541no—
307C2×C4C2×D3×D911yescharacteristic
308C8—32no—
309C8—272no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3Q16×D912yes—
313C2×C3#6350211yescharacteristic
314C2×C3D3×#21612yes—
315C2×C3#6397911yescharacteristic
316C22D9×#6711yescharacteristic
317C4—41no—
318C4—94no—
319C4—121no—
320C4—361no—
321C4—1081no—
322C4D3×#61111yescharacteristic
323C4#8866811yescharacteristic
324C3—21no—
325C3#1058886311yescharacteristic, complemented
326C3#1059162611yescharacteristic
327C2D9×#21612yes—
328C2#1172128911yescharacteristic
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#494
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
202. Dic9#216semidirect
210. C3×C9#494semidirect
293. C9#5092semidirect
325. C3#10588863semidirect

Maximal subgroups

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

subgroupindex
C2×#885472
C3×#635062
C2×C9×#2162
#117209242
C2×#884272
C2×#884182
C3⋊C8×Dic92
#105916263
#105888633