Finite Group Databasebeta

← #13137799

Subgroups — #13137799

Order 1728.

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

normal subgroups only

This group has 366 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
1C3×#63459C211yesmaximal, characteristic
2#11720944C211yesmaximal, characteristic
3C2×#88427C211yesmaximal, characteristic, complemented
4C2×#88413C211yesmaximal, characteristic
5#11720526C211yesmaximal, characteristic, complemented
6C9×#810C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10590520—31nomaximal
9#10588496—31nomaximal
10C2×C3×C4×D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C3×C4⋊Dic9—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20#88442—21no—
21#88427—22no—
22#88413—22no—
23C3×#897—31no—
24#63856—31no—
25C2×#3718—31no—
26C2×#3711—31no—
27#63635—31no—
28C3×#810D311yescharacteristic
29#63566—31no—
30#63508—31no—
31C2×#3674—31no—
32C2×#3663—31no—
33#63459D311yescharacteristic
34C9×#103—31no—
35#63376—31no—
36#63367—31no—
37#63367—61no—
38C22×C3×D9D411yescharacteristic
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C3×C4×D9—21no—
46C3×C4×D9—21no—
47C9×Dic6—21no—
48C9×Dic6—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3⋊(C9⋊C8)D412yes—
52C9×C3⋊C8—21no—
53#5052—91no—
54#4816—91no—
55C2×C3×C4×D3—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—31no—
60C2×C3⋊(C3⋊C8)—31no—
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—61no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64#3724—61no—
65(C2×D3)⋊Dic3—61no—
66#3718—62no—
67#3711—62no—
68C2×Q8×C9—31no—
69C2×C4×D9C2×D311yescharacteristic
70C2×Dic18C2×D311yescharacteristic
71C2×C8×C9—31no—
72C9×C4⋊C4—21no—
73C9×C4⋊C4C2×D311yescharacteristic
74#3674—62no—
75(C2×D9)⋊C4—21no—
76(C2×D9)⋊C4—61no—
77C4⋊Dic9—21no—
78C4⋊Dic9—31no—
79C4⋊Dic9—61no—
80Dic9⋊C4—61no—
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83#3663—62no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3×D9—21no—
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C3×C4×C9—41no—
89C3×C4×C9C2×D411yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C3⋊Dic9—41no—
92C9×Dic3—41no—
93C3×Dic9—21no—
94C3×Dic9—41no—
95C3×C22⋊Q8—91no—
96#946—91no—
97C2×#216—91no—
98C2×#216—91no—
99#924—91no—
100C2×#208—91no—
101C2×ΓL(1,𝔽25)—91no—
102#897—31no—
103C3×#103—31no—
104#835—91no—
105#816—91no—
106#810D911yescharacteristic, complemented
107#808—91no—
108#808—91no—
109#808—181no—
110C22×C3×D3—31no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—61no—
117C3×C4×D3—61no—
118C3×C4×D3—61no—
119C3×Dic6—21no—
120C3×Dic6—21no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C22×D9#6711yescharacteristic
124C3⋊(C3⋊C8)—32no—
125C3×C3⋊C8—21no—
126Q8×C9—61no—
127Q8×C9—61no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—41no—
131C2×C4×C9—61no—
132C2×C4×C9C22×D311yescharacteristic
133C2×Dic9—21no—
134C2×Dic9—61no—
135C2×Dic9—121no—
136C2×Dic9#6711yescharacteristic
137C4×D9—21no—
138C4×D9—21no—
139Dic18—21no—
140Dic18—21no—
141C8×C9—61no—
142C9⋊C8—32no—
143C9⋊C8—62no—
144#470—271noSylow
145C2×C3×C9#12111yescharacteristic
146C2×C3×C9#13711yescharacteristic
147C2×C3×C9#13711yescharacteristic
148C3×D9—41no—
149C2×C3×Q8—31no—
150C2×C3×Q8—91no—
151C22×C3×C4—91no—
152C2×C4×D3—31no—
153C2×Dic6—31no—
154C2×Dic6C2×D911yescharacteristic
155C2×C3×C8—31no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4C2×D911yescharacteristic
159C3×C4⋊C4D3211yescharacteristic
160C3×C22⋊C4—181no—
161#217—181no—
162#216—182no—
163#216—182no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—61no—
167C4⋊Dic3—91no—
168C4⋊Dic3—91no—
169C4⋊Dic3—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172#208—182no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C2×C3⋊C8C2×D911yescharacteristic
177ΓL(1,𝔽25)—182no—
178C22×C32D3×D411yescharacteristic
179C2×C3×D3—61no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C32×C4—41no—
183C32×C4#23511yescharacteristic
184C32×C4D3×D411yescharacteristic
185C3⋊Dic3—121no—
186C3×Dic3—41no—
187C3×Dic3—61no—
188C3×Dic3—121no—
189C22×C9—21no—
190C22×C9C2×#6711yescharacteristic
191C2×D9—21no—
192C2×D9—21no—
193C2×D9—21no—
194C4×C9—21no—
195C4×C9—21no—
196C4×C9—41no—
197C4×C9—81no—
198C4×C9—121no—
199C4×C9D3×D411yescharacteristic
200C4×C9#23711yescharacteristic
201Dic9—21no—
202Dic9—41no—
203Dic9—121no—
204Dic9—241no—
205C2×Q16—271no—
206C2×M16—271no—
207C22⋊Q8—91no—
208C22⋊Q8—271no—
209#106—271no—
210#103—31no—
211#103—271no—
212C3×C9#47011yescharacteristic, complemented, Sylow
213C23×C3—91no—
214C22×D3—31no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—31no—
232C2×Dic3—61no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—361no—
236C4×D3—61no—
237C4×D3—61no—
238Dic6—21no—
239Dic6—21no—
240Dic6—61no—
241Dic6—61no—
242C3×C8—61no—
243C3⋊C8—21no—
244C3⋊C8—92no—
245C3⋊C8—92no—
246C3⋊C8—182no—
247C2×C32#88411yescharacteristic
248C2×C32#91511yescharacteristic
249C2×C32#91811yescharacteristic
250C3×D3—121no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9—21no—
254C2×C9#93911yescharacteristic
255C2×C9#94211yescharacteristic
256C2×C9#95111yescharacteristic
257D9—41no—
258C2×Q8—31no—
259C2×Q8—91no—
260C22×C4—91no—
261Q16—542no—
262M16—542no—
263C2×C8—31no—
264C2×C8—271no—
265C4⋊C4—181no—
266C4⋊C4—271no—
267C4⋊C4—541no—
268C4⋊C4D3×D911yescharacteristic
269C22⋊C4—181no—
270C22⋊C4—541no—
271C22×C3—21no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3—181no—
275C22×C3D4×D911yescharacteristic
276C22×C3D3×#6711yescharacteristic
277C2×D3—61no—
278C2×D3—61no—
279C2×D3—61no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—81no—
285C3×C4—121no—
286C3×C4—181no—
287C3×C4—361no—
288C3×C4#369711yescharacteristic
289C3×C4D4×D911yescharacteristic
290C3×C4#379511yescharacteristic
291C3×C4#380211yescharacteristic
292Dic3—41no—
293Dic3—61no—
294Dic3—121no—
295Dic3—361no—
296Dic3—361no—
297Dic3—721no—
298C32#481611yescharacteristic
299C9—21no—
300C9#505211yescharacteristic, complemented
301C23—91no—
302Q8—61no—
303Q8—61no—
304Q8—181no—
305Q8—181no—
306C2×C4—21no—
307C2×C4—61no—
308C2×C4—91no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—541no—
313C2×C4C2×D3×D911yescharacteristic
314C8—61no—
315C8—272no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—361no—
320C2×C3#6344611yescharacteristic
321C2×C3#6347711yescharacteristic
322C2×C3#6348011yescharacteristic
323C2×C3#6388511yescharacteristic
324C2×C3#6392911yescharacteristic
325C2×C3#6394811yescharacteristic
326D3—121no—
327C22—181no—
328C22—181no—
329C22—181no—
330C22D9×#6711yescharacteristic
331C4—41no—
332C4—121no—
333C4—181no—
334C4—361no—
335C4—1081no—
336C4#8864111yescharacteristic
337C4#8864611yescharacteristic
338C3—21no—
339C3#1058849611yescharacteristic, complemented
340C3#1059052011yescharacteristic
341C2—361no—
342C2#1172098211yescharacteristic
343C2#1172118111yescharacteristic
344C2#1172122211yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#470
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#88427C2semidirect
5. #11720526C2semidirect
6. C9×#810C2semidirect
7. #11720276C2semidirect
106. #810D9semidirect
212. C3×C9#470semidirect
300. C9#5052semidirect
339. C3#10588496semidirect

Maximal subgroups

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

subgroupindex
C3×#634592
#117209442
C2×#884272
C2×#884132
#117205262
C9×#8102
#117202762
#105905203
#105884963