Finite Group Databasebeta

← #13137608

Subgroups — #13137608

Order 1728.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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×#63457C211yesmaximal, characteristic
2#11720945C211yesmaximal, characteristic, complemented
3C2×#88415C211yesmaximal, characteristic, complemented
4C2×#88413C211yesmaximal, characteristic, complemented
5#11720527C211yesmaximal, characteristic, complemented
6C9×#807C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590588—31nomaximal
9#10588566—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×D36—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C3×C4×D9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20Dic3⋊Dic9—21no—
21#88415—22no—
22#88413—22no—
23C3×#895—31no—
24#63857—31no—
25C2×#3716—31no—
26C2×#3711—31no—
27#63637—31no—
28C3×#807D311yescharacteristic
29#63562—31no—
30C2×#3664—31no—
31C2×#3663—31no—
32#63459—31no—
33#63457D311yescharacteristic
34C9×#107—31no—
35#63369—31no—
36#63368—31no—
37#63368—61no—
38C22×C3×D9—21no—
39C22×C3×D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9D411yescharacteristic
47C3×D36—21no—
48C3×D36—21no—
49C3×D36—42no—
50C3×C4×D9—21no—
51C3×C4×D9—21no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8D412yescomplemented
54#5042—91no—
55#4886—91no—
56C2×#634—31no—
57C2×C3×D12—31no—
58C2×C3×D12—61no—
59C2×C3×C4×D3—31no—
60C32×C4⋊C4C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)—31no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67#3716—62no—
68#3711—62no—
69C2×D36—21no—
70C2×D36C2×D311yescharacteristic
71C2×C4×D9C2×D311yescharacteristic
72C2×Dic18—31no—
73C2×Dic18—61no—
74C2×C8×C9—31no—
75C9×C4⋊C4—21no—
76C9×C4⋊C4—31no—
77C9×C4⋊C4C2×D311yescharacteristic
78(C2×D9)⋊C4—21no—
79(C2×D9)⋊C4—61no—
80C4⋊Dic9—61no—
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83#3664—62no—
84#3663—62no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C2×C3×D9—21no—
89C2×C3×D9—41no—
90C2×C3×D9—41no—
91C2×C3×D9—41no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3⋊Dic9—41no—
96C9×Dic3—41no—
97C3×Dic9—21no—
98C3×#121—91no—
99C2×#214—91no—
100#924—91no—
101C2×#208—91no—
102C2×#204—91no—
103C2×ΓL(1,𝔽25)—91no—
104#897—91no—
105#895—31no—
106C3×#107—31no—
107#832—91no—
108#810—91no—
109#810—91no—
110#810—181no—
111#809—91no—
112#807D911yescharacteristic, complemented
113C22×C3×D3—31no—
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118#634—61no—
119#634—61no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—31no—
122C3×D12—61no—
123C3×D12—61no—
124C3×D12—122no—
125C3×C4×D3—61no—
126C3×C4×D3—61no—
127C22×D9—21no—
128C22×D9#6711yescharacteristic
129C3⋊(C3⋊C8)—61no—
130C3×C3⋊C8D1212yes—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—41no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136C2×Dic9—61no—
137C2×Dic9—121no—
138C2×Dic9#6711yescharacteristic
139D36—21no—
140D36—21no—
141D36—42no—
142C4×D9—21no—
143C4×D9—21no—
144Dic18—62no—
145Dic18—61no—
146Dic18—61no—
147Dic18—121no—
148C8×C9—32no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#468—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
154C2×C3×C9#13711yescharacteristic
155C3×D9—41no—
156C3×D9—81no—
157C2×C3×D4—91no—
158C2×C3×D4—181no—
159C22×C3×C4—91no—
160C2×D12—31no—
161C2×D12—61no—
162C2×C4×D3—31no—
163C2×Dic6—91no—
164C2×Dic6—91no—
165C2×Dic6—181no—
166C2×C3×C8—31no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4C2×D911yescharacteristic
170C3×C4⋊C4D3211yescharacteristic
171C3×C22⋊C4—181no—
172#217—181no—
173#214—182no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—181no—
177C4⋊Dic3C2×D911yescharacteristic
178Dic3⋊C4—181no—
179#208—182no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—181no—
183C2×C3⋊C8C2×D911yescharacteristic
184#204—182no—
185ΓL(1,𝔽25)—182no—
186C22×C32D3×D411yescharacteristic
187C2×C3×D3—61no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C32×C4—41no—
194C32×C4C2×D1211yescharacteristic
195C32×C4#23911yescharacteristic
196C3⋊Dic3—121no—
197C3×Dic3—41no—
198C3×Dic3—61no—
199C22×C9—21no—
200C22×C9C2×#6711yescharacteristic
201C2×D9—21no—
202C2×D9—21no—
203C2×D9—21no—
204C2×D9—41no—
205C2×D9—41no—
206C2×D9—41no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—41no—
210C4×C9—81no—
211C4×C9—121no—
212C4×C9D3×D411yescharacteristic
213C4×C9#23711yescharacteristic
214Dic9—21no—
215Dic9—122no—
216Dic9—121no—
217C2×SD16—271no—
218C2×M16—271no—
219C22⋊Q8—271no—
220#121—91no—
221#107—31no—
222#103—271no—
223#102—271no—
224C3×C9#46811yescharacteristic, complemented, Sylow
225C23×C3—91no—
226C23×C3—181no—
227C22×D3—31no—
228C22×D3—61no—
229C3×D4—181no—
230C3×D4—181no—
231C3×D4—362no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C22×D911yescharacteristic
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—21no—
243C2×Dic3—31no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C2×Dic3—361no—
247D12—61no—
248D12—61no—
249D12—122no—
250C4×D3—61no—
251C4×D3—61no—
252Dic6—182no—
253Dic6—181no—
254Dic6—181no—
255Dic6—181no—
256Dic6—181no—
257Dic6—361no—
258C3×C8—32no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C3⋊C8—361no—
262C3⋊C8D3612yescomplemented
263C2×C32#89511yescharacteristic
264C2×C32#91111yescharacteristic
265C2×C32#91811yescharacteristic
266C3×D3—121no—
267C3×D3—241no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9#93211yescharacteristic
272C2×C9#93911yescharacteristic
273C2×C9#95111yescharacteristic
274D9—41no—
275D9—81no—
276C2×Q8—271no—
277C2×D4—91no—
278C2×D4—181no—
279C22×C4—91no—
280SD16—542no—
281M16—542no—
282C2×C8—31no—
283C2×C8—271no—
284C4⋊C4—31no—
285C4⋊C4—541no—
286C4⋊C4D3×D911yescharacteristic
287C22⋊C4—181no—
288C22⋊C4—541no—
289C22×C3—21no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D4×D911yescharacteristic
297C22×C3D3×#6711yescharacteristic
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—121no—
302C2×D3—121no—
303C2×D3—121no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—81no—
309C3×C4—121no—
310C3×C4—181no—
311C3×C4C2×D3611yescharacteristic
312C3×C4#370111yescharacteristic
313C3×C4#379611yescharacteristic
314C3×C4D3×D1211yescharacteristic
315Dic3—41no—
316Dic3—61no—
317Dic3—362no—
318Dic3—361no—
319Dic3—361no—
320C32#488611yescharacteristic
321C9—21no—
322C9#504211yescharacteristic, complemented
323C23—91no—
324C23—181no—
325Q8—541no—
326Q8—541no—
327D4—181no—
328D4—181no—
329D4—362no—
330C2×C4—21no—
331C2×C4—61no—
332C2×C4—91no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—541no—
336C2×C4C2×D3×D911yescharacteristic
337C8—32no—
338C8—541no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—361no—
343C2×C3—721no—
344C2×C3#6345711yescharacteristic
345C2×C3#6347311yescharacteristic
346C2×C3#6348011yescharacteristic
347C2×C3#6388611yescharacteristic
348C2×C3#6393211yescharacteristic
349C2×C3#6394311yescharacteristic
350D3—121no—
351D3—241no—
352C22—181no—
353C22—181no—
354C22—181no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22D9×#6711yescharacteristic
359C4—41no—
360C4—121no—
361C4—181no—
362C4—1081no—
363C4#8863411yescharacteristic
364C4D3×D3611yescharacteristic
365C3—21no—
366C3#1058856611yescharacteristic, complemented
367C3#1059058811yescharacteristic
368C2—361no—
369C2—721no—
370C2#1172098311yescharacteristic
371C2#1172117611yescharacteristic
372C2#1172121911yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC3×#63457C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#468
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11720945C2semidirect
3. C2×#88415C2semidirect
4. C2×#88413C2semidirect
5. #11720527C2semidirect
6. C9×#807C2semidirect
7. #11720266C2semidirect
53. C9×C3⋊C8D4semidirect
112. #807D9semidirect
224. C3×C9#468semidirect
262. C3⋊C8D36semidirect
322. C9#5042semidirect
366. C3#10588566semidirect

Maximal subgroups

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

subgroupindex
C3×#634572
#117209452
C2×#884152
C2×#884132
#117205272
C9×#8072
#117202662
#105905883
#105885663