Finite Group Databasebeta

← #10588468⋊C3 #13141961

Subgroups — #10588468⋊C3 #13141961

Order 1728.

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

normal subgroups only

This group has 387 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
1Q8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic
3#63455⋊C3C211yesmaximal, characteristic
43−1+2×#103C211yesmaximal, characteristic, complemented
5#63380⋊C3C211yesmaximal, characteristic
6C8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#4788—31nomaximal
9#10588468C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C4×AGL(1,ℤ/9)—21no—
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15#3675⋊C3C414yescomplemented
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)—21no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19C4×3−1+2⋊C4—21no—
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22C3×Q8×Dic3—31no—
23C2×C3×#215—31no—
24C3×#893—31no—
25C32×#103D311yescharacteristic
26C3×#821—31no—
27C3×C8×Dic3—31no—
28C3×#808—31no—
29Q8×Dic9C2×C311yescharacteristic, complemented
30C2×#3675C2×C311yescharacteristic
31#63455C2×C311yescharacteristic
32C9×#103—21no—
33C9×#103C2×C311yescharacteristic, complemented
34#63380C2×C311yescharacteristic
35C8×Dic9C2×C311yescharacteristic, complemented
36#63367C2×C311yescharacteristic, complemented
37C22×AGL(1,ℤ/9)—21no—
38Q8×3−1+2—21no—
39Q8×3−1+2C2×C412yescomplemented
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2C2311yescharacteristic
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4D411yescharacteristic
45C2×3−1+2⋊C4D411yescharacteristic
46C4⋊AGL(1,ℤ/9)—21no—
47C4⋊AGL(1,ℤ/9)C2×C412yes—
48C4×AGL(1,ℤ/9)—42no—
49C8×3−1+2—21no—
503−1+2⋊C8C2×C412yescomplemented
51C3×C4×SD16—91no—
52#4788—31no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C3×D12—31no—
55C2×C3×C4×D3—61no—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C3×#215—34no—
59C3×(C2×D3)⋊C4—61no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—61no—
62C3×C4×Dic3—31no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—31no—
65C2×Q8×C9—21no—
66C2×Q8×C9C22×C311yescharacteristic
67C2×D36C22×C311yescharacteristic
68C2×C4×D9—21no—
69C2×C8×C9—21no—
70C2×C8×C9C22×C311yescharacteristic
71C9×C4⋊C4—21no—
72C9×C4⋊C4C22×C311yescharacteristic
73#3675C3×C414yescomplemented
74(C2×D9)⋊C4—21no—
75C4⋊Dic9—21no—
76C4⋊Dic9C22×C311yescharacteristic
77C4×Dic9—21no—
78C4×Dic9C22×C311yescharacteristic
79C2×C9⋊C8C22×C311yescharacteristic
80C22×3−1+2C2×D411yescharacteristic
81C2×AGL(1,ℤ/9)—22no—
82C2×AGL(1,ℤ/9)—41no—
83C2×AGL(1,ℤ/9)—41no—
84C4×3−1+2—22no—
85C4×3−1+2—41no—
86C4×3−1+2C22×C411yescharacteristic
87C4×3−1+2#4111yescharacteristic
883−1+2⋊C4—21no—
893−1+2⋊C4—41no—
903−1+2⋊C4SD1612yescomplemented
91C2×C3×SD16—91no—
92C3×C4×Q8—91no—
93C3×C4×D4—91no—
94Q8×Dic3—31no—
95C2×#215—31no—
96#893—31no—
97C3×#106—91no—
98C3×#103—31no—
99C3×#103C3×D311yescharacteristic
100C3×#102—91no—
101C3×C4×C8—91no—
102#821—31no—
103C8×Dic3—31no—
104#808—31no—
105C22×C3×D3—61no—
106C32×Q8—21no—
107C32×Q8C4×D312yes—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3×Dic3—31no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—61no—
114C3×D12—32no—
115C3×D12—61no—
116C3×C4×D3—122no—
117C22×D9—21no—
118C32×C8—21no—
119C3×C3⋊C8—32no—
120Q8×C9—21no—
121Q8×C9—22no—
122Q8×C9—41no—
123Q8×C9C2×C3×C412yescomplemented
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—41no—
128C2×C4×C9—41no—
129C2×C4×C9C23×C311yescharacteristic
130C2×Dic9—21no—
131C2×Dic9C3×D411yescharacteristic
132C2×Dic9C3×D411yescharacteristic
133D36—21no—
134D36C2×C3×C412yes—
135C4×D9—42no—
136C8×C9—21no—
137C8×C9—41no—
138C9⋊C8C2×C3×C412yescomplemented
139C4×SD16—91noSylow
140C2×3−1+2C4×D411yescharacteristic
141C2×3−1+2C2×SD1611yescharacteristic
142C2×3−1+2#13511yescharacteristic
143AGL(1,ℤ/9)—42no—
144C2×C3×Q8—31no—
145C2×C3×Q8C2×C3×D311yescharacteristic
146C2×C3×D4—91no—
147C22×C3×C4—181no—
148C2×D12—31no—
149C2×C4×D3—61no—
150C3×SD16—94no—
151C2×C3×C8—31no—
152C2×C3×C8—91no—
153C2×C3×C8C2×C3×D311yescharacteristic
154C3×C4⋊C4—31no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—181no—
157C3×C4⋊C4C2×C3×D311yescharacteristic
158C3×C22⋊C4—181no—
159C3×C42—91no—
160C3×C42—181no—
161#215—34no—
162(C2×D3)⋊C4—61no—
163C4⋊Dic3—31no—
164C4⋊Dic3—61no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C2×C3⋊C8—31no—
168C22×C32D3×D411yescharacteristic
169C2×C3×D3—62no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C32×C4—22no—
173C32×C4—41no—
174C32×C4C2×C4×D311yescharacteristic
175C32×C4#23711yescharacteristic
176C3×Dic3—32no—
177C3×Dic3—61no—
178C3×Dic3—121no—
179C22×C9—21no—
180C22×C9C2×C3×D411yescharacteristic
181C2×D9—22no—
182C2×D9—41no—
183C2×D9—41no—
184C4×C9—22no—
185C4×C9—21no—
186C4×C9—21no—
187C4×C9—42no—
188C4×C9—41no—
189C4×C9—81no—
190C4×C9C22×C3×C411yescharacteristic
191C4×C9C3×#4111yescharacteristic
192Dic9—21no—
193Dic9—41no—
194Dic9C3×SD1612yescomplemented
195C2×SD16—91no—
196C4×Q8—91no—
197C4×D4—91no—
198#106—91no—
199#103AGL(1,ℤ/9)11yescharacteristic, complemented
200#102—91no—
201C4×C8—91no—
2023−1+2C4×SD1611yescharacteristic, complemented, Sylow
203C23×C3—181no—
204C22×D3—61no—
205C3×Q8—21no—
206C3×Q8—32no—
207C3×Q8—61no—
208C3×Q8C3×C4×D312yes—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—31no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—181no—
219C2×C3×C4—362no—
220C2×C3×C4C22×C3×D311yescharacteristic
221C2×Dic3—31no—
222C2×Dic3—31no—
223C2×Dic3—61no—
224D12—32no—
225D12—61no—
226C4×D3—122no—
227C3×C8—21no—
228C3×C8—61no—
229C3×C8—92no—
230C3⋊C8—32no—
231C2×C32#67⋊C411yescharacteristic
232C2×C32D3×SD1611yescharacteristic
233C2×C32#91911yescharacteristic
234C3×D3—122no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9C3×C4×D411yescharacteristic
239C2×C9C2×C3×SD1611yescharacteristic
240C2×C9C3×#13511yescharacteristic
241D9—42no—
242C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
243C2×D4—91no—
244C22×C4—181no—
245SD16—94no—
246C2×C8—91no—
247C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
248C4⋊C4—91no—
249C4⋊C4—181no—
250C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
251C22⋊C4—181no—
252C42—91no—
253C42—181no—
254C22×C3—31no—
255C22×C3—182no—
256C22×C3—361no—
257C22×C3—361no—
258C22×C3C3×D3×D411yescharacteristic
259C2×D3—62no—
260C2×D3—121no—
261C2×D3—121no—
262C3×C4—22no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—41no—
266C3×C4—62no—
267C3×C4—92no—
268C3×C4—121no—
269C3×C4—181no—
270C3×C4—361no—
271C3×C4C2×C3×C4×D311yescharacteristic
272C3×C4C3×#23711yescharacteristic
273Dic3—32no—
274Dic3—61no—
275Dic3—121no—
276C32#478811yescharacteristic
277C9—21no—
278C9C3×C4×SD1611yescharacteristic, complemented
279C23—181no—
280Q8—21no—
281Q8C4×AGL(1,ℤ/9)12yescomplemented
282D4—92no—
283D4—181no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—91no—
287C2×C4—91no—
288C2×C4—181no—
289C2×C4—362no—
290C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
291C8—21no—
292C8—92no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—362no—
297C2×C3C3×#67⋊C411yescharacteristic
298C2×C3C3×D3×SD1611yescharacteristic
299C2×C3C3×#91911yescharacteristic
300D3—122no—
301C22—182no—
302C22—361no—
303C22—361no—
304C22D4×AGL(1,ℤ/9)11yescharacteristic
305C4—22no—
306C4—41no—
307C4—92no—
308C4—181no—
309C4—361no—
310C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
311C4#3699⋊C311yescharacteristic
312C3—31no—
313C3C3×#478811yescharacteristic
314C2—362no—
315C2(#611⋊C3)⋊C411yescharacteristic
316C2SD16×AGL(1,ℤ/9)11yescharacteristic
317C2#63481⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
cosocleC2×C4×C9C23×C3
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×C4×SD16
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. Q8×3−1+2⋊C4C2semidirect
4. 3−1+2×#103C2semidirect
6. C8×3−1+2⋊C4C2semidirect
7. #63367⋊C3C2semidirect
9. #10588468C3semidirect
15. #3675⋊C3C4semidirect
29. Q8×Dic9C2×C3semidirect
33. C9×#103C2×C3semidirect
35. C8×Dic9C2×C3semidirect
36. #63367C2×C3semidirect
39. Q8×3−1+2C2×C4semidirect
50. 3−1+2⋊C8C2×C4semidirect
73. #3675C3×C4semidirect
90. 3−1+2⋊C4SD16semidirect
123. Q8×C9C2×C3×C4semidirect
138. C9⋊C8C2×C3×C4semidirect
194. Dic9C3×SD16semidirect
199. #103AGL(1,ℤ/9)semidirect
202. 3−1+2C4×SD16semidirect
278. C9C3×C4×SD16semidirect
281. Q8C4×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
Q8×3−1+2⋊C42
C2×#3675⋊C32
#63455⋊C32
3−1+2×#1032
#63380⋊C32
C8×3−1+2⋊C42
#63367⋊C32
C3×#47883
#105884683