Finite Group Databasebeta

← #13144101

Subgroups — #13144101

Order 1728.

620 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88524C211yesmaximal, characteristic, complemented
2C2×C8×C3⋊D9C211yesmaximal, characteristic, complemented
3#11721753C211yesmaximal, characteristic, complemented
4#11721751C211yesmaximal, characteristic
5C3×C9×#106C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592563—31nomaximal
9#10588544—33nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C4⋊C4C2211yescharacteristic
17#88536—21no—
18#88536—21no—
19C4⋊(C3⋊Dic9)—21no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#88524—22no—
22C8×C3⋊D9—22no—
23C2×#3744—31no—
24C2×C8×C3⋊D3—31no—
25#64106—31no—
26#64104—31no—
27C32×#106D311yescharacteristic
28#63637—31no—
29#63636—31no—
30C2×#3664—33no—
31C2×C8×D9—33no—
32#63459—33no—
33#63457—33no—
34C9×#106D313yescomplemented
35#63369—33no—
36#63368—33no—
37C22×C3⋊D9—21no—
38C22×C3⋊D9D411yescharacteristic
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—42no—
47C4×C3⋊D9—21no—
48C4×C3⋊D9—21no—
49#6226—21no—
50#6226—21no—
51C3×C8×C9D412yescomplemented
52C3⋊(C9⋊C8)—21no—
53#4864—93no—
54#4864—91no—
55C2×C4⋊(C3⋊D3)—31no—
56C2×C4⋊(C3⋊D3)—61no—
57C2×C4×C3⋊D3—31no—
58C2×#634—31no—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62#3752—61no—
63#3752—61no—
64C4⋊(C3⋊Dic3)—61no—
65C2×C3⋊(C3⋊C8)—31no—
66#3744—62no—
67C8×C3⋊D3—62no—
68C2×D36—33no—
69C2×D36—63no—
70C2×C4×D9—33no—
71C2×Dic18—33no—
72C2×C8×C9C2×D313yes—
73C9×C4⋊C4C2×D313yes—
74C9×C4⋊C4C2×D313yes—
75(C2×D9)⋊C4—63no—
76(C2×D9)⋊C4—63no—
77C4⋊Dic9—63no—
78C2×C9⋊C8—33no—
79#3664—66no—
80C8×D9—66no—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—21no—
83C2×C3⋊D9—21no—
84C2×C3⋊D9—41no—
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C2×C3⋊D9SD1612yescomplemented
88C3×C4×C9—41no—
89C3×C4×C9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3⋊Dic9—21no—
93C3⋊Dic9—41no—
94C2×#204—93no—
95C2×#204—91no—
96C2×D3×C8—93no—
97C2×D3×C8—91no—
98#897—93no—
99#897—91no—
100#895—93no—
101#895—91no—
102C3×#106D913yescomplemented
103C3×#106C3⋊D311yescharacteristic
104#810—93no—
105#810—91no—
106#809—93no—
107#809—91no—
108C22×C3⋊D3—31no—
109C22×C3⋊D3—61no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—61no—
115C4⋊(C3⋊D3)—61no—
116C4⋊(C3⋊D3)—61no—
117C4⋊(C3⋊D3)—122no—
118C4×C3⋊D3—61no—
119C4×C3⋊D3—61no—
120#634—61no—
121#634—61no—
122C22×D9—33no—
123C22×D9—63no—
124C32×C8D1212yes—
125C3⋊(C3⋊C8)—61no—
126C2×C4×C9—23no—
127C2×C4×C9—23no—
128C2×C4×C9C22×D313yes—
129C2×Dic9—33no—
130C2×Dic9—63no—
131D36—63no—
132D36—63no—
133D36—126no—
134C4×D9—63no—
135C4×D9—63no—
136Dic18—63no—
137Dic18—63no—
138C8×C9D1216yescomplemented
139C9⋊C8—63no—
140#465—271noSylow
141C2×C3×C9#12111yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9#13511yescharacteristic
144C3⋊D9—22no—
145C3⋊D9—81no—
146C2×D12—93no—
147C2×D12—91no—
148C2×D12—181no—
149C2×D12—183no—
150C2×C4×D3—93no—
151C2×C4×D3—91no—
152C2×Dic6—93no—
153C2×Dic6—91no—
154C2×C3×C8C2×D913yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4C2×D913yes—
157C3×C4⋊C4C2×D913yes—
158C3×C4⋊C4C2×C3⋊D311yescharacteristic
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—183no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—183no—
164C4⋊Dic3—181no—
165C4⋊Dic3—183no—
166C2×C3⋊C8—93no—
167C2×C3⋊C8—91no—
168#204—186no—
169#204—182no—
170D3×C8—186no—
171D3×C8—182no—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—32no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—121no—
177C2×C3⋊D3—121no—
178C2×C3⋊D3—121no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4C2×D1211yescharacteristic
182C32×C4#23911yescharacteristic
183C3⋊Dic3—61no—
184C3⋊Dic3—121no—
185C22×C9D3×D413yes—
186C2×D9—36no—
187C2×D9—63no—
188C2×D9—63no—
189C2×D9—123no—
190C2×D9—123no—
191C2×D9—123no—
192C4×C9—43no—
193C4×C9—43no—
194C4×C9C2×D1213yes—
195C4×C9#23913yes—
196Dic9—63no—
197Dic9—123no—
198C2×SD16—271no—
199C22×C8—271no—
200C22⋊Q8—271no—
201#121—271no—
202#106C3⋊D911yescharacteristic, complemented
203#103—271no—
204#102—271no—
205C3×C9#46511yescharacteristic, complemented, Sylow
206C22×D3—93no—
207C22×D3—91no—
208C22×D3—183no—
209C22×D3—181no—
210C2×C3×C4—23no—
211C2×C3×C4—23no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4C22×D913yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×Dic3—93no—
217C2×Dic3—91no—
218C2×Dic3—183no—
219C2×Dic3—181no—
220D12—183no—
221D12—183no—
222D12—181no—
223D12—181no—
224D12—362no—
225D12—366no—
226C4×D3—183no—
227C4×D3—183no—
228C4×D3—181no—
229C4×D3—181no—
230Dic6—183no—
231Dic6—183no—
232Dic6—181no—
233Dic6—181no—
234C3×C8D3616yescomplemented
235C3×C8C4⋊(C3⋊D3)12yes—
236C3⋊C8—183no—
237C3⋊C8—181no—
238C2×C32#89511yescharacteristic
239C2×C32D3×SD1611yescharacteristic
240C2×C32#91611yescharacteristic
241C3⋊D3—62no—
242C3⋊D3—241no—
243C2×C9#89513yes—
244C2×C9D3×SD1613yes—
245C2×C9#91613yes—
246D9—66no—
247D9—243no—
248C2×Q8—271no—
249C2×D4—271no—
250C2×D4—541no—
251C22×C4—271no—
252SD16—542no—
253C2×C8—271no—
254C2×C8—542no—
255C2×C8C2×C3⋊D911yescharacteristic
256C4⋊C4—541no—
257C4⋊C4C2×C3⋊D911yescharacteristic
258C4⋊C4C2×C3⋊D911yescharacteristic
259C22⋊C4—541no—
260C22⋊C4—541no—
261C22×C3D4×D913yes—
262C22×C3D4×C3⋊D311yescharacteristic
263C2×D3—92no—
264C2×D3—96no—
265C2×D3—181no—
266C2×D3—181no—
267C2×D3—183no—
268C2×D3—183no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—361no—
272C2×D3—363no—
273C2×D3—363no—
274C2×D3—363no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—43no—
278C3×C4—43no—
279C3×C4C2×D3613yes—
280C3×C4#370113yes—
281C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
282C3×C4#383211yescharacteristic
283Dic3—181no—
284Dic3—183no—
285Dic3—361no—
286Dic3—363no—
287C32#486411yescharacteristic
288C9#486413yescomplemented
289C23—271no—
290C23—541no—
291Q8—541no—
292Q8—541no—
293D4—541no—
294D4—541no—
295D4—1082no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—271no—
299C2×C4—541no—
300C2×C4—541no—
301C2×C4—541no—
302C2×C4C22×C3⋊D911yescharacteristic
303C8—541no—
304C8C4⋊(C3⋊D9)12yescomplemented
305C2×C3#6345713yes—
306C2×C3SD16×D913yes—
307C2×C3#6347813yes—
308C2×C3#6410411yescharacteristic
309C2×C3SD16×C3⋊D311yescharacteristic
310C2×C3#6412511yescharacteristic
311D3—182no—
312D3—186no—
313D3—721no—
314D3—723no—
315C22—272no—
316C22—541no—
317C22—541no—
318C22—1081no—
319C22—1081no—
320C22—1081no—
321C22D4×C3⋊D911yescharacteristic
322C4—41no—
323C4—41no—
324C4—541no—
325C4—1081no—
326C4C2×C4⋊(C3⋊D9)11yescharacteristic
327C4#8874611yescharacteristic
328C3#1058854413yescomplemented
329C3#1059256311yescharacteristic
330C2—542no—
331C2—2161no—
332C2#1172175111yescharacteristic
333C2SD16×C3⋊D911yescharacteristic
334C2#1172179811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radical#11721751C2
hypercenter#106C3⋊D9
hypocenterC3×C9#465
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
1. C2×#88524C2semidirect
2. C2×C8×C3⋊D9C2semidirect
3. #11721753C2semidirect
5. C3×C9×#106C2semidirect
6. #11720528C2semidirect
7. #11720527C2semidirect
34. C9×#106D3semidirect
51. C3×C8×C9D4semidirect
87. C2×C3⋊D9SD16semidirect
102. C3×#106D9semidirect
138. C8×C9D12semidirect
202. #106C3⋊D9semidirect
205. C3×C9#465semidirect
234. C3×C8D36semidirect
288. C9#4864semidirect
304. C8C4⋊(C3⋊D9)semidirect
328. C3#10588544semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
C2×C8×C3⋊D92
#117217532
#117217512
C3×C9×#1062
#117205282
#117205272
#105925633
#105885443