Finite Group Databasebeta

← #13139088

Subgroups — #13139088

Order 1728.

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

normal subgroups only

This group has 370 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×#63470C211yesmaximal, characteristic, complemented
2#11720873C211yesmaximal, characteristic, complemented
3C2×#88417C211yesmaximal, characteristic, complemented
4#11720858C211yesmaximal, characteristic, complemented
5#11720557C211yesmaximal, characteristic, complemented
6C9×#822C211yesmaximal, characteristic, complemented
7#11720287C211yesmaximal, characteristic, complemented
8#10590984—31nomaximal
9#10588577—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14#88532C2211yescharacteristic
15C9×#208C2211yescharacteristic
16C2×C9×C3⋊C8C2211yescharacteristic
17C3×D72—21no—
18C3×#3664—21no—
19#88417—21no—
20#88417—21no—
21#88417—21no—
22#88416—21no—
23#88413—21no—
24D9×C3⋊C8—21no—
25C3×#908—31no—
26#63825—31no—
27C2×#3714—31no—
28#63814—31no—
29#63650—31no—
30C3×#822D311yescharacteristic
31#63558—31no—
32#63470—31no—
33#63470D311yescharacteristic, complemented
34C2×D72—31no—
35#63464—31no—
36C9×#108—31no—
37#63383—31no—
38#63383—31no—
39#63383—61no—
40C22×C3⋊D9—21no—
41C22×C3×D9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C3×#611—21no—
46C3×D36—21no—
47C3×D36—21no—
48C3×D36D411yescharacteristic
49C3×C4×D9—21no—
50C3×Dic18D411yescharacteristic
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56#5199—91no—
57#4897—91no—
58C2×C4⋊(C3⋊D3)—31no—
59C3×#235—31no—
60C2×C3×D12—31no—
61C32×M16C2×D311yescharacteristic
62#3748—31no—
63C3×#208C2×D311yescharacteristic
64C2×C3×C3⋊C8C2×D311yescharacteristic
65C3×D24—61no—
66C3×#204—61no—
67#3717—61no—
68#3714—61no—
69#3714—61no—
70#3714—61no—
71#3711—61no—
72D3×C3⋊C8—61no—
73#3697C2×D311yescharacteristic
74C2×D36—31no—
75C2×D36—61no—
76C2×D36C2×D311yescharacteristic
77C9×M16—21no—
78C9×M16—31no—
79C9×M16C2×D311yescharacteristic
80C2×C8×C9—31no—
81#3667—31no—
82#3667—61no—
83D72—21no—
84D72—61no—
85D72—61no—
86D72—61no—
87#3664—21no—
88#3664—61no—
89#3663—61no—
90C8×D9—61no—
91C22×C3×C9#4111yescharacteristic
92C2×C3⋊D9—41no—
93C2×C3⋊D9—81no—
94C2×C3×D9—21no—
95C2×C3×D9—41no—
96C2×C3×D9—81no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3×Dic9—21no—
100C3×AGL(1,ℤ/8)—91no—
101#949—91no—
102#948—91no—
103C2×#213—91no—
104#908—31no—
105#908—91no—
106C2×D24—91no—
107#901—91no—
108C3×#108—31no—
109#833—91no—
110#823—91no—
111#823—91no—
112#823—91no—
113#823—181no—
114#822D911yescharacteristic, complemented
115C22×C3⋊D3—61no—
116C22×C3×D3—61no—
117C2×C32×C4C22×D311yescharacteristic
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—61no—
120C3×#67—61no—
121C3×D12—31no—
122C3×D12—61no—
123C3×D12—61no—
124C3×C4×D3—61no—
125C3×Dic6—31no—
126C22×D9—21no—
127C22×D9—62no—
128C22×D9—61no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—21no—
132C3×C3⋊C8D1211yescharacteristic
133C3×C3⋊C8D1211yescharacteristic
134C2×C4×C9—21no—
135C2×C4×C9C22×D311yescharacteristic
136#611—21no—
137D36—21no—
138D36—21no—
139D36—61no—
140D36—61no—
141D36—121no—
142D36—121no—
143D36#6711yescharacteristic
144C4×D9—21no—
145Dic18#6711yescharacteristic
146C8×C9—21no—
147C8×C9—31no—
148C8×C9—31no—
149C8×C9—41no—
150C8×C9—61no—
151C9⋊C8—61no—
152C9⋊C8—121no—
153#472—271noSylow
154C2×C3×C9—21no—
155C2×C3×C9#12111yescharacteristic
156C3⋊D9—81no—
157C3×D9—41no—
158C3×D9—81no—
159C3×#41—91no—
160C2×C3×D4—91no—
161#235—31no—
162C2×D12—31no—
163C2×D12—91no—
164C2×D12—91no—
165C2×D12—181no—
166C3×SD16—181no—
167C3×D8—181no—
168C3×M16—21no—
169C3×M16—31no—
170C3×M16C2×D911yescharacteristic
171C3×M16D3211yescharacteristic
172C2×C3×C8—31no—
173#215—181no—
174#213—181no—
175#213—181no—
176#213—181no—
177#208—91no—
178#208—91no—
179#208—181no—
180#208—181no—
181#208C2×D911yescharacteristic
182C2×C3⋊C8—181no—
183C2×C3⋊C8C2×D911yescharacteristic
184D24—61no—
185D24—181no—
186D24—181no—
187D24—181no—
188#204—61no—
189#204—181no—
190ΓL(1,𝔽25)—181no—
191D3×C8—181no—
192C22×C32#23911yescharacteristic
193C2×C3⋊D3—121no—
194C2×C3⋊D3—241no—
195C2×C3×D3—61no—
196C2×C3×D3—121no—
197C2×C3×D3—241no—
198C32×C4C2×D1211yescharacteristic
199C32×C4D3×D411yescharacteristic
200C3×Dic3—61no—
201C22×C9—21no—
202C22×C9#23711yescharacteristic
203C2×D9—21no—
204C2×D9—41no—
205C2×D9—81no—
206C2×D9—122no—
207C2×D9—121no—
208C2×D9—241no—
209C2×D9—242no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9D3×D411yescharacteristic
213C4×C9C2×#6711yescharacteristic
214Dic9—21no—
215AGL(1,ℤ/8)—91no—
216AGL(1,ℤ/8)—271no—
217C2×D8—271no—
218#131—271no—
219#108—31no—
220#100—271no—
221#100—271no—
222C3×C9#47211yescharacteristic, complemented, Sylow
223C23×C3—181no—
224C22×D3—61no—
225C22×D3—182no—
226C22×D3—181no—
227C22×D3—181no—
228C3×Q8—91no—
229C3×D4—91no—
230C3×D4—181no—
231C3×D4—181no—
232C3×D4—181no—
233C2×C3×C4—21no—
234C2×C3×C4—181no—
235C2×C3×C4C22×D911yescharacteristic
236C2×C3×C4C2×D3211yescharacteristic
237#67—61no—
238D12—31no—
239D12—61no—
240D12—61no—
241D12—181no—
242D12—181no—
243D12—181no—
244D12—181no—
245D12—361no—
246D12—361no—
247C4×D3—61no—
248Dic6—31no—
249C3×C8—21no—
250C3×C8—21no—
251C3×C8—31no—
252C3×C8—31no—
253C3×C8—41no—
254C3×C8—61no—
255C3⋊C8—21no—
256C3⋊C8—181no—
257C3⋊C8—181no—
258C3⋊C8—361no—
259C3⋊C8D3611yescharacteristic
260C3⋊C8D3611yescharacteristic
261C2×C32—21no—
262C2×C32#89511yescharacteristic
263C3⋊D3—241no—
264C3×D3—121no—
265C3×D3—241no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—41no—
269C2×C9#93911yescharacteristic
270D9—41no—
271D9—81no—
272D9—242no—
273D9—241no—
274#41—91no—
275C2×D4—91no—
276C2×D4—271no—
277SD16—181no—
278SD16—541no—
279D8—181no—
280D8—541no—
281D8—541no—
282D8—541no—
283M16—31no—
284M16—271no—
285M16—541no—
286M16D3×D911yescharacteristic
287C2×C8—31no—
288C2×C8—541no—
289C22×C3—21no—
290C22×C3—181no—
291C22×C3—361no—
292C22×C3—721no—
293C22×C3#370111yescharacteristic
294C22×C3#379611yescharacteristic
295C2×D3—61no—
296C2×D3—121no—
297C2×D3—241no—
298C2×D3—362no—
299C2×D3—361no—
300C2×D3—361no—
301C2×D3—721no—
302C2×D3—721no—
303C2×D3—722no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—181no—
307C3×C4C2×D3611yescharacteristic
308C3×C4D4×D911yescharacteristic
309C3×C4D3×D1211yescharacteristic
310C3×C4D3×#6711yescharacteristic
311Dic3—61no—
312C32#489711yescharacteristic
313C9—21no—
314C9#519911yescharacteristic, complemented
315C23—181no—
316C23—541no—
317Q8—91no—
318D4—91no—
319D4—181no—
320D4—181no—
321D4—181no—
322D4—541no—
323D4—541no—
324C2×C4—181no—
325C2×C4C2×D3×D911yescharacteristic
326C8—21no—
327C8—31no—
328C8—31no—
329C8—61no—
330C8—541no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—41no—
335C2×C3—361no—
336C2×C3—721no—
337C2×C3#6345711yescharacteristic
338C2×C3#6388611yescharacteristic
339D3—121no—
340D3—241no—
341D3—722no—
342D3—721no—
343D3—721no—
344C22—181no—
345C22—361no—
346C22—721no—
347C22—1081no—
348C22—2161no—
349C22#8863411yescharacteristic
350C4—181no—
351C4D3×D3611yescharacteristic
352C4D9×#6711yescharacteristic
353C3—21no—
354C3#1058857711yescharacteristic, complemented
355C3#1059098411yescharacteristic
356C2—21no—
357C2—361no—
358C2—721no—
359C2—2161no—
360C2#1172098311yescharacteristic
subgroupG/H
centerC2#11720983
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#895
hypercenterM16D3×D9
hypocenterC3×C9#472
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
1. C3×#63470C2semidirect
2. #11720873C2semidirect
3. C2×#88417C2semidirect
4. #11720858C2semidirect
5. #11720557C2semidirect
6. C9×#822C2semidirect
7. #11720287C2semidirect
33. #63470D3semidirect
114. #822D9semidirect
222. C3×C9#472semidirect
314. C9#5199semidirect
354. C3#10588577semidirect

Maximal subgroups

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

subgroupindex
C3×#634702
#117208732
C2×#884172
#117208582
#117205572
C9×#8222
#117202872
#105909843
#105885773