Finite Group Databasebeta

← #13140411

Subgroups — #13140411

Order 1728.

536 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic
2D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
3C2×#88420C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720297C211yesmaximal, characteristic, complemented
8#10591414—31nomaximal
9#10588916—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C22×C9×Dic3—21no—
13C9×#237—22no—
14C9×#237C414yescomplemented
15C2×C4×D3×C9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×D3×Dic9—21no—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C3×C4⋊Dic9C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22Dic3⋊Dic9—21no—
23#88420—22no—
24C2×C3×#237D311yescharacteristic
25D3×C4⋊Dic3—31no—
26C2×#3711—31no—
27C2×C9×#41—31no—
28#63659—31no—
29C3×#832—31no—
30#63566—31no—
31#63565—31no—
32C2×C4⋊Dic9—31no—
33C2×#3667—31no—
34#63395—31no—
35#63395—31no—
36#63392—31no—
37#63392—61no—
38#63392D311yescharacteristic, complemented
39C23×C3×C9—21no—
40C22×D3×C9D411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C412yescomplemented
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C9×#67—24no—
46C2×C9×Dic3—21no—
47C2×C9×Dic3—24no—
48C2×C9×Dic3D411yescharacteristic
49C2×C3×Dic9—21no—
50C4×D3×C9C2×C412yes—
51C4×D3×C9D412yes—
52C9×Dic6—21no—
53C9×Dic6C2×C412yescomplemented
54D3×Dic9—42no—
55C3⋊(C9⋊C8)—21no—
56C3×C9⋊C8—21no—
57#5232—91no—
58#4771—91no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C3×#67—21no—
61C22×C3×Dic3—21no—
62C3×#237—22no—
63C3×#237Dic314yes—
64C2×C3×C4×D3C2×D311yescharacteristic
65C2×C3×Dic6C2×D311yescharacteristic
66C2×D3×Dic3—61no—
67C4⋊(C3⋊Dic3)—31no—
68C2×C3⋊(C3⋊C8)—31no—
69C3×C4⋊Dic3—31no—
70C2×C3×C3⋊C8—31no—
71Dic3⋊Dic3—61no—
72#3711—62no—
73C9×#41—34no—
74C9×#41—62no—
75C2×Q8×C9—31no—
76C2×D4×C9—21no—
77C2×D4×C9—61no—
78C2×D4×C9C2×D311yescharacteristic
79C22×C4×C9—31no—
80C22×C4×C9—61no—
81C22×Dic9—61no—
82C4⋊Dic9—31no—
83C4⋊Dic9—61no—
84C4⋊Dic9—61no—
85C4⋊Dic9C2×D311yescharacteristic
86#3667—62no—
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89C2×C9⋊C8C2×D311yescharacteristic
90C22×C3×C9—22no—
91C22×C3×C9—41no—
92C22×C3×C9—41no—
93C22×C3×C9C2×D411yescharacteristic
94C2×D3×C9—21no—
95C2×D3×C9—21no—
96C2×D3×C9C22⋊C412yes—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—41no—
100C9×Dic3—22no—
101C9×Dic3C22⋊C412yes—
102C3×Dic9—41no—
103C2×C3×#41—31no—
104C2×#237D911yescharacteristic
105C2×C4⋊Dic3—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—91no—
108D3×C4⋊C4—91no—
109C3×#102—91no—
110#835—91no—
111#835—91no—
112#832—31no—
113#832—91no—
114#832—91no—
115#832—181no—
116#816—91no—
117#810—91no—
118C23×C32—21no—
119C22×C3×D3#6711yescharacteristic
120C32×D4—21no—
121C32×D4C2×Dic312yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—61no—
124C3×#67—24no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3#6711yescharacteristic
129C3×C4×D3C2×Dic312yes—
130C3×C4×D3#6712yes—
131C3×Dic6—21no—
132C3×Dic6C2×Dic312yes—
133D3×Dic3—122no—
134C23×C9—21no—
135C23×C9—31no—
136C23×C9—41no—
137C3⋊(C3⋊C8)—61no—
138C3×C3⋊C8—61no—
139Q8×C9—32no—
140Q8×C9—61no—
141D4×C9—21no—
142D4×C9—22no—
143D4×C9—41no—
144D4×C9—64no—
145D4×C9C4×D312yescomplemented
146C2×C4×C9—21no—
147C2×C4×C9—31no—
148C2×C4×C9—32no—
149C2×C4×C9—32no—
150C2×C4×C9—64no—
151C2×C4×C9—61no—
152C2×C4×C9C22×D311yescharacteristic
153C2×Dic9—21no—
154C2×Dic9—61no—
155C2×Dic9—121no—
156C2×Dic9—122no—
157C9⋊C8—21no—
158C9⋊C8—61no—
159C9⋊C8—121no—
160#417—271noSylow
161C2×C3×C9—42no—
162C2×C3×C9C2×C22⋊C411yescharacteristic
163C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
164C2×C3×C9#13711yescharacteristic
165D3×C9—22no—
166C3×#41—34no—
167C3×#41—62no—
168C2×C3×Q8—31no—
169C2×C3×D4—21no—
170C2×C3×D4—61no—
171C2×C3×D4C2×D911yescharacteristic
172C2×C3×D4D3211yescharacteristic
173C22×C3×C4—31no—
174C22×C3×C4—61no—
175C2×#67—21no—
176C22×Dic3—21no—
177C22×Dic3—181no—
178#237—22no—
179#237Dic914yescomplemented
180C2×C4×D3—181no—
181C2×C4×D3C2×D911yescharacteristic
182C2×Dic6C2×D911yescharacteristic
183C2×C3×C8—91no—
184C3×C4⋊C4—91no—
185C4⋊Dic3—31no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189C4⋊Dic3—181no—
190Dic3⋊C4—181no—
191#208—182no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—91no—
195C2×C3⋊C8—181no—
196ΓL(1,𝔽25)—182no—
197C22×C32—22no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32C2×#6711yescharacteristic
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3#21712yes—
204C32×C4C22×Dic311yescharacteristic
205C32×C4C2×#6711yescharacteristic
206C3⋊Dic3—121no—
207C3×Dic3—22no—
208C3×Dic3—121no—
209C3×Dic3#21712yes—
210C22×C9—22no—
211C22×C9—21no—
212C22×C9—32no—
213C22×C9—42no—
214C22×C9—41no—
215C22×C9—41no—
216C22×C9—61no—
217C22×C9—61no—
218C22×C9—81no—
219C22×C9—81no—
220C22×C9D3×D411yescharacteristic
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—32no—
224C4×C9—62no—
225C4×C9C2×C4×D311yescharacteristic
226C4×C9D3×D411yescharacteristic
227Dic9—41no—
228Dic9—121no—
229Dic9—241no—
230C2×#41—31no—
231C2×M16—271no—
232C2×C4⋊C4—271no—
233#103—271no—
234#103—271no—
235#102—91no—
236#102—271no—
237C3×C9#41711yescharacteristic, complemented, Sylow
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—31no—
241C23×C3—41no—
242C22×D3#61111yescharacteristic
243C3×Q8—32no—
244C3×Q8—61no—
245C3×D4—21no—
246C3×D4—21no—
247C3×D4—22no—
248C3×D4—41no—
249C3×D4—64no—
250C3×D4C2×Dic912yescomplemented
251C3×D4D3×Dic312yes—
252C2×C3×C4—21no—
253C2×C3×C4—31no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—64no—
257C2×C3×C4—61no—
258C2×C3×C4—181no—
259C2×C3×C4C22×D911yescharacteristic
260C2×C3×C4C2×D3211yescharacteristic
261#67—24no—
262C2×Dic3—21no—
263C2×Dic3—24no—
264C2×Dic3—61no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—361no—
268C2×Dic3—362no—
269C2×Dic3#61111yescharacteristic
270C4×D3—362no—
271C4×D3C2×Dic912yes—
272C4×D3#61112yes—
273Dic6—21no—
274Dic6C2×Dic912yescomplemented
275C3×C8—181no—
276C3⋊C8—61no—
277C3⋊C8—181no—
278C3⋊C8—181no—
279C3⋊C8—361no—
280C2×C32—42no—
281C2×C32#94911yescharacteristic
282C2×C32#95111yescharacteristic
283C2×C32C2×#21711yescharacteristic
284C3×D3—22no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—42no—
289C2×C9—62no—
290C2×C9—82no—
291C2×C9D3×C22⋊C411yescharacteristic
292C2×C9#91111yescharacteristic
293C2×C9#91511yescharacteristic
294#41—34no—
295#41—62no—
296C2×Q8—31no—
297C2×D4—61no—
298C2×D4D3×D911yescharacteristic
299C22×C4—31no—
300C22×C4—61no—
301C22×C4—541no—
302M16—542no—
303C2×C8—91no—
304C2×C8—271no—
305C4⋊C4—91no—
306C4⋊C4—271no—
307C4⋊C4—541no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—32no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3C2×#61111yescharacteristic
322C22×C3D3×#6711yescharacteristic
323C2×D3—21no—
324C2×D3—21no—
325C2×D3#367612yes—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—32no—
329C3×C4—62no—
330C3×C4—361no—
331C3×C4C22×Dic911yescharacteristic
332C3×C4C2×#61111yescharacteristic
333C3×C4C2×D3×Dic311yescharacteristic
334C3×C4D3×#6711yescharacteristic
335Dic3—22no—
336Dic3—121no—
337Dic3—361no—
338Dic3—361no—
339Dic3—721no—
340Dic3#367612yes—
341C32#523211yescharacteristic
342C9—21no—
343C9#477111yescharacteristic, complemented
344C23—21no—
345C23—31no—
346Q8—32no—
347Q8—61no—
348D4—21no—
349D4—64no—
350D4D3×Dic912yescomplemented
351C2×C4—31no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—64no—
355C2×C4—61no—
356C2×C4—181no—
357C2×C4—541no—
358C2×C4—1082no—
359C2×C4C2×D3×D911yescharacteristic
360C8—181no—
361C8—541no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—62no—
368C2×C3—82no—
369C2×C3#6351111yescharacteristic
370C2×C3#6351211yescharacteristic
371C2×C3C2×#367611yescharacteristic
372C2×C3#6392511yescharacteristic
373C2×C3#6392911yescharacteristic
374C2×C3D3×#21711yescharacteristic
375D3—22no—
376C22—22no—
377C22—32no—
378C22—41no—
379C22—41no—
380C22—61no—
381C22—61no—
382C22D3×#61111yescharacteristic
383C4—32no—
384C4—62no—
385C4—361no—
386C4—1081no—
387C4C2×D3×Dic911yescharacteristic
388C4D3×#61111yescharacteristic
389C3—21no—
390C3#1058891611yescharacteristic, complemented
391C3#1059141411yescharacteristic
392C2—42no—
393C2—62no—
394C2#1172116511yescharacteristic
395C2#1172116711yescharacteristic
396C2D3×#367611yescharacteristic
subgroupG/H
centerC22D3×#611
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×#237D3
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D3×C4⋊Dic9C2semidirect
3. C2×#88420C2semidirect
4. #11720576C2semidirect
5. C3×#63392C2semidirect
6. #11720298C2semidirect
7. #11720297C2semidirect
14. C9×#237C4semidirect
38. #63392D3semidirect
42. C3×D4×C9C2×C4semidirect
53. C9×Dic6C2×C4semidirect
145. D4×C9C4×D3semidirect
179. #237Dic9semidirect
237. C3×C9#417semidirect
250. C3×D4C2×Dic9semidirect
274. Dic6C2×Dic9semidirect
343. C9#4771semidirect
350. D4D3×Dic9semidirect
390. C3#10588916semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
D3×C4⋊Dic92
C2×#884202
#117205762
C3×#633922
#117202982
#117202972
#105914143
#105889163