Finite Group Databasebeta

← #13138234

Subgroups — #13138234

Order 1728.

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

normal subgroups only

This group has 508 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×#88525C211yesmaximal, characteristic, complemented
2C2×C3×#3664C211yesmaximal, characteristic, complemented
3#11721006C211yesmaximal, characteristic, complemented
4#11720952C211yesmaximal, characteristic, complemented
5C2×#88417C211yesmaximal, characteristic, complemented
6C2×#88416C211yesmaximal, characteristic, complemented
7C9×#815C211yesmaximal, characteristic, complemented
8#10590881—31nomaximal
9#10588393—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×#6192—22no—
15C2×C3×C8×C9C2211yescharacteristic
16#88525—22no—
17C4×C9×Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C3×#3664—22no—
20#88445—22no—
21#88417—22no—
22#88416—22no—
23C2×#3745—31no—
24C2×C3×#204—31no—
25#63911—31no—
26#63864—31no—
27C2×#3717—31no—
28C2×#3714—31no—
29C3×#815D311yescharacteristic
30C2×D72—31no—
31C2×D72—31no—
32C2×D72—61no—
33C2×#3664—31no—
34C2×#3664D311yescharacteristic, complemented
35#63437—31no—
36#63436—31no—
37C9×#97—31no—
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3×D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46C2×C9×Dic3D412yes—
47C2×C3×Dic9—21no—
48C3×D36—21no—
49C3×D36—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52#6192—44no—
53C3×C8×C9D412yescomplemented
54C9×C3⋊C8D412yescomplemented
55#5176—91no—
56#4712—91no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C3×D12—31no—
60C2×C3×Dic6—31no—
61C2×#626—62no—
62C2×C32×C8C2×D311yescharacteristic
63#3745—62no—
64C3×C4×Dic3C2×D311yescharacteristic
65C2×C3×C3⋊C8C2×D311yescharacteristic
66C3×#204—62no—
67#3722—62no—
68#3717—62no—
69#3714—62no—
70C2×D36—31no—
71C2×D36—31no—
72C2×D36—61no—
73C2×D36—61no—
74C2×D36—62no—
75C2×D36C2×D311yescharacteristic
76C2×Dic18C2×D311yescharacteristic
77C2×C8×C9—21no—
78C2×C8×C9—31no—
79C2×C8×C9C2×D311yescharacteristic
80C42×C9—31no—
81(C2×D9)⋊C4—62no—
82D72—64no—
83D72—62no—
84D72—62no—
85#3664—22no—
86#3664—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—41no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—42no—
91C2×C3⋊D9—42no—
92C2×C3×D9—41no—
93C2×C3×D9—42no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C9×Dic3—22no—
97C3×Dic9—41no—
98C2×C3×SD16—91no—
99#947—91no—
100C2×#215—91no—
101#940—91no—
102C2×#213—91no—
103C2×D24—91no—
104C2×D24—91no—
105C2×D24—91no—
106C2×D24—181no—
107C2×#204—31no—
108C2×#204—91no—
109#875—91no—
110#874—91no—
111C3×#97—31no—
112#815D911yescharacteristic, complemented
113C22×C3⋊D3—61no—
114C22×C3⋊D3—61no—
115C22×C3×D3—61no—
116C2×C32×C4C22×D311yescharacteristic
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—61no—
120C4⋊(C3⋊D3)—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3D1212yes—
123C3×D12—61no—
124C3×D12—61no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127#626—124no—
128C22×D9—21no—
129C22×D9—61no—
130C22×D9—61no—
131C22×D9—121no—
132C22×D9—121no—
133C32×C8D1212yes—
134C3×C3⋊C8D1212yes—
135C2×C4×C9—21no—
136C2×C4×C9—32no—
137C2×C4×C9C22×D311yescharacteristic
138C2×Dic9—21no—
139D36—21no—
140D36—21no—
141D36—62no—
142D36—62no—
143D36—61no—
144D36—61no—
145D36—61no—
146D36—61no—
147D36—121no—
148D36—121no—
149D36—124no—
150Dic18—21no—
151Dic18—21no—
152C8×C9—22no—
153C8×C9—32no—
154C8×C9#6712yescomplemented
155#496—271noSylow
156C2×C3×C9#12711yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)12yes—
158C3⋊D9—81no—
159C3⋊D9—81no—
160C3×D9—81no—
161C2×C3×Q8—91no—
162C2×C3×D4—91no—
163C2×#67—182no—
164C2×D12—31no—
165C2×D12—91no—
166C2×D12—91no—
167C2×D12—91no—
168C2×D12—91no—
169C2×D12—181no—
170C2×D12—181no—
171C2×D12—182no—
172C2×Dic6—31no—
173C3×SD16—182no—
174C2×C3×C8—21no—
175C2×C3×C8—31no—
176C2×C3×C8C2×D911yescharacteristic
177C2×C3×C8D3211yescharacteristic
178C3×C42—31no—
179#215—182no—
180#213—182no—
181(C2×D3)⋊C4—182no—
182(C2×D3)⋊C4—182no—
183C4×Dic3C2×D911yescharacteristic
184C2×C3⋊C8C2×D911yescharacteristic
185D24—184no—
186D24—182no—
187D24—182no—
188D24—182no—
189#204—62no—
190#204—182no—
191C22×C32C2×D1211yescharacteristic
192C2×C3⋊D3—121no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—122no—
195C2×C3⋊D3—122no—
196C2×C3×D3—121no—
197C2×C3×D3—122no—
198C32×C4C2×D1211yescharacteristic
199C32×C4C2×D1211yescharacteristic
200C3×Dic3—22no—
201C3×Dic3—121no—
202C22×C9—21no—
203C22×C9D3×D411yescharacteristic
204C2×D9—42no—
205C2×D9—41no—
206C2×D9—122no—
207C2×D9—122no—
208C2×D9—122no—
209C2×D9—122no—
210C2×D9—121no—
211C2×D9—121no—
212C2×D9—242no—
213C2×D9—242no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—62no—
217C4×C9D3×D411yescharacteristic
218C4×C9C2×#6711yescharacteristic
219Dic9—41no—
220C2×SD16—91no—
221C2×SD16—271no—
222C2×D8—271no—
223C2×D8—271no—
224#127—271no—
225#124—271no—
226#97—31no—
227C3×C9#49611yescharacteristic, complemented, Sylow
228C23×C3—181no—
229C22×D3—61no—
230C22×D3—181no—
231C22×D3—181no—
232C22×D3—181no—
233C22×D3—181no—
234C22×D3—361no—
235C22×D3—361no—
236C3×Q8—181no—
237C3×Q8—181no—
238C3×D4—181no—
239C3×D4—181no—
240C2×C3×C4—21no—
241C2×C3×C4—32no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245#67—364no—
246C2×Dic3—61no—
247C2×Dic3D3612yes—
248D12—61no—
249D12—61no—
250D12—182no—
251D12—182no—
252D12—181no—
253D12—181no—
254D12—181no—
255D12—181no—
256D12—181no—
257D12—181no—
258D12—181no—
259D12—181no—
260D12—361no—
261D12—361no—
262D12—364no—
263Dic6—61no—
264Dic6—61no—
265C3×C8—22no—
266C3×C8—32no—
267C3×C8D3612yescomplemented
268C3×C8#62612yes—
269C3⋊C8D3612yescomplemented
270C2×C32#87411yescharacteristic
271C2×C32#90812yes—
272C3⋊D3—241no—
273C3⋊D3—241no—
274C3×D3—241no—
275C2×C9—22no—
276C2×C9—21no—
277C2×C9#91412yes—
278C2×C9#94011yescharacteristic
279D9—81no—
280D9—242no—
281D9—242no—
282D9—241no—
283D9—241no—
284C2×Q8—91no—
285C2×D4—91no—
286C2×D4—271no—
287C2×D4—271no—
288C2×D4—542no—
289SD16—182no—
290SD16—542no—
291D8—542no—
292D8—542no—
293C2×C8—31no—
294C2×C8D3×D911yescharacteristic
295C22⋊C4—542no—
296C42—31no—
297C22×C3—21no—
298C22×C3—361no—
299C22×C3—362no—
300C22×C3C2×D3611yescharacteristic
301C22×C3D3×D1211yescharacteristic
302C2×D3—122no—
303C2×D3—121no—
304C2×D3—362no—
305C2×D3—362no—
306C2×D3—362no—
307C2×D3—362no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—362no—
313C2×D3—362no—
314C2×D3—722no—
315C2×D3—722no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—62no—
319C3×C4—361no—
320C3×C4C2×D3611yescharacteristic
321C3×C4C2×D3611yescharacteristic
322C3×C4D3×D1211yescharacteristic
323C3×C4C2×#62611yescharacteristic
324Dic3—22no—
325Dic3—121no—
326C32#471211yescharacteristic
327C9—21no—
328C9#517611yescharacteristic, complemented
329C23—181no—
330C23—541no—
331C23—541no—
332Q8—181no—
333Q8—181no—
334D4—181no—
335D4—181no—
336D4—541no—
337D4—541no—
338D4—541no—
339D4—541no—
340D4—1084no—
341C2×C4—32no—
342C2×C4—181no—
343C2×C4C2×D3×D911yescharacteristic
344C8—32no—
345C8#619212yescomplemented
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—721no—
349C2×C3#6343611yescharacteristic
350C2×C3#6347012yes—
351C2×C3#6379412yes—
352C2×C3#6391111yescharacteristic
353D3—241no—
354D3—722no—
355D3—722no—
356D3—721no—
357D3—721no—
358D3—721no—
359D3—721no—
360C22—361no—
361C22—362no—
362C22—1081no—
363C22—1081no—
364C22—1082no—
365C22—1082no—
366C22D3×D3611yescharacteristic
367C4—62no—
368C4—361no—
369C4D3×D3611yescharacteristic
370C4C2×#619211yescharacteristic
371C3—21no—
372C3#1058839311yescharacteristic, complemented
373C3#1059088111yescharacteristic
374C2—721no—
375C2—2161no—
376C2—2161no—
377C2#1172080412yes—
378C2#1172100611yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#496
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
1. C2×#88525C2semidirect
2. C2×C3×#3664C2semidirect
3. #11721006C2semidirect
4. #11720952C2semidirect
5. C2×#88417C2semidirect
6. C2×#88416C2semidirect
7. C9×#815C2semidirect
34. C2×#3664D3semidirect
53. C3×C8×C9D4semidirect
54. C9×C3⋊C8D4semidirect
112. #815D9semidirect
154. C8×C9#67semidirect
227. C3×C9#496semidirect
267. C3×C8D36semidirect
269. C3⋊C8D36semidirect
328. C9#5176semidirect
345. C8#6192semidirect
372. C3#10588393semidirect

Maximal subgroups

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

subgroupindex
C2×#885252
C2×C3×#36642
#117210062
#117209522
C2×#884172
C2×#884162
C9×#8152
#105908813
#105883933