Finite Group Databasebeta

← #13143921

Subgroups — #13143921

Order 1728.

600 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 135 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#11721838C211yesmaximal, characteristic
2C2×#88543C211yesmaximal, characteristic, complemented
3C2×#88523C211yesmaximal, characteristic, complemented
4#11721753C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6#11720539C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592479—31nomaximal
9#10588460—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88551—21no—
17#88543—22no—
18#88536—21no—
19C4⋊(C3⋊Dic9)—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22#88523—22no—
23#64147—31no—
24C2×#3754—31no—
25C2×#3743—31no—
26#64106—31no—
27C32×#102D311yescharacteristic
28#63643—31no—
29#63634—31no—
30#63500—33no—
31C2×#3672—33no—
32C2×#3663—33no—
33#63459—33no—
34C9×#102D313yescomplemented
35#63376—33no—
36#63366—33no—
37C23×C3×C9—21no—
38C22×C3⋊D9D411yescharacteristic
39C3×D4×C9—21no—
40C3×D4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—42no—
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C4×C3⋊D9—21no—
48C4×C3⋊D9—21no—
49#6226—21no—
50#6226—21no—
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)D412yescomplemented
53#4780—93no—
54#4780—91no—
55C2×C32×D4C2×D311yescharacteristic
56C2×#638—61no—
57C2×C4×C3⋊D3—31no—
58C2×#634—31no—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61#3757—61no—
62#3754—62no—
63#3752—61no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—61no—
66C2×C3⋊(C3⋊C8)—31no—
67#3743—62no—
68C2×D4×C9C2×D313yes—
69C2×#611—63no—
70C2×C4×D9—33no—
71C2×Dic18—33no—
72C2×C8×C9C2×D313yes—
73C9×C4⋊C4C2×D313yes—
74#3676—63no—
75#3672—66no—
76(C2×D9)⋊C4—63no—
77C4⋊Dic9—33no—
78C4⋊Dic9—63no—
79C2×C9⋊C8—33no—
80#3663—66no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9—41no—
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—21no—
92C3⋊Dic9—41no—
93C3⋊Dic9—41no—
94#938—93no—
95#938—91no—
96C2×#214—93no—
97C2×#214—91no—
98C2×ΓL(1,𝔽25)—93no—
99C2×ΓL(1,𝔽25)—91no—
100#897—93no—
101#897—91no—
102C3×#102D913yescomplemented
103C3×#102C3⋊D311yescharacteristic
104#816—93no—
105#816—91no—
106#807—93no—
107#807—91no—
108C23×C32—21no—
109C22×C3⋊D3—31no—
110C32×D4—21no—
111C32×D4—21no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114#638—122no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—61no—
118C4×C3⋊D3—61no—
119C4×C3⋊D3—61no—
120#634—61no—
121#634—61no—
122C23×C9—23no—
123C22×D9—33no—
124C32×C8—21no—
125C3⋊(C3⋊C8)—32no—
126D4×C9—23no—
127D4×C9—23no—
128C2×C4×C9—23no—
129C2×C4×C9C22×D313yes—
130#611—126no—
131C2×Dic9—33no—
132C2×Dic9—63no—
133C2×Dic9—63no—
134C4×D9—63no—
135C4×D9—63no—
136Dic18—63no—
137Dic18—63no—
138C8×C9—23no—
139C9⋊C8—36no—
140#468—271noSylow
141C2×C3×C9—81no—
142C2×C3×C9#12111yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144C2×C3×C9#13711yescharacteristic
145C3⋊D9—41no—
146C2×C3×D4C2×D913yes—
147C2×C3×D4C2×C3⋊D311yescharacteristic
148C2×#67—181no—
149C2×#67—183no—
150C2×C4×D3—93no—
151C2×C4×D3—91no—
152C2×Dic6—93no—
153C2×Dic6—91no—
154C2×C3×C8C2×D913yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4C2×D913yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158#217—181no—
159#217—183no—
160#214—182no—
161#214—186no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—183no—
164C4⋊Dic3—93no—
165C4⋊Dic3—91no—
166C4⋊Dic3—181no—
167C4⋊Dic3—183no—
168C2×C3⋊C8—93no—
169C2×C3⋊C8—91no—
170ΓL(1,𝔽25)—182no—
171ΓL(1,𝔽25)—186no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32D3×D411yescharacteristic
176C2×C3⋊D3—61no—
177C2×C3⋊D3—61no—
178C2×C3⋊D3—61no—
179C32×C4—41no—
180C32×C4#23511yescharacteristic
181C32×C4D3×D411yescharacteristic
182C3⋊Dic3—61no—
183C3⋊Dic3—121no—
184C3⋊Dic3—121no—
185C22×C9—43no—
186C22×C9—43no—
187C22×C9—43no—
188C22×C9D3×D413yes—
189C2×D9—63no—
190C2×D9—63no—
191C2×D9—63no—
192C4×C9—43no—
193C4×C9#23513yes—
194C4×C9D3×D413yes—
195Dic9—63no—
196Dic9—123no—
197Dic9—123no—
198C2×SD16—271no—
199C2×M16—271no—
200C22⋊Q8—271no—
201#121—271no—
202#107—271no—
203#103—271no—
204#102C3⋊D911yescharacteristic, complemented
205C3×C9#46811yescharacteristic, complemented, Sylow
206C23×C3—23no—
207C23×C3—21no—
208C22×D3—93no—
209C22×D3—91no—
210C3×D4—23no—
211C3×D4—23no—
212C3×D4—21no—
213C3×D4—21no—
214C2×C3×C4—23no—
215C2×C3×C4—21no—
216C2×C3×C4C22×D913yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218#67—362no—
219#67—366no—
220C2×Dic3—93no—
221C2×Dic3—91no—
222C2×Dic3—183no—
223C2×Dic3—183no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C4×D3—183no—
227C4×D3—183no—
228C4×D3—181no—
229C4×D3—181no—
230Dic6—183no—
231Dic6—183no—
232Dic6—181no—
233Dic6—181no—
234C3×C8—23no—
235C3×C8—21no—
236C3⋊C8—96no—
237C3⋊C8—92no—
238C2×C32—81no—
239C2×C32#88411yescharacteristic
240C2×C32#91111yescharacteristic
241C2×C32#91511yescharacteristic
242C3⋊D3—121no—
243C2×C9—83no—
244C2×C9#88413yes—
245C2×C9#91113yes—
246C2×C9#91513yes—
247D9—123no—
248C2×Q8—271no—
249C2×D4—541no—
250C2×D4C2×C3⋊D911yescharacteristic
251C22×C4—271no—
252SD16—542no—
253M16—542no—
254C2×C8—271no—
255C2×C8C2×C3⋊D911yescharacteristic
256C4⋊C4—271no—
257C4⋊C4—541no—
258C4⋊C4C2×C3⋊D911yescharacteristic
259C22⋊C4—541no—
260C22⋊C4—541no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—43no—
265C22×C3—43no—
266C22×C3—43no—
267C22×C3D4×D913yes—
268C22×C3D4×C3⋊D311yescharacteristic
269C2×D3—181no—
270C2×D3—181no—
271C2×D3—181no—
272C2×D3—183no—
273C2×D3—183no—
274C2×D3—183no—
275C3×C4—41no—
276C3×C4—43no—
277C3×C4#369713yes—
278C3×C4D4×D913yes—
279C3×C4#382811yescharacteristic
280C3×C4D4×C3⋊D311yescharacteristic
281Dic3—181no—
282Dic3—183no—
283Dic3—361no—
284Dic3—361no—
285Dic3—363no—
286Dic3—363no—
287C32#478011yescharacteristic
288C9#478013yescomplemented
289C23—21no—
290C23—271no—
291Q8—541no—
292Q8—541no—
293D4—21no—
294D4—21no—
295D4—1082no—
296C2×C4—21no—
297C2×C4—271no—
298C2×C4—541no—
299C2×C4—541no—
300C2×C4—541no—
301C2×C4—541no—
302C2×C4C22×C3⋊D911yescharacteristic
303C8—21no—
304C8—272no—
305C2×C3—83no—
306C2×C3—81no—
307C2×C3#6344613yes—
308C2×C3#6347313yes—
309C2×C3#6347713yes—
310C2×C3#6409311yescharacteristic
311C2×C3#6412011yescharacteristic
312C2×C3#6412411yescharacteristic
313D3—363no—
314D3—361no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—541no—
319C22—541no—
320C22—541no—
321C22D4×C3⋊D911yescharacteristic
322C4—41no—
323C4—541no—
324C4—1081no—
325C4—1081no—
326C4#8873611yescharacteristic
327C4D4×C3⋊D911yescharacteristic
328C3#1058846013yescomplemented
329C3#1059247911yescharacteristic
330C2—81no—
331C2—1081no—
332C2#1172173011yescharacteristic
333C2#1172179011yescharacteristic
334C2#1172179711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radical#11721838C2
hypercenter#102C3⋊D9
hypocenterC3×C9#468
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#88543C2semidirect
3. C2×#88523C2semidirect
4. #11721753C2semidirect
5. C3×C9×#102C2semidirect
6. #11720539C2semidirect
7. #11720525C2semidirect
34. C9×#102D3semidirect
52. C3⋊(C9⋊C8)D4semidirect
102. C3×#102D9semidirect
204. #102C3⋊D9semidirect
205. C3×C9#468semidirect
288. C9#4780semidirect
328. C3#10588460semidirect

Maximal subgroups

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

subgroupindex
#117218382
C2×#885432
C2×#885232
#117217532
C3×C9×#1022
#117205392
#117205252
#105924793
#105884603