Finite Group Databasebeta

← #13143900

Subgroups — #13143900

Order 1728.

640 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 137 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#11721840C211yesmaximal, characteristic
2C2×#88543C211yesmaximal, characteristic, complemented
3C2×#88524C211yesmaximal, characteristic, complemented
4#11721743C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6#11720537C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592461—31nomaximal
9#10588442—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×#6226C2211yescharacteristic
15C2×C3×C8×C9C2211yescharacteristic
16C3×C9×C4⋊C4C2211yescharacteristic
17#88543—22no—
18#88534—21no—
19C4×C3⋊Dic9—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22#88524—22no—
23#64149—31no—
24C2×#3754—31no—
25C2×#3744—31no—
26#64096—31no—
27C32×#102D311yescharacteristic
28#63641—31no—
29#63636—31no—
30#63502—33no—
31C2×#3672—33no—
32C2×#3664—33no—
33#63449—33no—
34C9×#102D313yescomplemented
35Dic9⋊C8—33no—
36#63369—33no—
37C23×C3×C9—21no—
38C22×C3⋊D9—21no—
39C3×D4×C9—21no—
40C3×D4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—42no—
44#6233—42no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C2×C3⋊Dic9D411yescharacteristic
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)—21no—
50#6226—21no—
51#6226D412yescomplemented
52C3×C8×C9—21no—
53C3⋊(C9⋊C8)—21no—
54#4762—93no—
55#4762—91no—
56C2×C32×D4C2×D311yescharacteristic
57C2×#638—61no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×#634—31no—
61C2×C32×C8C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63#3754—62no—
64#3750—61no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—61no—
67C2×C3⋊(C3⋊C8)—31no—
68#3744—62no—
69C2×D4×C9C2×D313yes—
70C2×#611—63no—
71C2×#611—63no—
72C2×D36—33no—
73C2×Dic18—33no—
74C2×C8×C9C2×D313yes—
75C9×C4⋊C4C2×D313yes—
76#3672—66no—
77Dic9⋊C4—63no—
78C4×Dic9—33no—
79C4×Dic9—63no—
80C2×C9⋊C8—33no—
81#3664—66no—
82C22×C3×C9—41no—
83C22×C3×C9—41no—
84C22×C3×C9—41no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C2×C3⋊D9—41no—
89C3×C4×C9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3⋊Dic9—22no—
93C3⋊Dic9—21no—
94C3⋊Dic9SD1612yescomplemented
95#940—93no—
96#940—91no—
97C2×#214—93no—
98C2×#214—91no—
99C2×#204—93no—
100C2×#204—91no—
101#887—93no—
102#887—91no—
103C3×#102D913yescomplemented
104C3×#102C3⋊D311yescharacteristic
105Dic3⋊C8—93no—
106Dic3⋊C8—91no—
107#809—93no—
108#809—91no—
109C23×C32—21no—
110C22×C3⋊D3—61no—
111C32×D4—21no—
112C32×D4—21no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115#638—122no—
116#638—122no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—61no—
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—61no—
122#634—32no—
123#634—61no—
124C23×C9—23no—
125C22×D9—63no—
126C32×C8—21no—
127C3⋊(C3⋊C8)—61no—
128D4×C9—23no—
129D4×C9—23no—
130C2×C4×C9—23no—
131C2×C4×C9C22×D313yes—
132#611—126no—
133#611—126no—
134C2×Dic9—33no—
135C2×Dic9—33no—
136C2×Dic9—63no—
137D36—63no—
138D36—63no—
139Dic18—36no—
140Dic18—63no—
141C8×C9—23no—
142C9⋊C8—63no—
143#460—271noSylow
144C2×C3×C9—81no—
145C2×C3×C9#12111yescharacteristic
146C2×C3×C9C2×SD1611yescharacteristic
147C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
148C3⋊D9—81no—
149C2×C3×D4C2×D913yes—
150C2×C3×D4C2×C3⋊D311yescharacteristic
151C2×#67—181no—
152C2×#67—181no—
153C2×#67—183no—
154C2×#67—183no—
155C2×D12—93no—
156C2×D12—91no—
157C2×Dic6—93no—
158C2×Dic6—91no—
159C2×C3×C8C2×D913yes—
160C2×C3×C8C2×C3⋊D311yescharacteristic
161C3×C4⋊C4C2×D913yes—
162C3×C4⋊C4C2×C3⋊D311yescharacteristic
163#214—182no—
164#214—186no—
165Dic3⋊C4—181no—
166Dic3⋊C4—183no—
167C4×Dic3—93no—
168C4×Dic3—91no—
169C4×Dic3—181no—
170C4×Dic3—183no—
171C2×C3⋊C8—93no—
172C2×C3⋊C8—91no—
173#204—182no—
174#204—186no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—121no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—121no—
182C32×C4—41no—
183C32×C4#23511yescharacteristic
184C32×C4D3×D411yescharacteristic
185C3⋊Dic3—32no—
186C3⋊Dic3—62no—
187C3⋊Dic3—61no—
188C22×C9—43no—
189C22×C9—43no—
190C22×C9—43no—
191C22×C9D3×D413yes—
192C2×D9—123no—
193C2×D9—123no—
194C2×D9—123no—
195C4×C9—43no—
196C4×C9#23513yes—
197C4×C9D3×D413yes—
198Dic9—36no—
199Dic9—66no—
200Dic9—63no—
201C2×SD16—271no—
202C2×SD16—271no—
203#127—271no—
204C4×Q8—271no—
205C4⋊C8—271no—
206#102—271no—
207#102C3⋊D911yescharacteristic, complemented
208C3×C9#46011yescharacteristic, complemented, Sylow
209C23×C3—23no—
210C23×C3—21no—
211C22×D3—183no—
212C22×D3—181no—
213C3×D4—23no—
214C3×D4—23no—
215C3×D4—21no—
216C3×D4—21no—
217C2×C3×C4—23no—
218C2×C3×C4—21no—
219C2×C3×C4C22×D913yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221#67—362no—
222#67—366no—
223#67—362no—
224#67—366no—
225C2×Dic3—93no—
226C2×Dic3—91no—
227C2×Dic3—93no—
228C2×Dic3—91no—
229C2×Dic3—183no—
230C2×Dic3—181no—
231D12—183no—
232D12—183no—
233D12—181no—
234D12—181no—
235Dic6—96no—
236Dic6—92no—
237Dic6—183no—
238Dic6—181no—
239C3×C8—23no—
240C3×C8—21no—
241C3⋊C8—183no—
242C3⋊C8—181no—
243C2×C32—81no—
244C2×C32#88411yescharacteristic
245C2×C32#91111yescharacteristic
246C2×C32D3×SD1611yescharacteristic
247C3⋊D3—241no—
248C2×C9—83no—
249C2×C9#88413yes—
250C2×C9#91113yes—
251C2×C9D3×SD1613yes—
252D9—243no—
253C2×Q8—271no—
254C2×D4—271no—
255C2×D4—541no—
256C2×D4—541no—
257C2×D4C2×C3⋊D911yescharacteristic
258SD16—542no—
259SD16—542no—
260C2×C8—271no—
261C2×C8C2×C3⋊D911yescharacteristic
262C4⋊C4—541no—
263C4⋊C4C2×C3⋊D911yescharacteristic
264C42—271no—
265C42—541no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—43no—
270C22×C3—43no—
271C22×C3—43no—
272C22×C3D4×D913yes—
273C22×C3D4×C3⋊D311yescharacteristic
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—363no—
278C2×D3—363no—
279C2×D3—363no—
280C3×C4—41no—
281C3×C4—43no—
282C3×C4#369713yes—
283C3×C4D4×D913yes—
284C3×C4#382811yescharacteristic
285C3×C4D4×C3⋊D311yescharacteristic
286Dic3—92no—
287Dic3—96no—
288Dic3—182no—
289Dic3—181no—
290Dic3—186no—
291Dic3—183no—
292C32#476211yescharacteristic
293C9#476213yescomplemented
294C23—21no—
295C23—541no—
296Q8—272no—
297Q8—541no—
298D4—21no—
299D4—21no—
300D4—541no—
301D4—541no—
302D4—1082no—
303D4—1082no—
304C2×C4—21no—
305C2×C4—271no—
306C2×C4—271no—
307C2×C4—541no—
308C2×C4C22×C3⋊D911yescharacteristic
309C8—21no—
310C8—541no—
311C2×C3—83no—
312C2×C3—81no—
313C2×C3#6344613yes—
314C2×C3#6347313yes—
315C2×C3SD16×D913yes—
316C2×C3#6409311yescharacteristic
317C2×C3#6412011yescharacteristic
318C2×C3SD16×C3⋊D311yescharacteristic
319D3—723no—
320D3—721no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—1081no—
325C22—1081no—
326C22—1081no—
327C22D4×C3⋊D911yescharacteristic
328C4—41no—
329C4—272no—
330C4—542no—
331C4—541no—
332C4#8873611yescharacteristic
333C4D4×C3⋊D911yescharacteristic
334C3#1058844213yescomplemented
335C3#1059246111yescharacteristic
336C2—81no—
337C2—2161no—
338C2#1172173011yescharacteristic
339C2#1172179011yescharacteristic
340C2SD16×C3⋊D911yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radical#11721840C2
hypercenter#102C3⋊D9
hypocenterC3×C9#460
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×#88524C2semidirect
4. #11721743C2semidirect
5. C3×C9×#102C2semidirect
6. #11720537C2semidirect
7. #11720528C2semidirect
34. C9×#102D3semidirect
51. #6226D4semidirect
94. C3⋊Dic9SD16semidirect
103. C3×#102D9semidirect
207. #102C3⋊D9semidirect
208. C3×C9#460semidirect
293. C9#4762semidirect
334. C3#10588442semidirect

Maximal subgroups

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

subgroupindex
#117218402
C2×#885432
C2×#885242
#117217432
C3×C9×#1022
#117205372
#117205282
#105924613
#105884423