Finite Group Databasebeta

← #13161150

Subgroups — #13161150

Order 1728.

888 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 216 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×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic
3D12×C3⋊Dic3C211yesmaximal, characteristic
4#11722205C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592863—31nomaximal
9#10591566—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#215C414yescomplemented
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C2×C32×SD16—31no—
23Q8×C3⋊Dic3D311yescharacteristic, complemented
24D4×C3⋊Dic3—31no—
25C3×Q8×Dic3—34no—
26C2×C3×#215D314yes—
27Dic3×D12—34no—
28#63662—31no—
29#63662—34no—
30#63662—64no—
31#63659—31no—
32#63645—31no—
33#63642—31no—
34#63575—34no—
35#63562—34no—
36#63554—34no—
37C22×C32×D3—21no—
38C33×Q8—21no—
39C33×Q8C2×C412yescomplemented
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C32×D12—21no—
47C32×D12C2×C412yes—
48D3×C3⋊Dic3—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#5164—94no—
52#5026—91no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C2×D314yes—
56C2×C32×D4—31no—
57C22×C3⋊Dic3—61no—
58C2×C3×D12C2×D314yes—
59C2×D3×Dic3—64no—
60C32×SD16—34no—
61C2×C32×C8—31no—
62#3757—61no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3—21no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C3×#215Dic3116yescomplemented
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—64no—
76C3×C4×Dic3—34no—
77C3×C4×Dic3—64no—
78C2×C3×C3⋊C8C2×D314yes—
79(C2×D3)⋊Dic3—64no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—22no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—21no—
90C3×C3⋊Dic3—41no—
91C2×C3×SD16—34no—
92C3×C4×Q8—91no—
93Q8×Dic3—34no—
94C2×#215C3⋊D311yescharacteristic
95D4×Dic3—94no—
96C4×D12—91no—
97#835—94no—
98#835—91no—
99#835—184no—
100#832—94no—
101#818—94no—
102#815—94no—
103#809—91no—
104#807—91no—
105#803—91no—
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×Q8—21no—
109C32×Q8—28no—
110C32×Q8—24no—
111C32×Q8—44no—
112C32×Q8C4×D312yescomplemented
113C32×Q8C2×Dic318yescomplemented
114C32×D4—32no—
115C32×D4—61no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—44no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—124no—
126C2×C3⋊Dic3—122no—
127C2×C3⋊Dic3#6711yescharacteristic
128C2×C3⋊Dic3#6711yescharacteristic
129C2×C3×Dic3—34no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—64no—
132C3×D12—24no—
133C3×D12C2×Dic318yes—
134D3×Dic3—128no—
135C32×C8—32no—
136C3⋊(C3⋊C8)—61no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—124no—
139C3×C3⋊C8C2×Dic318yescomplemented
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C32×D3—42no—
145C2×C3×Q8—24no—
146C2×C3×Q8D3214yes—
147C2×C3×Q8C2×C3⋊D311yescharacteristic
148C2×C3×D4—34no—
149C22×Dic3—184no—
150C2×D12C2×C3⋊D311yescharacteristic
151C2×C4×D3—181no—
152C3×SD16—316no—
153C2×C3×C8—34no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—181no—
156C3×C42—91no—
157C3×C42—181no—
158#217—184no—
159#215C3⋊Dic314yescomplemented
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—34no—
162C4⋊Dic3—64no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4×Dic3—34no—
167C4×Dic3—64no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171C2×C3⋊C8C2×C3⋊D311yescharacteristic
172C22×C32—24no—
173C22×C32—62no—
174C22×C32—121no—
175C22×C32—121no—
176C22×C32D3×D414yes—
177C22×C32C2×#6711yescharacteristic
178C2×C3×D3—28no—
179C2×C3×D3—44no—
180C2×C3×D3—44no—
181C32×C4—22no—
182C32×C4—24no—
183C32×C4—24no—
184C32×C4—28no—
185C32×C4—48no—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4#23511yescharacteristic
188C32×C4#23714yes—
189C32×C4C22×Dic314yes—
190C3⋊Dic3—21no—
191C3⋊Dic3—21no—
192C3⋊Dic3—41no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3×Dic3—64no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199C2×SD16—31no—
200C4×Q8—91no—
201C4×D4—271no—
202#107—271no—
203#103—271no—
204#102—271no—
205#97—271no—
206C33#44011yescharacteristic, complemented, Sylow
207C23×C3—64no—
208C22×D3—21no—
209C3×Q8—24no—
210C3×Q8—21no—
211C3×Q8—28no—
212C3×Q8—44no—
213C3×Q8D3×Dic318yescomplemented
214C3×Q8C2×C3⋊Dic312yescomplemented
215C3×D4—38no—
216C3×D4—64no—
217C2×C3×C4—24no—
218C2×C3×C4—21no—
219C2×C3×C4—24no—
220C2×C3×C4—44no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—181no—
224C2×C3×C4C2×D3214yes—
225C2×C3×C4C22×C3⋊D311yescharacteristic
226C2×Dic3—34no—
227C2×Dic3—34no—
228C2×Dic3—64no—
229C2×Dic3—184no—
230C2×Dic3—181no—
231C2×Dic3—364no—
232C2×Dic3—368no—
233D12—21no—
234D12C2×C3⋊Dic312yes—
235C4×D3—362no—
236C3×C8—38no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C3⋊C8—364no—
240C3⋊C8C2×C3⋊Dic312yescomplemented
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—122no—
245C2×C32#91414yes—
246C2×C32#91514yes—
247C2×C32C4×#6711yescharacteristic
248C2×C32D4×Dic314yes—
249C2×C32#94211yescharacteristic
250C2×C32#94911yescharacteristic
251C3×D3—48no—
252C2×Q8D3×C3⋊D311yescharacteristic
253C2×D4—31no—
254C22×C4—541no—
255SD16—34no—
256C2×C8—31no—
257C2×C8—271no—
258C4⋊C4—91no—
259C4⋊C4—181no—
260C4⋊C4—271no—
261C22⋊C4—541no—
262C42—91no—
263C42—181no—
264C22×C3—24no—
265C22×C3—68no—
266C22×C3—124no—
267C22×C3—124no—
268C22×C3D3×#6714yes—
269C22×C3D4×C3⋊D311yescharacteristic
270C2×D3—22no—
271C2×D3—41no—
272C2×D3—41no—
273C3×C4—22no—
274C3×C4—28no—
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—48no—
278C3×C4—181no—
279C3×C4—181no—
280C3×C4—361no—
281C3×C4C2×D3×Dic314yes—
282C3×C4#380414yes—
283C3×C4#383011yescharacteristic
284C3×C4C22×C3⋊Dic311yescharacteristic
285Dic3—64no—
286Dic3—64no—
287Dic3—124no—
288Dic3—361no—
289Dic3—364no—
290Dic3—724no—
291C32—24no—
292C32#502611yescharacteristic, complemented
293C32#516414yescomplemented
294C23—61no—
295Q8—21no—
296Q8D3×C3⋊Dic312yescomplemented
297D4—32no—
298D4—61no—
299C2×C4—21no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4—1082no—
305C2×C4C2×D3×C3⋊D311yescharacteristic
306C8—32no—
307C8—541no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—128no—
312C2×C3#6393914yes—
313C2×C3#6394614yes—
314C2×C3Dic3×#6714yes—
315C2×C3#6412311yescharacteristic
316C2×C3#6412411yescharacteristic
317C2×C3D4×C3⋊Dic311yescharacteristic
318D3—42no—
319C22—62no—
320C22—121no—
321C22—121no—
322C22C3⋊D3×#6711yescharacteristic
323C4—22no—
324C4—181no—
325C4—181no—
326C4—361no—
327C4—1081no—
328C4C2×D3×C3⋊Dic311yescharacteristic
329C4#8902811yescharacteristic
330C3—24no—
331C3#1059156614yescomplemented
332C3#1059286311yescharacteristic, complemented
333C2—122no—
334C2#1172357011yescharacteristic
335C2#1172358411yescharacteristic
336C2#67×C3⋊Dic311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C3⋊Dic3C2semidirect
4. #11722205C2semidirect
5. #11722055C2semidirect
7. #11722020C2semidirect
19. C32×#215C4semidirect
23. Q8×C3⋊Dic3D3semidirect
39. C33×Q8C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
73. C3×#215Dic3semidirect
112. C32×Q8C4×D3semidirect
113. C32×Q8C2×Dic3semidirect
139. C3×C3⋊C8C2×Dic3semidirect
159. #215C3⋊Dic3semidirect
206. C33#440semidirect
213. C3×Q8D3×Dic3semidirect
214. C3×Q8C2×C3⋊Dic3semidirect
240. C3⋊C8C2×C3⋊Dic3semidirect
292. C32#5026semidirect
293. C32#5164semidirect
296. Q8D3×C3⋊Dic3semidirect
331. C3#10591566semidirect
332. C3#10592863semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C2×C32×#2152
D12×C3⋊Dic32
#117222052
#117220552
#117220312
#117220202
#105928633
#105915663