Finite Group Databasebeta

← #13138152

Subgroups — #13138152

Order 1728.

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

normal subgroups only

This group has 440 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
1C2×#88524C211yesmaximal, characteristic
2C2×C3×Dic36C211yesmaximal, characteristic, complemented
3#11720952C211yesmaximal, characteristic
4#11720932C211yesmaximal, characteristic, complemented
5C2×#88418C211yesmaximal, characteristic, complemented
6C2×#88416C211yesmaximal, characteristic
7C9×#815C211yesmaximal, characteristic, complemented
8#10590899—31nomaximal
9#10588396—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15#88524—22no—
16C4×C9×Dic3C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C3×Dic36—22no—
19#88445—22no—
20Dic3⋊Dic9—22no—
21#88418—22no—
22#88416—22no—
23C2×#3744—31no—
24C2×C3×Dic12—31no—
25#63866—31no—
26#63864—31no—
27C2×#3719—31no—
28C2×#3717—31no—
29C3×#815D311yescharacteristic
30C2×#3664—31no—
31C2×#3664—31no—
32C2×#3664—61no—
33C2×Dic36—31no—
34C2×Dic36D311yescharacteristic, complemented
35#63437—31no—
36#63431—31no—
37C9×#97—31no—
38C22×C3⋊D9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)—21no—
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3D412yes—
46C2×C3×Dic9—21no—
47C2×C3×Dic9—21no—
48C3×Dic18—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C3×C8×C9D412yes—
53C9×C3⋊C8D412yes—
54#5190—91no—
55#4715—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×#634—31no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6—31no—
60C2×C32×C8C2×D311yescharacteristic
61#3744—62no—
62C3×C4×Dic3C2×D311yescharacteristic
63C2×C3×C3⋊C8C2×D311yescharacteristic
64C3×Dic12—62no—
65Dic3⋊Dic3—62no—
66#3722—62no—
67#3719—62no—
68#3717—62no—
69C2×D36—31no—
70C2×D36—61no—
71C2×Dic18—31no—
72C2×Dic18—61no—
73C2×Dic18C2×D311yescharacteristic
74C2×Dic18C2×D311yescharacteristic
75C2×C8×C9—21no—
76C2×C8×C9—31no—
77C2×C8×C9C2×D311yescharacteristic
78C42×C9—31no—
79(C2×D9)⋊C4—62no—
80C4⋊Dic9—62no—
81#3664—64no—
82#3664—62no—
83#3664—62no—
84Dic36—22no—
85Dic36—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—41no—
88C2×C3⋊D9—42no—
89C3×C4×C9C2×D411yescharacteristic
90C3×C4×C9C2×D411yescharacteristic
91C3⋊Dic9—41no—
92C9×Dic3—22no—
93C3×Dic9—41no—
94C3×Dic9—41no—
95C2×C3×Q16—91no—
96#947—91no—
97#944—91no—
98C2×#216—91no—
99C2×#215—91no—
100C2×Dic12—31no—
101C2×Dic12—91no—
102C2×#204—91no—
103C2×#204—91no—
104C2×#204—91no—
105C2×#204—181no—
106#875—91no—
107#869—91no—
108C3×#97—31no—
109#815D911yescharacteristic, complemented
110C22×C3⋊D3—61no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—61no—
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—61no—
115#634—61no—
116#634—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3D1212yes—
120C3×Dic6—61no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C22×D9—61no—
125C22×D9—121no—
126C32×C8D1212yes—
127C3×C3⋊C8D1212yes—
128C2×C4×C9—21no—
129C2×C4×C9—32no—
130C2×C4×C9C22×D311yescharacteristic
131C2×Dic9—21no—
132C2×Dic9—21no—
133C2×Dic9—61no—
134C2×Dic9—121no—
135D36—62no—
136D36—61no—
137D36—61no—
138D36—121no—
139Dic18—21no—
140Dic18—21no—
141Dic18—21no—
142Dic18—21no—
143Dic18—62no—
144Dic18—61no—
145Dic18—61no—
146Dic18—121no—
147C8×C9—22no—
148C8×C9—32no—
149C8×C9#6712yes—
150#497—271noSylow
151C2×C3×C9#12711yescharacteristic
152C2×C3×C9#13712yes—
153C3⋊D9—81no—
154C2×C3×Q8—91no—
155C2×C3×Q8—91no—
156C2×D12—91no—
157C2×D12—91no—
158C2×D12—181no—
159C2×Dic6—31no—
160C2×Dic6—31no—
161C2×Dic6—91no—
162C2×Dic6—91no—
163C2×Dic6—181no—
164C3×Q16—182no—
165C2×C3×C8—21no—
166C2×C3×C8—31no—
167C2×C3×C8C2×D911yescharacteristic
168C2×C3×C8D3211yescharacteristic
169C3×C42—31no—
170#216—182no—
171#215—182no—
172(C2×D3)⋊C4—182no—
173(C2×D3)⋊C4—182no—
174C4⋊Dic3—182no—
175Dic3⋊C4—182no—
176C4×Dic3C2×D911yescharacteristic
177C2×C3⋊C8C2×D911yescharacteristic
178Dic12—62no—
179Dic12—182no—
180#204—184no—
181#204—182no—
182#204—182no—
183#204—182no—
184C22×C32C2×D1211yescharacteristic
185C2×C3⋊D3—121no—
186C2×C3⋊D3—122no—
187C32×C4C2×D1211yescharacteristic
188C32×C4C2×D1211yescharacteristic
189C3⋊Dic3—121no—
190C3×Dic3—22no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C22×C9—21no—
194C22×C9D3×D411yescharacteristic
195C2×D9—122no—
196C2×D9—122no—
197C2×D9—121no—
198C2×D9—242no—
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—62no—
202C4×C9D3×D411yescharacteristic
203C4×C9C2×#6711yescharacteristic
204Dic9—41no—
205Dic9—41no—
206Dic9—122no—
207Dic9—121no—
208C2×Q16—91no—
209C2×Q16—271no—
210C2×SD16—271no—
211C2×SD16—271no—
212C4⋊Q8—271no—
213#124—271no—
214#97—31no—
215C3×C9#49711yescharacteristic, complemented, Sylow
216C22×D3—181no—
217C22×D3—181no—
218C22×D3—361no—
219C3×Q8—181no—
220C3×Q8—181no—
221C3×Q8—181no—
222C3×Q8—181no—
223C2×C3×C4—21no—
224C2×C3×C4—32no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4C22×D911yescharacteristic
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233C2×Dic3—361no—
234C2×Dic3D3612yes—
235D12—182no—
236D12—181no—
237D12—181no—
238D12—181no—
239D12—181no—
240D12—361no—
241Dic6—61no—
242Dic6—61no—
243Dic6—61no—
244Dic6—61no—
245Dic6—182no—
246Dic6—181no—
247Dic6—181no—
248Dic6—181no—
249Dic6—181no—
250Dic6—361no—
251C3×C8—22no—
252C3×C8—32no—
253C3×C8D3612yes—
254C3×C8#62612yes—
255C3⋊C8D3612yes—
256C2×C32#87411yescharacteristic
257C2×C32#90912yes—
258C3⋊D3—241no—
259C2×C9—22no—
260C2×C9—21no—
261C2×C9#91812yes—
262C2×C9#94011yescharacteristic
263D9—242no—
264D9—241no—
265C2×Q8—91no—
266C2×Q8—91no—
267C2×Q8—271no—
268C2×D4—271no—
269Q16—182no—
270Q16—542no—
271SD16—542no—
272SD16—542no—
273C2×C8—31no—
274C2×C8D3×D911yescharacteristic
275C4⋊C4—542no—
276C22⋊C4—542no—
277C42—31no—
278C22×C3—21no—
279C22×C3C2×D3611yescharacteristic
280C22×C3D3×D1211yescharacteristic
281C2×D3—362no—
282C2×D3—362no—
283C2×D3—361no—
284C2×D3—361no—
285C2×D3—362no—
286C2×D3—722no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—62no—
290C3×C4—361no—
291C3×C4—361no—
292C3×C4C2×D3611yescharacteristic
293C3×C4C2×D3611yescharacteristic
294C3×C4D3×D1211yescharacteristic
295C3×C4C2×#62611yescharacteristic
296Dic3—22no—
297Dic3—121no—
298Dic3—121no—
299Dic3—362no—
300Dic3—361no—
301Dic3—361no—
302C32#471511yescharacteristic
303C9—21no—
304C9#519011yescharacteristic, complemented
305C23—541no—
306Q8—181no—
307Q8—181no—
308Q8—181no—
309Q8—181no—
310Q8—541no—
311Q8—541no—
312D4—541no—
313D4—541no—
314C2×C4—32no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4—541no—
318C2×C4C2×D3×D911yescharacteristic
319C8—32no—
320C8#619212yes—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3#6343611yescharacteristic
324C2×C3#6347112yes—
325C2×C3#6380112yes—
326C2×C3#6391111yescharacteristic
327D3—722no—
328D3—721no—
329D3—721no—
330C22—1081no—
331C22—1082no—
332C22D3×D3611yescharacteristic
333C4—62no—
334C4—361no—
335C4—361no—
336C4—1081no—
337C4D3×D3611yescharacteristic
338C4C2×#619211yescharacteristic
339C3—21no—
340C3#1058839611yescharacteristic, complemented
341C3#1059089911yescharacteristic
342C2—2161no—
343C2#1172079712yes—
344C2#1172100611yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×C8D3×D9
hypocenterC3×C9#497
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
2. C2×C3×Dic36C2semidirect
4. #11720932C2semidirect
5. C2×#88418C2semidirect
7. C9×#815C2semidirect
34. C2×Dic36D3semidirect
109. #815D9semidirect
215. C3×C9#497semidirect
304. C9#5190semidirect
340. C3#10588396semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
C2×C3×Dic362
#117209522
#117209322
C2×#884182
C2×#884162
C9×#8152
#105908993
#105883963