Finite Group Databasebeta

← #13139078

Subgroups — #13139078

Order 1728.

356 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 85 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×#63469C211yesmaximal, characteristic, complemented
2#11720867C211yesmaximal, characteristic
3#11720861C211yesmaximal, characteristic, complemented
4#11720558C211yesmaximal, characteristic, complemented
5C9×#822C211yesmaximal, characteristic, complemented
6#11720291C211yesmaximal, characteristic, complemented
7#11720275C211yesmaximal, characteristic, complemented
8#10590994—31nomaximal
9#10588574—31nomaximal
10#88736C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C3×C9×M16C2211yescharacteristic
13C4×C3⋊Dic9C2211yescharacteristic
14C9×#208C2211yescharacteristic
15C2×C9×C3⋊C8C2211yescharacteristic
16C2×C3×C9⋊C8C2211yescharacteristic
17C3×#3663—21no—
18C3×C8×D9—21no—
19#88418C411yescharacteristic, complemented
20#88417C411yescharacteristic, complemented
21#88416C411yescharacteristic, complemented
22#88415C411yescharacteristic, complemented
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25C3×#907—31no—
26#63829—31no—
27#63817—31no—
28#63652—31no—
29C3×#822D311yescharacteristic
30#63568—31no—
31#63555—31no—
32#63469—31no—
33#63469D311yescharacteristic, complemented
34#63467—31no—
35C9×#108—31no—
36#63384—31no—
37#63384—31no—
38#63384—61no—
39#63375—31no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—21no—
42C2×C3⋊Dic9—21no—
43C4⋊(C3⋊D9)C2×C411yescharacteristic
44C4×C3⋊D9—21no—
45#6226C2×C411yescharacteristic
46C3×#611—21no—
47C3×D36C2×C411yescharacteristic
48C3×C4×D9—21no—
49C3×Dic18C2×C411yescharacteristic
50C3×C8×C9—21no—
51C9×C3⋊C8—21no—
52C9×C3⋊C8C2×C411yescharacteristic
53C9×C3⋊C8C2×C411yescharacteristic
54C3×C9⋊C8D411yescharacteristic
55C3×C9⋊C8D411yescharacteristic
56#5145—91no—
57#4894—91no—
58#3828—31no—
59C3×#235—31no—
60C32×M16C2×D311yescharacteristic
61C4×C3⋊Dic3—31no—
62C3×#208C2×D311yescharacteristic
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65C3×ΓL(1,𝔽25)—61no—
66C3×D3×C8—61no—
67#3719—31no—
68#3717—31no—
69#3716—31no—
70#3714—31no—
71#3712—61no—
72#3710—61no—
73#3697—31no—
74#3697—61no—
75#3697C2×D311yescharacteristic
76C9×M16—21no—
77C9×M16—31no—
78C9×M16C2×D311yescharacteristic
79C2×C8×C9—31no—
80C4×Dic9—31no—
81C4×Dic9—61no—
82C2×C9⋊C8C2×D311yescharacteristic
83D72—31no—
84#3664—31no—
85#3664—31no—
86#3663—21no—
87#3663—61no—
88C8×D9—21no—
89C8×D9—61no—
90Dic36—31no—
91C22×C3×C9#4111yescharacteristic
92C2×C3⋊D9—21no—
93C2×C3×D9—21no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—21no—
97C3⋊Dic9—41no—
98C3×Dic9—21no—
99C3×#131—91no—
100#950—91no—
101#907—31no—
102#907—91no—
103#904—91no—
104#901—91no—
105C3×#108—31no—
106#837—91no—
107#825—91no—
108#825—91no—
109#825—91no—
110#825—181no—
111#822D911yescharacteristic, complemented
112#815—91no—
113#815—91no—
114C2×C32×C4C22×D311yescharacteristic
115#638—61no—
116C2×C3⋊Dic3—61no—
117C4⋊(C3⋊D3)—31no—
118C4×C3⋊D3—61no—
119#634—31no—
120C3×#67—61no—
121C3×D12—31no—
122C3×C4×D3—61no—
123C3×Dic6—31no—
124C32×C8—21no—
125C3×C3⋊C8—21no—
126C3×C3⋊C8—31no—
127C3×C3⋊C8—31no—
128C3×C3⋊C8C4×D311yescharacteristic
129C3×C3⋊C8C4×D311yescharacteristic
130C2×C4×C9—21no—
131C2×C4×C9C22×D311yescharacteristic
132#611—21no—
133#611—61no—
134#611—121no—
135C2×Dic9—61no—
136C2×Dic9—121no—
137D36—31no—
138D36—61no—
139D36C4×D311yescharacteristic
140C4×D9—21no—
141C4×D9—61no—
142C4×D9—121no—
143Dic18—31no—
144Dic18—61no—
145Dic18C4×D311yescharacteristic
146C8×C9—21no—
147C8×C9—31no—
148C8×C9—31no—
149C8×C9—41no—
150C8×C9—61no—
151C9⋊C8#6711yescharacteristic
152C9⋊C8#6711yescharacteristic
153#444—271noSylow
154C2×C3×C9—21no—
155C2×C3×C9C4×D411yescharacteristic
156C3⋊D9—41no—
157C3×D9—41no—
158C3×#41—91no—
159#235—31no—
160#235—91no—
161#235—91no—
162#235—181no—
163C3×M16—21no—
164C3×M16—31no—
165C3×M16—181no—
166C3×M16C2×D911yescharacteristic
167C3×M16D3211yescharacteristic
168C2×C3×C8—31no—
169C2×C3×C8—91no—
170C2×C3×C8—181no—
171#216—91no—
172#215—91no—
173#214—91no—
174#213—91no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—181no—
178#208C2×D911yescharacteristic
179C2×C3⋊C8—31no—
180C2×C3⋊C8C2×D911yescharacteristic
181Dic12—91no—
182D24—91no—
183#204—91no—
184#204—91no—
185ΓL(1,𝔽25)—61no—
186ΓL(1,𝔽25)—181no—
187ΓL(1,𝔽25)—181no—
188D3×C8—61no—
189D3×C8—181no—
190D3×C8—181no—
191C22×C32#23911yescharacteristic
192C2×C3⋊D3—61no—
193C2×C3×D3—61no—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4D3×D411yescharacteristic
196C3⋊Dic3—61no—
197C3⋊Dic3—121no—
198C3×Dic3—61no—
199C22×C9—21no—
200C22×C9#23511yescharacteristic
201C2×D9—21no—
202C2×D9—61no—
203C2×D9—121no—
204C4×C9—21no—
205C4×C9—21no—
206C4×C9C2×C4×D311yescharacteristic
207C4×C9C2×#6711yescharacteristic
208Dic9—21no—
209Dic9—61no—
210Dic9—121no—
211Dic9—121no—
212Dic9—122no—
213#135—271no—
214#131—91no—
215#131—271no—
216#108—31no—
217#104—271no—
218#104—271no—
219#97—271no—
220C3×C9#44411yescharacteristic, complemented, Sylow
221C3×Q8—91no—
222C3×D4—91no—
223C3×D4—181no—
224C2×C3×C4—21no—
225C2×C3×C4—181no—
226C2×C3×C4C22×D911yescharacteristic
227C2×C3×C4C2×D3211yescharacteristic
228#67—61no—
229#67—181no—
230#67—181no—
231#67—361no—
232C2×Dic3—181no—
233C2×Dic3—181no—
234C2×Dic3—361no—
235D12—31no—
236D12—91no—
237D12—91no—
238D12—181no—
239C4×D3—61no—
240C4×D3—181no—
241C4×D3—181no—
242C4×D3—361no—
243Dic6—31no—
244Dic6—91no—
245Dic6—91no—
246Dic6—181no—
247C3×C8—21no—
248C3×C8—21no—
249C3×C8—31no—
250C3×C8—31no—
251C3×C8—41no—
252C3×C8—61no—
253C3×C8—91no—
254C3×C8—91no—
255C3⋊C8—21no—
256C3⋊C8—31no—
257C3⋊C8—31no—
258C3⋊C8C4×D911yescharacteristic
259C3⋊C8C4×D911yescharacteristic
260C2×C32—21no—
261C2×C32#89311yescharacteristic
262C3⋊D3—121no—
263C3×D3—121no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—41no—
267C2×C9C4×#6711yescharacteristic
268D9—41no—
269D9—121no—
270D9—241no—
271#41—91no—
272#41—271no—
273Q16—271no—
274SD16—271no—
275SD16—271no—
276D8—271no—
277M16—31no—
278M16—181no—
279M16—541no—
280M16D3×D911yescharacteristic
281C2×C8—31no—
282C2×C8—91no—
283C2×C8—181no—
284C2×C8—541no—
285C42—271no—
286C22×C3—21no—
287C22×C3—181no—
288C22×C3#370111yescharacteristic
289C22×C3#379911yescharacteristic
290C2×D3—61no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—361no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—181no—
297C3×C4C2×C4×D911yescharacteristic
298C3×C4D4×D911yescharacteristic
299C3×C4C4×D3211yescharacteristic
300C3×C4D3×#6711yescharacteristic
301Dic3—61no—
302Dic3—181no—
303Dic3—181no—
304Dic3—361no—
305Dic3—361no—
306Dic3—361no—
307Dic3—362no—
308C32#489411yescharacteristic
309C9—21no—
310C9#514511yescharacteristic, complemented
311Q8—91no—
312Q8—271no—
313D4—91no—
314D4—181no—
315D4—271no—
316D4—541no—
317C2×C4—181no—
318C2×C4—541no—
319C2×C4—541no—
320C2×C4C2×D3×D911yescharacteristic
321C8—21no—
322C8—31no—
323C8—31no—
324C8—61no—
325C8—91no—
326C8—91no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3—361no—
332C2×C3#6345511yescharacteristic
333C2×C3#6390411yescharacteristic
334D3—121no—
335D3—361no—
336D3—361no—
337D3—721no—
338C22—181no—
339C22—541no—
340C22#8863511yescharacteristic
341C4—181no—
342C4—541no—
343C4—1081no—
344C4C4×D3×D911yescharacteristic
345C4D9×#6711yescharacteristic
346C3—21no—
347C3#1058857411yescharacteristic, complemented
348C3#1059099411yescharacteristic
349C2—21no—
350C2—361no—
351C2—1081no—
352C2#1172100211yescharacteristic
subgroupG/H
centerC4D9×#67
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. C3×#63469C2semidirect
3. #11720861C2semidirect
4. #11720558C2semidirect
5. C9×#822C2semidirect
6. #11720291C2semidirect
7. #11720275C2semidirect
19. #88418C4semidirect
20. #88417C4semidirect
21. #88416C4semidirect
22. #88415C4semidirect
33. #63469D3semidirect
111. #822D9semidirect
220. C3×C9#444semidirect
310. C9#5145semidirect
347. C3#10588574semidirect

Maximal subgroups

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

subgroupindex
C3×#634692
#117208672
#117208612
#117205582
C9×#8222
#117202912
#117202752
#105909943
#105885743