Finite Group Databasebeta

← #13138638

Subgroups — #13138638

Order 1728.

486 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 95 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×D3×Dic18C211yesmaximal, characteristic
2C2×C3×Dic36C211yesmaximal, characteristic, complemented
3#11720960C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic, complemented
5#11720539C211yesmaximal, characteristic, complemented
6C9×(C2×D3)⋊C8C211yesmaximal, characteristic
7#11720276C211yesmaximal, characteristic, complemented
8#10590949—31nomaximal
9#10588405—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×D3×Dic9—21no—
15C2×#6189—21no—
16D3×Dic18—22no—
17D3×Dic18—22no—
18C2×C3×C8×C9C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C3×Dic36—22no—
22(C2×D3)⋊Dic9—21no—
23#88442—21no—
24#88418—22no—
25C2×D3×Dic6—31no—
26C2×C3×Dic12—31no—
27#63856—31no—
28C2×#3719—31no—
29C22×Dic18—31no—
30#63643—31no—
31C3×(C2×D3)⋊C8D311yescharacteristic
32#63566—31no—
33#63486—31no—
34C2×Dic36—31no—
35C2×Dic36D311yescharacteristic, complemented
36C9×C22⋊C8—31no—
37#63376—31no—
38#63376—31no—
39#63376—61no—
40C22×D3×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—21no—
44#6226—21no—
45#6226D412yes—
46C2×C9×Dic3D411yescharacteristic
47C2×C3×Dic9—21no—
48C2×C3×Dic9—21no—
49C4×D3×C9—21no—
50C4×D3×C9—21no—
51C3×Dic18—21no—
52C3×Dic18—21no—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55D3×Dic9—24no—
56#6189—24no—
57C3×C8×C9—21no—
58C9×C3⋊C8—21no—
59#5186—91no—
60#4739—91no—
61C2×#634—31no—
62C2×C3×C4×D3C2×D311yescharacteristic
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×#627—61no—
66C2×D3×Dic3—61no—
67D3×Dic6—62no—
68D3×Dic6—62no—
69C2×C32×C8C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)—31no—
71C2×C3×C3⋊C8C2×D311yescharacteristic
72C3×Dic12—62no—
73#3724—61no—
74(C2×D3)⋊Dic3—61no—
75#3719—62no—
76C22×C4×C9—31no—
77C22×Dic9—61no—
78C2×Dic18—31no—
79C2×Dic18—62no—
80C2×Dic18—61no—
81C2×Dic18—62no—
82C2×Dic18—61no—
83C2×Dic18C2×D311yescharacteristic
84C2×Dic18C2×D311yescharacteristic
85C2×C8×C9—21no—
86C2×C8×C9—31no—
87C2×C8×C9C2×D311yescharacteristic
88#3676—61no—
89C4⋊Dic9—31no—
90C4⋊Dic9—61no—
91Dic9⋊C4—61no—
92Dic36—22no—
93Dic36—62no—
94C22×C3×C9C2×D411yescharacteristic
95C2×D3×C9—21no—
96C2×D3×C9—21no—
97C2×D3×C9Q1612yescomplemented
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3⋊Dic9—22no—
101C3⋊Dic9—41no—
102C9×Dic3—21no—
103C3×Dic9—22no—
104C3×Dic9—41no—
105C2×D3×Q8—91no—
106C22×Dic6—91no—
107C2×C3×Q16—91no—
108#946—91no—
109C2×#216—91no—
110#924—91no—
111C2×Dic12—31no—
112C2×Dic12—91no—
113C3×C22⋊C8—31no—
114#835—91no—
115(C2×D3)⋊C8D911yescharacteristic
116#816—91no—
117#816—91no—
118#816—91no—
119#816—181no—
120C22×C3×D3D1211yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124#634—32no—
125#634—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3D1211yescharacteristic
129C3×C4×D3—21no—
130C3×C4×D3—21no—
131C3×Dic6—32no—
132C3×Dic6—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135#627—64no—
136D3×Dic3—64no—
137C23×C9—31no—
138C32×C8—21no—
139C3×C3⋊C8—21no—
140C2×C4×C9—21no—
141C2×C4×C9—31no—
142C2×C4×C9—61no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145C2×Dic9—21no—
146C2×Dic9—21no—
147C2×Dic9—64no—
148C2×Dic9—61no—
149C2×Dic9—61no—
150C2×Dic9—121no—
151C2×Dic9—121no—
152Dic18—21no—
153Dic18—21no—
154Dic18—21no—
155Dic18—32no—
156Dic18—62no—
157Dic18—64no—
158Dic18—61no—
159Dic18—121no—
160Dic18#6712yes—
161C8×C9—21no—
162C8×C9—41no—
163C8×C9—61no—
164#451—271noSylow
165C2×C3×C9#12011yescharacteristic
166C2×C3×C9C2×Q1611yescharacteristic
167C2×C3×C9#13711yescharacteristic
168D3×C9—22no—
169C2×C3×Q8—91no—
170C2×C3×Q8—91no—
171C22×C3×C4—31no—
172C22×Dic3—181no—
173D3×Q8—182no—
174D3×Q8—182no—
175C2×C4×D3—181no—
176C2×C4×D3C2×D911yescharacteristic
177C2×Dic6—31no—
178C2×Dic6—31no—
179C2×Dic6—91no—
180C2×Dic6—91no—
181C2×Dic6—181no—
182C2×Dic6—182no—
183C2×Dic6—182no—
184C2×Dic6—181no—
185C2×Dic6—181no—
186C3×Q16—182no—
187C2×C3×C8—21no—
188C2×C3×C8—31no—
189C2×C3×C8C2×D911yescharacteristic
190C2×C3×C8D3211yescharacteristic
191#217—181no—
192#216—182no—
193(C2×D3)⋊C4—181no—
194C4⋊Dic3—91no—
195C4⋊Dic3—91no—
196C4⋊Dic3—181no—
197Dic3⋊C4—181no—
198Dic3⋊C4—181no—
199C2×C3⋊C8C2×D911yescharacteristic
200Dic12—62no—
201Dic12—182no—
202C22×C32C2×D1211yescharacteristic
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3Dic1212yes—
206C32×C4D3×D411yescharacteristic
207C32×C4D3×D411yescharacteristic
208C3⋊Dic3—62no—
209C3⋊Dic3—121no—
210C3×Dic3—21no—
211C3×Dic3—62no—
212C3×Dic3—121no—
213C22×C9—21no—
214C22×C9—32no—
215C22×C9—61no—
216C22×C9—61no—
217C22×C9D3×D411yescharacteristic
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—61no—
221C4×C9D3×D411yescharacteristic
222C4×C9C2×#6711yescharacteristic
223Dic9—22no—
224Dic9—41no—
225Dic9—62no—
226Dic9—122no—
227Dic9—121no—
228Dic9—241no—
229C22×Q8—271no—
230C2×Q16—91no—
231C2×Q16—271no—
232C22⋊Q8—271no—
233#103—271no—
234#103—271no—
235C22⋊C8—31no—
236C3×C9#45111yescharacteristic, complemented, Sylow
237C23×C3—31no—
238C22×D3D3611yescharacteristic
239C3×Q8—92no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×Q8—181no—
243C2×C3×C4—21no—
244C2×C3×C4—31no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C22×D911yescharacteristic
250C2×C3×C4C2×D3211yescharacteristic
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—184no—
254C2×Dic3—181no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C2×Dic3—361no—
259C2×Dic3—361no—
260C2×Dic3D3611yescharacteristic
261C4×D3—21no—
262C4×D3—21no—
263C4×D3—184no—
264Dic6—32no—
265Dic6—61no—
266Dic6—61no—
267Dic6—61no—
268Dic6—92no—
269Dic6—92no—
270Dic6—182no—
271Dic6—184no—
272Dic6—184no—
273Dic6—181no—
274Dic6—181no—
275Dic6—361no—
276C3×C8—21no—
277C3×C8—21no—
278C3×C8—41no—
279C3×C8—61no—
280C3⋊C8—21no—
281C2×C32#88211yescharacteristic
282C2×C32C2×Dic1211yescharacteristic
283C2×C32#90911yescharacteristic
284C3×D3—22no—
285C2×C9—21no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—62no—
289C2×C9D3×Q1611yescharacteristic
290C2×C9#91811yescharacteristic
291C2×C9#93711yescharacteristic
292C2×Q8—91no—
293C2×Q8—91no—
294C2×Q8—271no—
295C2×Q8—541no—
296C2×Q8—542no—
297C2×Q8—542no—
298C22×C4—31no—
299C22×C4—541no—
300Q16—182no—
301Q16—542no—
302C2×C8—31no—
303C2×C8D3×D911yescharacteristic
304C4⋊C4—271no—
305C4⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—21no—
308C22×C3—32no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3C2×D3611yescharacteristic
312C22×C3D3×D1211yescharacteristic
313C2×D3—21no—
314C2×D3—21no—
315C2×D3Dic3612yescomplemented
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—61no—
319C3×C4—182no—
320C3×C4—361no—
321C3×C4D4×D911yescharacteristic
322C3×C4D4×D911yescharacteristic
323C3×C4#380211yescharacteristic
324C3×C4D3×#6711yescharacteristic
325Dic3—21no—
326Dic3—62no—
327Dic3—121no—
328Dic3—182no—
329Dic3—182no—
330Dic3—362no—
331Dic3—361no—
332Dic3—361no—
333Dic3—721no—
334C32#473911yescharacteristic
335C9—21no—
336C9#518611yescharacteristic, complemented
337C23—31no—
338Q8—92no—
339Q8—181no—
340Q8—181no—
341Q8—181no—
342Q8—272no—
343Q8—544no—
344Q8—541no—
345C2×C4—31no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—544no—
351C2×C4—541no—
352C2×C4—541no—
353C2×C4C2×D3×D911yescharacteristic
354C8—21no—
355C8—61no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—62no—
360C2×C3#6344411yescharacteristic
361C2×C3C2×Dic3611yescharacteristic
362C2×C3#6347111yescharacteristic
363C2×C3D3×Dic1211yescharacteristic
364C2×C3#6380111yescharacteristic
365C2×C3#6392211yescharacteristic
366D3—22no—
367C22—32no—
368C22—61no—
369C22—61no—
370C22D3×D3611yescharacteristic
371C4—61no—
372C4—182no—
373C4—361no—
374C4—542no—
375C4—1081no—
376C4#8864611yescharacteristic
377C4D9×#6711yescharacteristic
378C3—21no—
379C3#1058840511yescharacteristic, complemented
380C3#1059094911yescharacteristic
381C2—62no—
382C2D3×Dic3611yescharacteristic
383C2#1172079711yescharacteristic
384C2#1172104911yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC22×C3×D3D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×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×Dic36C2semidirect
4. C2×#88418C2semidirect
5. #11720539C2semidirect
7. #11720276C2semidirect
35. C2×Dic36D3semidirect
97. C2×D3×C9Q16semidirect
236. C3×C9#451semidirect
315. C2×D3Dic36semidirect
336. C9#5186semidirect
379. C3#10588405semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C2×C3×Dic362
#117209602
C2×#884182
#117205392
C9×(C2×D3)⋊C82
#117202762
#105909493
#105884053