Finite Group Databasebeta

← #10588436⋊C3 #13141835

Subgroups — #10588436⋊C3 #13141835

Order 1728.

427 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 116 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
1D4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
2C2×#3673⋊C3C211yesmaximal, characteristic
3#63455⋊C3C211yesmaximal, characteristic, complemented
43−1+2×#102C211yesmaximal, characteristic, complemented
5#63380⋊C3C211yesmaximal, characteristic, complemented
6C8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
7#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#4756—31nomaximal
9#10588436C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4—21no—
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13C2×C4×AGL(1,ℤ/9)—21no—
14C2×C8×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16#3676⋊C3—21no—
17#3673⋊C3C414yescomplemented
18(C2×AGL(1,ℤ/9))⋊C4—21no—
19C4⋊(3−1+2⋊C4)C2211yescharacteristic
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22C3×D4×Dic3—31no—
23C2×C3×#213—31no—
24C3×#893—31no—
25C32×#102D311yescharacteristic
26C3×#821—31no—
27C3×C8×Dic3—31no—
28C3×#807—31no—
29D4×Dic9C2×C311yescharacteristic, complemented
30C2×#3673C2×C311yescharacteristic
31#63455C2×C311yescharacteristic, complemented
32C9×#102—21no—
33C9×#102C2×C311yescharacteristic, complemented
34#63380C2×C311yescharacteristic, complemented
35C8×Dic9C2×C311yescharacteristic, complemented
36#63366C2×C311yescharacteristic, complemented
37C23×3−1+2—21no—
38C22×AGL(1,ℤ/9)—21no—
39D4×3−1+2—21no—
40D4×3−1+2C2×C412yescomplemented
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2C2311yescharacteristic
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—42no—
45C2×3−1+2⋊C4D411yescharacteristic
46C2×3−1+2⋊C4D411yescharacteristic
47C4⋊AGL(1,ℤ/9)—21no—
48C4⋊AGL(1,ℤ/9)C2×C412yescomplemented
49C4×AGL(1,ℤ/9)—42no—
50C8×3−1+2—21no—
513−1+2⋊C8C2×C412yescomplemented
52C3×C4×D8—91no—
53#4756—31no—
54C2×C32×D4C2×D311yescharacteristic
55C22×C3×Dic3—61no—
56C2×C3×D12—31no—
57C2×C3×C4×D3—61no—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C3×#217—61no—
61C3×#213—34no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C2×D4×C9—21no—
67C2×D4×C9C22×C311yescharacteristic
68C22×Dic9—21no—
69C2×D36C22×C311yescharacteristic
70C2×C4×D9—21no—
71C2×C8×C9—21no—
72C2×C8×C9C22×C311yescharacteristic
73C9×C4⋊C4—21no—
74C9×C4⋊C4C22×C311yescharacteristic
75#3676—21no—
76#3673C3×C414yescomplemented
77(C2×D9)⋊C4—21no—
78C4⋊Dic9C22×C311yescharacteristic
79C4×Dic9C22×C311yescharacteristic
80C2×C9⋊C8C22×C311yescharacteristic
81C22×3−1+2—22no—
82C22×3−1+2—41no—
83C22×3−1+2—41no—
84C22×3−1+2C2×D411yescharacteristic
85C2×AGL(1,ℤ/9)—22no—
86C2×AGL(1,ℤ/9)—41no—
87C2×AGL(1,ℤ/9)—41no—
88C4×3−1+2—41no—
89C4×3−1+2C22×C411yescharacteristic
90C4×3−1+2#4111yescharacteristic
913−1+2⋊C4—21no—
923−1+2⋊C4—41no—
933−1+2⋊C4D812yescomplemented
94C2×C3×D8—91no—
95C3×C4×D4—91no—
96C3×C4×D4—91no—
97D4×Dic3—31no—
98C2×#213—31no—
99#893—31no—
100C3×#107—91no—
101C3×#102—31no—
102C3×#102—91no—
103C3×#102C3×D311yescharacteristic
104C3×C4×C8—91no—
105#821—31no—
106C8×Dic3—31no—
107#807—31no—
108C23×C32—21no—
109C22×C3×D3—61no—
110C32×D4—21no—
111C32×D4C4×D312yes—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—122no—
118C3×D12—32no—
119C3×D12—61no—
120C3×C4×D3—122no—
121C23×C9—21no—
122C23×C9—41no—
123C22×D9—21no—
124C32×C8—21no—
125C3×C3⋊C8—32no—
126D4×C9—21no—
127D4×C9—22no—
128D4×C9—41no—
129D4×C9C2×C3×C412yescomplemented
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—41no—
133C2×C4×C9C23×C311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—42no—
136C2×Dic9C3×D411yescharacteristic
137C2×Dic9C3×D411yescharacteristic
138D36—21no—
139D36C2×C3×C412yescomplemented
140C4×D9—42no—
141C8×C9—21no—
142C8×C9—41no—
143C9⋊C8C2×C3×C412yescomplemented
144C4×D8—91noSylow
145C2×3−1+2—42no—
146C2×3−1+2C4×D411yescharacteristic
147C2×3−1+2C2×D811yescharacteristic
148C2×3−1+2#13511yescharacteristic
149AGL(1,ℤ/9)—42no—
150C2×C3×D4—31no—
151C2×C3×D4—91no—
152C2×C3×D4C2×C3×D311yescharacteristic
153C22×C3×C4—181no—
154C22×C3×C4—181no—
155C22×Dic3—61no—
156C2×D12—31no—
157C2×C4×D3—61no—
158C3×D8—94no—
159C2×C3×C8—31no—
160C2×C3×C8—91no—
161C2×C3×C8C2×C3×D311yescharacteristic
162C3×C4⋊C4—31no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4C2×C3×D311yescharacteristic
165C3×C22⋊C4—181no—
166C3×C22⋊C4—181no—
167C3×C42—91no—
168#217—61no—
169#213—34no—
170(C2×D3)⋊C4—61no—
171C4⋊Dic3—31no—
172C4×Dic3—31no—
173C2×C3⋊C8—31no—
174C22×C32—22no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32D3×D411yescharacteristic
178C2×C3×D3—62no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C32×C4—41no—
182C32×C4C2×C4×D311yescharacteristic
183C32×C4#23711yescharacteristic
184C3×Dic3—32no—
185C3×Dic3—61no—
186C3×Dic3—121no—
187C22×C9—22no—
188C22×C9—21no—
189C22×C9—42no—
190C22×C9—41no—
191C22×C9—41no—
192C22×C9—81no—
193C22×C9—81no—
194C22×C9C2×C3×D411yescharacteristic
195C2×D9—22no—
196C2×D9—41no—
197C2×D9—41no—
198C4×C9—21no—
199C4×C9—21no—
200C4×C9—41no—
201C4×C9—81no—
202C4×C9C22×C3×C411yescharacteristic
203C4×C9C3×#4111yescharacteristic
204Dic9—21no—
205Dic9—41no—
206Dic9C3×D812yescomplemented
207C2×D8—91no—
208C4×D4—91no—
209C4×D4—91no—
210#107—91no—
211#102—91no—
212#102AGL(1,ℤ/9)11yescharacteristic, complemented
213C4×C8—91no—
2143−1+2C4×D811yescharacteristic, complemented, Sylow
215C23×C3—21no—
216C23×C3—61no—
217C23×C3—181no—
218C22×D3—61no—
219C3×D4—21no—
220C3×D4—32no—
221C3×D4—61no—
222C3×D4—92no—
223C3×D4—181no—
224C3×D4C3×C4×D312yes—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—181no—
231C2×C3×C4—362no—
232C2×C3×C4—362no—
233C2×C3×C4C22×C3×D311yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—61no—
237C2×Dic3—122no—
238D12—32no—
239D12—61no—
240C4×D3—122no—
241C3×C8—21no—
242C3×C8—61no—
243C3×C8—92no—
244C3⋊C8—32no—
245C2×C32—42no—
246C2×C32#67⋊C411yescharacteristic
247C2×C32D3×D811yescharacteristic
248C2×C32#91611yescharacteristic
249C3×D3—122no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9—42no—
254C2×C9—82no—
255C2×C9C3×C4×D411yescharacteristic
256C2×C9C2×C3×D811yescharacteristic
257C2×C9C3×#13511yescharacteristic
258D9—42no—
259C2×D4—91no—
260C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
261C22×C4—181no—
262C22×C4—181no—
263D8—94no—
264C2×C8—91no—
265C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
266C4⋊C4—91no—
267C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
268C22⋊C4—181no—
269C22⋊C4—181no—
270C42—91no—
271C22×C3—22no—
272C22×C3—31no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—62no—
276C22×C3—121no—
277C22×C3—121no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3C3×D3×D411yescharacteristic
282C2×D3—62no—
283C2×D3—121no—
284C2×D3—121no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—41no—
288C3×C4—92no—
289C3×C4—121no—
290C3×C4—181no—
291C3×C4—361no—
292C3×C4C2×C3×C4×D311yescharacteristic
293C3×C4C3×#23711yescharacteristic
294Dic3—32no—
295Dic3—61no—
296Dic3—121no—
297C32#475611yescharacteristic
298C9—21no—
299C9C3×C4×D811yescharacteristic, complemented
300C23—21no—
301C23—181no—
302D4—21no—
303D4—92no—
304D4—181no—
305D4C4×AGL(1,ℤ/9)12yescomplemented
306C2×C4—21no—
307C2×C4—91no—
308C2×C4—91no—
309C2×C4—181no—
310C2×C4—362no—
311C2×C4—362no—
312C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
313C8—21no—
314C8—92no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—42no—
319C2×C3—122no—
320C2×C3—362no—
321C2×C3C3×#67⋊C411yescharacteristic
322C2×C3C3×D3×D811yescharacteristic
323C2×C3C3×#91611yescharacteristic
324D3—122no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—182no—
329C22—361no—
330C22—361no—
331C22D4×AGL(1,ℤ/9)11yescharacteristic
332C4—41no—
333C4—92no—
334C4—181no—
335C4—361no—
336C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
337C4#3699⋊C311yescharacteristic
338C3—31no—
339C3C3×#475611yescharacteristic
340C2—42no—
341C2—362no—
342C2(#611⋊C3)⋊C411yescharacteristic
343C2D8×AGL(1,ℤ/9)11yescharacteristic
344C2#63478⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×#3673⋊C3C2
cosocleC2×C4×C9C23×C3
hypercenter#102AGL(1,ℤ/9)
hypocenterC9C3×C4×D8
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. D4×3−1+2⋊C4C2semidirect
3. #63455⋊C3C2semidirect
4. 3−1+2×#102C2semidirect
5. #63380⋊C3C2semidirect
6. C8×3−1+2⋊C4C2semidirect
7. #63366⋊C3C2semidirect
9. #10588436C3semidirect
17. #3673⋊C3C4semidirect
29. D4×Dic9C2×C3semidirect
31. #63455C2×C3semidirect
33. C9×#102C2×C3semidirect
34. #63380C2×C3semidirect
35. C8×Dic9C2×C3semidirect
36. #63366C2×C3semidirect
40. D4×3−1+2C2×C4semidirect
48. C4⋊AGL(1,ℤ/9)C2×C4semidirect
51. 3−1+2⋊C8C2×C4semidirect
76. #3673C3×C4semidirect
93. 3−1+2⋊C4D8semidirect
129. D4×C9C2×C3×C4semidirect
139. D36C2×C3×C4semidirect
143. C9⋊C8C2×C3×C4semidirect
206. Dic9C3×D8semidirect
212. #102AGL(1,ℤ/9)semidirect
214. 3−1+2C4×D8semidirect
299. C9C3×C4×D8semidirect
305. D4C4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
D4×3−1+2⋊C42
C2×#3673⋊C32
#63455⋊C32
3−1+2×#1022
#63380⋊C32
C8×3−1+2⋊C42
#63366⋊C32
C3×#47563
#105884363