Finite Group Databasebeta

← #13137192

Subgroups — #13137192

Order 1728.

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

normal subgroups only

This group has 466 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
1#11721712C211yesmaximal, characteristic, complemented
2C3×#63437C211yesmaximal, characteristic, complemented
3C2×#88418C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic
5C2×#88416C211yesmaximal, characteristic, complemented
6C2×#88415C211yesmaximal, characteristic, complemented
7C4×C9×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590370—31nomaximal
9#10588378—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C3×C42×C9C2211yescharacteristic
15#88536—22no—
16C3×(C2×D9)⋊C4—22no—
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88418—22no—
20#88417—22no—
21#88416—22no—
22#88415—22no—
23#64084—31no—
24C3×#875—31no—
25C2×#3719—31no—
26C2×#3717—31no—
27C2×#3716—31no—
28C2×#3714—31no—
29C3×C4×C3⋊C8D311yescharacteristic
30C2×D72—31no—
31C2×#3664—31no—
32C2×#3664—31no—
33C2×Dic36—31no—
34#63437—31no—
35#63437—61no—
36#63437D311yescharacteristic, complemented
37C4×C8×C9—31no—
38C22×C3⋊D9—21no—
39C22×C3×D9—21no—
40C2×C3×C4×C9D412yes—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45#6226—21no—
46#6226—21no—
47C2×C3×Dic9—21no—
48C3×D36—21no—
49C3×D36—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C9×C3⋊C8D412yes—
53C9×C3⋊C8D412yes—
54#5059—91no—
55#4696—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×#634—31no—
58C2×C3×D12—31no—
59C2×C3×Dic6—31no—
60C32×C42C2×D311yescharacteristic
61#3752—62no—
62C3×(C2×D3)⋊C4—62no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65#3719—62no—
66#3717—62no—
67#3716—62no—
68#3714—62no—
69C2×D36—31no—
70C2×D36—61no—
71C2×D36C2×D311yescharacteristic
72C2×Dic18—31no—
73C2×Dic18—61no—
74C2×Dic18C2×D311yescharacteristic
75C2×C8×C9—31no—
76C2×C8×C9—31no—
77C42×C9—21no—
78C42×C9C2×D311yescharacteristic
79(C2×D9)⋊C4—22no—
80(C2×D9)⋊C4—64no—
81(C2×D9)⋊C4—62no—
82D72—62no—
83#3664—62no—
84#3664—62no—
85Dic36—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—41no—
88C2×C3⋊D9—42no—
89C2×C3×D9—41no—
90C2×C3×D9—42no—
91C3×C4×C9—22no—
92C3×C4×C9C2×D411yescharacteristic
93C3×C4×C9C2×D411yescharacteristic
94C3⋊Dic9—41no—
95C3×Dic9—41no—
96C3×#124—91no—
97C2×#216—91no—
98C2×#215—91no—
99C2×#214—91no—
100C2×#213—91no—
101C2×Dic12—91no—
102C2×D24—91no—
103C2×#204—91no—
104C2×#204—91no—
105#875—31no—
106#875—91no—
107#875—91no—
108#875—181no—
109C3×C4×C8—31no—
110C4×C3⋊C8D911yescharacteristic, complemented
111C22×C3⋊D3—61no—
112C22×C3×D3—61no—
113C2×C32×C4D1212yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—61no—
116C4⋊(C3⋊D3)—61no—
117C4⋊(C3⋊D3)—61no—
118#634—61no—
119#634—61no—
120C2×C3×Dic3—61no—
121C3×D12—61no—
122C3×D12—61no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125C22×D9—21no—
126C22×D9—62no—
127C22×D9—61no—
128C3×C3⋊C8D1212yes—
129C3×C3⋊C8D1212yes—
130C2×C4×C9—22no—
131C2×C4×C9—21no—
132C2×C4×C9#6712yes—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—62no—
136C2×Dic9—61no—
137D36—21no—
138D36—21no—
139D36—61no—
140D36—61no—
141D36—121no—
142D36—121no—
143Dic18—21no—
144Dic18—21no—
145Dic18—61no—
146Dic18—61no—
147Dic18—121no—
148Dic18—121no—
149C8×C9—32no—
150C8×C9—32no—
151#495—271noSylow
152C2×C3×C9#12711yescharacteristic
153C2×C3×C9#13512yes—
154C3⋊D9—81no—
155C3×D9—81no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C2×D12—31no—
159C2×D12—91no—
160C2×D12—91no—
161C2×D12—181no—
162C2×Dic6—31no—
163C2×Dic6—91no—
164C2×Dic6—91no—
165C2×Dic6—181no—
166C2×C3×C8—31no—
167C2×C3×C8—31no—
168C3×C22⋊C4—182no—
169C3×C42—21no—
170C3×C42C2×D911yescharacteristic
171C3×C42D3211yescharacteristic
172#216—182no—
173#215—182no—
174#214—182no—
175#213—182no—
176(C2×D3)⋊C4—62no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—182no—
179(C2×D3)⋊C4—182no—
180C2×C3⋊C8C2×D911yescharacteristic
181C2×C3⋊C8C2×D911yescharacteristic
182Dic12—182no—
183D24—182no—
184#204—182no—
185#204—182no—
186C22×C32C2×D1211yescharacteristic
187C2×C3⋊D3—121no—
188C2×C3⋊D3—122no—
189C2×C3×D3—121no—
190C2×C3×D3—122no—
191C32×C4—22no—
192C32×C4C2×D1211yescharacteristic
193C32×C4C2×D1211yescharacteristic
194C3⋊Dic3—121no—
195C3×Dic3—121no—
196C22×C9—21no—
197C22×C9C2×#6711yescharacteristic
198C2×D9—41no—
199C2×D9—42no—
200C2×D9—122no—
201C2×D9—124no—
202C2×D9—121no—
203C2×D9—122no—
204C4×C9—22no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—42no—
208C4×C9D3×D411yescharacteristic
209C4×C9D3×D411yescharacteristic
210Dic9—41no—
211Dic9—122no—
212Dic9—121no—
213C2×Q16—271no—
214C2×SD16—271no—
215C2×SD16—271no—
216C2×D8—271no—
217#124—91no—
218#124—271no—
219C4×C8—31no—
220C3×C9#49511yescharacteristic, complemented, Sylow
221C23×C3—181no—
222C22×D3—61no—
223C22×D3—182no—
224C22×D3—181no—
225C22×D3—181no—
226C3×Q8—181no—
227C3×Q8—181no—
228C3×D4—181no—
229C3×D4—181no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—181no—
233C2×C3×C4D3612yes—
234C2×C3×C4C22×D911yescharacteristic
235C2×C3×C4#62612yes—
236C2×C3×C4C2×D3211yescharacteristic
237C2×Dic3—61no—
238C2×Dic3—182no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241D12—61no—
242D12—61no—
243D12—181no—
244D12—181no—
245D12—181no—
246D12—181no—
247D12—361no—
248D12—361no—
249Dic6—61no—
250Dic6—61no—
251Dic6—181no—
252Dic6—181no—
253Dic6—181no—
254Dic6—181no—
255Dic6—361no—
256Dic6—361no—
257C3×C8—32no—
258C3×C8—32no—
259C3⋊C8D3612yes—
260C3⋊C8D3612yes—
261C2×C32#87411yescharacteristic
262C2×C32#90412yes—
263C3⋊D3—241no—
264C3×D3—241no—
265C2×C9—22no—
266C2×C9—21no—
267C2×C9#94011yescharacteristic
268C2×C9#95012yes—
269D9—81no—
270D9—242no—
271D9—241no—
272C2×Q8—91no—
273C2×Q8—271no—
274C2×D4—91no—
275C2×D4—271no—
276Q16—542no—
277SD16—542no—
278SD16—542no—
279D8—542no—
280C2×C8—31no—
281C2×C8—31no—
282C22⋊C4—182no—
283C22⋊C4—542no—
284C42D3×D911yescharacteristic
285C22×C3—21no—
286C22×C3—361no—
287C22×C3—362no—
288C22×C3C2×D3611yescharacteristic
289C22×C3C2×#62611yescharacteristic
290C2×D3—122no—
291C2×D3—121no—
292C2×D3—362no—
293C2×D3—364no—
294C2×D3—362no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—362no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—42no—
303C3×C4—361no—
304C3×C4C2×D3611yescharacteristic
305C3×C4C2×D3611yescharacteristic
306C3×C4D3×D1211yescharacteristic
307C3×C4D3×D1211yescharacteristic
308Dic3—121no—
309Dic3—362no—
310Dic3—361no—
311Dic3—361no—
312C32#469611yescharacteristic
313C9—21no—
314C9#505911yescharacteristic, complemented
315C23—181no—
316C23—541no—
317Q8—181no—
318Q8—181no—
319Q8—541no—
320Q8—541no—
321D4—181no—
322D4—181no—
323D4—541no—
324D4—541no—
325C2×C4—181no—
326C2×C4—541no—
327C2×C4#619212yes—
328C2×C4C2×D3×D911yescharacteristic
329C8—32no—
330C8—32no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—721no—
334C2×C3#6343611yescharacteristic
335C2×C3#6346712yes—
336C2×C3#6382912yes—
337C2×C3#6391111yescharacteristic
338D3—241no—
339D3—722no—
340D3—721no—
341D3—721no—
342C22—361no—
343C22—362no—
344C22—1081no—
345C22—1082no—
346C22C2×#619211yescharacteristic
347C4—22no—
348C4—361no—
349C4—1081no—
350C4D3×D3611yescharacteristic
351C4D3×D3611yescharacteristic
352C3—21no—
353C3#1058837811yescharacteristic, complemented
354C3#1059037011yescharacteristic
355C2—721no—
356C2—2161no—
357C2#1172086712yes—
358C2#1172100611yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C42×C9C22
socleC22×C32C2×D12
prime radicalC2×#88417C2
hypercenterC42D3×D9
hypocenterC3×C9#495
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721712C2semidirect
2. C3×#63437C2semidirect
3. C2×#88418C2semidirect
5. C2×#88416C2semidirect
6. C2×#88415C2semidirect
7. C4×C9×C3⋊C8C2semidirect
36. #63437D3semidirect
110. C4×C3⋊C8D9semidirect
220. C3×C9#495semidirect
314. C9#5059semidirect
353. C3#10588378semidirect

Maximal subgroups

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

subgroupindex
#117217122
C3×#634372
C2×#884182
C2×#884172
C2×#884162
C2×#884152
C4×C9×C3⋊C82
#105903703
#105883783