Finite Group Databasebeta

← #13139080

Subgroups — #13139080

Order 1728.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721782C211yesmaximal, characteristic, complemented
2#11720867C211yesmaximal, characteristic
3#11720859C211yesmaximal, characteristic, complemented
4C3×#63384C211yesmaximal, characteristic, complemented
5C9×#822C211yesmaximal, characteristic, complemented
6#11720292C211yesmaximal, characteristic, complemented
7#11720274C211yesmaximal, characteristic, complemented
8#10590978—31nomaximal
9#10588574—31nomaximal
10#88736C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C3×C9×M16C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14#88523—21no—
15C8×C3⋊D9—21no—
16C3×C4×Dic9C2211yescharacteristic
17C9×#208C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88418C411yescharacteristic, complemented
20#88417C411yescharacteristic, complemented
21#88416C411yescharacteristic, complemented
22#88415C411yescharacteristic, complemented
23#88413—21no—
24D9×C3⋊C8—21no—
25#64116—31no—
26#63829—31no—
27#63815—31no—
28C3×#825—31no—
29C3×#822D311yescharacteristic
30#63570—31no—
31#63554—31no—
32#63469—31no—
33#63469—31no—
34#63469—61no—
35#63467—31no—
36C9×#108—31no—
37#63384—31no—
38#63384D311yescharacteristic, complemented
39#63375—31no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—21no—
42C4⋊(C3⋊D9)C2×C411yescharacteristic
43C4×C3⋊D9—21no—
44#6226C2×C411yescharacteristic
45C3×#611—21no—
46C2×C3×Dic9—21no—
47C3×D36C2×C411yescharacteristic
48C3×C4×D9—21no—
49C3×Dic18C2×C411yescharacteristic
50C3×C8×C9—21no—
51C3⋊(C9⋊C8)D411yescharacteristic
52C3⋊(C9⋊C8)D411yescharacteristic
53C9×C3⋊C8—21no—
54C9×C3⋊C8C2×C411yescharacteristic
55C9×C3⋊C8C2×C411yescharacteristic
56#4894—91no—
57#4823—91no—
58#3828—31no—
59C3×#235—31no—
60C32×M16C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)—31no—
62#3743—61no—
63C8×C3⋊D3—61no—
64C3×C4×Dic3—31no—
65C3×#208C2×D311yescharacteristic
66C2×C3×C3⋊C8C2×D311yescharacteristic
67#3719—31no—
68#3717—31no—
69#3716—31no—
70#3714—31no—
71#3711—61no—
72D3×C3⋊C8—61no—
73#3697—31no—
74#3697—61no—
75#3697C2×D311yescharacteristic
76C9×M16—21no—
77C9×M16—31no—
78C9×M16C2×D311yescharacteristic
79C2×C8×C9—31no—
80C4×Dic9C2×D311yescharacteristic
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83D72—31no—
84#3664—31no—
85#3664—31no—
86#3663—61no—
87#3663—61no—
88#3663—62no—
89C8×D9—61no—
90C8×D9—61no—
91C8×D9—62no—
92Dic36—31no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—21no—
95C2×C3×D9—21no—
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—21no—
99C3×Dic9—21no—
100C3×Dic9—41no—
101#950—91no—
102#948—91no—
103#907—91no—
104#907—91no—
105#907—91no—
106#907—181no—
107#904—91no—
108C3×#108—31no—
109C3×#104—91no—
110#825—31no—
111#825—91no—
112#822D911yescharacteristic, complemented
113#815—91no—
114#805—91no—
115#803—91no—
116C2×C32×C4C22×D311yescharacteristic
117#638—61no—
118C4⋊(C3⋊D3)—31no—
119C4×C3⋊D3—61no—
120#634—31no—
121C3×#67—61no—
122C2×C3×Dic3—61no—
123C3×D12—31no—
124C3×C4×D3—61no—
125C3×Dic6—31no—
126C32×C8—21no—
127C3⋊(C3⋊C8)—31no—
128C3⋊(C3⋊C8)—31no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8C4×D311yescharacteristic
131C3×C3⋊C8C4×D311yescharacteristic
132C2×C4×C9—21no—
133C2×C4×C9C22×D311yescharacteristic
134#611—21no—
135#611—62no—
136#611—61no—
137C2×Dic9—21no—
138D36—31no—
139D36—61no—
140D36C4×D311yescharacteristic
141C4×D9—21no—
142C4×D9—62no—
143C4×D9—61no—
144Dic18—31no—
145Dic18—61no—
146Dic18C4×D311yescharacteristic
147C8×C9—22no—
148C8×C9—21no—
149C8×C9—31no—
150C8×C9—31no—
151C8×C9—61no—
152C9⋊C8—31no—
153C9⋊C8—31no—
154C9⋊C8—61no—
155C9⋊C8—61no—
156#444—271noSylow
157C2×C3×C9—21no—
158C2×C3×C9C4×D411yescharacteristic
159C3⋊D9—41no—
160C3×D9—41no—
161C3×#41—91no—
162#235—31no—
163#235—91no—
164#235—91no—
165#235—181no—
166C3×M16—21no—
167C3×M16—31no—
168C3×M16C2×D911yescharacteristic
169C3×M16D3211yescharacteristic
170C2×C3×C8—31no—
171C3×C42—91no—
172#216—91no—
173#215—91no—
174#214—91no—
175#213—91no—
176C4×Dic3—31no—
177#208—181no—
178#208C2×D911yescharacteristic
179C2×C3⋊C8—91no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—181no—
182C2×C3⋊C8—181no—
183C2×C3⋊C8C2×D911yescharacteristic
184Dic12—91no—
185D24—91no—
186#204—91no—
187#204—91no—
188ΓL(1,𝔽25)—181no—
189ΓL(1,𝔽25)—181no—
190ΓL(1,𝔽25)—181no—
191ΓL(1,𝔽25)—182no—
192D3×C8—181no—
193D3×C8—181no—
194D3×C8—181no—
195D3×C8—182no—
196C22×C32#23911yescharacteristic
197C2×C3⋊D3—61no—
198C2×C3×D3—61no—
199C32×C4C2×C4×D311yescharacteristic
200C32×C4D3×D411yescharacteristic
201C3⋊Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—121no—
204C22×C9—21no—
205C22×C9#23711yescharacteristic
206C2×D9—21no—
207C2×D9—62no—
208C2×D9—61no—
209C4×C9—21no—
210C4×C9—21no—
211C4×C9C2×C4×D311yescharacteristic
212C4×C9D3×D411yescharacteristic
213Dic9—21no—
214Dic9—41no—
215Dic9—62no—
216Dic9—61no—
217#135—271no—
218#131—271no—
219#131—271no—
220#108—31no—
221#104—91no—
222#104—271no—
223#97—271no—
224C3×C9#44411yescharacteristic, complemented, Sylow
225C3×Q8—91no—
226C3×D4—91no—
227C3×D4—181no—
228C2×C3×C4—21no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4C22×D911yescharacteristic
232C2×C3×C4C2×D3211yescharacteristic
233#67—61no—
234#67—182no—
235#67—181no—
236#67—181no—
237C2×Dic3—61no—
238D12—31no—
239D12—91no—
240D12—91no—
241D12—181no—
242C4×D3—61no—
243C4×D3—182no—
244C4×D3—181no—
245C4×D3—181no—
246Dic6—31no—
247Dic6—91no—
248Dic6—91no—
249Dic6—181no—
250C3×C8—22no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—31no—
254C3×C8—31no—
255C3×C8—61no—
256C3⋊C8—21no—
257C3⋊C8—91no—
258C3⋊C8—91no—
259C3⋊C8—91no—
260C3⋊C8—91no—
261C3⋊C8—181no—
262C3⋊C8—181no—
263C3⋊C8C4×D911yescharacteristic
264C3⋊C8C4×D911yescharacteristic
265C2×C32—21no—
266C2×C32#89311yescharacteristic
267C3⋊D3—121no—
268C3×D3—121no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—41no—
272C2×C9#67⋊C411yescharacteristic
273D9—41no—
274D9—122no—
275D9—121no—
276#41—91no—
277#41—271no—
278Q16—271no—
279SD16—271no—
280SD16—271no—
281D8—271no—
282M16—31no—
283M16—541no—
284M16—541no—
285M16D3×D911yescharacteristic
286C2×C8—31no—
287C2×C8—271no—
288C2×C8—541no—
289C2×C8—541no—
290C42—91no—
291C22×C3—21no—
292C22×C3—181no—
293C22×C3#370111yescharacteristic
294C22×C3#379611yescharacteristic
295C2×D3—61no—
296C2×D3—182no—
297C2×D3—181no—
298C2×D3—181no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—181no—
302C3×C4—361no—
303C3×C4C2×C4×D911yescharacteristic
304C3×C4D4×D911yescharacteristic
305C3×C4C4×D3211yescharacteristic
306C3×C4#381111yescharacteristic
307Dic3—61no—
308Dic3—121no—
309Dic3—182no—
310Dic3—181no—
311Dic3—181no—
312C32#489411yescharacteristic
313C9—21no—
314C9#482311yescharacteristic, complemented
315Q8—91no—
316Q8—271no—
317D4—91no—
318D4—181no—
319D4—271no—
320D4—541no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—541no—
324C2×C4C2×D3×D911yescharacteristic
325C8—21no—
326C8—31no—
327C8—31no—
328C8—61no—
329C8—271no—
330C8—271no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—41no—
335C2×C3—361no—
336C2×C3#6345511yescharacteristic
337C2×C3#6388111yescharacteristic
338D3—121no—
339D3—362no—
340D3—361no—
341D3—361no—
342C22—181no—
343C22—541no—
344C22#8863411yescharacteristic
345C4—181no—
346C4—361no—
347C4—541no—
348C4C4×D3×D911yescharacteristic
349C4#8866811yescharacteristic
350C3—21no—
351C3#1058857411yescharacteristic, complemented
352C3#1059097811yescharacteristic
353C2—21no—
354C2—361no—
355C2—1081no—
356C2#1172097811yescharacteristic
subgroupG/H
centerC4#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#893
prime radical#11720867C2
cosocleC2×C3×C4×C9C23
hypercenterM16D3×D9
hypocenterC3×C9#444
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11721782C2semidirect
3. #11720859C2semidirect
4. C3×#63384C2semidirect
5. C9×#822C2semidirect
6. #11720292C2semidirect
7. #11720274C2semidirect
19. #88418C4semidirect
20. #88417C4semidirect
21. #88416C4semidirect
22. #88415C4semidirect
38. #63384D3semidirect
112. #822D9semidirect
224. C3×C9#444semidirect
314. C9#4823semidirect
351. C3#10588574semidirect

Maximal subgroups

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

subgroupindex
#117217822
#117208672
#117208592
C3×#633842
C9×#8222
#117202922
#117202742
#105909783
#105885743