Finite Group Databasebeta

← #13140672

Subgroups — #13140672

Order 1728.

434 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88548C211yesmaximal, characteristic
2Q8×C9×Dic3C211yesmaximal, characteristic, complemented
3#11720993C211yesmaximal, characteristic
4C3×#63395C211yesmaximal, characteristic, complemented
5#11720267C211yesmaximal, characteristic
6#11720265C211yesmaximal, characteristic, complemented
7Dic3×C9⋊C8C211yesmaximal, characteristic, complemented
8#10591536—31nomaximal
9C4×#3675—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×(C3⋊D9)⋊C4—21no—
13#88548C414yescomplemented
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C4×C9×Dic3—21no—
18C4×C9×Dic3C2211yescharacteristic
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88445—21no—
22C2×#3755—31no—
23C3×Q8×Dic3D311yescharacteristic
24#63894—31no—
25C3×#835—31no—
26#63574—31no—
27#63564—31no—
28Dic3×C3⋊C8—31no—
29C4×Q8×C9—31no—
30C2×#3675—31no—
31C2×#3675—61no—
32C4×D36—31no—
33#63395D311yescharacteristic, complemented
34#63369—31no—
35#63367—31no—
36C4×C9⋊C8—31no—
37C22×C3⋊D9—21no—
38C3×Q8×C9—21no—
39C3×Q8×C9C2×C412yescomplemented
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)C2×C412yes—
44C2×C9×Dic3—21no—
45C2×C9×Dic3D411yescharacteristic
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48(C3⋊D9)⋊C4—42no—
49C3⋊(C9⋊C8)C2×C412yescomplemented
50C3×C9⋊C8—21no—
51C4×#215—91no—
52#4788—91no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C4⋊(C3⋊D3)—31no—
55C2×#624—61no—
56#3755—34no—
57C2×C3⋊(C3⋊C8)—31no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×C4×Dic3—21no—
62C3×C4×Dic3C2×D311yescharacteristic
63C2×C3×C3⋊C8—31no—
64#3722—61no—
65C2×Q8×C9—21no—
66C2×Q8×C9C2×D311yescharacteristic
67C2×D36—31no—
68C2×D36—61no—
69C2×C4×D9—61no—
70C9×C4⋊C4—31no—
71C9×C4⋊C4—61no—
72C42×C9—31no—
73C42×C9—61no—
74#3675—34no—
75#3675—64no—
76(C2×D9)⋊C4—61no—
77C4⋊Dic9C2×D311yescharacteristic
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80C2×C9⋊C8C2×D311yescharacteristic
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C3×C4×C9—22no—
86C3×C4×C9C22×C411yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C9×Dic3—21no—
89C9×Dic3—41no—
90C9×Dic3SD1612yescomplemented
91C3×Dic9—41no—
92C3×C4×Q8—31no—
93Q8×Dic3D911yescharacteristic, complemented
94C2×#215—91no—
95C2×#215—91no—
96C2×#215—181no—
97#893—91no—
98C4×D12—91no—
99C3×#103—91no—
100#835—31no—
101#821—91no—
102C8×Dic3—91no—
103#809—91no—
104#808—91no—
105#808—91no—
106C4×C3⋊C8—91no—
107C22×C3⋊D3—61no—
108C32×Q8—21no—
109C32×Q8C4×D312yes—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C4⋊(C3⋊D3)—32no—
113C4⋊(C3⋊D3)—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3#6711yescharacteristic
117C2×C3×Dic3#6711yescharacteristic
118#624—122no—
119C22×D9—61no—
120C22×D9—121no—
121C3⋊(C3⋊C8)—32no—
122C3×C3⋊C8—61no—
123Q8×C9—21no—
124Q8×C9—22no—
125Q8×C9—41no—
126Q8×C9C4×D312yescomplemented
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—31no—
130C2×C4×C9—31no—
131C2×C4×C9—41no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135D36—32no—
136D36—62no—
137D36—61no—
138D36—121no—
139C4×D9—122no—
140C9⋊C8—21no—
141C9⋊C8—32no—
142C9⋊C8—62no—
143C4×SD16—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9C2×SD1611yescharacteristic
146C2×C3×C9#13511yescharacteristic
147C3⋊D9—42no—
148C2×C3×Q8—21no—
149C2×C3×Q8C2×D911yescharacteristic
150C2×C3×Q8D3211yescharacteristic
151C2×D12—91no—
152C2×D12—91no—
153C2×D12—181no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C2×C3×C8—91no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—91no—
160C3×C42—31no—
161C3×C42—61no—
162#215—94no—
163#215—94no—
164#215—184no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—21no—
168C4⋊Dic3—31no—
169C4⋊Dic3C2×D911yescharacteristic
170C4×Dic3—21no—
171C4×Dic3C2×D911yescharacteristic
172C2×C3⋊C8—31no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C22×C32C2×#6711yescharacteristic
177C2×C3⋊D3—62no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C32×C4—22no—
181C32×C4C2×C4×D311yescharacteristic
182C32×C4#23511yescharacteristic
183C3×Dic3—21no—
184C3×Dic3—41no—
185C3×Dic3—121no—
186C3×Dic3#21512yes—
187C22×C9—21no—
188C22×C9D3×D411yescharacteristic
189C2×D9—62no—
190C2×D9—122no—
191C2×D9—121no—
192C2×D9—121no—
193C2×D9—241no—
194C2×D9—241no—
195C4×C9—22no—
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—32no—
199C4×C9—42no—
200C4×C9—61no—
201C4×C9—121no—
202C4×C9C2×C4×D311yescharacteristic
203C4×C9#23711yescharacteristic
204Dic9—41no—
205C2×SD16—271no—
206C4×Q8—31no—
207C4×D4—271no—
208#106—271no—
209#103—91no—
210#102—271no—
211C4×C8—271no—
212C3×C9C4×SD1611yescharacteristic, complemented, Sylow
213C22×D3—181no—
214C22×D3—181no—
215C22×D3—361no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—22no—
219C3×Q8—41no—
220C3×Q8C4×D912yescomplemented
221C3×Q8#62412yes—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—41no—
228C2×C3×C4—61no—
229C2×C3×C4—181no—
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—21no—
233C2×Dic3—61no—
234C2×Dic3#61111yescharacteristic
235C2×Dic3#61111yescharacteristic
236D12—92no—
237D12—92no—
238D12—182no—
239D12—181no—
240D12—181no—
241D12—361no—
242C4×D3—362no—
243C4×D3—362no—
244C3×C8—181no—
245C3⋊C8—61no—
246C3⋊C8—92no—
247C3⋊C8—92no—
248C3⋊C8—182no—
249C2×C32C4×#6711yescharacteristic
250C2×C32C2×#21511yescharacteristic
251C2×C32#95011yescharacteristic
252C3⋊D3—122no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9#67⋊C411yescharacteristic
257C2×C9D3×SD1611yescharacteristic
258C2×C9#91911yescharacteristic
259D9—122no—
260D9—242no—
261C2×Q8D3×D911yescharacteristic
262C2×D4—271no—
263C22×C4—541no—
264SD16—274no—
265C2×C8—91no—
266C2×C8—271no—
267C4⋊C4—31no—
268C4⋊C4—61no—
269C4⋊C4—91no—
270C22⋊C4—541no—
271C42—31no—
272C42—61no—
273C22×C3—21no—
274C22×C3C2×#61111yescharacteristic
275C22×C3D3×#6711yescharacteristic
276C2×D3—182no—
277C2×D3—182no—
278C2×D3—362no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—721no—
284C2×D3—721no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—32no—
290C3×C4—42no—
291C3×C4—61no—
292C3×C4—121no—
293C3×C4—361no—
294C3×C4C2×C4×D911yescharacteristic
295C3×C4#369711yescharacteristic
296C3×C4#380411yescharacteristic
297C3×C4C2×#62411yescharacteristic
298Dic3—21no—
299Dic3—41no—
300Dic3—121no—
301Dic3#367512yescomplemented
302C32C4×#21511yescharacteristic
303C9—21no—
304C9#478811yescharacteristic, complemented
305C23—541no—
306Q8—21no—
307Q8(C3⋊D9)⋊C412yescomplemented
308D4—272no—
309D4—541no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—181no—
315C2×C4—1082no—
316C2×C4C2×D3×D911yescharacteristic
317C8—181no—
318C8—272no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3C4×#61111yescharacteristic
323C2×C3C2×#367511yescharacteristic
324C2×C3#6351311yescharacteristic
325C2×C3D3×#21511yescharacteristic
326C2×C3#6394911yescharacteristic
327C2×C3#6395211yescharacteristic
328D3—362no—
329D3—362no—
330D3—722no—
331C22—542no—
332C22—1081no—
333C22—1081no—
334C22D3×#61111yescharacteristic
335C4—22no—
336C4—32no—
337C4—61no—
338C4—121no—
339C4—361no—
340C4#8865811yescharacteristic
341C4C2×(C3⋊D9)⋊C411yescharacteristic
342C3—21no—
343C3C4×#367511yescharacteristic, complemented
344C3#1059153611yescharacteristic
345C2—1082no—
346C2D3×#367511yescharacteristic
347C2#1172120911yescharacteristic
348C2#1172124011yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C2×#67
prime radicalC2×C4⋊(C3⋊D9)C22
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9C4×SD16
LCSG ▷ 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
2. Q8×C9×Dic3C2semidirect
4. C3×#63395C2semidirect
6. #11720265C2semidirect
7. Dic3×C9⋊C8C2semidirect
13. #88548C4semidirect
33. #63395D3semidirect
39. C3×Q8×C9C2×C4semidirect
49. C3⋊(C9⋊C8)C2×C4semidirect
90. C9×Dic3SD16semidirect
93. Q8×Dic3D9semidirect
126. Q8×C9C4×D3semidirect
212. C3×C9C4×SD16semidirect
220. C3×Q8C4×D9semidirect
301. Dic3#3675semidirect
304. C9#4788semidirect
307. Q8(C3⋊D9)⋊C4semidirect
343. C3C4×#3675semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
Q8×C9×Dic32
#117209932
C3×#633952
#117202672
#117202652
Dic3×C9⋊C82
#105915363
C4×#36753