Finite Group Databasebeta

← #13139112

Subgroups — #13139112

Order 1728.

348 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C9×#907C211yesmaximal, characteristic, complemented
2#11720881C211yesmaximal, characteristic
3#11720858C211yesmaximal, characteristic, complemented
4#11720555C211yesmaximal, characteristic, complemented
5C3×#63384C211yesmaximal, characteristic, complemented
6#11720301C211yesmaximal, characteristic, complemented
7C3⋊C8×Dic9C211yesmaximal, characteristic, complemented
8#10590975—31nomaximal
9#10588824—31nomaximal
10C9×#235C2211yescharacteristic
11C3×#3697C2211yescharacteristic
12C3×C9×M16C2211yescharacteristic
13#88532C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×C4×Dic9C2211yescharacteristic
16C2×C9×C3⋊C8C2211yescharacteristic
17C9×ΓL(1,𝔽25)—21no—
18D3×C8×C9—21no—
19#88427C411yescharacteristic, complemented
20#88425C411yescharacteristic, complemented
21#88423C411yescharacteristic, complemented
22#88421C411yescharacteristic, complemented
23#88413—21no—
24D9×C3⋊C8—21no—
25C3×#907D311yescharacteristic
26#63822—31no—
27#63814—31no—
28#63649—31no—
29C3×#825—31no—
30#63569—31no—
31Dic3×C3⋊C8—31no—
32C9×#131—31no—
33#63513—31no—
34#63464—31no—
35#63396—31no—
36#63384D311yescharacteristic, complemented
37#63381—31no—
38#63381—61no—
39C8×Dic9—31no—
40C2×C3×C4×C9C2311yescharacteristic
41C9×#67—21no—
42C3×#611—21no—
43C2×C3×Dic9—21no—
44C9×D12C2×C411yescharacteristic
45C4×D3×C9—21no—
46C3×D36C2×C411yescharacteristic
47C3×C4×D9—21no—
48C9×Dic6C2×C411yescharacteristic
49C3×Dic18C2×C411yescharacteristic
50C3×C8×C9—21no—
51C3⋊(C9⋊C8)—21no—
52C3⋊(C9⋊C8)C2×C411yescharacteristic
53C3⋊(C9⋊C8)C2×C411yescharacteristic
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56#5144—91no—
57#4824—91no—
58C3×#235—31no—
59C3×#235C2×D311yescharacteristic
60C32×M16C2×D311yescharacteristic
61#3748—31no—
62C2×C3⋊(C3⋊C8)—31no—
63C3×C4×Dic3—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65C3×ΓL(1,𝔽25)—21no—
66C3×D3×C8—21no—
67#3718—31no—
68#3715—31no—
69#3715—31no—
70#3713—31no—
71#3711—61no—
72D3×C3⋊C8—61no—
73C9×#41—31no—
74#3697C2×D311yescharacteristic
75C9×M16—21no—
76C9×M16—61no—
77C9×M16C2×D311yescharacteristic
78C2×C8×C9—31no—
79C2×C8×C9—61no—
80#3675—31no—
81#3674—31no—
82#3673—31no—
83#3672—31no—
84C4×Dic9C2×D311yescharacteristic
85#3667—31no—
86#3667—61no—
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89#3663—61no—
90C8×D9—61no—
91C22×C3×C9#4111yescharacteristic
92C2×D3×C9—21no—
93C2×C3×D9—21no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C9×Dic3—21no—
97C3×Dic9—22no—
98C3×Dic9—21no—
99C3×#131—31no—
100#950—91no—
101#950—91no—
102#948—91no—
103#907D911yescharacteristic, complemented
104#901—91no—
105C3×#104—91no—
106#837—91no—
107#825—31no—
108#822—91no—
109#822—91no—
110#822—181no—
111C8×Dic3—91no—
112#805—91no—
113C4×C3⋊C8—91no—
114C2×C32×C4C22×D311yescharacteristic
115C3×#67—21no—
116C3×#67—61no—
117C2×C3×Dic3—61no—
118C3×D12—31no—
119C3×D12C4×D311yescharacteristic
120C3×C4×D3—21no—
121C3×C4×D3—61no—
122C3×Dic6—31no—
123C3×Dic6C4×D311yescharacteristic
124C32×C8—21no—
125C3⋊(C3⋊C8)—31no—
126C3⋊(C3⋊C8)—31no—
127C3⋊(C3⋊C8)—61no—
128C3×C3⋊C8#6711yescharacteristic
129C3×C3⋊C8#6711yescharacteristic
130Q8×C9—31no—
131D4×C9—31no—
132D4×C9—61no—
133C2×C4×C9—21no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136#611—21no—
137C2×Dic9—21no—
138D36C4×D311yescharacteristic
139C4×D9—21no—
140Dic18C4×D311yescharacteristic
141C8×C9—21no—
142C8×C9—31no—
143C8×C9—31no—
144C8×C9—41no—
145C9⋊C8—31no—
146C9⋊C8—31no—
147C9⋊C8—61no—
148C9⋊C8—61no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#443—271noSylow
152C2×C3×C9—21no—
153C2×C3×C9C4×D411yescharacteristic
154D3×C9—41no—
155C3×D9—41no—
156C3×#41—31no—
157C3×#41—91no—
158#235—31no—
159#235C2×D911yescharacteristic
160C3×M16—21no—
161C3×M16—61no—
162C3×M16C2×D911yescharacteristic
163C3×M16D3211yescharacteristic
164C2×C3×C8—31no—
165C2×C3×C8—61no—
166C3×C42—91no—
167#216—91no—
168#216—91no—
169#215—91no—
170#215—91no—
171#214—91no—
172#214—91no—
173#213—91no—
174#213—91no—
175C4×Dic3—31no—
176#208—91no—
177#208—91no—
178#208—181no—
179#208—181no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—181no—
183C2×C3⋊C8—181no—
184C2×C3⋊C8C2×D911yescharacteristic
185ΓL(1,𝔽25)—21no—
186ΓL(1,𝔽25)—181no—
187D3×C8—21no—
188D3×C8—181no—
189C22×C32#23511yescharacteristic
190C2×C3×D3—21no—
191C2×C3×D3—61no—
192C32×C4C2×C4×D311yescharacteristic
193C32×C4C2×#6711yescharacteristic
194C3×Dic3—21no—
195C3×Dic3—62no—
196C3×Dic3—61no—
197C22×C9—21no—
198C22×C9—61no—
199C22×C9#23711yescharacteristic
200C2×D9—21no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—61no—
204C4×C9C2×C4×D311yescharacteristic
205C4×C9D3×D411yescharacteristic
206Dic9—22no—
207Dic9—21no—
208#135—271no—
209#131—31no—
210#131—271no—
211#108—271no—
212#104—91no—
213#104—271no—
214C4×C8—271no—
215C3×C9#44311yescharacteristic, complemented, Sylow
216C3×Q8—31no—
217C3×Q8—91no—
218C3×D4—31no—
219C3×D4—61no—
220C3×D4—91no—
221C3×D4—181no—
222C2×C3×C4—21no—
223C2×C3×C4—61no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4C22×D911yescharacteristic
227C2×C3×C4C2×D3211yescharacteristic
228#67—21no—
229#67—61no—
230C2×Dic3—61no—
231D12—31no—
232D12C4×D911yescharacteristic
233C4×D3—21no—
234C4×D3—61no—
235Dic6—31no—
236Dic6C4×D911yescharacteristic
237C3×C8—21no—
238C3×C8—21no—
239C3×C8—31no—
240C3×C8—31no—
241C3×C8—41no—
242C3⋊C8—91no—
243C3⋊C8—91no—
244C3⋊C8—91no—
245C3⋊C8—91no—
246C3⋊C8—181no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—181no—
250C3⋊C8—361no—
251C3⋊C8#61111yescharacteristic
252C3⋊C8#61111yescharacteristic
253C2×C32—21no—
254C2×C32C4×#6711yescharacteristic
255C3×D3—41no—
256C3×D3—121no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—41no—
260C2×C9—121no—
261C2×C9#67⋊C411yescharacteristic
262D9—41no—
263#41—31no—
264#41—91no—
265Q16—271no—
266SD16—271no—
267SD16—271no—
268D8—271no—
269M16—61no—
270M16—271no—
271M16—541no—
272M16D3×D911yescharacteristic
273C2×C8—31no—
274C2×C8—61no—
275C2×C8—271no—
276C2×C8—541no—
277C42—91no—
278C22×C3—21no—
279C22×C3—61no—
280C22×C3—181no—
281C22×C3#369711yescharacteristic
282C22×C3#379511yescharacteristic
283C2×D3—21no—
284C2×D3—61no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—61no—
288C3×C4—182no—
289C3×C4—181no—
290C3×C4C2×C4×D911yescharacteristic
291C3×C4C2×#61111yescharacteristic
292C3×C4C4×D3211yescharacteristic
293C3×C4D3×#6711yescharacteristic
294Dic3—21no—
295Dic3—62no—
296Dic3—61no—
297C32#514411yescharacteristic
298C9—21no—
299C9#482411yescharacteristic, complemented
300Q8—31no—
301Q8—91no—
302D4—31no—
303D4—61no—
304D4—91no—
305D4—181no—
306C2×C4—61no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4C2×D3×D911yescharacteristic
310C8—21no—
311C8—31no—
312C8—31no—
313C8—271no—
314C8—271no—
315C8—541no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—41no—
320C2×C3—121no—
321C2×C3—361no—
322C2×C3C4×#61111yescharacteristic
323C2×C3#6387911yescharacteristic
324D3—41no—
325D3—121no—
326C22—61no—
327C22—181no—
328C22#8864111yescharacteristic
329C4—61no—
330C4—182no—
331C4—181no—
332C4C4×D3×D911yescharacteristic
333C4D3×#61111yescharacteristic
334C3—21no—
335C3#1058882411yescharacteristic, complemented
336C3#1059097511yescharacteristic
337C2—21no—
338C2—121no—
339C2—361no—
340C2#1172097611yescharacteristic
subgroupG/H
centerC4D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32C4×#67
prime radical#11720881C2
cosocleC2×C3×C4×C9C23
hypercenterM16D3×D9
hypocenterC3×C9#443
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. C9×#907C2semidirect
3. #11720858C2semidirect
4. #11720555C2semidirect
5. C3×#63384C2semidirect
6. #11720301C2semidirect
7. C3⋊C8×Dic9C2semidirect
19. #88427C4semidirect
20. #88425C4semidirect
21. #88423C4semidirect
22. #88421C4semidirect
36. #63384D3semidirect
103. #907D9semidirect
215. C3×C9#443semidirect
299. C9#4824semidirect
335. C3#10588824semidirect

Maximal subgroups

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

subgroupindex
C9×#9072
#117208812
#117208582
#117205552
C3×#633842
#117203012
C3⋊C8×Dic92
#105909753
#105888243