Finite Group Databasebeta

← #13159152

Subgroups — #13159152

Order 1728.

888 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 150 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×#64106C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic
3C2×#88792C211yesmaximal, characteristic, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
5#11722178C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592477—31nomaximal
9#10590501—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88792—22no—
22C3⋊D3×C3⋊C8—22no—
23#64147—31no—
24C2×#3754—31no—
25C2×C8×C3⋊D3—31no—
26#64106D311yescharacteristic, complemented
27C3×#897—34no—
28#63908—34no—
29C2×#3715—34no—
30C2×D3×C3⋊C8—34no—
31C32×#102—31no—
32#63643—31no—
33#63635—31no—
34#63635—34no—
35#63635—64no—
36C3×#809D314yescomplemented
37#63566—34no—
38C22×C3×C3⋊D3D411yescharacteristic
39C22×C32×D3—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6290—42no—
52C3⋊(C3⋊(C3⋊C8))D412yescomplemented
53C32×C3⋊C8—21no—
54#5048—91no—
55#4778—94no—
56C2×C32×D4—31no—
57C2×#638—61no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12C2×D314yes—
61C2×C3×C4×D3—34no—
62C2×C3×Dic6—34no—
63C2×#625—64no—
64C2×C32×C8—31no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D311yescharacteristic
68#3757—61no—
69#3754—62no—
70#3752—21no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C8×C3⋊D3—62no—
79C3×(C2×D3)⋊C4—64no—
80C3×C4⋊Dic3—64no—
81C2×C3×C3⋊C8C2×D314yes—
82(C2×D3)⋊Dic3—64no—
83#3715—68no—
84D3×C3⋊C8—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3SD1612yescomplemented
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—41no—
98C3×C22⋊Q8—91no—
99C2×#215—91no—
100#938—94no—
101C2×#214—94no—
102#930—91no—
103C22×C3⋊C8—91no—
104C2×D3×C8—94no—
105#897—34no—
106C3×#102—34no—
107#835—91no—
108#816—94no—
109#809C3⋊D311yescharacteristic, complemented
110#808—94no—
111#808—91no—
112#808—184no—
113C23×C32—61no—
114C22×C3⋊D3#6711yescharacteristic
115C22×C3×D3—24no—
116C22×C3×D3—34no—
117C32×D4—61no—
118C32×D4—61no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—44no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125#638—122no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3#6711yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—64no—
137C3×D12—24no—
138C3×D12—24no—
139C3×C4×D3—64no—
140C3×C4×D3—64no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143#625—128no—
144C32×C8—61no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148C3×C3⋊C8—24no—
149#465—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13511yescharacteristic
153C3×C3⋊D3—22no—
154C32×D3—81no—
155C2×C3×Q8—91no—
156C2×C3×D4—34no—
157C22×C3×C4—91no—
158C2×#67—181no—
159C2×#67—184no—
160C2×D12C2×C3⋊D311yescharacteristic
161C2×C4×D3—34no—
162C2×Dic6—34no—
163C2×C3×C8—34no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4D3214yes—
167C3×C4⋊C4C2×C3⋊D311yescharacteristic
168C3×C22⋊C4—181no—
169#217—184no—
170#215—182no—
171#214—188no—
172(C2×D3)⋊C4—64no—
173(C2×D3)⋊C4—181no—
174C4⋊Dic3—64no—
175C4⋊Dic3—94no—
176C4⋊Dic3—91no—
177C4⋊Dic3—184no—
178C2×C3⋊C8—94no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—184no—
181C2×C3⋊C8—182no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183D3×C8—188no—
184C22×C32—24no—
185C22×C32—121no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32D3×D414yes—
189C22×C32C2×#6711yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3#21512yescomplemented
193C2×C3×D3—38no—
194C2×C3×D3—44no—
195C2×C3×D3—44no—
196C2×C3×D3—44no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—41no—
202C32×C4—44no—
203C32×C4—84no—
204C32×C4#23514yes—
205C32×C4D3×D414yes—
206C32×C4D3×D411yescharacteristic
207C32×C4#23711yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3—41no—
210C3⋊Dic3—121no—
211C3⋊Dic3—124no—
212C3⋊Dic3—244no—
213C3×Dic3—64no—
214C3×Dic3—124no—
215C2×SD16—271no—
216C22×C8—271no—
217C22⋊Q8—91no—
218#121—271no—
219#106—271no—
220#103—271no—
221#102—31no—
222C33#46511yescharacteristic, complemented, Sylow
223C23×C3—64no—
224C23×C3—91no—
225C22×D3—21no—
226C22×D3—34no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×D4—64no—
230C3×D4—64no—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—24no—
234C2×C3×C4—44no—
235C2×C3×C4—91no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241#67—368no—
242#67—362no—
243C2×Dic3—34no—
244C2×Dic3—64no—
245C2×Dic3—184no—
246C2×Dic3—181no—
247C2×Dic3—364no—
248D12—21no—
249D12—21no—
250C4×D3—64no—
251C4×D3—64no—
252Dic6—64no—
253Dic6—64no—
254C3×C8—64no—
255C3⋊C8—21no—
256C3⋊C8—98no—
257C3⋊C8—92no—
258C3⋊C8—188no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32—241no—
263C2×C32#88414yes—
264C2×C32#91214yes—
265C2×C32D3×SD1614yes—
266C2×C32#93911yescharacteristic
267C2×C32C2×#21511yescharacteristic
268C2×C32#95011yescharacteristic
269C3⋊D3—22no—
270C3×D3—68no—
271C3×D3—84no—
272C2×Q8—91no—
273C2×D4—31no—
274C2×D4—541no—
275C22×C4—91no—
276SD16—542no—
277C2×C8—31no—
278C2×C8—271no—
279C2×C8—542no—
280C4⋊C4—181no—
281C4⋊C4—271no—
282C4⋊C4D3×C3⋊D311yescharacteristic
283C22⋊C4—181no—
284C22⋊C4—541no—
285C22×C3—24no—
286C22×C3—92no—
287C22×C3—124no—
288C22×C3—124no—
289C22×C3—124no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3D3×#6714yes—
293C22×C3D4×C3⋊D311yescharacteristic
294C2×D3—38no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—64no—
299C2×D3—64no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—41no—
303C3×C4—44no—
304C3×C4—84no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4#379514yes—
308C3×C4#380214yes—
309C3×C4#382811yescharacteristic
310C3×C4D4×C3⋊D311yescharacteristic
311Dic3—64no—
312Dic3—124no—
313Dic3—361no—
314Dic3—364no—
315Dic3—724no—
316C32—24no—
317C32#477814yescomplemented
318C32#504811yescharacteristic, complemented
319C23—61no—
320C23—91no—
321Q8—181no—
322Q8—181no—
323D4—61no—
324D4—61no—
325D4—1082no—
326C2×C4—21no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—541no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—61no—
334C8—272no—
335C2×C3—24no—
336C2×C3—24no—
337C2×C3—24no—
338C2×C3—182no—
339C2×C3—244no—
340C2×C3#6388514yes—
341C2×C3#6393314yes—
342C2×C3D3×#21514yes—
343C2×C3#6409311yescharacteristic
344C2×C3#6412111yescharacteristic
345C2×C3SD16×C3⋊D311yescharacteristic
346D3—68no—
347D3—81no—
348C22—92no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22—181no—
353C22—181no—
354C22C3⋊D3×#6711yescharacteristic
355C4—41no—
356C4—181no—
357C4—361no—
358C4—1081no—
359C4#8902111yescharacteristic
360C4#8902611yescharacteristic
361C3—24no—
362C3#1059050114yescomplemented
363C3#1059247711yescharacteristic, complemented
364C2—182no—
365C2—241no—
366C2#1172351911yescharacteristic
367C2#1172355911yescharacteristic
368C2C3⋊D3×#21511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723522C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#465
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
1. C3×#64106C2semidirect
3. C2×#88792C2semidirect
4. C2×C3⋊D3×C3⋊C8C2semidirect
5. #11722178C2semidirect
6. C32×#809C2semidirect
7. #11722040C2semidirect
26. #64106D3semidirect
36. C3×#809D3semidirect
52. C3⋊(C3⋊(C3⋊C8))D4semidirect
88. C2×C3×C3⋊D3SD16semidirect
109. #809C3⋊D3semidirect
192. C2×C3⋊D3#215semidirect
222. C33#465semidirect
317. C32#4778semidirect
318. C32#5048semidirect
362. C3#10590501semidirect
363. C3#10592477semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
#117235222
C2×#887922
C2×C3⋊D3×C3⋊C82
#117221782
C32×#8092
#117220402
#105924773
#105905013