Finite Group Databasebeta

← #13137652

Subgroups — #13137652

Order 1728.

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

normal subgroups only

This group has 460 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
1#11721749C211yesmaximal, characteristic
2#11720910C211yesmaximal, characteristic, complemented
3C2×#88416C211yesmaximal, characteristic
4C3×#63368C211yesmaximal, characteristic, complemented
5C9×#808C211yesmaximal, characteristic, complemented
6#11720271C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic
8#10590483—31nomaximal
9#10588532—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14#88536—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#88416C414yescomplemented
22#64102—31no—
23#63843—31no—
24C2×#3717—31no—
25C3×#810—31no—
26C3×#808D311yescharacteristic
27#63564—31no—
28#63553—31no—
29C2×#3664—31no—
30#63455—31no—
31#63455—61no—
32#63449—31no—
33C9×#106—31no—
34C8×Dic9—31no—
35#63369—31no—
36#63368D311yescharacteristic, complemented
37C22×C3⋊D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)C2×C412yes—
44C4×C3⋊D9—42no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C3×Dic18—21no—
48C3×Dic18C2×C412yescomplemented
49C9×C3⋊C8C2×C412yescomplemented
50C3×C9⋊C8—21no—
51#4852—91no—
52#4788—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4×C3⋊D3—61no—
55C2×C3×Dic6—31no—
56C32×C4⋊C4C2×D311yescharacteristic
57#3752—61no—
58C4×C3⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63Dic32—61no—
64#3717—34no—
65C2×D36—31no—
66C2×D36—61no—
67C2×C4×D9—61no—
68C2×C4×D9—62no—
69C2×Dic18C2×D311yescharacteristic
70C2×C8×C9—31no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4—31no—
73C9×C4⋊C4C2×D311yescharacteristic
74(C2×D9)⋊C4—61no—
75(C2×D9)⋊C4—62no—
76Dic9⋊C4—61no—
77C4×Dic9—31no—
78C4×Dic9—61no—
79C4×Dic9—61no—
80C2×C9⋊C8C2×D311yescharacteristic
81#3664—34no—
82C22×C3×C9C2×D411yescharacteristic
83C2×C3⋊D9—22no—
84C2×C3⋊D9—41no—
85C2×C3⋊D9—41no—
86C3×C4×C9—41no—
87C3×C4×C9C22×C411yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C3⋊Dic9SD1612yescomplemented
91C9×Dic3—41no—
92C3×Dic9—22no—
93Q8×Dic3—91no—
94C2×#215—91no—
95C2×#204—91no—
96#893—91no—
97#893—91no—
98#893—181no—
99#887—91no—
100C3×#106—31no—
101C3×#103—91no—
102#821—91no—
103C8×Dic3—91no—
104C8×Dic3—91no—
105#810—31no—
106#809—91no—
107#808D911yescharacteristic, complemented
108C22×C3⋊D3—61no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—31no—
113C4⋊(C3⋊D3)—32no—
114C4⋊(C3⋊D3)—61no—
115C4×C3⋊D3—122no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—61no—
118C3×Dic6—32no—
119C3×Dic6—61no—
120C22×D9—62no—
121C22×D9—61no—
122C3×C3⋊C8—61no—
123C3×C3⋊C8C4×D312yes—
124C2×C4×C9—22no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—61no—
128C2×C4×C9C22×D311yescharacteristic
129C2×Dic9—21no—
130C2×Dic9—31no—
131C2×Dic9—31no—
132C2×Dic9—61no—
133C2×Dic9—61no—
134D36—32no—
135D36—62no—
136D36—62no—
137D36—61no—
138C4×D9—122no—
139C4×D9—124no—
140Dic18—21no—
141Dic18C4×D312yescomplemented
142C8×C9—32no—
143C9⋊C8—21no—
144C4×SD16—271noSylow
145C2×C3×C9C4×D411yescharacteristic
146C2×C3×C9C2×SD1611yescharacteristic
147C2×C3×C9#13511yescharacteristic
148C3⋊D9—42no—
149C2×C3×Q8—91no—
150C2×D12—91no—
151C2×D12—91no—
152C2×D12—181no—
153C2×C4×D3—181no—
154C2×C4×D3—181no—
155C2×C4×D3—182no—
156C2×Dic6—31no—
157C2×C3×C8—31no—
158C2×C3×C8—91no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4C2×D911yescharacteristic
162C3×C4⋊C4D3211yescharacteristic
163#215—94no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—182no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×D911yescharacteristic
169Dic3⋊C4—181no—
170C4×Dic3—91no—
171C4×Dic3—91no—
172C4×Dic3—181no—
173C4×Dic3—181no—
174C4×Dic3—181no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8C2×D911yescharacteristic
177#204—94no—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—62no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—121no—
182C32×C4—41no—
183C32×C4C2×C4×D311yescharacteristic
184C32×C4#23911yescharacteristic
185C3⋊Dic3—32no—
186C3⋊Dic3—61no—
187C3×Dic3—41no—
188C3×Dic3—62no—
189C22×C9—21no—
190C22×C9D3×D411yescharacteristic
191C2×D9—64no—
192C2×D9—62no—
193C2×D9—122no—
194C2×D9—122no—
195C2×D9—121no—
196C2×D9—121no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—42no—
200C4×C9—41no—
201C4×C9—121no—
202C4×C9C2×C4×D311yescharacteristic
203C4×C9#23711yescharacteristic
204Dic9—22no—
205Dic9—32no—
206Dic9—62no—
207Dic9—61no—
208Dic9—121no—
209C2×SD16—271no—
210C4×Q8—271no—
211C4×D4—271no—
212#106—31no—
213#103—91no—
214#102—271no—
215C4×C8—271no—
216C3×C9C4×SD1611yescharacteristic, complemented, Sylow
217C22×D3—182no—
218C22×D3—181no—
219C22×D3—181no—
220C3×Q8—92no—
221C3×Q8—181no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—61no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—61no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238D12—92no—
239D12—92no—
240D12—182no—
241D12—182no—
242D12—181no—
243D12—181no—
244C4×D3—362no—
245C4×D3—362no—
246C4×D3—364no—
247Dic6—32no—
248Dic6—61no—
249C3×C8—32no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8C4×D912yescomplemented
253C2×C32#89311yescharacteristic
254C2×C32D3×SD1611yescharacteristic
255C2×C32#91611yescharacteristic
256C3⋊D3—122no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9#67⋊C411yescharacteristic
261C2×C9D3×SD1611yescharacteristic
262C2×C9#91911yescharacteristic
263D9—124no—
264D9—122no—
265C2×Q8—91no—
266C2×D4—271no—
267C22×C4—541no—
268SD16—274no—
269C2×C8—31no—
270C2×C8—91no—
271C4⋊C4—31no—
272C4⋊C4—541no—
273C4⋊C4D3×D911yescharacteristic
274C22⋊C4—541no—
275C42—271no—
276C42—541no—
277C22×C3—21no—
278C22×C3D4×D911yescharacteristic
279C22×C3#381111yescharacteristic
280C2×D3—184no—
281C2×D3—182no—
282C2×D3—182no—
283C2×D3—362no—
284C2×D3—362no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—361no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—42no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—121no—
295C3×C4—182no—
296C3×C4C2×C4×D911yescharacteristic
297C3×C4#370111yescharacteristic
298C3×C4#379611yescharacteristic
299C3×C4C4×D3211yescharacteristic
300Dic3—41no—
301Dic3—62no—
302Dic3—92no—
303Dic3—92no—
304Dic3—182no—
305Dic3—181no—
306Dic3—181no—
307Dic3—361no—
308C32#485211yescharacteristic
309C9—21no—
310C9#478811yescharacteristic, complemented
311C23—541no—
312Q8—92no—
313Q8—181no—
314D4—272no—
315D4—541no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—181no—
319C2×C4—271no—
320C2×C4—271no—
321C2×C4—1082no—
322C2×C4C2×D3×D911yescharacteristic
323C8—32no—
324C8—181no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#6345511yescharacteristic
329C2×C3SD16×D911yescharacteristic
330C2×C3#6347811yescharacteristic
331C2×C3#6388111yescharacteristic
332C2×C3#6393011yescharacteristic
333C2×C3#6395011yescharacteristic
334D3—364no—
335D3—362no—
336D3—362no—
337C22—542no—
338C22—1081no—
339C22—1081no—
340C22#8866811yescharacteristic
341C4—41no—
342C4—121no—
343C4—182no—
344C4—272no—
345C4—541no—
346C4#8863411yescharacteristic
347C4C4×D3×D911yescharacteristic
348C3—21no—
349C3#1058853211yescharacteristic, complemented
350C3#1059048311yescharacteristic
351C2—1082no—
352C2#1172097811yescharacteristic
353C2#1172118211yescharacteristic
354C2#1172121611yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×SD16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11720910C2semidirect
4. C3×#63368C2semidirect
5. C9×#808C2semidirect
6. #11720271C2semidirect
21. #88416C4semidirect
36. #63368D3semidirect
48. C3×Dic18C2×C4semidirect
49. C9×C3⋊C8C2×C4semidirect
90. C3⋊Dic9SD16semidirect
107. #808D9semidirect
141. Dic18C4×D3semidirect
216. C3×C9C4×SD16semidirect
252. C3⋊C8C4×D9semidirect
310. C9#4788semidirect
349. C3#10588532semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117209102
C2×#884162
C3×#633682
C9×#8082
#117202712
#117202672
#105904833
#105885323