Finite Group Databasebeta

← #13140728

Subgroups — #13140728

Order 1728.

430 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 107 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×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×C3×#3675C211yesmaximal, characteristic
3#11720998C211yesmaximal, characteristic
4C9×#835C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic
6#11720275C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10591572—31nomaximal
9#10588497—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×Dic3×D9—21no—
13C4⋊(C3⋊Dic9)—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C4×C3⋊Dic9—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C3×#3675C414yescomplemented
18C9×C4⋊Dic3C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D9)⋊Dic3—21no—
22Q8×C3⋊Dic3—31no—
23C2×C3×#215—31no—
24#63898—31no—
25C3×#835D311yescharacteristic
26#63575—31no—
27#63563—31no—
28#63555—31no—
29Q8×Dic9—31no—
30Q8×Dic9—61no—
31C2×#3675D311yescharacteristic
32#63455—31no—
33C9×#103—31no—
34#63380—31no—
35#63375—31no—
36#63366—31no—
37C22×C3×D9—21no—
38C3×Q8×C9—21no—
39C3×Q8×C9C2×C412yescomplemented
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C2×C9×Dic3—21no—
46C3×D36—21no—
47C3×D36C2×C412yes—
48Dic3×D9—42no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8C2×C412yescomplemented
51#5164—91no—
52#4817—91no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C3×D12—31no—
55C2×D3×Dic3—61no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—61no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3—61no—
60C3×#215—34no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64(C2×D3)⋊Dic3—61no—
65C2×Q8×C9—21no—
66C2×Q8×C9C2×D311yescharacteristic
67C2×D36C2×D311yescharacteristic
68C2×C4×D9—61no—
69C2×C8×C9—31no—
70C9×C4⋊C4—31no—
71#3675Dic314yescomplemented
72(C2×D9)⋊C4—61no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75C4⋊Dic9—61no—
76C4⋊Dic9—62no—
77C4×Dic9—31no—
78C4×Dic9—61no—
79C4×Dic9—61no—
80C4×Dic9—62no—
81C2×C9⋊C8C2×D311yescharacteristic
82C22×C3×C9C2×D411yescharacteristic
83C2×C3×D9—22no—
84C2×C3×D9—41no—
85C2×C3×D9—41no—
86C3×C4×C9—22no—
87C3×C4×C9C22×C411yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C3⋊Dic9—21no—
91C3⋊Dic9—41no—
92C9×Dic3—41no—
93C2×C3×SD16—91no—
94Q8×Dic3—91no—
95Q8×Dic3—91no—
96Q8×Dic3—181no—
97C2×#215—31no—
98D4×Dic3—91no—
99#893—91no—
100C3×#103—31no—
101#835D911yescharacteristic, complemented
102#832—91no—
103#821—91no—
104#818—91no—
105#815—91no—
106#815—91no—
107#807—91no—
108C22×C3×D3—61no—
109C32×Q8—21no—
110C32×Q8C4×D312yes—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—61no—
116C2×C3×Dic3—21no—
117C3×D12—32no—
118C3×D12—61no—
119D3×Dic3—122no—
120C22×D9—21no—
121C3×C3⋊C8—21no—
122C3×C3⋊C8—32no—
123Q8×C9—22no—
124Q8×C9—21no—
125Q8×C9—22no—
126Q8×C9C2×Dic312yescomplemented
127C2×C4×C9—22no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—61no—
131C2×C4×C9C22×D311yescharacteristic
132C2×Dic9—31no—
133C2×Dic9—31no—
134C2×Dic9—61no—
135C2×Dic9—61no—
136C2×Dic9—62no—
137C2×Dic9—61no—
138D36—21no—
139D36C2×Dic312yes—
140C4×D9—122no—
141C8×C9—61no—
142C9⋊C8C2×Dic312yescomplemented
143#440—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C9#13711yescharacteristic
147C3×D9—42no—
148C2×C3×Q8—21no—
149C2×C3×Q8C2×D911yescharacteristic
150C2×C3×Q8D3211yescharacteristic
151C2×C3×D4—91no—
152C22×Dic3—181no—
153C2×D12—31no—
154C2×C4×D3—181no—
155C3×SD16—94no—
156C2×C3×C8—31no—
157C2×C3×C8—91no—
158C3×C4⋊C4—31no—
159#217—181no—
160#215—34no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—181no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167C4⋊Dic3—182no—
168C4⋊Dic3C2×D911yescharacteristic
169C4×Dic3—91no—
170C4×Dic3—91no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C4×Dic3—181no—
174C4×Dic3—182no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8C2×D911yescharacteristic
177C22×C32D3×D411yescharacteristic
178C2×C3×D3—62no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C32×C4—22no—
182C32×C4C2×C4×D311yescharacteristic
183C32×C4#23711yescharacteristic
184C3⋊Dic3—61no—
185C3⋊Dic3—61no—
186C3⋊Dic3—121no—
187C3×Dic3—41no—
188C22×C9—21no—
189C22×C9D3×D411yescharacteristic
190C2×D9—22no—
191C2×D9—41no—
192C2×D9—41no—
193C4×C9—24no—
194C4×C9—22no—
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—121no—
198C4×C9#23711yescharacteristic
199C4×C9C22×Dic311yescharacteristic
200Dic9—62no—
201Dic9—61no—
202Dic9—61no—
203Dic9—121no—
204Dic9—122no—
205Dic9—121no—
206C2×SD16—91no—
207C4×Q8—271no—
208C4×D4—271no—
209#107—271no—
210#103—31no—
211#102—271no—
212#97—271no—
213C3×C9#44011yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C3×Q8—22no—
217C3×Q8—21no—
218C3×Q8—21no—
219C3×Q8—22no—
220C3×Q8C4×D912yescomplemented
221C3×Q8D3×Dic312yes—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—61no—
229C2×C3×C4C22×D911yescharacteristic
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238C2×Dic3—182no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241C2×Dic3—362no—
242D12—32no—
243D12—61no—
244C4×D3—362no—
245C3×C8—61no—
246C3×C8—92no—
247C3⋊C8—21no—
248C3⋊C8—32no—
249C2×C32#67⋊C411yescharacteristic
250C2×C32#91411yescharacteristic
251C2×C32#91811yescharacteristic
252C3×D3—122no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9#91411yescharacteristic
257C2×C9#91511yescharacteristic
258C2×C9D4×Dic311yescharacteristic
259D9—42no—
260C2×Q8D3×D911yescharacteristic
261C2×D4—91no—
262C22×C4—541no—
263SD16—94no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—31no—
267C4⋊C4—271no—
268C4⋊C4—541no—
269C22⋊C4—541no—
270C42—271no—
271C42—541no—
272C22×C3—21no—
273C22×C3—182no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3D4×D911yescharacteristic
277C22×C3#381111yescharacteristic
278C2×D3—62no—
279C2×D3—121no—
280C2×D3—121no—
281C3×C4—24no—
282C3×C4—22no—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—121no—
287C3×C4C2×C4×D911yescharacteristic
288C3×C4#369911yescharacteristic
289C3×C4C2×D3×Dic311yescharacteristic
290C3×C4#380511yescharacteristic
291Dic3—41no—
292Dic3—182no—
293Dic3—181no—
294Dic3—181no—
295Dic3—181no—
296Dic3—181no—
297Dic3—361no—
298Dic3—362no—
299Dic3—361no—
300Dic3—361no—
301C32#481711yescharacteristic
302C9—21no—
303C9#516411yescharacteristic, complemented
304C23—181no—
305Q8—21no—
306Q8Dic3×D912yescomplemented
307D4—92no—
308D4—181no—
309C2×C4—21no—
310C2×C4—61no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4C2×D3×D911yescharacteristic
316C8—61no—
317C8—92no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—362no—
322C2×C3#611⋊C411yescharacteristic
323C2×C3#6347611yescharacteristic
324C2×C3#6348011yescharacteristic
325C2×C3#6394111yescharacteristic
326C2×C3#6395111yescharacteristic
327C2×C3#67⋊Dic311yescharacteristic
328D3—122no—
329C22—182no—
330C22—361no—
331C22—361no—
332C22#8866811yescharacteristic
333C4—22no—
334C4—121no—
335C4—541no—
336C4—541no—
337C4—1081no—
338C4C2×Dic3×D911yescharacteristic
339C4#8866311yescharacteristic
340C3—21no—
341C3#1058849711yescharacteristic, complemented
342C3#1059157211yescharacteristic
343C2—362no—
344C2#1172121311yescharacteristic
345C2#1172122511yescharacteristic
346C2#611⋊Dic311yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#440
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊Dic9C2semidirect
4. C9×#835C2semidirect
6. #11720275C2semidirect
7. #11720262C2semidirect
17. C3×#3675C4semidirect
39. C3×Q8×C9C2×C4semidirect
50. C3×C9⋊C8C2×C4semidirect
71. #3675Dic3semidirect
101. #835D9semidirect
126. Q8×C9C2×Dic3semidirect
142. C9⋊C8C2×Dic3semidirect
213. C3×C9#440semidirect
220. C3×Q8C4×D9semidirect
303. C9#5164semidirect
306. Q8Dic3×D9semidirect
341. C3#10588497semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
C2×C3×#36752
#117209982
C9×#8352
#117202832
#117202752
#117202622
#105915723
#105884973