Finite Group Databasebeta

← #13140735

Subgroups — #13140735

Order 1728.

504 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 101 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×C3×Q8×D9C211yesmaximal, characteristic
2C2×#88547C211yesmaximal, characteristic, complemented
3#11720945C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic, complemented
8#10591595—31nomaximal
9#10588489—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C3×Q8×D9—22no—
13C3×Q8×D9—22no—
14C2×C3×C4×D9—21no—
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×Dic18—21no—
17C2×C3×Dic18C2211yescharacteristic
18#88547—22no—
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23Dic3⋊Dic9—21no—
24#88418—22no—
25C2×C3×D3×Q8—31no—
26C2×#3756—31no—
27#63857—31no—
28C2×#3719—31no—
29C2×Q8×D9D311yescharacteristic
30C3×#835D311yescharacteristic
31#63565—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3674—31no—
34C2×#3674—61no—
35C2×Dic36—31no—
36#63459—31no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63368—31no—
40C22×C3×D9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9D412yes—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46#6226—21no—
47#6226—21no—
48C2×C9×Dic3—21no—
49C2×C3×Dic9—21no—
50C2×C3×Dic9D411yescharacteristic
51C3×C4×D9—24no—
52C3×C4×D9—21no—
53C3×C4×D9—21no—
54C3×Dic18—24no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C3⋊(C9⋊C8)—21no—
58C9×C3⋊C8—21no—
59#5228—91no—
60#4809—91no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×#634—31no—
63C3×D3×Q8—62no—
64C3×D3×Q8—62no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69#3756—62no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75#3719—62no—
76C2×Q8×C9—21no—
77C2×Q8×C9C2×D311yescharacteristic
78Q8×D9—22no—
79Q8×D9—22no—
80C2×C4×D9—21no—
81C2×C4×D9C2×D311yescharacteristic
82C2×Dic18—21no—
83C2×Dic18—31no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C2×C8×C9—31no—
87C9×C4⋊C4—31no—
88#3674—64no—
89#3674—62no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—61no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94Dic36—62no—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3×D9—21no—
97C2×C3×D9—21no—
98C2×C3×D9Q1612yescomplemented
99C3×C4×C9—22no—
100C3×C4×C9C2×D411yescharacteristic
101C3×C4×C9C2×D411yescharacteristic
102C3⋊Dic9—41no—
103C9×Dic3—41no—
104C3×Dic9—21no—
105C3×Dic9—22no—
106C22×C3×Q8—91no—
107C2×D3×Q8—31no—
108C2×#216—91no—
109C2×#216—91no—
110C2×#216—91no—
111C2×#216—181no—
112#924—91no—
113C2×Dic12—91no—
114#897—91no—
115C3×#103—31no—
116#835—91no—
117#835D911yescharacteristic, complemented
118#830—91no—
119(C2×D3)⋊C8—91no—
120#810—91no—
121C22×C3×D3—31no—
122C32×Q8—21no—
123C32×Q8D1212yes—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127#634—61no—
128#634—61no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C3×C4×D3—64no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×Dic6—32no—
136C3×Dic6—64no—
137C3×Dic6—61no—
138C22×D9#6711yescharacteristic
139C3⋊(C3⋊C8)—61no—
140C3×C3⋊C8—21no—
141Q8×C9—21no—
142Q8×C9—22no—
143Q8×C9—41no—
144Q8×C9#6712yes—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—41no—
148C2×C4×C9—61no—
149C2×C4×C9C22×D311yescharacteristic
150C2×Dic9—21no—
151C2×Dic9—61no—
152C2×Dic9—121no—
153C2×Dic9#6711yescharacteristic
154C4×D9—24no—
155C4×D9—21no—
156C4×D9—21no—
157Dic18—24no—
158Dic18—21no—
159Dic18—62no—
160Dic18—61no—
161Dic18—61no—
162Dic18—121no—
163Dic18#6712yes—
164C8×C9—61no—
165C9⋊C8—62no—
166C9⋊C8—61no—
167#451—271noSylow
168C2×C3×C9#12011yescharacteristic
169C2×C3×C9C2×Q1611yescharacteristic
170C2×C3×C9#13711yescharacteristic
171C3×D9—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—91no—
174C2×C3×Q8—181no—
175C2×C3×Q8—182no—
176C2×C3×Q8—182no—
177C2×C3×Q8C2×D911yescharacteristic
178C2×C3×Q8D3211yescharacteristic
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181D3×Q8—62no—
182D3×Q8—62no—
183C2×C4×D3—31no—
184C2×C4×D3—61no—
185C2×Dic6—31no—
186C2×Dic6—61no—
187C2×Dic6—91no—
188C2×Dic6—91no—
189C2×Dic6—181no—
190C2×C3×C8—31no—
191C3×C4⋊C4—31no—
192#217—181no—
193#216—182no—
194#216—184no—
195#216—182no—
196#216—182no—
197(C2×D3)⋊C4—181no—
198C4⋊Dic3—181no—
199C4⋊Dic3C2×D911yescharacteristic
200Dic3⋊C4—181no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—181no—
204C2×C3⋊C8C2×D911yescharacteristic
205Dic12—182no—
206C22×C32D3×D411yescharacteristic
207C2×C3×D3—32no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C32×C4—22no—
211C32×C4C2×D1211yescharacteristic
212C32×C4D3×D411yescharacteristic
213C3⋊Dic3—121no—
214C3×Dic3—41no—
215C3×Dic3—62no—
216C3×Dic3—61no—
217C22×C9—21no—
218C22×C9C2×#6711yescharacteristic
219C2×D9—21no—
220C2×D9—21no—
221C2×D9#21612yescomplemented
222C4×C9—22no—
223C4×C9—21no—
224C4×C9—21no—
225C4×C9—42no—
226C4×C9—121no—
227C4×C9C2×#6711yescharacteristic
228C4×C9C2×#6711yescharacteristic
229Dic9—21no—
230Dic9—22no—
231Dic9—122no—
232Dic9—121no—
233C22×Q8—91no—
234C2×Q16—271no—
235C2×Q16—271no—
236C22⋊Q8—271no—
237#103—31no—
238#103—271no—
239C22⋊C8—271no—
240C3×C9#45111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C22×D3—31no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—22no—
246C3×Q8—41no—
247C3×Q8—92no—
248C3×Q8—184no—
249C3×Q8—181no—
250C3×Q8D3612yes—
251C3×Q8#62612yes—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—61no—
257C2×C3×C4—91no—
258C2×C3×C4—184no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C22×D911yescharacteristic
263C2×C3×C4C2×D3211yescharacteristic
264C2×Dic3—21no—
265C2×Dic3—31no—
266C2×Dic3—61no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C2×Dic3—361no—
270C4×D3—64no—
271C4×D3—61no—
272C4×D3—61no—
273Dic6—32no—
274Dic6—64no—
275Dic6—61no—
276Dic6—182no—
277Dic6—181no—
278Dic6—181no—
279Dic6—181no—
280Dic6—181no—
281Dic6—361no—
282C3×C8—61no—
283C3⋊C8—21no—
284C3⋊C8—182no—
285C3⋊C8—181no—
286C3⋊C8—181no—
287C2×C32#88211yescharacteristic
288C2×C32#91511yescharacteristic
289C2×C32D3×Q1611yescharacteristic
290C3×D3—62no—
291C2×C9—21no—
292C2×C9—21no—
293C2×C9—21no—
294C2×C9#94211yescharacteristic
295C2×C9C2×#21611yescharacteristic
296C2×C9#95311yescharacteristic
297D9—22no—
298C2×Q8—91no—
299C2×Q8—181no—
300C2×Q8—182no—
301C2×Q8—182no—
302C2×Q8—271no—
303C2×Q8D3×D911yescharacteristic
304C22×C4—91no—
305C22×C4—181no—
306Q16—542no—
307Q16—542no—
308C2×C8—31no—
309C2×C8—271no—
310C4⋊C4—31no—
311C4⋊C4—541no—
312C22⋊C4—541no—
313C22×C3—21no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3D4×D911yescharacteristic
318C22×C3D3×#6711yescharacteristic
319C2×D3—32no—
320C2×D3—61no—
321C2×D3—61no—
322C3×C4—22no—
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—42no—
327C3×C4—121no—
328C3×C4—181no—
329C3×C4—182no—
330C3×C4C2×D3611yescharacteristic
331C3×C4D4×D911yescharacteristic
332C3×C4C2×#62611yescharacteristic
333C3×C4D3×#6711yescharacteristic
334Dic3—41no—
335Dic3—62no—
336Dic3—61no—
337Dic3—362no—
338Dic3—361no—
339Dic3—361no—
340C32#480911yescharacteristic
341C9—21no—
342C9#522811yescharacteristic, complemented
343C23—91no—
344Q8—21no—
345Q8—92no—
346Q8—184no—
347Q8—181no—
348Q8—541no—
349Q8—541no—
350Q8#619212yes—
351C2×C4—21no—
352C2×C4—61no—
353C2×C4—91no—
354C2×C4—184no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—541no—
359C2×C4C2×D3×D911yescharacteristic
360C8—61no—
361C8—541no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—182no—
366C2×C3#6344411yescharacteristic
367C2×C3#6347711yescharacteristic
368C2×C3Q16×D911yescharacteristic
369C2×C3D3×#21611yescharacteristic
370C2×C3#6394611yescharacteristic
371C2×C3#6397711yescharacteristic
372D3—62no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22D9×#6711yescharacteristic
377C4—22no—
378C4—121no—
379C4—181no—
380C4—182no—
381C4—1081no—
382C4C2×#619211yescharacteristic
383C4D9×#6711yescharacteristic
384C3—21no—
385C3#1058848911yescharacteristic, complemented
386C3#1059159511yescharacteristic
387C2—182no—
388C2#1172121411yescharacteristic
389C2D9×#21611yescharacteristic
390C2#1172127611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88547C2semidirect
4. C2×#88418C2semidirect
5. C9×#835C2semidirect
7. #11720268C2semidirect
98. C2×C3×D9Q16semidirect
117. #835D9semidirect
221. C2×D9#216semidirect
240. C3×C9#451semidirect
342. C9#5228semidirect
385. C3#10588489semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
C2×#885472
#117209452
C2×#884182
C9×#8352
(C2×D9)⋊(C3⋊C8)2
#117202682
#105915953
#105884893