Finite Group Databasebeta

← #13159961

Subgroups — #13159961

Order 1728.

888 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×C3×#3744C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3#11723443C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic
5#11722186C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592540—31nomaximal
9#10590752—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×#3744—22no—
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88803—21no—
22#88796—22no—
23C2×#3744—31no—
24C2×#3744D311yescharacteristic
25C4×C4⋊(C3⋊D3)—31no—
26#64078—31no—
27C2×C3×#204—34no—
28Dic3×D12—34no—
29#63866—34no—
30C2×#3716—34no—
31C32×C4⋊C8—31no—
32#63643—31no—
33#63643—31no—
34#63643—34no—
35#63643—64no—
36C3×Dic3⋊C8D314yescomplemented
37#63566—34no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)D412yes—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51Dic3×C3⋊D3—42no—
52C33×C8—21no—
53C32×C3⋊C8—21no—
54#5162—91no—
55#4841—94no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—61no—
58C2×#634—31no—
59C2×#634—34no—
60C2×#634—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—34no—
63C2×C3×Dic6—34no—
64C2×D3×Dic3—64no—
65C2×C32×C8—24no—
66C2×C32×C8—31no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C2×D311yescharacteristic
69C32×C42—31no—
70#3752—61no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)—64no—
76#3744—22no—
77#3744—62no—
78C3×C4×Dic3C2×D314yes—
79C2×C3×C3⋊C8C2×D314yes—
80C3×#204—68no—
81Dic3⋊Dic3—64no—
82Dic3⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3716—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—21no—
95C32×Dic3SD1612yescomplemented
96C2×C3×SD16—91no—
97#944—91no—
98D4×Dic3—91no—
99C2×#214—91no—
100C2×#204—34no—
101C2×#204—94no—
102C4×D12—94no—
103#869—94no—
104C3×C4⋊C8—34no—
105#835—91no—
106#816—94no—
107#816—94no—
108#816—91no—
109#816—184no—
110Dic3⋊C8C3⋊D311yescharacteristic, complemented
111C22×C3⋊D3—21no—
112C22×C3×D3—64no—
113C2×C32×C4—24no—
114C2×C32×C4—31no—
115C2×C32×C4—31no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—124no—
124C2×C3⋊Dic3—124no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)#6712yes—
127C4×C3⋊D3—122no—
128#634—21no—
129#634—21no—
130#634—68no—
131#634—61no—
132#634—64no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3D1214yes—
138C2×C3×Dic3D1214yes—
139C3×D12—38no—
140C3×D12—64no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143D3×Dic3—128no—
144C32×C8—21no—
145C32×C8—24no—
146C32×C8—44no—
147C32×C8—61no—
148C3×C3⋊C8—24no—
149#461—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13711yescharacteristic
153C3×C3⋊D3—42no—
154C2×C3×Q8—91no—
155C2×C3×D4—91no—
156C22×Dic3—181no—
157C2×D12—34no—
158C2×C4×D3—184no—
159C2×Dic6—34no—
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C3×SD16—182no—
164C2×C3×C8—24no—
165C2×C3×C8—34no—
166C2×C3×C8D3214yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168C3×C42—34no—
169#217—181no—
170#214—182no—
171(C2×D3)⋊C4—184no—
172C4⋊Dic3—94no—
173C4⋊Dic3—91no—
174C4⋊Dic3—184no—
175C4⋊Dic3—184no—
176C4⋊Dic3—184no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3C2×C3⋊D311yescharacteristic
180C2×C3⋊C8C2×C3⋊D311yescharacteristic
181#204—68no—
182#204—188no—
183C22×C32—24no—
184C22×C32C2×D1214yes—
185C22×C32D3×D411yescharacteristic
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—32no—
195C32×C4—61no—
196C32×C4D3×D414yes—
197C32×C4#23711yescharacteristic
198C32×C4#23914yes—
199C32×C4C2×#6711yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—128no—
202C3⋊Dic3—124no—
203C3⋊Dic3—121no—
204C3⋊Dic3—121no—
205C3⋊Dic3—124no—
206C3⋊Dic3—244no—
207C3×Dic3—24no—
208C3×Dic3—124no—
209C3×Dic3#20418yescomplemented
210C2×SD16—91no—
211C2×SD16—271no—
212C4⋊Q8—271no—
213C4×D4—271no—
214C4⋊C8—31no—
215#103—271no—
216#103—271no—
217C33#46111yescharacteristic, complemented, Sylow
218C23×C3—181no—
219C22×D3—64no—
220C3×Q8—181no—
221C3×Q8—181no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—24no—
225C2×C3×C4—34no—
226C2×C3×C4—34no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—64no—
231C2×Dic3—184no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—364no—
236C2×Dic3—364no—
237C2×Dic3—362no—
238C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
239C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
240D12—38no—
241D12—64no—
242C4×D3—368no—
243Dic6—64no—
244Dic6—64no—
245Dic6—188no—
246Dic6—184no—
247Dic6—184no—
248Dic6—181no—
249Dic6—181no—
250Dic6—364no—
251C3×C8—24no—
252C3×C8—21no—
253C3×C8—44no—
254C3×C8—64no—
255C3⋊C8—21no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32#89514yes—
260C2×C32C2×#20414yes—
261C2×C32#90914yes—
262C2×C32D3×SD1611yescharacteristic
263C2×C32#91511yescharacteristic
264C2×C32#93911yescharacteristic
265C3⋊D3—42no—
266C3×D3—128no—
267C2×Q8—91no—
268C2×Q8—271no—
269C2×D4—91no—
270C22×C4—541no—
271SD16—182no—
272SD16—542no—
273C2×C8—31no—
274C2×C8D3×C3⋊D311yescharacteristic
275C4⋊C4—271no—
276C4⋊C4—541no—
277C4⋊C4—541no—
278C22⋊C4—541no—
279C42—31no—
280C22×C3—24no—
281C22×C3—182no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3D3×D1214yes—
285C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
286C2×D3—68no—
287C2×D3—124no—
288C2×D3—124no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—38no—
292C3×C4—64no—
293C3×C4—361no—
294C3×C4#379614yes—
295C3×C4D3×#6714yes—
296C3×C4D4×C3⋊D311yescharacteristic
297C3×C4#383211yescharacteristic
298Dic3—21no—
299Dic3—124no—
300Dic3—368no—
301Dic3—364no—
302Dic3—361no—
303Dic3—361no—
304Dic3—364no—
305Dic3—724no—
306Dic3#374412yescomplemented
307C32—24no—
308C32#484114yescomplemented
309C32#516211yescharacteristic, complemented
310C23—181no—
311Q8—181no—
312Q8—181no—
313Q8—541no—
314Q8—541no—
315D4—92no—
316D4—181no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—181no—
320C2×C4—541no—
321C2×C4—541no—
322C2×C4—1082no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—21no—
325C8—61no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—362no—
330C2×C3D3×#20414yes—
331C2×C3#6380014yes—
332C2×C3#6388614yes—
333C2×C3#6410411yescharacteristic
334C2×C3C2×#374411yescharacteristic
335C2×C3#6411811yescharacteristic
336D3—128no—
337C22—182no—
338C22—361no—
339C22—361no—
340C22D3×C4⋊(C3⋊D3)11yescharacteristic
341C4—32no—
342C4—61no—
343C4—361no—
344C4—1081no—
345C4—1081no—
346C4#8901711yescharacteristic
347C4C3⋊D3×#6711yescharacteristic
348C3—24no—
349C3#1059075214yescomplemented
350C3#1059254011yescharacteristic, complemented
351C2—362no—
352C2D3×#374411yescharacteristic
353C2#1172332611yescharacteristic
354C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11723443C2semidirect
5. #11722186C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722040C2semidirect
36. C3×Dic3⋊C8D3semidirect
95. C32×Dic3SD16semidirect
110. Dic3⋊C8C3⋊D3semidirect
209. C3×Dic3#204semidirect
217. C33#461semidirect
306. Dic3#3744semidirect
308. C32#4841semidirect
309. C32#5162semidirect
349. C3#10590752semidirect
350. C3#10592540semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
Dic3×C4⋊(C3⋊D3)2
#117234432
C2×#887962
#117221862
C32×Dic3⋊C82
#117220402
#105925403
#105907523