Finite Group Databasebeta

← #13137724

Subgroups — #13137724

Order 1728.

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

normal subgroups only

This group has 436 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#11721743C211yesmaximal, characteristic, complemented
2#11721025C211yesmaximal, characteristic
3C2×#88424C211yesmaximal, characteristic
4C9×#809C211yesmaximal, characteristic
5C3×#63367C211yesmaximal, characteristic, complemented
6#11720277C211yesmaximal, characteristic, complemented
7#11720271C211yesmaximal, characteristic, complemented
8#10590460—31nomaximal
9#10588438—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×D3×Dic9—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14#88534—21no—
15C4×C3⋊Dic9—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21#88424C414yescomplemented
22#64096—31no—
23#63898—31no—
24C2×#3716—31no—
25C3×#809D311yescharacteristic
26C3×#808—31no—
27#63567—31no—
28#63553—31no—
29D4×Dic9—31no—
30C2×#3672—31no—
31#63449—31no—
32#63449—61no—
33C9×#102—31no—
34#63376—31no—
35C8×Dic9—31no—
36#63367D311yescharacteristic, complemented
37C22×D3×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43#6226—21no—
44#6226C2×C412yescomplemented
45C2×C3×Dic9—21no—
46C9×D12—21no—
47C9×D12C2×C412yes—
48D3×Dic9—42no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8C2×C412yescomplemented
51#4852—91no—
52#4758—91no—
53C2×#634—31no—
54C2×C3×D12C2×D311yescharacteristic
55C2×D3×Dic3—61no—
56C32×C4⋊C4C2×D311yescharacteristic
57#3750—61no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3—61no—
60C3×C4⋊Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63(C2×D3)⋊Dic3—61no—
64#3716—34no—
65C2×D4×C9—31no—
66C22×Dic9—61no—
67C2×Dic18—31no—
68C2×Dic18—61no—
69C2×C8×C9—31no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4C2×D311yescharacteristic
72#3676—61no—
73#3672—34no—
74C4⋊Dic9C2×D311yescharacteristic
75Dic9⋊C4—61no—
76Dic9⋊C4—62no—
77C4×Dic9—31no—
78C4×Dic9—61no—
79C4×Dic9—61no—
80C4×Dic9—62no—
81C2×C9⋊C8C2×D311yescharacteristic
82C22×C3×C9C2×D411yescharacteristic
83C2×D3×C9—22no—
84C2×D3×C9—41no—
85C2×D3×C9—41no—
86C3×C4×C9—41no—
87C3×C4×C9C22×C411yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—22no—
90C3⋊Dic9—21no—
91C3⋊Dic9SD1612yescomplemented
92C3×Dic9—41no—
93D4×Dic3—91no—
94C2×#214—91no—
95C2×#204—91no—
96#893—91no—
97#887—91no—
98#887—91no—
99#887—181no—
100C3×#106—91no—
101C3×#102—31no—
102#816—91no—
103C8×Dic3—91no—
104C8×Dic3—91no—
105#810—91no—
106#809D911yescharacteristic
107#808—31no—
108C22×C3×D3—21no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—61no—
114#634—32no—
115#634—61no—
116C2×C3×Dic3—61no—
117C3×D12—21no—
118C3×D12C4×D312yes—
119D3×Dic3—122no—
120C23×C9—61no—
121C3×C3⋊C8—21no—
122C3×C3⋊C8—32no—
123D4×C9—32no—
124D4×C9—61no—
125C2×C4×C9—22no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9C22×D311yescharacteristic
129C2×Dic9—21no—
130C2×Dic9—31no—
131C2×Dic9—31no—
132C2×Dic9—61no—
133C2×Dic9—61no—
134C2×Dic9—62no—
135C2×Dic9—61no—
136C2×Dic9—122no—
137Dic18—32no—
138Dic18—62no—
139Dic18—62no—
140Dic18—61no—
141C8×C9—61no—
142C9⋊C8C4×D312yescomplemented
143C4×SD16—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9C2×SD1611yescharacteristic
146C2×C3×C9#13511yescharacteristic
147D3×C9—42no—
148C2×C3×D4—31no—
149C22×Dic3—181no—
150C2×D12C2×D911yescharacteristic
151C2×C4×D3—181no—
152C2×Dic6—91no—
153C2×Dic6—91no—
154C2×Dic6—181no—
155C2×C3×C8—31no—
156C2×C3×C8—91no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4C2×D911yescharacteristic
160C3×C4⋊C4D3211yescharacteristic
161#217—181no—
162#214—94no—
163(C2×D3)⋊C4—181no—
164C4⋊Dic3—31no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—182no—
168C4×Dic3—91no—
169C4×Dic3—91no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C4×Dic3—182no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8C2×D911yescharacteristic
176#204—94no—
177C22×C32D3×D411yescharacteristic
178C2×C3×D3—22no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C32×C4—41no—
182C32×C4C2×C4×D311yescharacteristic
183C32×C4#23711yescharacteristic
184C3⋊Dic3—32no—
185C3⋊Dic3—62no—
186C3⋊Dic3—61no—
187C3×Dic3—121no—
188C22×C9—21no—
189C22×C9—62no—
190C22×C9—121no—
191C22×C9—121no—
192C22×C9D3×D411yescharacteristic
193C4×C9—21no—
194C4×C9—21no—
195C4×C9—42no—
196C4×C9—41no—
197C4×C9C2×C4×D311yescharacteristic
198C4×C9#23911yescharacteristic
199Dic9—32no—
200Dic9—41no—
201Dic9—62no—
202Dic9—64no—
203Dic9—62no—
204Dic9—61no—
205Dic9—121no—
206C2×SD16—271no—
207C4×Q8—271no—
208C4×D4—271no—
209#106—91no—
210#103—271no—
211#102—31no—
212C4×C8—271no—
213C3×C9C4×SD1611yescharacteristic, complemented, Sylow
214C23×C3—61no—
215C22×D3—21no—
216C3×D4—32no—
217C3×D4—61no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—181no—
223C2×C3×C4C22×D911yescharacteristic
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—61no—
226C2×Dic3—91no—
227C2×Dic3—91no—
228C2×Dic3—91no—
229C2×Dic3—91no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232C2×Dic3—182no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—362no—
236D12—21no—
237D12C4×D912yes—
238C4×D3—362no—
239Dic6—92no—
240Dic6—92no—
241Dic6—182no—
242Dic6—182no—
243Dic6—181no—
244Dic6—181no—
245C3×C8—61no—
246C3×C8—92no—
247C3⋊C8—21no—
248C3⋊C8—32no—
249C2×C32#67⋊C411yescharacteristic
250C2×C32#91211yescharacteristic
251C2×C32D3×SD1611yescharacteristic
252C3×D3—42no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—122no—
257C2×C9#89311yescharacteristic
258C2×C9D3×SD1611yescharacteristic
259C2×C9#91611yescharacteristic
260C2×Q8—271no—
261C2×D4—31no—
262C22×C4—541no—
263SD16—274no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—91no—
267C4⋊C4—541no—
268C4⋊C4D3×D911yescharacteristic
269C22⋊C4—541no—
270C42—271no—
271C42—541no—
272C22×C3—21no—
273C22×C3—62no—
274C22×C3—121no—
275C22×C3—121no—
276C22×C3D4×D911yescharacteristic
277C22×C3#381111yescharacteristic
278C2×D3—22no—
279C2×D3—41no—
280C2×D3—41no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—42no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—361no—
287C3×C4C2×C4×D911yescharacteristic
288C3×C4#369911yescharacteristic
289C3×C4#379611yescharacteristic
290C3×C4C4×D3211yescharacteristic
291Dic3—92no—
292Dic3—92no—
293Dic3—121no—
294Dic3—182no—
295Dic3—184no—
296Dic3—182no—
297Dic3—181no—
298Dic3—182no—
299Dic3—181no—
300Dic3—361no—
301C32#475811yescharacteristic
302C9—21no—
303C9#485211yescharacteristic, complemented
304C23—61no—
305Q8—272no—
306Q8—541no—
307D4—32no—
308D4—61no—
309C2×C4—21no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4C2×D3×D911yescharacteristic
316C8—61no—
317C8—92no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—122no—
322C2×C3#611⋊C411yescharacteristic
323C2×C3#6347411yescharacteristic
324C2×C3SD16×D911yescharacteristic
325C2×C3#6388111yescharacteristic
326C2×C3#6393611yescharacteristic
327C2×C3#6394011yescharacteristic
328D3—42no—
329C22—62no—
330C22—121no—
331C22—121no—
332C22#8866811yescharacteristic
333C4—41no—
334C4—272no—
335C4—361no—
336C4—542no—
337C4—541no—
338C4#8863911yescharacteristic
339C4C4×D3×D911yescharacteristic
340C3—21no—
341C3#1058843811yescharacteristic, complemented
342C3#1059046011yescharacteristic
343C2—122no—
344C2#1172101411yescharacteristic
345C2#1172117811yescharacteristic
346C2#1172121211yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
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
1. #11721743C2semidirect
5. C3×#63367C2semidirect
6. #11720277C2semidirect
7. #11720271C2semidirect
21. #88424C4semidirect
36. #63367D3semidirect
44. #6226C2×C4semidirect
50. C3×C9⋊C8C2×C4semidirect
91. C3⋊Dic9SD16semidirect
142. C9⋊C8C4×D3semidirect
213. C3×C9C4×SD16semidirect
303. C9#4852semidirect
341. C3#10588438semidirect

Maximal subgroups

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

subgroupindex
#117217432
#117210252
C2×#884242
C9×#8092
C3×#633672
#117202772
#117202712
#105904603
#105884383