Finite Group Databasebeta

← #13138682

Subgroups — #13138682

Order 1728.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#88638C211yesmaximal, characteristic
2C2×C3×#3663C211yesmaximal, characteristic, complemented
3D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11720539C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590670—31nomaximal
9#10588451—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×#6194—21no—
14C2×D3×Dic9—21no—
15C2×Dic3×D9—21no—
16#88638—22no—
17#88638C414yescomplemented
18C2×C3×C8×C9C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C3×#3663—22no—
23Dic9⋊Dic3—21no—
24C2×#3795—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26D3×C4⋊Dic3—31no—
27C2×#3699—31no—
28#63643—31no—
29C3×#821D311yescharacteristic
30#63567—31no—
31#63563—31no—
32C2×#3663D311yescharacteristic, complemented
33C4⋊C4×D9—31no—
34C9×#102—31no—
35#63392—31no—
36#63376—31no—
37#63376—61no—
38#63368—31no—
39C22×D3×C9—21no—
40C22×C3×D9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—21no—
44#6226—21no—
45#6226C2×C412yescomplemented
46C2×C9×Dic3—21no—
47C2×C3×Dic9D411yescharacteristic
48C9×D12—21no—
49C9×D12C2×C412yescomplemented
50C3×C4×D9C2×C412yes—
51C3×C4×D9D412yes—
52#6194—24no—
53D3×Dic9—24no—
54Dic3×D9—42no—
55C3×C8×C9—21no—
56C3×C9⋊C8—21no—
57#5133—91no—
58#4771—91no—
59C2×#634—31no—
60C2×C3×D12C2×D311yescharacteristic
61C2×C3×C4×D3—31no—
62C2×#625—61no—
63C2×D3×Dic3—61no—
64C2×D3×Dic3—61no—
65#3795—34no—
66#3795—62no—
67C2×C32×C8C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)—31no—
69C3×C4⋊Dic3C2×D311yescharacteristic
70C2×C3×C3⋊C8—31no—
71C3×ΓL(1,𝔽25)—62no—
72Dic3⋊Dic3—61no—
73C2×D4×C9—31no—
74C2×#611—61no—
75C22×Dic9—61no—
76#3699—34no—
77#3699—62no—
78C2×C4×D9—61no—
79C2×C4×D9C2×D311yescharacteristic
80C2×Dic18—31no—
81C2×Dic18—61no—
82C2×C8×C9—21no—
83C2×C8×C9C2×D311yescharacteristic
84C9×C4⋊C4—31no—
85C4⋊Dic9—31no—
86C4⋊Dic9—61no—
87Dic9⋊C4—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89#3663—22no—
90C22×C3×C9C2×D411yescharacteristic
91C2×D3×C9—22no—
92C2×D3×C9—41no—
93C2×D3×C9—41no—
94C2×C3×D9—21no—
95C2×C3×D9—21no—
96C2×C3×D9C22⋊C412yes—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—22no—
100C3⋊Dic9—41no—
101C9×Dic3—41no—
102C3×Dic9C22⋊C412yes—
103C2×#237—91no—
104C2×#235—91no—
105C2×C3×M16—91no—
106C2×C4⋊Dic3—91no—
107C2×ΓL(1,𝔽25)—31no—
108D3×C4⋊C4—91no—
109C3×#102—31no—
110#832—91no—
111#821—91no—
112#821D911yescharacteristic, complemented
113#816—91no—
114#816—91no—
115#816—91no—
116#816—181no—
117#810—91no—
118C22×C3×D3—21no—
119C22×C3×D3—31no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—61no—
123#634—32no—
124#634—61no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—31no—
127C3×D12—21no—
128C3×D12C4×D312yes—
129C3×C4×D3—32no—
130C3×C4×D3—32no—
131#625—64no—
132D3×Dic3—64no—
133D3×Dic3—122no—
134C23×C9—61no—
135C22×D9D1211yescharacteristic
136C32×C8—21no—
137C3×C3⋊C8—61no—
138D4×C9—32no—
139D4×C9—61no—
140C2×C4×C9—21no—
141C2×C4×C9—61no—
142C2×C4×C9C22×D311yescharacteristic
143#611—64no—
144C2×Dic9—61no—
145C2×Dic9—64no—
146C2×Dic9—61no—
147C2×Dic9—121no—
148C2×Dic9—121no—
149C2×Dic9D1211yescharacteristic
150C4×D9—122no—
151C4×D9C4×D312yes—
152C4×D9#6712yes—
153Dic18—32no—
154Dic18—62no—
155Dic18—61no—
156Dic18—121no—
157C8×C9—21no—
158C8×C9—41no—
159C9⋊C8—21no—
160#417—271noSylow
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
163C2×C3×C9#13711yescharacteristic
164D3×C9—42no—
165C3×D9—22no—
166C2×C3×D4—31no—
167C22×C3×C4—91no—
168C2×#67—181no—
169C2×#67—181no—
170C22×Dic3—181no—
171C22×Dic3—181no—
172#237—94no—
173#237—182no—
174#235—94no—
175#235—182no—
176C2×D12C2×D911yescharacteristic
177C2×C4×D3—31no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×Dic6—91no—
181C2×Dic6—91no—
182C2×Dic6—181no—
183C3×M16—182no—
184C2×C3×C8—21no—
185C2×C3×C8—91no—
186C2×C3×C8C2×D911yescharacteristic
187C2×C3×C8D3211yescharacteristic
188C3×C4⋊C4—31no—
189C4⋊Dic3—91no—
190C4⋊Dic3—91no—
191C4⋊Dic3—181no—
192C4⋊Dic3—181no—
193C4⋊Dic3C2×D911yescharacteristic
194Dic3⋊C4—181no—
195C2×C3⋊C8—31no—
196ΓL(1,𝔽25)—62no—
197C22×C32D3×D411yescharacteristic
198C2×C3×D3—22no—
199C2×C3×D3—32no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4D3×D411yescharacteristic
206C3⋊Dic3—62no—
207C3⋊Dic3—121no—
208C3×Dic3—32no—
209C3×Dic3—41no—
210C22×C9—21no—
211C22×C9—62no—
212C22×C9—121no—
213C22×C9—121no—
214C22×C9C2×D1211yescharacteristic
215C2×D9—21no—
216C2×D9—21no—
217C2×D9(C2×D3)⋊C412yes—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—121no—
221C4×C9C2×C4×D311yescharacteristic
222C4×C9C2×#6711yescharacteristic
223Dic9—62no—
224Dic9—122no—
225Dic9—121no—
226Dic9—241no—
227Dic9(C2×D3)⋊C412yes—
228C2×#41—271no—
229C2×M16—91no—
230C2×C4⋊C4—271no—
231#103—271no—
232#103—271no—
233#102—31no—
234#102—271no—
235C3×C9#41711yescharacteristic, complemented, Sylow
236C23×C3—61no—
237C23×C3—91no—
238C22×D3—21no—
239C22×D3—31no—
240C3×D4—32no—
241C3×D4—61no—
242C2×C3×C4—21no—
243C2×C3×C4—61no—
244C2×C3×C4—91no—
245C2×C3×C4—92no—
246C2×C3×C4—92no—
247C2×C3×C4C22×D911yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249#67—184no—
250#67—184no—
251C2×Dic3—21no—
252C2×Dic3—31no—
253C2×Dic3—184no—
254C2×Dic3—181no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C2×Dic3—361no—
259C2×Dic3—361no—
260C2×Dic3—362no—
261D12—21no—
262D12C4×D912yescomplemented
263C4×D3—32no—
264C4×D3—32no—
265C4×D3—184no—
266C4×D3—362no—
267Dic6—92no—
268Dic6—92no—
269Dic6—182no—
270Dic6—181no—
271Dic6—181no—
272Dic6—361no—
273C3×C8—21no—
274C3×C8—21no—
275C3×C8—41no—
276C3×C8—181no—
277C3⋊C8—61no—
278C2×C32D3×C22⋊C411yescharacteristic
279C2×C32#91111yescharacteristic
280C2×C32#91511yescharacteristic
281C3×D3—42no—
282C3×D3—62no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—122no—
287C2×C9#90811yescharacteristic
288C2×C9#90911yescharacteristic
289C2×C9C2×(C2×D3)⋊C411yescharacteristic
290D9—22no—
291#41—274no—
292#41—542no—
293C2×Q8—271no—
294C2×D4—31no—
295C2×D4—541no—
296C22×C4—91no—
297C22×C4—541no—
298C22×C4—541no—
299M16—182no—
300C2×C8—91no—
301C2×C8D3×D911yescharacteristic
302C4⋊C4—31no—
303C4⋊C4—271no—
304C4⋊C4—541no—
305C22×C3—21no—
306C22×C3—62no—
307C22×C3—92no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3D4×D911yescharacteristic
313C22×C3D3×D1211yescharacteristic
314C2×D3—22no—
315C2×D3—32no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—61no—
319C2×D3—61no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—92no—
323C3×C4—121no—
324C3×C4C2×C4×D911yescharacteristic
325C3×C4D4×D911yescharacteristic
326C3×C4C4×D3211yescharacteristic
327C3×C4D3×#6711yescharacteristic
328Dic3—32no—
329Dic3—41no—
330Dic3—182no—
331Dic3—182no—
332Dic3—362no—
333Dic3—361no—
334Dic3—361no—
335Dic3—721no—
336C32#477111yescharacteristic
337C9—21no—
338C9#513311yescharacteristic, complemented
339C23—61no—
340C23—91no—
341Q8—272no—
342Q8—541no—
343D4—32no—
344D4—61no—
345D4—544no—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—544no—
351C2×C4—541no—
352C2×C4—541no—
353C2×C4—1082no—
354C2×C4C2×D3×D911yescharacteristic
355C8—21no—
356C8—181no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—122no—
361C2×C3—182no—
362C2×C3C22⋊C4×D911yescharacteristic
363C2×C3#6347311yescharacteristic
364C2×C3#6347711yescharacteristic
365C2×C3#6379511yescharacteristic
366C2×C3#6380011yescharacteristic
367C2×C3D3×(C2×D3)⋊C411yescharacteristic
368D3—42no—
369D3—62no—
370C22—62no—
371C22—92no—
372C22—121no—
373C22—121no—
374C22—181no—
375C22—181no—
376C22D9×D1211yescharacteristic
377C4—92no—
378C4—121no—
379C4—542no—
380C4—1081no—
381C4C4×D3×D911yescharacteristic
382C4D9×#6711yescharacteristic
383C3—21no—
384C3#1058845111yescharacteristic, complemented
385C3#1059067011yescharacteristic
386C2—122no—
387C2—182no—
388C2#1172081611yescharacteristic
389C2#1172082711yescharacteristic
390C2D9×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88638C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3663C2semidirect
3. D9×C4⋊Dic3C2semidirect
4. #11720539C2semidirect
5. C9×#821C2semidirect
6. #11720295C2semidirect
7. #11720269C2semidirect
17. #88638C4semidirect
32. C2×#3663D3semidirect
45. #6226C2×C4semidirect
49. C9×D12C2×C4semidirect
112. #821D9semidirect
235. C3×C9#417semidirect
262. D12C4×D9semidirect
338. C9#5133semidirect
384. C3#10588451semidirect

Maximal subgroups

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

subgroupindex
C2×#886382
C2×C3×#36632
D9×C4⋊Dic32
#117205392
C9×#8212
#117202952
#117202692
#105906703
#105884513