Finite Group Databasebeta

← #13159167

Subgroups — #13159167

Order 1728.

888 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C32×#897C211yesmaximal, characteristic, complemented
2#11723515C211yesmaximal, characteristic
3C2×#88793C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5#11722178C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592747—31nomaximal
9#10590501—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6290—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3—21no—
20#88803—21no—
21#88793—22no—
22D3×C3⋊(C3⋊C8)—22no—
23C32×C22⋊Q8—31no—
24C2×#3755—31no—
25#64139—31no—
26C22×C3⋊(C3⋊C8)—31no—
27C3×#897D314yescomplemented
28#63908—34no—
29C2×#3715—34no—
30C2×D3×C3⋊C8—34no—
31#63662—31no—
32#63636D311yescharacteristic, complemented
33#63635—31no—
34#63635—34no—
35#63635—64no—
36C3×#809—34no—
37#63566—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C32×C4×D3—21no—
48C32×C4×D3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6290—42no—
52C3⋊(C3⋊(C3⋊C8))D412yescomplemented
53C3×C3⋊(C3⋊C8)—21no—
54#5048—94no—
55#4778—91no—
56C2×C32×Q8—31no—
57C22×C32×C4—31no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×C4×D3C2×D314yes—
62C2×C3×Dic6C2×D314yes—
63C2×#625—64no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4—61no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C22⋊C4—61no—
69#3755—62no—
70#3752—61no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—64no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—62no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×(C2×D3)⋊C4—24no—
80C3×C4⋊Dic3—24no—
81C2×C3×C3⋊C8—34no—
82(C2×D3)⋊Dic3—64no—
83#3715—68no—
84D3×C3⋊C8—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3SD1612yescomplemented
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C32×Dic3—21no—
97C32×Dic3—41no—
98C3×C22⋊Q8—34no—
99C2×#215—94no—
100#938—91no—
101C2×#214—91no—
102#930—94no—
103C22×C3⋊C8—94no—
104C2×D3×C8—91no—
105#897C3⋊D311yescharacteristic, complemented
106C3×#102—91no—
107#835—94no—
108#816—91no—
109#809—34no—
110#808—94no—
111#808—91no—
112#808—184no—
113C23×C32—31no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C22×C3×D3#6714yes—
117C32×Q8—61no—
118C32×Q8—61no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—31no—
123C2×C32×C4—44no—
124C2×C32×C4—61no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129#638—122no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—124no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3#6714yes—
137C3×D12—64no—
138C3×D12—64no—
139C3×C4×D3—24no—
140C3×C4×D3—24no—
141C3×Dic6—24no—
142C3×Dic6—24no—
143#625—128no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148C3×C3⋊C8—64no—
149#465—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13511yescharacteristic
153C3×C3⋊D3—81no—
154C32×D3—22no—
155C2×C3×Q8—34no—
156C2×C3×D4—91no—
157C22×C3×C4—34no—
158C2×#67—184no—
159C2×#67—181no—
160C2×D12—34no—
161C2×C4×D3C2×C3⋊D311yescharacteristic
162C2×Dic6C2×C3⋊D311yescharacteristic
163C2×C3×C8—91no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—64no—
166C3×C4⋊C4D3214yes—
167C3×C4⋊C4C2×C3⋊D311yescharacteristic
168C3×C22⋊C4—64no—
169#217—181no—
170#215—188no—
171#214—182no—
172(C2×D3)⋊C4—21no—
173(C2×D3)⋊C4—184no—
174C4⋊Dic3—21no—
175C4⋊Dic3—94no—
176C4⋊Dic3—91no—
177C4⋊Dic3—184no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8—188no—
183D3×C8—182no—
184C22×C32—24no—
185C22×C32—32no—
186C22×C32—61no—
187C22×C32—61no—
188C22×C32D3×D411yescharacteristic
189C22×C32C2×#6714yes—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3×D3—24no—
194C2×C3×D3—24no—
195C2×C3×D3—124no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3#21518yescomplemented
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—41no—
202C32×C4—44no—
203C32×C4—61no—
204C32×C4—84no—
205C32×C4—121no—
206C32×C4#23511yescharacteristic
207C32×C4D3×D414yes—
208C32×C4D3×D411yescharacteristic
209C32×C4#23714yes—
210C3⋊Dic3—121no—
211C3⋊Dic3—124no—
212C3⋊Dic3—244no—
213C3×Dic3—24no—
214C3×Dic3—44no—
215C2×SD16—271no—
216C22×C8—271no—
217C22⋊Q8—31no—
218#121—271no—
219#106—271no—
220#103—271no—
221#102—91no—
222C33#46511yescharacteristic, complemented, Sylow
223C23×C3—34no—
224C23×C3—181no—
225C22×D3—64no—
226C22×D3#63811yescharacteristic
227C3×Q8—64no—
228C3×Q8—64no—
229C3×D4—181no—
230C3×D4—181no—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—24no—
234C2×C3×C4—34no—
235C2×C3×C4—44no—
236C2×C3×C4—64no—
237C2×C3×C4—64no—
238C2×C3×C4—64no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241#67—368no—
242#67—362no—
243C2×Dic3—21no—
244C2×Dic3—184no—
245C2×Dic3—181no—
246C2×Dic3—364no—
247C2×Dic3#63811yescharacteristic
248D12—64no—
249D12—64no—
250C4×D3—21no—
251C4×D3—21no—
252Dic6—21no—
253Dic6—21no—
254C3×C8—181no—
255C3⋊C8—64no—
256C3⋊C8—98no—
257C3⋊C8—92no—
258C3⋊C8—188no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32—62no—
263C2×C32#88411yescharacteristic
264C2×C32#91211yescharacteristic
265C2×C32D3×SD1611yescharacteristic
266C2×C32#93914yes—
267C2×C32C2×#21514yes—
268C2×C32#95014yes—
269C3⋊D3—81no—
270C3×D3—28no—
271C3×D3—244no—
272C2×Q8—31no—
273C2×D4—91no—
274C2×D4—541no—
275C22×C4—31no—
276SD16—542no—
277C2×C8—91no—
278C2×C8—271no—
279C2×C8—542no—
280C4⋊C4—61no—
281C4⋊C4—271no—
282C4⋊C4D3×C3⋊D311yescharacteristic
283C22⋊C4—61no—
284C22⋊C4—541no—
285C22×C3—24no—
286C22×C3—38no—
287C22×C3—64no—
288C22×C3—64no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3D3×#6714yes—
293C22×C3C2×#63811yescharacteristic
294C2×D3—21no—
295C2×D3—21no—
296C2×D3—124no—
297C2×D3—124no—
298C2×D3—124no—
299C2×D3#375512yescomplemented
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—41no—
303C3×C4—44no—
304C3×C4—64no—
305C3×C4—84no—
306C3×C4—124no—
307C3×C4#379514yes—
308C3×C4#380214yes—
309C3×C4D4×C3⋊D311yescharacteristic
310C3×C4#383011yescharacteristic
311Dic3—21no—
312Dic3—41no—
313Dic3—361no—
314Dic3—364no—
315Dic3—724no—
316C32—24no—
317C32#477811yescharacteristic, complemented
318C32#504814yescomplemented
319C23—31no—
320C23—181no—
321Q8—61no—
322Q8—61no—
323D4—181no—
324D4—181no—
325D4—1082no—
326C2×C4—21no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—541no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—181no—
334C8—272no—
335C2×C3—24no—
336C2×C3—24no—
337C2×C3—24no—
338C2×C3—68no—
339C2×C3—721no—
340C2×C3#6388514yes—
341C2×C3#6393314yes—
342C2×C3D3×#21514yes—
343C2×C3#6414811yescharacteristic
344C2×C3C2×#375511yescharacteristic
345C2×C3#6415911yescharacteristic
346D3—22no—
347D3—244no—
348C22—32no—
349C22—61no—
350C22—61no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22D3×#63811yescharacteristic
355C4—41no—
356C4—61no—
357C4—121no—
358C4—1081no—
359C4#8901411yescharacteristic
360C4#8902611yescharacteristic
361C3—24no—
362C3#1059050114yescomplemented
363C3#1059274711yescharacteristic, complemented
364C2—62no—
365C2—721no—
366C2#1172347411yescharacteristic
367C2#1172355611yescharacteristic
368C2D3×#375511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723515C2
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. C32×#897C2semidirect
3. C2×#88793C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. #11722178C2semidirect
6. C3×#63636C2semidirect
7. #11722038C2semidirect
27. C3×#897D3semidirect
32. #63636D3semidirect
52. C3⋊(C3⋊(C3⋊C8))D4semidirect
91. C2×C32×D3SD16semidirect
105. #897C3⋊D3semidirect
198. C2×C3×D3#215semidirect
222. C33#465semidirect
299. C2×D3#3755semidirect
317. C32#4778semidirect
318. C32#5048semidirect
362. C3#10590501semidirect
363. C3#10592747semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
#117235152
C2×#887932
C2×D3×C3⋊(C3⋊C8)2
#117221782
C3×#636362
#117220382
#105927473
#105905013