Finite Group Databasebeta

← #13159287

Subgroups — #13159287

Order 1728.

888 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 152 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×#64102C211yesmaximal, characteristic
2#11723455C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic
4C2×#88793C211yesmaximal, characteristic
5#11722180C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722051C211yesmaximal, characteristic, complemented
8#10592517—31nomaximal
9#10590559—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88807—21no—
21#88796—22no—
22#88793—22no—
23#64153—31no—
24C2×#3755—31no—
25C2×#3744—31no—
26#64102D311yescharacteristic
27C3×#893—34no—
28#63866—34no—
29C2×#3716—34no—
30C2×#3715—34no—
31C32×#103—31no—
32#63641—31no—
33#63637—31no—
34#63637—34no—
35#63637—64no—
36C3×#810D314yescomplemented
37Dic3⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)D412yes—
48C3×C4×C3⋊D3—42no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5019—91no—
55#4818—94no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×D12—34no—
63C2×C3×C4×D3—64no—
64C2×C3×Dic6C2×D314yes—
65C2×C32×C8—31no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C2×D311yescharacteristic
69#3755—62no—
70#3752—21no—
71#3750—61no—
72#3750—61no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77#3744—62no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4×Dic3—34no—
80C2×C3×C3⋊C8C2×D314yes—
81Dic3⋊Dic3—64no—
82Dic3⋊Dic3—64no—
83#3716—68no—
84#3715—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3SD1612yescomplemented
95C32×Dic3—41no—
96C3×C4×D4—91no—
97#944—94no—
98C2×#215—94no—
99C2×#214—91no—
100C2×#214—91no—
101C2×#204—94no—
102#893—34no—
103#869—91no—
104C3×#103—34no—
105Dic3⋊C8—94no—
106#810—94no—
107#810—91no—
108#810—184no—
109#810C3⋊D311yescharacteristic, complemented
110C4⋊(C3⋊C8)—91no—
111C22×C3⋊D3—21no—
112C22×C3×D3—64no—
113C32×Q8—61no—
114C32×Q8—61no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—44no—
119C2×C32×C4—61no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—124no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)D1212yes—
129C4×C3⋊D3—42no—
130#634—68no—
131#634—61no—
132#634—64no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C3×D12—38no—
140C3×D12—64no—
141C3×C4×D3—128no—
142C3×Dic6—24no—
143C3×Dic6—24no—
144C32×C8—61no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148C3×C3⋊C8—24no—
149#461—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13711yescharacteristic
153C3×C3⋊D3—42no—
154C2×C3×Q8—34no—
155C2×C3×D4—91no—
156C22×C3×C4—181no—
157C2×D12—34no—
158C2×C4×D3—64no—
159C2×Dic6—94no—
160C2×Dic6—91no—
161C2×Dic6—184no—
162C2×Dic6C2×C3⋊D311yescharacteristic
163C2×C3×C8—34no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4D3214yes—
166C3×C4⋊C4C2×C3⋊D311yescharacteristic
167C3×C22⋊C4—181no—
168C3×C42—91no—
169#215—188no—
170#214—182no—
171#214—182no—
172(C2×D3)⋊C4—64no—
173C4⋊Dic3—181no—
174C4⋊Dic3—181no—
175Dic3⋊C4—184no—
176Dic3⋊C4—184no—
177C4×Dic3—34no—
178C2×C3⋊C8—94no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—184no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182#204—188no—
183C22×C32—24no—
184C22×C32D3×D414yes—
185C22×C32C2×#6711yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—68no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—41no—
195C32×C4—44no—
196C32×C4—84no—
197C32×C4—121no—
198C32×C4C2×D1211yescharacteristic
199C32×C4#23514yes—
200C32×C4#23511yescharacteristic
201C32×C4D3×D414yes—
202C3⋊Dic3—21no—
203C3⋊Dic3—128no—
204C3⋊Dic3—124no—
205C3⋊Dic3—121no—
206C3⋊Dic3#21412yescomplemented
207C3×Dic3—38no—
208C3×Dic3—44no—
209C3×Dic3—64no—
210C2×SD16—271no—
211C2×SD16—271no—
212C4⋊Q8—271no—
213C4×D4—91no—
214C4⋊C8—271no—
215#103—31no—
216#103—271no—
217C33#46111yescharacteristic, complemented, Sylow
218C23×C3—181no—
219C22×D3—64no—
220C3×Q8—64no—
221C3×Q8—64no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—44no—
228C2×C3×C4—64no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—362no—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—34no—
236C2×Dic3—34no—
237C2×Dic3—184no—
238C2×Dic3—181no—
239C2×Dic3—364no—
240D12—38no—
241D12—64no—
242C4×D3—128no—
243Dic6—21no—
244Dic6—21no—
245Dic6—188no—
246Dic6—184no—
247Dic6—184no—
248Dic6—181no—
249Dic6—181no—
250Dic6—364no—
251C3×C8—64no—
252C3⋊C8—21no—
253C3⋊C8—184no—
254C3⋊C8—181no—
255C3⋊C8—364no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32#88414yes—
260C2×C32D3×SD1614yes—
261C2×C32#91814yes—
262C2×C32#93011yescharacteristic
263C2×C32C2×#21411yescharacteristic
264C2×C32#95111yescharacteristic
265C3⋊D3—42no—
266C3×D3—128no—
267C2×Q8—31no—
268C2×Q8—271no—
269C2×D4—91no—
270C22×C4—181no—
271SD16—542no—
272SD16—542no—
273C2×C8—31no—
274C2×C8—271no—
275C4⋊C4—541no—
276C4⋊C4—541no—
277C4⋊C4D3×C3⋊D311yescharacteristic
278C22⋊C4—181no—
279C42—91no—
280C22×C3—24no—
281C22×C3—182no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3D3×#6714yes—
285C22×C3D4×C3⋊D311yescharacteristic
286C2×D3—68no—
287C2×D3—124no—
288C2×D3—124no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—41no—
292C3×C4—44no—
293C3×C4—84no—
294C3×C4—92no—
295C3×C4—124no—
296C3×C4—181no—
297C3×C4#379814yes—
298C3×C4D3×D1214yes—
299C3×C4#382811yescharacteristic
300C3×C4D4×C3⋊D311yescharacteristic
301Dic3—38no—
302Dic3—41no—
303Dic3—64no—
304Dic3—368no—
305Dic3—361no—
306Dic3—364no—
307C32—24no—
308C32#481814yescomplemented
309C32#501911yescharacteristic, complemented
310C23—181no—
311Q8—61no—
312Q8—61no—
313Q8—541no—
314Q8—541no—
315D4—92no—
316D4—181no—
317C2×C4—21no—
318C2×C4—61no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—362no—
322C2×C4—541no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—61no—
325C8—541no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—362no—
330C2×C3#6390614yes—
331C2×C3D3×#21414yes—
332C2×C3#6394314yes—
333C2×C3#6409311yescharacteristic
334C2×C3SD16×C3⋊D311yescharacteristic
335C2×C3#6412711yescharacteristic
336D3—128no—
337C22—182no—
338C22—361no—
339C22—361no—
340C22C3⋊D3×#6711yescharacteristic
341C4—41no—
342C4—92no—
343C4—121no—
344C4—181no—
345C4—1081no—
346C4#8902011yescharacteristic
347C4C3⋊D3×D1211yescharacteristic
348C3—24no—
349C3#1059055914yescomplemented
350C3#1059251711yescharacteristic, complemented
351C2—362no—
352C2#1172352011yescharacteristic
353C2C3⋊D3×#21411yescharacteristic
354C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723455C2semidirect
5. #11722180C2semidirect
6. C32×#810C2semidirect
7. #11722051C2semidirect
36. C3×#810D3semidirect
94. C3×C3⋊Dic3SD16semidirect
109. #810C3⋊D3semidirect
206. C3⋊Dic3#214semidirect
217. C33#461semidirect
308. C32#4818semidirect
309. C32#5019semidirect
349. C3#10590559semidirect
350. C3#10592517semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
#117234552
C2×#887962
C2×#887932
#117221802
C32×#8102
#117220512
#105925173
#105905593