Finite Group Databasebeta

← #13160923

Subgroups — #13160923

Order 1728.

1168 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 214 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
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic
3#11723499C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592851—31nomaximal
9#10591436—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)—21no—
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14#88864—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#213C414yescomplemented
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C2×C32×D8—31no—
23D4×C3⋊Dic3—31no—
24D4×C3⋊Dic3—31no—
25D4×C3⋊Dic3—34no—
26D4×C3⋊Dic3—64no—
27C2×C3×#213D314yes—
28#63898—34no—
29#63659—31no—
30#63659D311yescharacteristic, complemented
31#63644—31no—
32#63642—31no—
33C3×#832—34no—
34#63573—34no—
35#63563—34no—
36#63555—34no—
37C23×C33—21no—
38C22×C32×D3—21no—
39C33×D4—21no—
40C33×D4C2×C412yescomplemented
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—42no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C32×D12—21no—
48C32×D12C2×C412yescomplemented
49D3×C3⋊Dic3—42no—
50C3×C3⋊(C3⋊C8)—21no—
51C32×C3⋊C8C2×C412yescomplemented
52#5152—94no—
53#4785—91no—
54C2×C32×D4—24no—
55C2×C32×D4—31no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C2×D314yes—
58C22×C3⋊Dic3—61no—
59C22×C3⋊Dic3—61no—
60C22×C3⋊Dic3—64no—
61C22×C3⋊Dic3—68no—
62C2×C3×D12C2×D314yes—
63C2×D3×Dic3—64no—
64C32×D8—34no—
65C2×C32×C8—31no—
66#3757—61no—
67#3757—61no—
68#3757—64no—
69#3757—68no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—34no—
72C4⋊(C3⋊Dic3)—64no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×#213Dic3116yescomplemented
79C3×C4⋊Dic3—34no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8C2×D314yes—
82(C2×D3)⋊Dic3—64no—
83C22×C33—22no—
84C22×C33—41no—
85C22×C33—41no—
86C22×C33C2×D411yescharacteristic
87C2×C32×D3—22no—
88C2×C32×D3—41no—
89C2×C32×D3—41no—
90C33×C4C22×C411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—21no—
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C2×C3×D8—34no—
97D4×Dic3—94no—
98D4×Dic3—94no—
99D4×Dic3—91no—
100D4×Dic3—184no—
101C2×#213C3⋊D311yescharacteristic
102#893—91no—
103C3×#102—91no—
104#832—34no—
105#832—94no—
106#821—91no—
107#817—94no—
108#815—94no—
109#815—91no—
110#808—91no—
111C23×C32—28no—
112C23×C32—24no—
113C23×C32—21no—
114C23×C32—61no—
115C22×C3×D3—24no—
116C32×D4—28no—
117C32×D4—24no—
118C32×D4—21no—
119C32×D4—28no—
120C32×D4—32no—
121C32×D4—61no—
122C32×D4C4×D312yescomplemented
123C32×D4C2×Dic318yescomplemented
124C2×C32×C4—24no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—34no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—68no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—122no—
138C2×C3⋊Dic3—1216no—
139C2×C3⋊Dic3—128no—
140C2×C3⋊Dic3—122no—
141C2×C3×Dic3—64no—
142C3×D12—24no—
143C3×D12C2×Dic318yescomplemented
144D3×Dic3—128no—
145C32×C8—32no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—64no—
148C3×C3⋊C8C2×Dic318yescomplemented
149#442—271noSylow
150C2×C33—42no—
151C2×C33C4×D411yescharacteristic
152C2×C33AGL(1,ℤ/8)11yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C32×D3—42no—
155C2×C3×D4—24no—
156C2×C3×D4—34no—
157C2×C3×D4D3214yes—
158C2×C3×D4C2×C3⋊D311yescharacteristic
159C22×Dic3—184no—
160C22×Dic3—184no—
161C22×Dic3—181no—
162C22×Dic3—188no—
163C2×D12C2×C3⋊D311yescharacteristic
164C2×C4×D3—181no—
165C3×D8—316no—
166C2×C3×C8—34no—
167C2×C3×C8—91no—
168C3×C4⋊C4—91no—
169#217—184no—
170#217—184no—
171#217—181no—
172#217—188no—
173#213C3⋊Dic314yescomplemented
174(C2×D3)⋊C4—181no—
175C4⋊Dic3—34no—
176C4⋊Dic3—94no—
177C4⋊Dic3—91no—
178C4⋊Dic3—184no—
179C4×Dic3—94no—
180C4×Dic3—91no—
181C4×Dic3—184no—
182C2×C3⋊C8—34no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184C22×C32—216no—
185C22×C32—22no—
186C22×C32—28no—
187C22×C32—24no—
188C22×C32—48no—
189C22×C32—48no—
190C22×C32—44no—
191C22×C32—44no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—62no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32D3×D414yes—
198C22×C32D3×D411yescharacteristic
199C2×C3×D3—28no—
200C2×C3×D3—44no—
201C2×C3×D3—44no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4#23714yes—
206C32×C4#23711yescharacteristic
207C32×C4C22×Dic314yes—
208C3⋊Dic3—41no—
209C3⋊Dic3—68no—
210C3⋊Dic3—64no—
211C3⋊Dic3—61no—
212C3⋊Dic3—61no—
213C3⋊Dic3—64no—
214C3⋊Dic3—124no—
215C3⋊Dic3—128no—
216C3⋊Dic3—124no—
217C3⋊Dic3—121no—
218C3×Dic3—124no—
219C2×D8—31no—
220C4×D4—271no—
221C4×D4—271no—
222#106—271no—
223#102—91no—
224#102—271no—
225#97—271no—
226C33#44211yescharacteristic, complemented, Sylow
227C23×C3—28no—
228C23×C3—24no—
229C23×C3—21no—
230C23×C3—64no—
231C22×D3—21no—
232C3×D4—28no—
233C3×D4—24no—
234C3×D4—21no—
235C3×D4—28no—
236C3×D4—38no—
237C3×D4—64no—
238C3×D4D3×Dic318yescomplemented
239C3×D4C2×C3⋊Dic312yescomplemented
240C2×C3×C4—24no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×Dic3—64no—
245C2×Dic3—94no—
246C2×Dic3—94no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—184no—
250C2×Dic3—184no—
251C2×Dic3—188no—
252C2×Dic3—184no—
253C2×Dic3—181no—
254C2×Dic3—368no—
255C2×Dic3—3616no—
256C2×Dic3—362no—
257C2×Dic3—368no—
258D12—21no—
259D12C2×C3⋊Dic312yescomplemented
260C4×D3—362no—
261C3×C8—38no—
262C3×C8—181no—
263C3⋊C8—64no—
264C3⋊C8C2×C3⋊Dic312yescomplemented
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—416no—
269C2×C32—48no—
270C2×C32—42no—
271C2×C32—122no—
272C2×C32#67⋊C411yescharacteristic
273C2×C32#91114yes—
274C2×C32#91114yes—
275C2×C32#91111yescharacteristic
276C2×C32#91411yescharacteristic
277C2×C32D4×Dic314yes—
278C3×D3—48no—
279C2×D4—31no—
280C2×D4D3×C3⋊D311yescharacteristic
281C22×C4—541no—
282C22×C4—541no—
283D8—34no—
284C2×C8—31no—
285C2×C8—91no—
286C4⋊C4—91no—
287C4⋊C4—271no—
288C22⋊C4—541no—
289C22⋊C4—541no—
290C42—271no—
291C22×C3—216no—
292C22×C3—22no—
293C22×C3—28no—
294C22×C3—24no—
295C22×C3—48no—
296C22×C3—48no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—44no—
300C22×C3—44no—
301C22×C3—68no—
302C22×C3—124no—
303C22×C3—124no—
304C22×C3#381114yes—
305C22×C3D4×C3⋊D311yescharacteristic
306C2×D3—22no—
307C2×D3—41no—
308C2×D3—41no—
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—361no—
312C3×C4C2×D3×Dic314yes—
313C3×C4#380514yes—
314C3×C4#383011yescharacteristic
315C3×C4C22×C3⋊Dic311yescharacteristic
316Dic3—124no—
317Dic3—188no—
318Dic3—181no—
319Dic3—181no—
320Dic3—184no—
321Dic3—184no—
322Dic3—364no—
323Dic3—368no—
324Dic3—361no—
325Dic3—364no—
326C32—24no—
327C32#478511yescharacteristic, complemented
328C32#515214yescomplemented
329C23—21no—
330C23—61no—
331D4—21no—
332D4—32no—
333D4—61no—
334D4D3×C3⋊Dic312yescomplemented
335C2×C4—181no—
336C2×C4—271no—
337C2×C4—271no—
338C2×C4—541no—
339C2×C4—1082no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—32no—
343C8—181no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—416no—
348C2×C3—42no—
349C2×C3—48no—
350C2×C3—128no—
351C2×C3#6392714yes—
352C2×C3#6393714yes—
353C2×C3#67⋊Dic314yes—
354C2×C3#6412011yescharacteristic
355C2×C3#6412011yescharacteristic
356C2×C3D4×C3⋊Dic311yescharacteristic
357D3—42no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—62no—
362C22—121no—
363C22—121no—
364C22#8903911yescharacteristic
365C4—361no—
366C4—541no—
367C4—541no—
368C4—1081no—
369C4C2×D3×C3⋊Dic311yescharacteristic
370C4#8902911yescharacteristic
371C3—24no—
372C3#1059143614yescomplemented
373C3#1059285111yescharacteristic, complemented
374C2—42no—
375C2—122no—
376C2#1172354811yescharacteristic
377C2#1172356411yescharacteristic
378C2#67⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#213C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊(C3⋊Dic3)C2semidirect
3. #11723499C2semidirect
4. C3×#63659C2semidirect
5. #11722052C2semidirect
6. #11722032C2semidirect
7. #11722021C2semidirect
19. C32×#213C4semidirect
30. #63659D3semidirect
40. C33×D4C2×C4semidirect
48. C32×D12C2×C4semidirect
51. C32×C3⋊C8C2×C4semidirect
78. C3×#213Dic3semidirect
122. C32×D4C4×D3semidirect
123. C32×D4C2×Dic3semidirect
143. C3×D12C2×Dic3semidirect
148. C3×C3⋊C8C2×Dic3semidirect
173. #213C3⋊Dic3semidirect
226. C33#442semidirect
238. C3×D4D3×Dic3semidirect
239. C3×D4C2×C3⋊Dic3semidirect
259. D12C2×C3⋊Dic3semidirect
264. C3⋊C8C2×C3⋊Dic3semidirect
327. C32#4785semidirect
328. C32#5152semidirect
334. D4D3×C3⋊Dic3semidirect
372. C3#10591436semidirect
373. C3#10592851semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C2×C32×#2132
#117234992
C3×#636592
#117220522
#117220322
#117220212
#105928513
#105914363