Finite Group Databasebeta

← #13139089

Subgroups — #13139089

Order 1728.

338 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 75 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×#63471C211yesmaximal, characteristic
2C2×#88418C211yesmaximal, characteristic, complemented
3#11720869C211yesmaximal, characteristic
4#11720858C211yesmaximal, characteristic
5#11720556C211yesmaximal, characteristic, complemented
6C9×#822C211yesmaximal, characteristic, complemented
7#11720289C211yesmaximal, characteristic, complemented
8#10590983—31nomaximal
9#10588576—31nomaximal
10C2×#6226C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14#88532C2211yescharacteristic
15C9×#208C2211yescharacteristic
16C2×C9×C3⋊C8C2211yescharacteristic
17C3×#3664—21no—
18C3×Dic36—21no—
19#88418—21no—
20#88418—21no—
21#88418—21no—
22#88415—21no—
23#88413—21no—
24D9×C3⋊C8—21no—
25C3×#909—31no—
26C2×#3719—31no—
27#63831—31no—
28#63814—31no—
29#63651—31no—
30C3×#822D311yescharacteristic
31#63560—31no—
32#63471—31no—
33#63471D311yescharacteristic
34#63464—31no—
35C2×Dic36—31no—
36C9×#108—31no—
37#63382—31no—
38#63382—31no—
39#63382—61no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42#6226—21no—
43#6226—21no—
44C3×#611—21no—
45C2×C3×Dic9—21no—
46C3×D36D411yescharacteristic
47C3×C4×D9—21no—
48C3×Dic18—21no—
49C3×Dic18—21no—
50C3×Dic18D411yescharacteristic
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56#5204—91no—
57#4896—91no—
58C2×#634—31no—
59C3×#235—31no—
60C2×C3×Dic6—31no—
61C32×M16C2×D311yescharacteristic
62#3748—31no—
63C3×#208C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65C3×Dic12—61no—
66C3×#204—61no—
67#3719—61no—
68#3719—61no—
69#3719—61no—
70#3716—61no—
71#3711—61no—
72D3×C3⋊C8—61no—
73#3697C2×D311yescharacteristic
74C2×Dic18—31no—
75C2×Dic18—61no—
76C2×Dic18C2×D311yescharacteristic
77C9×M16—21no—
78C9×M16—31no—
79C9×M16C2×D311yescharacteristic
80C2×C8×C9—31no—
81#3667—31no—
82#3667—61no—
83#3664—21no—
84#3664—61no—
85#3663—61no—
86C8×D9—61no—
87Dic36—21no—
88Dic36—61no—
89Dic36—61no—
90Dic36—61no—
91C22×C3×C9#4111yescharacteristic
92C2×C3×D9—21no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9C2×D411yescharacteristic
95C3⋊Dic9—41no—
96C3×Dic9—21no—
97C3×Dic9—41no—
98C3×#137—91no—
99#951—91no—
100#948—91no—
101C2×#216—91no—
102#909—31no—
103#909—91no—
104C2×Dic12—91no—
105#901—91no—
106C3×#108—31no—
107#836—91no—
108#824—91no—
109#824—91no—
110#824—91no—
111#824—181no—
112#822D911yescharacteristic, complemented
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—61no—
115#634—61no—
116#634—61no—
117C3×#67—61no—
118C2×C3×Dic3—61no—
119C3×D12—31no—
120C3×C4×D3—61no—
121C3×Dic6—31no—
122C3×Dic6—61no—
123C3×Dic6—61no—
124C32×C8—21no—
125C3⋊(C3⋊C8)—61no—
126C3×C3⋊C8—21no—
127C3×C3⋊C8D1211yescharacteristic
128C3×C3⋊C8D1211yescharacteristic
129C2×C4×C9—21no—
130C2×C4×C9C22×D311yescharacteristic
131#611—21no—
132C2×Dic9—21no—
133C2×Dic9—62no—
134C2×Dic9—61no—
135D36#6711yescharacteristic
136C4×D9—21no—
137Dic18—21no—
138Dic18—21no—
139Dic18—61no—
140Dic18—61no—
141Dic18—121no—
142Dic18—121no—
143Dic18#6711yescharacteristic
144C8×C9—21no—
145C8×C9—31no—
146C8×C9—31no—
147C8×C9—41no—
148C8×C9—61no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#473—271noSylow
152C2×C3×C9—21no—
153C2×C3×C9#12111yescharacteristic
154C3×D9—41no—
155C3×#41—91no—
156C2×C3×Q8—91no—
157#235—31no—
158C2×Dic6—31no—
159C2×Dic6—91no—
160C2×Dic6—91no—
161C2×Dic6—181no—
162C3×Q16—181no—
163C3×SD16—181no—
164C3×M16—21no—
165C3×M16—31no—
166C3×M16C2×D911yescharacteristic
167C3×M16D3211yescharacteristic
168C2×C3×C8—31no—
169#216—181no—
170#216—181no—
171#216—181no—
172#214—181no—
173#208—91no—
174#208—91no—
175#208—181no—
176#208—181no—
177#208C2×D911yescharacteristic
178C2×C3⋊C8—181no—
179C2×C3⋊C8C2×D911yescharacteristic
180Dic12—61no—
181Dic12—181no—
182Dic12—181no—
183Dic12—181no—
184#204—61no—
185#204—181no—
186ΓL(1,𝔽25)—181no—
187D3×C8—181no—
188C22×C32#23911yescharacteristic
189C2×C3×D3—61no—
190C32×C4C2×D1211yescharacteristic
191C32×C4D3×D411yescharacteristic
192C3⋊Dic3—121no—
193C3×Dic3—61no—
194C3×Dic3—121no—
195C22×C9—21no—
196C22×C9#23711yescharacteristic
197C2×D9—21no—
198C4×C9—21no—
199C4×C9—21no—
200C4×C9D3×D411yescharacteristic
201C4×C9C2×#6711yescharacteristic
202Dic9—21no—
203Dic9—41no—
204Dic9—122no—
205Dic9—121no—
206#137—91no—
207#137—271no—
208C2×Q16—271no—
209#131—271no—
210#108—31no—
211#101—271no—
212#101—271no—
213C3×C9#47311yescharacteristic, complemented, Sylow
214C3×Q8—91no—
215C3×Q8—181no—
216C3×Q8—181no—
217C3×D4—91no—
218C3×D4—181no—
219C2×C3×C4—21no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4C22×D911yescharacteristic
223C2×C3×C4C2×D3211yescharacteristic
224#67—61no—
225C2×Dic3—61no—
226C2×Dic3—182no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229D12—31no—
230C4×D3—61no—
231Dic6—31no—
232Dic6—61no—
233Dic6—61no—
234Dic6—181no—
235Dic6—181no—
236Dic6—181no—
237Dic6—181no—
238Dic6—361no—
239Dic6—361no—
240C3×C8—21no—
241C3×C8—21no—
242C3×C8—31no—
243C3×C8—31no—
244C3×C8—41no—
245C3×C8—61no—
246C3⋊C8—21no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C3⋊C8D3611yescharacteristic
251C3⋊C8D3611yescharacteristic
252C2×C32—21no—
253C2×C32#89511yescharacteristic
254C3×D3—121no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—41no—
258C2×C9#93911yescharacteristic
259D9—41no—
260#41—91no—
261C2×Q8—91no—
262C2×Q8—271no—
263Q16—181no—
264Q16—541no—
265Q16—541no—
266Q16—541no—
267SD16—181no—
268SD16—541no—
269M16—31no—
270M16—271no—
271M16—541no—
272M16D3×D911yescharacteristic
273C2×C8—31no—
274C2×C8—541no—
275C22×C3—21no—
276C22×C3—181no—
277C22×C3#370111yescharacteristic
278C22×C3#379611yescharacteristic
279C2×D3—61no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—181no—
283C3×C4—361no—
284C3×C4C2×D3611yescharacteristic
285C3×C4D4×D911yescharacteristic
286C3×C4D3×D1211yescharacteristic
287C3×C4D3×#6711yescharacteristic
288Dic3—61no—
289Dic3—121no—
290Dic3—362no—
291Dic3—361no—
292Dic3—361no—
293C32#489611yescharacteristic
294C9—21no—
295C9#520411yescharacteristic, complemented
296Q8—91no—
297Q8—181no—
298Q8—181no—
299Q8—541no—
300Q8—541no—
301D4—91no—
302D4—181no—
303C2×C4—181no—
304C2×C4—181no—
305C2×C4—541no—
306C2×C4C2×D3×D911yescharacteristic
307C8—21no—
308C8—31no—
309C8—31no—
310C8—61no—
311C8—541no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—361no—
317C2×C3#6345711yescharacteristic
318C2×C3#6388611yescharacteristic
319D3—121no—
320C22—181no—
321C22#8863411yescharacteristic
322C4—181no—
323C4—361no—
324C4—1081no—
325C4D3×D3611yescharacteristic
326C4D9×#6711yescharacteristic
327C3—21no—
328C3#1058857611yescharacteristic, complemented
329C3#1059098311yescharacteristic
330C2—21no—
331C2—361no—
332C2#1172098311yescharacteristic
subgroupG/H
centerC2#11720983
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#895
prime radicalC3×#3697C22
hypercenterM16D3×D9
hypocenterC3×C9#473
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C2×#88418C2semidirect
5. #11720556C2semidirect
6. C9×#822C2semidirect
7. #11720289C2semidirect
112. #822D9semidirect
213. C3×C9#473semidirect
295. C9#5204semidirect
328. C3#10588576semidirect

Maximal subgroups

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

subgroupindex
C3×#634712
C2×#884182
#117208692
#117208582
#117205562
C9×#8222
#117202892
#105909833
#105885763