Finite Group Databasebeta

← #10588908⋊C3 #13143608

Subgroups — #10588908⋊C3 #13143608

Order 1728.

552 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C22×Q8×3−1+2C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C212yesmaximal, complemented
3#63492⋊C3C211yesmaximal, characteristic
4#63395⋊C3C212yesmaximal, complemented
5#63389⋊C3C211yesmaximal, characteristic, complemented
6C3×#5227—31nomaximal
7#10588908C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2—24no—
9C2×Q8×3−1+2—21no—
10C2×Q8×3−1+2C2212yes—
11C22×C4×3−1+2—21no—
12C22×C4×3−1+2C2211yescharacteristic
13C2×#611⋊C3—21no—
14C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
15#3675⋊C3—24no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18C2×3−1+2⋊C8C2212yes—
19C22×C32×Q8D311yescharacteristic
20C2×C3×#215—32no—
21C3×#930—31no—
22C22×Q8×C9—21no—
23C22×Q8×C9C2×C311yescharacteristic, complemented
24C3×#835—32no—
25C3×#830—31no—
26C2×#3675C2×C312yescomplemented
27#63492C2×C311yescharacteristic
28#63395C2×C312yescomplemented
29#63389C2×C311yescharacteristic, complemented
30C23×3−1+2D411yescharacteristic
31C22×AGL(1,ℤ/9)—21no—
32Q8×3−1+2—24no—
33Q8×3−1+2—22no—
34Q8×3−1+2D414yescomplemented
35C2×C4×3−1+2—24no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2—22no—
38C2×C4×3−1+2D411yescharacteristic
39C2×C4×3−1+2C2311yescharacteristic
40#611⋊C3—42no—
41C2×3−1+2⋊C4—21no—
42C4⋊AGL(1,ℤ/9)—22no—
433−1+2⋊C8—22no—
44C3×#448—91no—
45#5227—31no—
46C2×C32×Q8—24no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D312yes—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C2×C3×#67—61no—
52C2×C3×D12—31no—
53C3×#215—64no—
54C3×(C2×D3)⋊C4—61no—
55C3×C4⋊Dic3—31no—
56C2×C3×C3⋊C8—32no—
57C2×Q8×C9—28no—
58C2×Q8×C9—22no—
59C2×Q8×C9—22no—
60C2×Q8×C9—24no—
61C2×Q8×C9—21no—
62C2×Q8×C9C22×C312yes—
63C22×C4×C9—22no—
64C22×C4×C9—21no—
65C22×C4×C9—21no—
66C22×C4×C9C22×C311yescharacteristic
67C2×#611—21no—
68C2×D36C22×C311yescharacteristic
69#3675—24no—
70(C2×D9)⋊C4—21no—
71C4⋊Dic9C22×C311yescharacteristic
72C2×C9⋊C8C22×C312yes—
73C22×3−1+2—21no—
74C22×3−1+2—21no—
75C22×3−1+2SD1612yescomplemented
76C22×3−1+2C2×D411yescharacteristic
77C2×AGL(1,ℤ/9)—41no—
78C2×AGL(1,ℤ/9)—41no—
79C2×AGL(1,ℤ/9)—41no—
80C4×3−1+2—24no—
81C4×3−1+2—21no—
82C4×3−1+2C2×D412yes—
833−1+2⋊C4—41no—
84C22×C3×Q8—31no—
85C22×C3×Q8C3×D311yescharacteristic
86C2×C3×SD16—92no—
87C3×#121—91no—
88C2×#215—32no—
89#930—31no—
90C3×#103—92no—
91C3×C22⋊C8—91no—
92#835—32no—
93#830—31no—
94C23×C32#6711yescharacteristic
95C22×C3×D3—61no—
96C32×Q8—24no—
97C32×Q8—22no—
98C32×Q8#6714yes—
99C2×C32×C4—24no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4#6711yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C3×#67—122no—
105C2×C3×Dic3—61no—
106C3×D12—62no—
107C23×C9—21no—
108C23×C9C3×D411yescharacteristic
109C22×D9—21no—
110C3×C3⋊C8—62no—
111Q8×C9—28no—
112Q8×C9—24no—
113Q8×C9—24no—
114Q8×C9—22no—
115Q8×C9—24no—
116Q8×C9C3×D414yescomplemented
117C2×C4×C9—28no—
118C2×C4×C9—24no—
119C2×C4×C9—24no—
120C2×C4×C9—24no—
121C2×C4×C9—22no—
122C2×C4×C9—22no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9C3×D411yescharacteristic
126C2×C4×C9C23×C311yescharacteristic
127#611—42no—
128C2×Dic9—21no—
129D36—22no—
130C9⋊C8—22no—
131#448—91noSylow
132C2×3−1+2—22no—
133C2×3−1+2#12011yescharacteristic
134C2×3−1+2C2×SD1611yescharacteristic
135C2×3−1+2#13711yescharacteristic
136AGL(1,ℤ/9)—81no—
137C2×C3×Q8—24no—
138C2×C3×Q8—21no—
139C2×C3×Q8—32no—
140C2×C3×Q8—61no—
141C2×C3×Q8—64no—
142C2×C3×Q8C2×C3×D312yes—
143C2×C3×D4—91no—
144C2×C3×D4—181no—
145C22×C3×C4—21no—
146C22×C3×C4—31no—
147C22×C3×C4—61no—
148C22×C3×C4C2×C3×D311yescharacteristic
149C2×#67—61no—
150C2×D12—31no—
151C3×SD16—184no—
152C2×C3×C8—92no—
153C3×C4⋊C4—91no—
154C3×C22⋊C4—181no—
155#215—64no—
156(C2×D3)⋊C4—61no—
157C4⋊Dic3—31no—
158C2×C3⋊C8—32no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32#21512yes—
162C22×C32C2×#6711yescharacteristic
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C32×C4—24no—
167C32×C4—21no—
168C32×C4C2×#6712yes—
169C3×Dic3—121no—
170C22×C9—22no—
171C22×C9—22no—
172C22×C9—21no—
173C22×C9—21no—
174C22×C9—22no—
175C22×C9—21no—
176C22×C9C3×SD1612yescomplemented
177C22×C9C2×C3×D411yescharacteristic
178C2×D9—41no—
179C2×D9—41no—
180C2×D9—41no—
181C4×C9—28no—
182C4×C9—22no—
183C4×C9—24no—
184C4×C9—21no—
185C4×C9—22no—
186C4×C9C2×C3×D412yes—
187Dic9—41no—
188C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
189C2×SD16—92no—
190#121—91no—
191#103—92no—
192C22⋊C8—91no—
1933−1+2#44811yescharacteristic, complemented, Sylow
194C23×C3—31no—
195C23×C3—181no—
196C23×C3C3×#6711yescharacteristic
197C22×D3—61no—
198C3×Q8—24no—
199C3×Q8—22no—
200C3×Q8—34no—
201C3×Q8—62no—
202C3×Q8—64no—
203C3×Q8C3×#6714yes—
204C3×D4—182no—
205C3×D4—362no—
206C2×C3×C4—24no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—64no—
214C2×C3×C4—181no—
215C2×C3×C4C3×#6711yescharacteristic
216C2×C3×C4C22×C3×D311yescharacteristic
217#67—122no—
218C2×Dic3—61no—
219D12—62no—
220C3×C8—182no—
221C3⋊C8—62no—
222C2×C32—22no—
223C2×C32C2×#21511yescharacteristic
224C2×C32#94211yescharacteristic
225C2×C32#95311yescharacteristic
226C3×D3—241no—
227C2×C9—24no—
228C2×C9—22no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9—21no—
232C2×C9C3×#12011yescharacteristic
233C2×C9C2×C3×SD1611yescharacteristic
234C2×C9C3×#13711yescharacteristic
235D9—81no—
236C2×Q8—24no—
237C2×Q8—21no—
238C2×Q8C2×AGL(1,ℤ/9)12yes—
239C2×D4—91no—
240C2×D4—181no—
241C22×C4—21no—
242C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
243SD16—184no—
244C2×C8—92no—
245C4⋊C4—91no—
246C22⋊C4—181no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—361no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3C3×#21512yes—
257C22×C3C2×C3×#6711yescharacteristic
258C2×D3—121no—
259C2×D3—121no—
260C2×D3—121no—
261C3×C4—24no—
262C3×C4—21no—
263C3×C4—32no—
264C3×C4—61no—
265C3×C4—64no—
266C3×C4—361no—
267C3×C4C2×C3×#6712yes—
268Dic3—121no—
269C32#522711yescharacteristic
270C9—21no—
271C9C3×#44811yescharacteristic, complemented
272C23—181no—
273C23#611⋊C311yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8#611⋊C314yescomplemented
277D4—182no—
278D4—362no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—181no—
283C2×C4#611⋊C311yescharacteristic
284C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
285C8—182no—
286C2×C3—22no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—62no—
291C2×C3—721no—
292C2×C3C2×C3×#21511yescharacteristic
293C2×C3C3×#94211yescharacteristic
294C2×C3C3×#95311yescharacteristic
295D3—241no—
296C22—21no—
297C22—21no—
298C22—361no—
299C22—361no—
300C22—361no—
301C22#3675⋊C312yescomplemented
302C22C2×#611⋊C311yescharacteristic
303C4—24no—
304C4—21no—
305C4—361no—
306C4C2×#611⋊C312yes—
307C3—31no—
308C3C3×#522711yescharacteristic
309C2—22no—
310C2—721no—
311C2C2×#3675⋊C311yescharacteristic
312C2#63505⋊C311yescharacteristic
313C2#1172170411yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×Q8×3−1+2C2
socleC22×C3C2×C3×#67
prime radical#63492⋊C3C2
hypercenterC22×Q8AGL(1,ℤ/9)
hypocenterC9C3×#448
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×3−1+2C2semidirect
2. C2×#3675⋊C3C2semidirect
4. #63395⋊C3C2semidirect
5. #63389⋊C3C2semidirect
7. #10588908C3semidirect
23. C22×Q8×C9C2×C3semidirect
26. C2×#3675C2×C3semidirect
28. #63395C2×C3semidirect
29. #63389C2×C3semidirect
34. Q8×3−1+2D4semidirect
75. C22×3−1+2SD16semidirect
116. Q8×C9C3×D4semidirect
176. C22×C9C3×SD16semidirect
188. C22×Q8AGL(1,ℤ/9)semidirect
193. 3−1+2#448semidirect
271. C9C3×#448semidirect
276. Q8#611⋊C3semidirect
301. C22#3675⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×3−1+22
C2×#3675⋊C32
#63492⋊C32
#63395⋊C32
#63389⋊C32
C3×#52273
#105889083