Finite Group Databasebeta

← #13144856

Subgroups — #13144856

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×#41C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3C2×#88544C211yesmaximal, characteristic, complemented
4#11721819C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720576C211yesmaximal, characteristic, complemented
7#11720571C211yesmaximal, characteristic, complemented
8#10592937—31nomaximal
9#10588919—33nomaximal
10C3×C9×#41—22no—
11C3×C9×#41—22no—
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×D4×C9—21no—
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9—21no—
16C22×C3×C4×C9C2211yescharacteristic
17C2×#6233—21no—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19#88548—22no—
20#88544—22no—
21#88536—21no—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C2×C3⋊(C9⋊C8)C2211yescharacteristic
24C2×C3⋊(C9⋊C8)C2211yescharacteristic
25C2×C32×#41D311yescharacteristic
26C2×#3755—31no—
27C2×#3753—31no—
28#64139—31no—
29C2×C9×#41D313yescomplemented
30#63662—31no—
31#63659—31no—
32#63657—31no—
33C2×#3675—33no—
34C2×#3673—33no—
35#63492—33no—
36#63395—33no—
37#63392—33no—
38#63389—33no—
39C23×C3×C9—21no—
40C23×C3×C9D411yescharacteristic
41C22×C3⋊D9—21no—
42C3×Q8×C9—21no—
43C3×Q8×C9D412yescomplemented
44C3×D4×C9—24no—
45C3×D4×C9—21no—
46C3×D4×C9D412yescomplemented
47C2×C3×C4×C9—24no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9D411yescharacteristic
52C2×C3×C4×C9C2311yescharacteristic
53#6233—42no—
54C2×C3⋊Dic9—21no—
55C4⋊(C3⋊D9)—21no—
56C4⋊(C3⋊D9)—21no—
57C3⋊(C9⋊C8)—21no—
58C3⋊(C9⋊C8)—21no—
59#5238—93no—
60#5238—91no—
61C32×#41—22no—
62C32×#41—22no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×D4—21no—
65C2×C32×D4C2×D311yescharacteristic
66C22×C32×C4—21no—
67C22×C32×C4C2×D311yescharacteristic
68C2×#638—61no—
69C2×C4⋊(C3⋊D3)—31no—
70#3755—62no—
71#3753—62no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—31no—
76C9×#41—26no—
77C9×#41—26no—
78C2×Q8×C9C2×D313yes—
79C2×D4×C9—23no—
80C2×D4×C9C2×D313yes—
81C22×C4×C9—23no—
82C22×C4×C9C2×D313yes—
83C2×#611—63no—
84C2×D36—33no—
85#3675—66no—
86#3673—66no—
87(C2×D9)⋊C4—63no—
88C4⋊Dic9—33no—
89C2×C9⋊C8—33no—
90C2×C9⋊C8—33no—
91C22×C3×C9—21no—
92C22×C3×C9—22no—
93C22×C3×C9—21no—
94C22×C3×C9—21no—
95C22×C3×C9—41no—
96C22×C3×C9—41no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C3×C4×C9—21no—
102C3×C4×C9—22no—
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9—41no—
106C2×C3×#41D913yescomplemented
107C2×C3×#41C3⋊D311yescharacteristic
108C2×#215—93no—
109C2×#215—91no—
110C2×#213—93no—
111C2×#213—91no—
112#930—93no—
113#930—91no—
114#835—93no—
115#835—91no—
116#832—93no—
117#832—91no—
118#830—93no—
119#830—91no—
120C23×C32—21no—
121C23×C32#6711yescharacteristic
122C22×C3⋊D3—61no—
123C32×Q8—21no—
124C32×Q8#6712yes—
125C32×D4—24no—
126C32×D4—21no—
127C32×D4#6712yes—
128C2×C32×C4—24no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4#6711yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135C2×C3⋊Dic3—61no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—61no—
138C23×C9—23no—
139C23×C9#6713yes—
140C22×D9—63no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—61no—
143Q8×C9—23no—
144Q8×C9#6716yescomplemented
145D4×C9—212no—
146D4×C9—23no—
147D4×C9#6716yescomplemented
148C2×C4×C9—212no—
149C2×C4×C9—23no—
150C2×C4×C9—23no—
151C2×C4×C9—23no—
152C2×C4×C9#6713yes—
153C2×C4×C9C22×D313yes—
154#611—126no—
155C2×Dic9—63no—
156D36—63no—
157D36—63no—
158C9⋊C8—63no—
159C9⋊C8—63no—
160#449—271noSylow
161C2×C3×C9—41no—
162C2×C3×C9—42no—
163C2×C3×C9#12011yescharacteristic
164C2×C3×C9#13511yescharacteristic
165C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
166C3⋊D9—81no—
167C3×#41—26no—
168C3×#41—22no—
169C3×#41—26no—
170C3×#41—22no—
171C2×C3×Q8C2×D913yes—
172C2×C3×Q8C2×C3⋊D311yescharacteristic
173C2×C3×D4—23no—
174C2×C3×D4—21no—
175C2×C3×D4C2×D913yes—
176C2×C3×D4C2×C3⋊D311yescharacteristic
177C22×C3×C4—23no—
178C22×C3×C4—21no—
179C22×C3×C4C2×D913yes—
180C22×C3×C4C2×C3⋊D311yescharacteristic
181C2×#67—183no—
182C2×#67—181no—
183C2×D12—93no—
184C2×D12—91no—
185#215—186no—
186#215—182no—
187#213—186no—
188#213—182no—
189(C2×D3)⋊C4—183no—
190(C2×D3)⋊C4—181no—
191C4⋊Dic3—93no—
192C4⋊Dic3—91no—
193C2×C3⋊C8—93no—
194C2×C3⋊C8—93no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—91no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32C2×#6711yescharacteristic
204C2×C3⋊D3—121no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C32×C4—21no—
208C32×C4—22no—
209C32×C4C2×#6711yescharacteristic
210C32×C4C2×#6711yescharacteristic
211C3⋊Dic3—121no—
212C22×C9—23no—
213C22×C9—23no—
214C22×C9—23no—
215C22×C9—26no—
216C22×C9—43no—
217C22×C9—43no—
218C22×C9C2×#6713yes—
219C2×D9—123no—
220C2×D9—123no—
221C2×D9—123no—
222C4×C9—23no—
223C4×C9—26no—
224C4×C9C2×#6713yes—
225C4×C9C2×#6713yes—
226Dic9—123no—
227C2×#41C3⋊D911yescharacteristic, complemented
228C2×SD16—271no—
229C2×D8—271no—
230#121—271no—
231#103—271no—
232#102—271no—
233C22⋊C8—271no—
234C3×C9#44911yescharacteristic, complemented, Sylow
235C23×C3—23no—
236C23×C3—21no—
237C23×C3#61113yes—
238C23×C3#63811yescharacteristic
239C22×D3—183no—
240C22×D3—181no—
241C3×Q8—23no—
242C3×Q8—21no—
243C3×Q8#61116yescomplemented
244C3×Q8#63812yes—
245C3×D4—24no—
246C3×D4—212no—
247C3×D4—23no—
248C3×D4—21no—
249C3×D4#61116yescomplemented
250C3×D4#63812yes—
251C2×C3×C4—24no—
252C2×C3×C4—212no—
253C2×C3×C4—23no—
254C2×C3×C4—21no—
255C2×C3×C4—23no—
256C2×C3×C4—23no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4#61113yes—
260C2×C3×C4C22×D913yes—
261C2×C3×C4#63811yescharacteristic
262C2×C3×C4C22×C3⋊D311yescharacteristic
263#67—362no—
264#67—366no—
265C2×Dic3—183no—
266C2×Dic3—181no—
267D12—183no—
268D12—183no—
269D12—181no—
270D12—181no—
271C3⋊C8—183no—
272C3⋊C8—183no—
273C3⋊C8—181no—
274C3⋊C8—181no—
275C2×C32—41no—
276C2×C32—42no—
277C2×C32#94911yescharacteristic
278C2×C32#95011yescharacteristic
279C2×C32#95311yescharacteristic
280C3⋊D3—241no—
281C2×C9—43no—
282C2×C9—46no—
283C2×C9#94913yes—
284C2×C9#95013yes—
285C2×C9#95313yes—
286D9—243no—
287#41—22no—
288#41—22no—
289C2×Q8C2×C3⋊D911yescharacteristic
290C2×D4—21no—
291C2×D4—271no—
292C2×D4—541no—
293C2×D4C2×C3⋊D911yescharacteristic
294C22×C4—21no—
295C22×C4C2×C3⋊D911yescharacteristic
296SD16—542no—
297D8—542no—
298C2×C8—271no—
299C2×C8—271no—
300C4⋊C4—271no—
301C22⋊C4—541no—
302C22×C3—23no—
303C22×C3—23no—
304C22×C3—23no—
305C22×C3—26no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—43no—
311C22×C3—43no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3C2×#61113yes—
315C22×C3C2×#63811yescharacteristic
316C2×D3—363no—
317C2×D3—363no—
318C2×D3—363no—
319C2×D3—361no—
320C2×D3—361no—
321C2×D3—361no—
322C3×C4—23no—
323C3×C4—26no—
324C3×C4—21no—
325C3×C4—22no—
326C3×C4C2×#61113yes—
327C3×C4C2×#61113yes—
328C3×C4C2×#63811yescharacteristic
329C3×C4C2×#63811yescharacteristic
330Dic3—363no—
331Dic3—361no—
332C32#523811yescharacteristic
333C9#523813yescomplemented
334C23—21no—
335C23—541no—
336C23#623311yescharacteristic
337Q8—21no—
338Q8#623312yescomplemented
339D4—24no—
340D4—21no—
341D4—541no—
342D4—541no—
343D4—1082no—
344D4#623312yescomplemented
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—541no—
350C2×C4#623311yescharacteristic
351C2×C4C22×C3⋊D911yescharacteristic
352C8—541no—
353C8—541no—
354C2×C3—41no—
355C2×C3—42no—
356C2×C3—43no—
357C2×C3—46no—
358C2×C3#6351213yes—
359C2×C3#6351313yes—
360C2×C3#6351513yes—
361C2×C3#6415811yescharacteristic
362C2×C3#6415911yescharacteristic
363C2×C3#6416211yescharacteristic
364D3—721no—
365D3—723no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—1081no—
373C22—1081no—
374C22—1081no—
375C22C2×#623311yescharacteristic
376C4—21no—
377C4—22no—
378C4—1081no—
379C4C2×#623311yescharacteristic
380C4C2×#623311yescharacteristic
381C3#1058891913yescomplemented
382C3#1059293711yescharacteristic
383C2—41no—
384C2—42no—
385C2—2161no—
386C2#1172186711yescharacteristic
387C2#1172186811yescharacteristic
388C2#1172187411yescharacteristic
subgroupG/H
centerC22C2×#6233
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C3×C9×#41C2
socleC22×C32C2×#67
hypercenterC2×#41C3⋊D9
hypocenterC3×C9#449
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×#41C2semidirect
2. C2×#88548C2semidirect
3. C2×#88544C2semidirect
4. #11721819C2semidirect
5. #11720582C2semidirect
6. #11720576C2semidirect
7. #11720571C2semidirect
29. C2×C9×#41D3semidirect
43. C3×Q8×C9D4semidirect
46. C3×D4×C9D4semidirect
106. C2×C3×#41D9semidirect
144. Q8×C9#67semidirect
147. D4×C9#67semidirect
227. C2×#41C3⋊D9semidirect
234. C3×C9#449semidirect
243. C3×Q8#611semidirect
249. C3×D4#611semidirect
333. C9#5238semidirect
338. Q8#6233semidirect
344. D4#6233semidirect
381. C3#10588919semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×#412
C2×#885482
C2×#885442
#117218192
#117205822
#117205762
#117205712
#105929373
#105889193