Finite Group Databasebeta

← #13140472

Subgroups — #13140472

Order 1728.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C3×D4×Dic9C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic
3Dic6×Dic9C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic
5#11720297C211yesmaximal, characteristic, complemented
6#11720279C211yesmaximal, characteristic, complemented
7#11720274C211yesmaximal, characteristic, complemented
8#10591440—31nomaximal
9#10588844—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3×Dic9—21no—
12C2×C9×Dic6C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×#3676—21no—
16C9×#214C414yescomplemented
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic9—21no—
21Dic3×Dic9—21no—
22C3×D4×Dic3—31no—
23C2×C3×#214D311yescharacteristic
24Dic3×Dic6—31no—
25#63659—31no—
26#63575—31no—
27#63565—31no—
28#63554—31no—
29C2×C9×SD16—31no—
30Q8×Dic9—31no—
31D4×Dic9D311yescharacteristic
32#63395—31no—
33#63392—31no—
34#63392—61no—
35#63378—31no—
36#63375—31no—
37C23×C3×C9—21no—
38C3×D4×C9—21no—
39C3×D4×C9C2×C412yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—42no—
45C2×C3×Dic9D411yescharacteristic
46C2×C3×Dic9D411yescharacteristic
47C9×Dic6—21no—
48C9×Dic6C2×C412yescomplemented
49C3⋊(C9⋊C8)—21no—
50C9×C3⋊C8C2×C412yescomplemented
51#5164—91no—
52#5018—91no—
53C2×C32×D4C2×D311yescharacteristic
54C22×C3×Dic3—61no—
55C2×C3×Dic6C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57C2×C3⋊(C3⋊C8)—31no—
58C3×#217—61no—
59C3×#214Dic314yes—
60C3×C4⋊Dic3—31no—
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63Dic3⋊Dic3—61no—
64Dic32—61no—
65C2×Q8×C9—31no—
66C2×D4×C9—21no—
67C2×D4×C9C2×D311yescharacteristic
68C22×Dic9—21no—
69C9×SD16—34no—
70C2×C8×C9—31no—
71#3676—21no—
72C4⋊Dic9—31no—
73C4⋊Dic9—61no—
74C4⋊Dic9—61no—
75C4⋊Dic9C2×D311yescharacteristic
76C4×Dic9—61no—
77C4×Dic9C2×D311yescharacteristic
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80C22×C3×C9—22no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9C2×D411yescharacteristic
84C3×C4×C9C22×C411yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—41no—
87C9×Dic3—22no—
88C3×Dic9—21no—
89C3×Dic9—21no—
90C3×Dic9—41no—
91C2×C3×SD16—31no—
92C3×C4×D4—91no—
93Q8×Dic3—91no—
94D4×Dic3—31no—
95C2×#214D911yescharacteristic
96C4×Dic6—91no—
97#835—91no—
98#832—91no—
99#832—91no—
100#832—181no—
101#818—91no—
102#815—91no—
103#810—91no—
104#807—91no—
105#803—91no—
106C23×C32—21no—
107C32×D4—21no—
108C32×D4C2×Dic312yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—122no—
116C3×Dic6—21no—
117C3×Dic6C2×Dic312yes—
118C23×C9—21no—
119C23×C9—41no—
120C3⋊(C3⋊C8)—61no—
121C3×C3⋊C8C2×Dic312yes—
122Q8×C9—32no—
123Q8×C9—61no—
124D4×C9—21no—
125D4×C9—22no—
126D4×C9—41no—
127D4×C9C4×D312yes—
128C2×C4×C9—21no—
129C2×C4×C9—61no—
130C2×C4×C9C22×D311yescharacteristic
131C2×Dic9—21no—
132C2×Dic9—42no—
133C2×Dic9—61no—
134C2×Dic9—121no—
135C2×Dic9#6711yescharacteristic
136C2×Dic9#6711yescharacteristic
137C8×C9—32no—
138C9⋊C8—61no—
139C9⋊C8—121no—
140#440—271noSylow
141C2×C3×C9—42no—
142C2×C3×C9C4×D411yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144C2×C3×C9#13711yescharacteristic
145C2×C3×Q8—31no—
146C2×C3×D4—21no—
147C2×C3×D4C2×D911yescharacteristic
148C2×C3×D4D3211yescharacteristic
149C22×C3×C4—181no—
150C22×Dic3—61no—
151C2×Dic6C2×D911yescharacteristic
152C3×SD16—34no—
153C2×C3×C8—31no—
154C3×C4⋊C4—91no—
155C3×C22⋊C4—181no—
156C3×C42—91no—
157#217—61no—
158#214Dic914yescomplemented
159C4⋊Dic3—31no—
160C4⋊Dic3—91no—
161C4⋊Dic3—91no—
162C4⋊Dic3—181no—
163C4⋊Dic3—181no—
164Dic3⋊C4—181no—
165C4×Dic3—31no—
166C4×Dic3—181no—
167C4×Dic3—181no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—181no—
171C2×C3⋊C8C2×D911yescharacteristic
172C22×C32—22no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32D3×D411yescharacteristic
176C32×C4#23711yescharacteristic
177C32×C4C22×Dic311yescharacteristic
178C3⋊Dic3—121no—
179C3×Dic3—22no—
180C3×Dic3—61no—
181C3×Dic3—61no—
182C3×Dic3—121no—
183C22×C9—22no—
184C22×C9—21no—
185C22×C9—42no—
186C22×C9—41no—
187C22×C9—41no—
188C22×C9—81no—
189C22×C9—81no—
190C22×C9C2×#6711yescharacteristic
191C4×C9—21no—
192C4×C9—21no—
193C4×C9—62no—
194C4×C9C2×C4×D311yescharacteristic
195C4×C9#23511yescharacteristic
196Dic9—21no—
197Dic9—21no—
198Dic9—41no—
199Dic9—121no—
200Dic9—241no—
201C2×SD16—31no—
202C4×Q8—271no—
203C4×D4—91no—
204#107—271no—
205#103—271no—
206#102—271no—
207#97—271no—
208C3×C9#44011yescharacteristic, complemented, Sylow
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—41no—
212C3×Q8—32no—
213C3×Q8—61no—
214C3×D4—21no—
215C3×D4—21no—
216C3×D4—22no—
217C3×D4—41no—
218C3×D4C2×Dic912yes—
219C3×D4D3×Dic312yes—
220C2×C3×C4—21no—
221C2×C3×C4—61no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—181no—
225C2×C3×C4—362no—
226C2×C3×C4C22×D911yescharacteristic
227C2×C3×C4C2×D3211yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—31no—
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—122no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—361no—
236Dic6—21no—
237Dic6C2×Dic912yescomplemented
238C3×C8—32no—
239C3⋊C8—181no—
240C3⋊C8—181no—
241C3⋊C8—361no—
242C3⋊C8C2×Dic912yescomplemented
243C2×C32—42no—
244C2×C32#91411yescharacteristic
245C2×C32#91511yescharacteristic
246C2×C32D4×Dic311yescharacteristic
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—21no—
250C2×C9—42no—
251C2×C9—82no—
252C2×C9C4×#6711yescharacteristic
253C2×C9#93211yescharacteristic
254C2×C9#95111yescharacteristic
255C2×Q8—31no—
256C2×D4D3×D911yescharacteristic
257C22×C4—181no—
258SD16—34no—
259C2×C8—31no—
260C2×C8—271no—
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C22⋊C4—181no—
265C42—91no—
266C42—541no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—42no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3D4×D911yescharacteristic
278C22×C3D3×#6711yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—62no—
282C3×C4—181no—
283C3×C4—181no—
284C3×C4—361no—
285C3×C4#369911yescharacteristic
286C3×C4C22×Dic911yescharacteristic
287C3×C4C2×D3×Dic311yescharacteristic
288C3×C4#380411yescharacteristic
289Dic3—22no—
290Dic3—61no—
291Dic3—61no—
292Dic3—121no—
293Dic3—361no—
294Dic3—361no—
295Dic3—721no—
296C32#516411yescharacteristic
297C9—21no—
298C9#501811yescharacteristic, complemented
299C23—21no—
300Q8—32no—
301Q8—61no—
302D4—21no—
303D4D3×Dic912yes—
304C2×C4—61no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—181no—
308C2×C4—362no—
309C2×C4—541no—
310C2×C4C2×D3×D911yescharacteristic
311C8—32no—
312C8—541no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—42no—
317C2×C3—42no—
318C2×C3—82no—
319C2×C3#6347611yescharacteristic
320C2×C3#6347711yescharacteristic
321C2×C3D4×Dic911yescharacteristic
322C2×C3#6392911yescharacteristic
323C2×C3#6393411yescharacteristic
324C2×C3Dic3×#6711yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22D9×#6711yescharacteristic
329C4—62no—
330C4—181no—
331C4—181no—
332C4—361no—
333C4—1081no—
334C4C2×D3×Dic911yescharacteristic
335C4#8866211yescharacteristic
336C3—21no—
337C3#1058884411yescharacteristic, complemented
338C3#1059144011yescharacteristic
339C2—42no—
340C2#1172118111yescharacteristic
341C2#1172118411yescharacteristic
342C2#67×Dic911yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#440
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. Dic6×Dic9C2semidirect
5. #11720297C2semidirect
6. #11720279C2semidirect
7. #11720274C2semidirect
16. C9×#214C4semidirect
48. C9×Dic6C2×C4semidirect
50. C9×C3⋊C8C2×C4semidirect
158. #214Dic9semidirect
208. C3×C9#440semidirect
237. Dic6C2×Dic9semidirect
242. C3⋊C8C2×Dic9semidirect
298. C9#5018semidirect
337. C3#10588844semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Dic92
C2×C9×#2142
Dic6×Dic92
#117205762
#117202972
#117202792
#117202742
#105914403
#105888443