Finite Group Databasebeta

← #13160925

Subgroups — #13160925

Order 1728.

968 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 160 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×#64149C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592862—31nomaximal
9#10591438—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×#638—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#214—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88793—22no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25C2×#3755—31no—
26#64149D311yescharacteristic
27C3×#940—34no—
28C2×C3×#214D314yescomplemented
29Dic3×Dic6—34no—
30C2×#3715—34no—
31#63659—31no—
32#63659—34no—
33#63659—64no—
34#63648—31no—
35#63641—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63563—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C3×#638—42no—
45C3×#638—42no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6D412yescomplemented
53C3⋊(C3⋊(C3⋊C8))—21no—
54C32×C3⋊C8—21no—
55#5163—94no—
56#5075—91no—
57C2×C32×Q8—31no—
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×#638—21no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×#67—64no—
65C2×C3×#67—64no—
66C2×C3×D12—34no—
67C2×C3×Dic6C2×D314yes—
68C32×SD16—62no—
69C2×C32×C8—31no—
70#3755—62no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—64no—
75C4×C3⋊Dic3—61no—
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80C3×#214—28no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3715—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3SD1612yescomplemented
98C32×Dic3—22no—
99C2×C3×SD16—34no—
100C3×#127—91no—
101Q8×Dic3—94no—
102C2×#215—94no—
103#940—34no—
104C2×#214—91no—
105C2×#214C3⋊D311yescharacteristic, complemented
106C4×Dic6—91no—
107#832—94no—
108#832—91no—
109#832—91no—
110#832—184no—
111#821—94no—
112Dic3⋊C8—94no—
113C4⋊(C3⋊C8)—91no—
114C23×C32—21no—
115C23×C32—24no—
116C23×C32—44no—
117C22×C3⋊D3—21no—
118C22×C3×D3—64no—
119C32×Q8—32no—
120C32×Q8—61no—
121C32×D4—28no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—21no—
125C32×D4—24no—
126C32×D4—44no—
127C2×C32×C4—24no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—42no—
132#638—42no—
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—124no—
136C2×C3⋊Dic3#6711yescharacteristic
137C2×C3⋊Dic3#6711yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C3×#67—128no—
141C3×#67—128no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—34no—
145C3×D12—64no—
146C3×D12—64no—
147C3×Dic6—24no—
148C3×Dic6#6718yescomplemented
149C32×C8—61no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—124no—
153C3×C3⋊C8—24no—
154#460—271noSylow
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C2×C33C2×SD1611yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C3×C3⋊D3—81no—
160C2×C3×Q8—34no—
161C2×C3×D4—24no—
162C2×C3×D4—91no—
163C2×C3×D4—181no—
164C2×C3×D4—181no—
165C2×C3×D4D3214yes—
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C2×#67—64no—
168C2×#67—64no—
169C2×D12—34no—
170C2×Dic6C2×C3⋊D311yescharacteristic
171C3×SD16—68no—
172C2×C3×C8—34no—
173C3×C42—91no—
174#215—188no—
175#214—22no—
176#214—182no—
177C4⋊Dic3—94no—
178C4⋊Dic3—91no—
179C4⋊Dic3—184no—
180C4⋊Dic3—184no—
181Dic3⋊C4—181no—
182C4×Dic3—34no—
183C4×Dic3—181no—
184C4×Dic3—184no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189C22×C32—24no—
190C22×C32—48no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—44no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32—84no—
198C22×C32—84no—
199C22×C32D3×D414yes—
200C22×C32C2×#6711yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—62no—
210C32×C4D3×D411yescharacteristic
211C32×C4#23714yes—
212C32×C4#23711yescharacteristic
213C32×C4C2×#6714yes—
214C3⋊Dic3—21no—
215C3⋊Dic3—121no—
216C3⋊Dic3—124no—
217C3⋊Dic3—244no—
218C3⋊Dic3#21412yescomplemented
219C3×Dic3—28no—
220C3×Dic3—38no—
221C3×Dic3—64no—
222C2×SD16—31no—
223C2×SD16—271no—
224#127—91no—
225C4×Q8—271no—
226C4⋊C8—271no—
227#102—271no—
228#102—271no—
229C33#46011yescharacteristic, complemented, Sylow
230C23×C3—24no—
231C23×C3—21no—
232C23×C3—44no—
233C23×C3—181no—
234C22×D3—64no—
235C3×Q8—38no—
236C3×Q8—64no—
237C3×D4—28no—
238C3×D4—24no—
239C3×D4—24no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—44no—
243C3×D4—181no—
244C3×D4—181no—
245C3×D4—362no—
246C3×D4—362no—
247C2×C3×C4—24no—
248C2×C3×C4—64no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—128no—
254#67—128no—
255C2×Dic3—21no—
256C2×Dic3—34no—
257C2×Dic3—34no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261D12—64no—
262D12—64no—
263Dic6—21no—
264Dic6#63812yescomplemented
265C3×C8—64no—
266C3⋊C8—21no—
267C3⋊C8—184no—
268C3⋊C8—181no—
269C3⋊C8—364no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—88no—
274C2×C32—84no—
275C2×C32—81no—
276C2×C32D3×SD1614yes—
277C2×C32#91414yes—
278C2×C32#93211yescharacteristic
279C2×C32C2×#21411yescharacteristic
280C2×C32#93811yescharacteristic
281C2×C32#93914yes—
282C3⋊D3—81no—
283C3×D3—244no—
284C2×Q8—31no—
285C2×D4—91no—
286C2×D4—181no—
287C2×D4—181no—
288C2×D4D3×C3⋊D311yescharacteristic
289SD16—62no—
290SD16—542no—
291C2×C8—31no—
292C2×C8—271no—
293C4⋊C4—271no—
294C4⋊C4—541no—
295C42—91no—
296C42—541no—
297C22×C3—24no—
298C22×C3—48no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—44no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—84no—
306C22×C3—84no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3—361no—
310C22×C3D3×#6714yes—
311C22×C3D4×C3⋊D311yescharacteristic
312C2×D3—124no—
313C2×D3—124no—
314C2×D3—124no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—68no—
318C3×C4—92no—
319C3×C4—181no—
320C3×C4#380514yes—
321C3×C4D3×#6714yes—
322C3×C4#383011yescharacteristic
323C3×C4C2×#63811yescharacteristic
324Dic3—22no—
325Dic3—38no—
326Dic3—64no—
327Dic3—361no—
328Dic3—364no—
329Dic3—724no—
330C32—24no—
331C32#507511yescharacteristic, complemented
332C32#516314yescomplemented
333C23—21no—
334C23—181no—
335Q8—32no—
336Q8—61no—
337D4—21no—
338D4—21no—
339D4—181no—
340D4—181no—
341D4—362no—
342D4—362no—
343C2×C4—61no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—541no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—61no—
349C8—541no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—88no—
354C2×C3—81no—
355C2×C3—84no—
356C2×C3—721no—
357C2×C3D3×#21414yes—
358C2×C3#6393414yes—
359C2×C3#6397014yes—
360C2×C3SD16×C3⋊D311yescharacteristic
361C2×C3#6412311yescharacteristic
362C2×C3#6414811yescharacteristic
363D3—244no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—361no—
368C22—361no—
369C22—361no—
370C22C3⋊D3×#6711yescharacteristic
371C4—62no—
372C4—92no—
373C4—181no—
374C4—1081no—
375C4#8902911yescharacteristic
376C4D3×#63811yescharacteristic
377C3—24no—
378C3#1059143814yescomplemented
379C3#1059286211yescharacteristic, complemented
380C2—81no—
381C2—721no—
382C2C3⋊D3×#21411yescharacteristic
383C2#1172356211yescharacteristic
384C2#1172361511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#460
LCSG ▷ C2×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
2. C2×C32×#214C2semidirect
3. Dic6×C3⋊Dic3C2semidirect
4. C2×#88793C2semidirect
5. #11722202C2semidirect
6. #11722051C2semidirect
7. #11722034C2semidirect
28. C2×C3×#214D3semidirect
52. C32×Dic6D4semidirect
97. C3×C3⋊Dic3SD16semidirect
105. C2×#214C3⋊D3semidirect
148. C3×Dic6#67semidirect
218. C3⋊Dic3#214semidirect
229. C33#460semidirect
264. Dic6#638semidirect
331. C32#5075semidirect
332. C32#5163semidirect
378. C3#10591438semidirect
379. C3#10592862semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C32×#2142
Dic6×C3⋊Dic32
C2×#887932
#117222022
#117220512
#117220342
#105928623
#105914383