Finite Group Databasebeta

← #13139090

Subgroups — #13139090

Order 1728.

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

normal subgroups only

This group has 350 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
2#11720869C211yesmaximal, characteristic, complemented
3C2×#88415C211yesmaximal, characteristic, complemented
4#11720858C211yesmaximal, characteristic, complemented
5#11720556C211yesmaximal, characteristic, complemented
6C9×#822C211yesmaximal, characteristic, complemented
7#11720287C211yesmaximal, characteristic, complemented
8#10590986—31nomaximal
9#10588578—31nomaximal
10C2×#6226C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14#88532C2211yescharacteristic
15C9×#208C2211yescharacteristic
16C2×C9×C3⋊C8C2211yescharacteristic
17C3×D72—21no—
18C3×#3664—21no—
19#88418—21no—
20#88415—21no—
21#88415—21no—
22#88415—21no—
23#88413—21no—
24D9×C3⋊C8—21no—
25C3×#908—31no—
26#63831—31no—
27C2×#3716—31no—
28#63814—31no—
29#63651—31no—
30C3×#822D311yescharacteristic
31#63558—31no—
32#63471—31no—
33#63470D311yescharacteristic
34C2×#3664—31no—
35#63464—31no—
36C9×#108—31no—
37#63383—31no—
38#63382—31no—
39#63382—61no—
40C22×C3×D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43#6226—21no—
44#6226—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#5200—91no—
57#4898—91no—
58C2×#634—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#3719—61no—
68#3716—61no—
69#3716—61no—
70#3716—61no—
71#3711—61no—
72D3×C3⋊C8—61no—
73#3697C2×D311yescharacteristic
74C2×D36C2×D311yescharacteristic
75C2×Dic18—31no—
76C2×Dic18—61no—
77C9×M16—21no—
78C9×M16—31no—
79C9×M16C2×D311yescharacteristic
80C2×C8×C9—31no—
81#3667—31no—
82#3667—61no—
83D72—21no—
84#3664—21no—
85#3664—61no—
86#3664—61no—
87#3664—61no—
88#3663—61no—
89C8×D9—61no—
90Dic36—61no—
91C22×C3×C9#4111yescharacteristic
92C2×C3×D9—21no—
93C2×C3×D9—41no—
94C2×C3×D9—81no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C3⋊Dic9—41no—
98C3×Dic9—21no—
99C3×AGL(1,ℤ/8)—91no—
100#951—91no—
101#948—91no—
102C2×#214—91no—
103#909—91no—
104#908—31no—
105C2×#204—91no—
106#901—91no—
107C3×#108—31no—
108#833—91no—
109#824—91no—
110#824—91no—
111#824—181no—
112#823—91no—
113#822D911yescharacteristic, complemented
114C22×C3×D3—61no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—61no—
117#634—61no—
118#634—61no—
119C3×#67—61no—
120C3×D12—31no—
121C3×D12—61no—
122C3×D12—61no—
123C3×C4×D3—61no—
124C3×Dic6—31no—
125C22×D9—21no—
126C32×C8—21no—
127C3⋊(C3⋊C8)—61no—
128C3×C3⋊C8—21no—
129C3×C3⋊C8D1211yescharacteristic
130C3×C3⋊C8D1211yescharacteristic
131C2×C4×C9—21no—
132C2×C4×C9C22×D311yescharacteristic
133#611—21no—
134C2×Dic9—62no—
135C2×Dic9—61no—
136D36—21no—
137D36—21no—
138D36#6711yescharacteristic
139C4×D9—21no—
140Dic18—61no—
141Dic18—61no—
142Dic18—121no—
143Dic18—121no—
144Dic18#6711yescharacteristic
145C8×C9—21no—
146C8×C9—31no—
147C8×C9—31no—
148C8×C9—41no—
149C8×C9—61no—
150C9⋊C8—61no—
151C9⋊C8—121no—
152#471—271noSylow
153C2×C3×C9—21no—
154C2×C3×C9#12111yescharacteristic
155C3×D9—41no—
156C3×D9—81no—
157C3×#41—91no—
158C2×C3×D4—91no—
159#235—31no—
160C2×D12—31no—
161C2×Dic6—91no—
162C2×Dic6—91no—
163C2×Dic6—181no—
164C3×SD16—181no—
165C3×D8—181no—
166C3×M16—21no—
167C3×M16—31no—
168C3×M16C2×D911yescharacteristic
169C3×M16D3211yescharacteristic
170C2×C3×C8—31no—
171#216—181no—
172#214—181no—
173#214—181no—
174#214—181no—
175#208—91no—
176#208—91no—
177#208—181no—
178#208—181no—
179#208C2×D911yescharacteristic
180C2×C3⋊C8—181no—
181C2×C3⋊C8C2×D911yescharacteristic
182Dic12—181no—
183D24—61no—
184#204—61no—
185#204—181no—
186#204—181no—
187#204—181no—
188ΓL(1,𝔽25)—181no—
189D3×C8—181no—
190C22×C32#23911yescharacteristic
191C2×C3×D3—61no—
192C2×C3×D3—121no—
193C2×C3×D3—241no—
194C32×C4C2×D1211yescharacteristic
195C32×C4D3×D411yescharacteristic
196C3⋊Dic3—121no—
197C3×Dic3—61no—
198C22×C9—21no—
199C22×C9#23711yescharacteristic
200C2×D9—21no—
201C2×D9—41no—
202C2×D9—81no—
203C4×C9—21no—
204C4×C9—21no—
205C4×C9D3×D411yescharacteristic
206C4×C9C2×#6711yescharacteristic
207Dic9—21no—
208Dic9—122no—
209Dic9—121no—
210#137—271no—
211AGL(1,ℤ/8)—91no—
212C2×SD16—271no—
213#131—271no—
214#108—31no—
215#101—271no—
216#100—271no—
217C3×C9#47111yescharacteristic, complemented, Sylow
218C23×C3—181no—
219C22×D3—61no—
220C3×Q8—91no—
221C3×D4—91no—
222C3×D4—181no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—21no—
226C2×C3×C4—181no—
227C2×C3×C4C22×D911yescharacteristic
228C2×C3×C4C2×D3211yescharacteristic
229#67—61no—
230C2×Dic3—182no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233D12—31no—
234D12—61no—
235D12—61no—
236C4×D3—61no—
237Dic6—31no—
238Dic6—181no—
239Dic6—181no—
240Dic6—181no—
241Dic6—181no—
242Dic6—361no—
243Dic6—361no—
244C3×C8—21no—
245C3×C8—21no—
246C3×C8—31no—
247C3×C8—31no—
248C3×C8—41no—
249C3×C8—61no—
250C3⋊C8—21no—
251C3⋊C8—181no—
252C3⋊C8—181no—
253C3⋊C8—361no—
254C3⋊C8D3611yescharacteristic
255C3⋊C8D3611yescharacteristic
256C2×C32—21no—
257C2×C32#89511yescharacteristic
258C3×D3—121no—
259C3×D3—241no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—41no—
263C2×C9#93911yescharacteristic
264D9—41no—
265D9—81no—
266#41—91no—
267C2×Q8—271no—
268C2×D4—91no—
269Q16—541no—
270SD16—181no—
271SD16—541no—
272SD16—541no—
273SD16—541no—
274D8—181no—
275M16—31no—
276M16—271no—
277M16—541no—
278M16D3×D911yescharacteristic
279C2×C8—31no—
280C2×C8—541no—
281C22×C3—21no—
282C22×C3—181no—
283C22×C3—361no—
284C22×C3—721no—
285C22×C3#370111yescharacteristic
286C22×C3#379611yescharacteristic
287C2×D3—61no—
288C2×D3—121no—
289C2×D3—241no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—181no—
293C3×C4C2×D3611yescharacteristic
294C3×C4D4×D911yescharacteristic
295C3×C4D3×D1211yescharacteristic
296C3×C4D3×#6711yescharacteristic
297Dic3—61no—
298Dic3—362no—
299Dic3—361no—
300Dic3—361no—
301C32#489811yescharacteristic
302C9—21no—
303C9#520011yescharacteristic, complemented
304C23—181no—
305Q8—91no—
306Q8—541no—
307Q8—541no—
308D4—91no—
309D4—181no—
310D4—181no—
311D4—181no—
312C2×C4—181no—
313C2×C4—541no—
314C2×C4C2×D3×D911yescharacteristic
315C8—21no—
316C8—31no—
317C8—31no—
318C8—61no—
319C8—541no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—41no—
324C2×C3—361no—
325C2×C3—721no—
326C2×C3#6345711yescharacteristic
327C2×C3#6388611yescharacteristic
328D3—121no—
329D3—241no—
330C22—181no—
331C22—361no—
332C22—721no—
333C22#8863411yescharacteristic
334C4—181no—
335C4—1081no—
336C4D3×D3611yescharacteristic
337C4D9×#6711yescharacteristic
338C3—21no—
339C3#1058857811yescharacteristic, complemented
340C3#1059098611yescharacteristic
341C2—21no—
342C2—361no—
343C2—721no—
344C2#1172098311yescharacteristic
subgroupG/H
centerC2#11720983
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#895
prime radicalC3×#63470C2
hypercenterM16D3×D9
hypocenterC3×C9#471
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. #11720869C2semidirect
3. C2×#88415C2semidirect
4. #11720858C2semidirect
5. #11720556C2semidirect
6. C9×#822C2semidirect
7. #11720287C2semidirect
113. #822D9semidirect
217. C3×C9#471semidirect
303. C9#5200semidirect
339. C3#10588578semidirect

Maximal subgroups

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

subgroupindex
C3×#634702
#117208692
C2×#884152
#117208582
#117205562
C9×#8222
#117202872
#105909863
#105885783