Finite Group Databasebeta

← #13140741

Subgroups — #13140741

Order 1728.

516 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 121 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×#3701C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic, complemented
3C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
4#11720582C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10591556—31nomaximal
9#10588484—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C3×#3701—22no—
12C3×#3701C414yescomplemented
13C2×C3×D36—21no—
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9—21no—
16C2×C3×C4×D9C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21(C2×D9)⋊Dic3—21no—
22Dic3×Dic9—21no—
23D9×C3⋊C8—22no—
24C2×C3×#239—31no—
25#63841—31no—
26C2×D3×C3⋊C8—31no—
27C2×#3701D311yescharacteristic
28#63662—31no—
29C3×#835D311yescharacteristic
30#63563—31no—
31#63562—31no—
32C2×C8×D9—31no—
33#63454—31no—
34C9×#103—31no—
35#63395—31no—
36#63395—61no—
37#63380—31no—
38#63369—31no—
39C22×C3×D9—21no—
40C22×C3×D9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9C2×C412yes—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×D36—24no—
49C3×D36—21no—
50C3×D36C2×C412yes—
51C3×C4×D9—24no—
52C3×C4×D9C2×C412yes—
53C3×C4×D9D412yes—
54C3⋊(C9⋊C8)—21no—
55C9×C3⋊C8—21no—
56#5233—91no—
57#4804—91no—
58C2×C32×Q8C2×D311yescharacteristic
59C3×#239—34no—
60C3×#239—62no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C4⋊(C3⋊Dic3)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C3×C4⋊Dic3C2×D311yescharacteristic
68C2×C3×C3⋊C8C2×D311yescharacteristic
69(C2×D3)⋊Dic3—61no—
70Dic32—61no—
71D3×C3⋊C8—62no—
72C2×Q8×C9—21no—
73C2×Q8×C9C2×D311yescharacteristic
74#3701—22no—
75#3701Dic314yescomplemented
76C2×D36—21no—
77C2×D36C2×D311yescharacteristic
78C2×C4×D9—21no—
79C2×C4×D9C2×D311yescharacteristic
80C2×C8×C9—31no—
81C9×C4⋊C4—31no—
82(C2×D9)⋊C4—61no—
83C4⋊Dic9—31no—
84C4⋊Dic9—61no—
85C4×Dic9—61no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88C8×D9—62no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9—22no—
93C2×C3×D9—41no—
94C2×C3×D9—41no—
95C2×C3×D9C22⋊C412yes—
96C3×C4×C9—22no—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—41no—
100C9×Dic3—41no—
101C3×Dic9C22⋊C412yes—
102C2×C3×#41—91no—
103C2×#239—31no—
104#926—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—91no—
107#892—91no—
108C3×#103—31no—
109#835—91no—
110#835—91no—
111#835—181no—
112#835D911yescharacteristic, complemented
113#832—91no—
114#832—91no—
115#821—91no—
116#809—91no—
117C22×C3×D3—31no—
118C22×C3×D3—61no—
119C32×Q8—21no—
120C32×Q8C4×D312yes—
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—61no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—31no—
126C3×D12—32no—
127C3×D12—64no—
128C3×D12—61no—
129C3×C4×D3—32no—
130C3×C4×D3—32no—
131C3×C4×D3—64no—
132C22×D9—21no—
133C22×D9#6711yescharacteristic
134C3⋊(C3⋊C8)—61no—
135C3×C3⋊C8—21no—
136Q8×C9—21no—
137Q8×C9—22no—
138Q8×C9—41no—
139Q8×C9C2×Dic312yes—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—41no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145C2×Dic9—61no—
146C2×Dic9—121no—
147C2×Dic9#6711yescharacteristic
148D36—24no—
149D36—21no—
150D36C2×Dic312yes—
151C4×D9—24no—
152C4×D9C2×Dic312yes—
153C4×D9#6712yes—
154C8×C9—61no—
155C9⋊C8—61no—
156C9⋊C8—121no—
157#416—271noSylow
158C2×C3×C9C2×C22⋊C411yescharacteristic
159C2×C3×C9#13511yescharacteristic
160C2×C3×C9#13511yescharacteristic
161C3×D9—22no—
162C3×D9—42no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×Q8—21no—
166C2×C3×Q8C2×D911yescharacteristic
167C2×C3×Q8D3211yescharacteristic
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172#239—34no—
173#239—62no—
174C2×D12—31no—
175C2×D12—61no—
176C2×C4×D3—31no—
177C2×C4×D3—61no—
178C2×C3×C8—31no—
179C3×C4⋊C4—31no—
180#217—181no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—91no—
183C4⋊Dic3—91no—
184C4⋊Dic3—181no—
185C4⋊Dic3C2×D911yescharacteristic
186C4×Dic3—181no—
187C4×Dic3—181no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—181no—
191C2×C3⋊C8—182no—
192C2×C3⋊C8C2×D911yescharacteristic
193D3×C8—182no—
194C22×C32D3×D411yescharacteristic
195C2×C3×D3—32no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—62no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—22no—
202C32×C4C2×C4×D311yescharacteristic
203C32×C4D3×D411yescharacteristic
204C3⋊Dic3—121no—
205C3×Dic3—32no—
206C3×Dic3—41no—
207C22×C9—21no—
208C22×C9C2×#6711yescharacteristic
209C2×D9—21no—
210C2×D9—21no—
211C2×D9—22no—
212C2×D9—41no—
213C2×D9—41no—
214C2×D9#21712yes—
215C4×C9—22no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—42no—
219C4×C9—121no—
220C4×C9C22×Dic311yescharacteristic
221C4×C9C2×#6711yescharacteristic
222Dic9—121no—
223Dic9—241no—
224Dic9#21712yes—
225C2×#41—91no—
226C22×C8—271no—
227#117—271no—
228#103—31no—
229#103—271no—
230#102—271no—
231#102—271no—
232C3×C9#41611yescharacteristic, complemented, Sylow
233C23×C3—91no—
234C23×C3—181no—
235C22×D3—31no—
236C22×D3—61no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—22no—
240C3×Q8—41no—
241C3×Q8C4×D912yes—
242C3×Q8D3×Dic312yes—
243C3×D4—92no—
244C3×D4—184no—
245C3×D4—181no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—61no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—92no—
254C2×C3×C4—184no—
255C2×C3×C4C22×D911yescharacteristic
256C2×C3×C4C2×D3211yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—31no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261C2×Dic3—361no—
262D12—32no—
263D12—64no—
264D12—61no—
265C4×D3—32no—
266C4×D3—32no—
267C4×D3—64no—
268C3×C8—61no—
269C3⋊C8—21no—
270C3⋊C8—181no—
271C3⋊C8—181no—
272C3⋊C8—361no—
273C2×C32D3×C22⋊C411yescharacteristic
274C2×C32#91611yescharacteristic
275C2×C32#91911yescharacteristic
276C3×D3—62no—
277C3×D3—122no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—21no—
281C2×C9#95011yescharacteristic
282C2×C9#95011yescharacteristic
283C2×C9C2×#21711yescharacteristic
284D9—22no—
285D9—42no—
286#41—94no—
287#41—182no—
288C2×Q8D3×D911yescharacteristic
289C2×D4—91no—
290C2×D4—181no—
291C22×C4—91no—
292C22×C4—181no—
293C2×C8—31no—
294C2×C8—271no—
295C2×C8—542no—
296C4⋊C4—31no—
297C4⋊C4—271no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—21no—
301C22×C3—92no—
302C22×C3—181no—
303C22×C3—181no—
304C22×C3—182no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3D4×D911yescharacteristic
308C22×C3D3×#6711yescharacteristic
309C2×D3—32no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—62no—
313C2×D3—121no—
314C2×D3—121no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—92no—
321C3×C4—121no—
322C3×C4C2×C4×D911yescharacteristic
323C3×C4D4×D911yescharacteristic
324C3×C4C2×D3×Dic311yescharacteristic
325C3×C4D3×#6711yescharacteristic
326Dic3—32no—
327Dic3—41no—
328Dic3—361no—
329Dic3—361no—
330Dic3—721no—
331C32#480411yescharacteristic
332C9—21no—
333C9#523311yescharacteristic, complemented
334C23—91no—
335C23—181no—
336Q8—21no—
337Q8Dic3×D912yes—
338D4—92no—
339D4—184no—
340D4—181no—
341C2×C4—21no—
342C2×C4—61no—
343C2×C4—91no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—184no—
347C2×C4—541no—
348C2×C4C2×D3×D911yescharacteristic
349C8—61no—
350C8—541no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—182no—
355C2×C3—362no—
356C2×C3C22⋊C4×D911yescharacteristic
357C2×C3#6347811yescharacteristic
358C2×C3#6348111yescharacteristic
359C2×C3#6394711yescharacteristic
360C2×C3#6394911yescharacteristic
361C2×C3D3×#21711yescharacteristic
362D3—62no—
363D3—122no—
364C22—92no—
365C22—181no—
366C22—181no—
367C22—182no—
368C22—361no—
369C22—361no—
370C22D9×#6711yescharacteristic
371C4—22no—
372C4—92no—
373C4—121no—
374C4—1081no—
375C4C2×Dic3×D911yescharacteristic
376C4D9×#6711yescharacteristic
377C3—21no—
378C3#1058848411yescharacteristic, complemented
379C3#1059155611yescharacteristic
380C2—182no—
381C2—362no—
382C2#1172121511yescharacteristic
383C2#1172122311yescharacteristic
384C2D9×#21711yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#3701C2
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#416
LCSG ▷ 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. #11720907C2semidirect
3. C2×D9×C3⋊C8C2semidirect
4. #11720582C2semidirect
5. C9×#835C2semidirect
6. #11720283C2semidirect
7. #11720266C2semidirect
12. C3×#3701C4semidirect
75. #3701Dic3semidirect
112. #835D9semidirect
232. C3×C9#416semidirect
333. C9#5233semidirect
378. C3#10588484semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37012
#117209072
C2×D9×C3⋊C82
#117205822
C9×#8352
#117202832
#117202662
#105915563
#105884843