Finite Group Databasebeta

← #13159962

Subgroups — #13159962

Order 1728.

912 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 156 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×#3745C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3#11723439C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592543—31nomaximal
9#10590755—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×#3745—22no—
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20#88803—21no—
21#88803—21no—
22#88796—22no—
23C2×#3745D311yescharacteristic
24C2×#3744—31no—
25#64084—31no—
26C4×C4⋊(C3⋊D3)—31no—
27C2×C3×D24—34no—
28Dic3×D12—34no—
29#63860—34no—
30C2×#3716—34no—
31C32×C4⋊C8—31no—
32#63648—31no—
33#63643—31no—
34#63643—34no—
35#63643—64no—
36C3×Dic3⋊C8D314yescomplemented
37#63563—34no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)D412yes—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51Dic3×C3⋊D3—42no—
52C33×C8—21no—
53C32×C3⋊C8—21no—
54#5153—91no—
55#4844—94no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—61no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×D12—34no—
63C2×C3×D12—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—
71#3752—61no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—64no—
76#3745—22no—
77#3744—62no—
78C3×C4×Dic3C2×D314yes—
79C2×C3×C3⋊C8C2×D314yes—
80C3×D24—68no—
81(C2×D3)⋊Dic3—64no—
82(C2×D3)⋊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—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C3⋊(C3⋊Dic3)—41no—
96C32×Dic3—21no—
97C32×Dic3—21no—
98C2×C3×D8—91no—
99#936—91no—
100D4×Dic3—91no—
101C2×#214—91no—
102C2×D24—34no—
103C2×#204—94no—
104#875—94no—
105C4×D12—94no—
106C3×C4⋊C8—34no—
107#832—91no—
108#821—94no—
109#816—94no—
110#816—91no—
111#816—184no—
112Dic3⋊C8C3⋊D311yescharacteristic, complemented
113C22×C3⋊D3—21no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C22×C3×D3—64no—
117C2×C32×C4—24no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—124no—
127C2×C3⋊Dic3—124no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)#6712yes—
132C4×C3⋊D3—122no—
133#634—68no—
134#634—61no—
135#634—64no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3D1214yes—
140C2×C3×Dic3D1214yes—
141C3×D12—38no—
142C3×D12—64no—
143C3×D12—64no—
144C3×D12—64no—
145D3×Dic3—128no—
146C32×C8—21no—
147C32×C8—24no—
148C32×C8—44no—
149C32×C8—61no—
150C3×C3⋊C8—24no—
151#463—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C3×C3⋊D3—42no—
156C3×C3⋊D3—81no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C22×Dic3—181no—
160C2×D12—34no—
161C2×D12—34no—
162C2×C4×D3—184no—
163C2×Dic6—94no—
164C2×Dic6—91no—
165C2×Dic6—184no—
166C3×D8—182no—
167C2×C3×C8—24no—
168C2×C3×C8—34no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C42—34no—
172#217—181no—
173#217—181no—
174#217—181no—
175#214—182no—
176(C2×D3)⋊C4—184no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—184no—
179C4⋊Dic3—94no—
180C4⋊Dic3—91no—
181C4⋊Dic3—184no—
182C4×Dic3C2×C3⋊D311yescharacteristic
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184D24—68no—
185#204—188no—
186C22×C32—24no—
187C22×C32C2×D1214yes—
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—22no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3×D3—68no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C2×C3×D3—124no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—61no—
204C32×C4—61no—
205C32×C4D3×D414yes—
206C32×C4#23711yescharacteristic
207C32×C4#23914yes—
208C32×C4C2×#6711yescharacteristic
209C3⋊Dic3—128no—
210C3⋊Dic3—124no—
211C3⋊Dic3—121no—
212C3⋊Dic3—121no—
213C3⋊Dic3—124no—
214C3⋊Dic3—244no—
215C3×Dic3—24no—
216C3×Dic3—24no—
217C2×SD16—271no—
218C2×D8—91no—
219#124—271no—
220C4×D4—271no—
221C4⋊C8—31no—
222#103—271no—
223#102—271no—
224C33#46311yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C23×C3—181no—
227C22×D3—64no—
228C22×D3—64no—
229C3×D4—92no—
230C3×D4—181no—
231C3×D4—181no—
232C3×D4—181no—
233C2×C3×C4—24no—
234C2×C3×C4—34no—
235C2×C3×C4—34no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—184no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C2×Dic3—364no—
243C2×Dic3—364no—
244C2×Dic3—362no—
245C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
246C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
247D12—38no—
248D12—64no—
249D12—64no—
250D12—64no—
251C4×D3—368no—
252Dic6—188no—
253Dic6—184no—
254Dic6—184no—
255Dic6—181no—
256Dic6—181no—
257Dic6—364no—
258C3×C8—24no—
259C3×C8—21no—
260C3×C8—44no—
261C3×C8—64no—
262C3⋊C8—21no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32#89514yes—
267C2×C32#90414yes—
268C2×C32#90814yes—
269C2×C32#91111yescharacteristic
270C2×C32#91211yescharacteristic
271C2×C32#93911yescharacteristic
272C3⋊D3—42no—
273C3⋊D3—81no—
274C3×D3—128no—
275C3×D3—244no—
276C2×Q8—271no—
277C2×D4—91no—
278C2×D4—91no—
279C22×C4—541no—
280SD16—542no—
281D8—182no—
282C2×C8—31no—
283C2×C8D3×C3⋊D311yescharacteristic
284C4⋊C4—271no—
285C22⋊C4—541no—
286C22⋊C4—541no—
287C22⋊C4—541no—
288C42—31no—
289C22×C3—24no—
290C22×C3—182no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D3×D1214yes—
297C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
298C2×D3—68no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—124no—
302C2×D3—124no—
303C2×D3—124no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—64no—
307C3×C4—64no—
308C3×C4#379614yes—
309C3×C4D3×#6714yes—
310C3×C4D4×C3⋊D311yescharacteristic
311C3×C4#383211yescharacteristic
312Dic3—21no—
313Dic3—21no—
314Dic3—368no—
315Dic3—364no—
316Dic3—361no—
317Dic3—361no—
318Dic3—364no—
319Dic3—724no—
320C32—24no—
321C32#484414yescomplemented
322C32#515311yescharacteristic, complemented
323C23—181no—
324C23—181no—
325Q8—541no—
326Q8—541no—
327D4—92no—
328D4—181no—
329D4—181no—
330D4—181no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4—1082no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—21no—
338C8—61no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—362no—
343C2×C3—721no—
344C2×C3#6379514yes—
345C2×C3#6380714yes—
346C2×C3#6388614yes—
347C2×C3#6410411yescharacteristic
348C2×C3#6411311yescharacteristic
349C2×C3#6411711yescharacteristic
350D3—128no—
351D3—244no—
352C22—182no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22D3×C4⋊(C3⋊D3)11yescharacteristic
359C4—61no—
360C4—61no—
361C4—1081no—
362C4—1081no—
363C4#8901711yescharacteristic
364C4C3⋊D3×#6711yescharacteristic
365C3—24no—
366C3#1059075514yescomplemented
367C3#1059254311yescharacteristic, complemented
368C2—362no—
369C2—721no—
370C2#1172331611yescharacteristic
371C2#1172333611yescharacteristic
372C2#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×#3745C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#463
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
2. Dic3×C4⋊(C3⋊D3)C2semidirect
3. #11723439C2semidirect
4. C2×#88796C2semidirect
5. #11722186C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722034C2semidirect
36. C3×Dic3⋊C8D3semidirect
112. Dic3⋊C8C3⋊D3semidirect
224. C33#463semidirect
321. C32#4844semidirect
322. C32#5153semidirect
366. C3#10590755semidirect
367. C3#10592543semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37452
Dic3×C4⋊(C3⋊D3)2
#117234392
C2×#887962
#117221862
C32×Dic3⋊C82
#117220342
#105925433
#105907553