Finite Group Databasebeta

← #13139084

Subgroups — #13139084

Order 1728.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#11721783C211yesmaximal, characteristic
2#11720895C211yesmaximal, characteristic, complemented
3C2×#88426C211yesmaximal, characteristic, complemented
4#11720861C211yesmaximal, characteristic, complemented
5C9×#824C211yesmaximal, characteristic, complemented
6C3×#63381C211yesmaximal, characteristic, complemented
7#11720288C211yesmaximal, characteristic, complemented
8#10590990—31nomaximal
9#10588434—31nomaximal
10#88736C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14#88525—21no—
15#88524—21no—
16C9×#208C2211yescharacteristic
17C3×#3667C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88428—21no—
20#88426—21no—
21#88426—21no—
22#88426—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64117—31no—
26#63833—31no—
27C2×#3717—31no—
28#63817—31no—
29C3×#824D311yescharacteristic
30C3×#822—31no—
31#63559—31no—
32#63505—31no—
33C2×#3675—31no—
34#63470—31no—
35#63470—61no—
36#63469—31no—
37C9×#101—31no—
38#63383—31no—
39#63381D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42#6233—21no—
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)D411yescharacteristic
46C4×C3⋊D9—21no—
47#6226D411yescharacteristic
48C2×C9×Dic3—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53C3×C9⋊C8—21no—
54C3×C9⋊C8D411yescharacteristic
55C3×C9⋊C8D411yescharacteristic
56#4898—91no—
57#4754—91no—
58#3828—31no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C32×M16C2×D311yescharacteristic
62#3745—61no—
63#3744—61no—
64C3×#208—31no—
65C3×#208C2×D311yescharacteristic
66C2×C3×C3⋊C8—31no—
67#3719—61no—
68#3717—61no—
69#3717—61no—
70#3717—61no—
71#3712—61no—
72#3710—61no—
73C2×Q8×C9—31no—
74#3697—31no—
75#3697—61no—
76C2×D36—31no—
77C2×D36—61no—
78C9×M16—21no—
79C9×M16—31no—
80C9×M16C2×D311yescharacteristic
81#3675—61no—
82#3675—61no—
83#3675—61no—
84#3674—61no—
85#3667C2×D311yescharacteristic
86C2×C9⋊C8C2×D311yescharacteristic
87D72—61no—
88D72—62no—
89#3664—61no—
90#3664—62no—
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—21no—
95C2×C3⋊D9—41no—
96C2×C3⋊D9—81no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—21no—
100C9×Dic3—41no—
101#942—91no—
102C2×#215—91no—
103#909—91no—
104#908—91no—
105#908—91no—
106#908—181no—
107#907—91no—
108C2×#204—91no—
109#901—91no—
110C3×#108—91no—
111C3×#101—31no—
112#824D911yescharacteristic, complemented
113#823—91no—
114#823—91no—
115#822—31no—
116C22×C3⋊D3—61no—
117C2×C32×C4C22×D311yescharacteristic
118#638—61no—
119C4⋊(C3⋊D3)—31no—
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—61no—
122C4×C3⋊D3—61no—
123#634—31no—
124C2×C3×Dic3—21no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C22×D9—61no—
128C22×D9—121no—
129C32×C8—21no—
130C3×C3⋊C8—21no—
131C3×C3⋊C8—31no—
132C3×C3⋊C8—31no—
133C3×C3⋊C8—61no—
134Q8×C9—61no—
135Q8×C9—61no—
136C2×C4×C9—21no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139#611—61no—
140#611—121no—
141D36—31no—
142D36—61no—
143D36—62no—
144D36—61no—
145D36—61no—
146D36—121no—
147C4×D9—61no—
148C4×D9—121no—
149Dic18—31no—
150Dic18—61no—
151C8×C9—22no—
152C8×C9—21no—
153C8×C9—61no—
154C9⋊C8—21no—
155C9⋊C8D1211yescharacteristic
156C9⋊C8D1211yescharacteristic
157#471—271noSylow
158C2×C3×C9—21no—
159C2×C3×C9#12111yescharacteristic
160C3⋊D9—41no—
161C3⋊D9—81no—
162C2×C3×Q8—31no—
163#235—91no—
164#235—91no—
165#235—181no—
166C2×D12—91no—
167C2×D12—91no—
168C2×D12—181no—
169C2×Dic6C2×D911yescharacteristic
170C3×M16—21no—
171C3×M16—31no—
172C3×M16—91no—
173C3×M16C2×D911yescharacteristic
174C3×M16D3211yescharacteristic
175C2×C3×C8—91no—
176#216—181no—
177#215—181no—
178#215—181no—
179#215—181no—
180#208—31no—
181#208C2×D911yescharacteristic
182C2×C3⋊C8—31no—
183Dic12—181no—
184D24—181no—
185D24—181no—
186D24—182no—
187#204—181no—
188#204—181no—
189#204—182no—
190#204—181no—
191#204—181no—
192#204—181no—
193ΓL(1,𝔽25)—181no—
194ΓL(1,𝔽25)—181no—
195D3×C8—181no—
196D3×C8—181no—
197C22×C32#23511yescharacteristic
198C2×C3⋊D3—61no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—241no—
201C32×C4D3×D411yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—61no—
204C3×Dic3—41no—
205C22×C9—21no—
206C22×C9#23911yescharacteristic
207C2×D9—61no—
208C2×D9—121no—
209C2×D9—122no—
210C2×D9—121no—
211C2×D9—242no—
212C2×D9—241no—
213C4×C9—21no—
214C4×C9—21no—
215C4×C9—121no—
216C4×C9C2×D1211yescharacteristic
217C4×C9D3×D411yescharacteristic
218Dic9—61no—
219Dic9—121no—
220#137—271no—
221AGL(1,ℤ/8)—271no—
222C2×SD16—271no—
223#131—271no—
224#108—91no—
225#101—31no—
226#100—271no—
227C3×C9#47111yescharacteristic, complemented, Sylow
228C22×D3—181no—
229C22×D3—181no—
230C22×D3—361no—
231C3×Q8—61no—
232C3×Q8—61no—
233C2×C3×C4—21no—
234C2×C3×C4—61no—
235C2×C3×C4C22×D911yescharacteristic
236C2×C3×C4C2×D3211yescharacteristic
237#67—181no—
238#67—181no—
239#67—361no—
240C2×Dic3—21no—
241D12—91no—
242D12—91no—
243D12—181no—
244D12—182no—
245D12—181no—
246D12—181no—
247D12—181no—
248D12—181no—
249D12—361no—
250C4×D3—181no—
251C4×D3—181no—
252C4×D3—361no—
253Dic6—21no—
254Dic6—21no—
255Dic6—91no—
256Dic6—91no—
257Dic6—181no—
258C3×C8—22no—
259C3×C8—21no—
260C3×C8—21no—
261C3×C8—61no—
262C3×C8—91no—
263C3×C8—91no—
264C3×C8—181no—
265C3⋊C8—21no—
266C3⋊C8—31no—
267C3⋊C8—31no—
268C3⋊C8—61no—
269C2×C32—21no—
270C2×C32#88411yescharacteristic
271C3⋊D3—121no—
272C3⋊D3—241no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—41no—
276C2×C9#89511yescharacteristic
277D9—121no—
278D9—241no—
279D9—242no—
280D9—241no—
281#41—271no—
282C2×Q8—31no—
283C2×D4—271no—
284Q16—541no—
285SD16—541no—
286SD16—541no—
287SD16—541no—
288SD16—541no—
289D8—541no—
290M16—31no—
291M16—91no—
292M16—541no—
293M16D3×D911yescharacteristic
294C2×C8—91no—
295C2×C8—541no—
296C22×C3—21no—
297C22×C3#369711yescharacteristic
298C22×C3#379911yescharacteristic
299C2×D3—181no—
300C2×D3—181no—
301C2×D3—361no—
302C2×D3—362no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—722no—
306C2×D3—721no—
307C2×D3—721no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—121no—
311C3×C4D4×D911yescharacteristic
312C3×C4D4×D911yescharacteristic
313C3×C4D3×D1211yescharacteristic
314C3×C4#381111yescharacteristic
315Dic3—41no—
316Dic3—181no—
317Dic3—181no—
318Dic3—361no—
319C32#475411yescharacteristic
320C9—21no—
321C9#489811yescharacteristic, complemented
322C23—541no—
323Q8—61no—
324Q8—61no—
325Q8—271no—
326D4—271no—
327D4—541no—
328D4—541no—
329D4—541no—
330C2×C4—61no—
331C2×C4—541no—
332C2×C4C2×D3×D911yescharacteristic
333C8—21no—
334C8—61no—
335C8—91no—
336C8—91no—
337C8—181no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3#6344611yescharacteristic
343C2×C3#6391011yescharacteristic
344D3—361no—
345D3—361no—
346D3—721no—
347D3—722no—
348D3—721no—
349D3—721no—
350C22—541no—
351C22—1081no—
352C22—2161no—
353C22#8864211yescharacteristic
354C4—121no—
355C4—541no—
356C4D9×D1211yescharacteristic
357C4#8866811yescharacteristic
358C3—21no—
359C3#1058843411yescharacteristic, complemented
360C3#1059099011yescharacteristic
361C2—21no—
362C2—1081no—
363C2—2161no—
364C2#1172103911yescharacteristic
subgroupG/H
centerC2#11721039
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#884
prime radical#11721783C2
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. #11720895C2semidirect
3. C2×#88426C2semidirect
4. #11720861C2semidirect
5. C9×#824C2semidirect
6. C3×#63381C2semidirect
7. #11720288C2semidirect
39. #63381D3semidirect
112. #824D9semidirect
227. C3×C9#471semidirect
321. C9#4898semidirect
359. C3#10588434semidirect

Maximal subgroups

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

subgroupindex
#117217832
#117208952
C2×#884262
#117208612
C9×#8242
C3×#633812
#117202882
#105909903
#105884343