Finite Group Databasebeta

← #13140746

Subgroups — #13140746

Order 1728.

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

normal subgroups only

This group has 492 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×Q8×D9C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic
3C2×#88413C211yesmaximal, characteristic
4#11720582C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6#11720276C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10591557—31nomaximal
9#10588482—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C3×Q8×D9—22no—
12C3×Q8×D9C414yescomplemented
13C2×C3×C4×D9—21no—
14C2×C3×C4×D9C2211yescharacteristic
15C2×C3×Dic18—21no—
16C2×C3×Dic18C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21(C2×D9)⋊Dic3—21no—
22Dic3×Dic9—21no—
23#88413—22no—
24C2×C3×D3×Q8—31no—
25#63841—31no—
26C2×#3711—31no—
27C2×Q8×D9D311yescharacteristic
28#63662—31no—
29C3×#835D311yescharacteristic
30#63566—31no—
31#63565—31no—
32C2×#3663—31no—
33#63454—31no—
34C9×#103—31no—
35#63395—31no—
36#63395—61no—
37#63376—31no—
38#63368—31no—
39C22×C3×D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9C2×C412yes—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×C4×D9—24no—
49C3×C4×D9C2×C412yes—
50C3×C4×D9D412yes—
51C3×Dic18—21no—
52C3×Dic18—24no—
53C3×Dic18C2×C412yes—
54C3⋊(C9⋊C8)—21no—
55C9×C3⋊C8—21no—
56#5225—91no—
57#4802—91no—
58C2×C32×Q8C2×D311yescharacteristic
59C3×D3×Q8—34no—
60C3×D3×Q8—62no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—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—
71#3711—62no—
72C2×Q8×C9—21no—
73C2×Q8×C9C2×D311yescharacteristic
74Q8×D9—22no—
75Q8×D9Dic314yescomplemented
76C2×C4×D9—21no—
77C2×C4×D9C2×D311yescharacteristic
78C2×Dic18—21no—
79C2×Dic18C2×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—
88#3663—62no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9C22⋊C412yes—
93C3×C4×C9—22no—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9—41no—
97C9×Dic3—41no—
98C3×Dic9—22no—
99C3×Dic9C22⋊C412yes—
100C22×C3×Q8—91no—
101C2×D3×Q8—31no—
102#926—91no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—91no—
105#892—91no—
106C3×#103—31no—
107#835—91no—
108#835—91no—
109#835—91no—
110#835—91no—
111#835—181no—
112#835D911yescharacteristic, complemented
113#816—91no—
114#810—91no—
115C22×C3×D3—31no—
116C32×Q8—21no—
117C32×Q8C4×D312yes—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—61no—
124C3×C4×D3—32no—
125C3×C4×D3—32no—
126C3×C4×D3—64no—
127C3×Dic6—32no—
128C3×Dic6—64no—
129C3×Dic6—61no—
130C22×D9#6711yescharacteristic
131C3⋊(C3⋊C8)—61no—
132C3×C3⋊C8—21no—
133Q8×C9—21no—
134Q8×C9—22no—
135Q8×C9—41no—
136Q8×C9C2×Dic312yes—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9—61no—
141C2×C4×C9C22×D311yescharacteristic
142C2×Dic9—21no—
143C2×Dic9—61no—
144C2×Dic9—121no—
145C2×Dic9#6711yescharacteristic
146C4×D9—24no—
147C4×D9C2×Dic312yes—
148C4×D9#6712yes—
149Dic18—21no—
150Dic18—24no—
151Dic18C2×Dic312yes—
152C8×C9—61no—
153C9⋊C8—61no—
154C9⋊C8—121no—
155#419—271noSylow
156C2×C3×C9C2×C22⋊C411yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C2×C3×C9#13711yescharacteristic
159C3×D9—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—94no—
162C2×C3×Q8—91no—
163C2×C3×Q8—181no—
164C2×C3×Q8—182no—
165C2×C3×Q8C2×D911yescharacteristic
166C2×C3×Q8D3211yescharacteristic
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169D3×Q8—34no—
170D3×Q8—62no—
171C2×C4×D3—31no—
172C2×C4×D3—61no—
173C2×Dic6—31no—
174C2×Dic6—61no—
175C2×C3×C8—31no—
176C3×C4⋊C4—31no—
177#217—181no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—91no—
180C4⋊Dic3—91no—
181C4⋊Dic3—181no—
182C4⋊Dic3C2×D911yescharacteristic
183C4×Dic3—181no—
184C4×Dic3—181no—
185#208—182no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—181no—
189C2×C3⋊C8C2×D911yescharacteristic
190ΓL(1,𝔽25)—182no—
191C22×C32D3×D411yescharacteristic
192C2×C3×D3—32no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C32×C4—22no—
196C32×C4C2×C4×D311yescharacteristic
197C32×C4D3×D411yescharacteristic
198C3⋊Dic3—121no—
199C3×Dic3—32no—
200C3×Dic3—41no—
201C3×Dic3—62no—
202C22×C9—21no—
203C22×C9C2×#6711yescharacteristic
204C2×D9—21no—
205C2×D9—21no—
206C2×D9#21712yes—
207C4×C9—22no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—42no—
211C4×C9—121no—
212C4×C9C22×Dic311yescharacteristic
213C4×C9C2×#6711yescharacteristic
214Dic9—22no—
215Dic9—121no—
216Dic9—241no—
217Dic9#21712yes—
218C22×Q8—91no—
219C2×M16—271no—
220#117—271no—
221#103—31no—
222#103—271no—
223#103—271no—
224#103—271no—
225C3×C9#41911yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—31no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—22no—
231C3×Q8—41no—
232C3×Q8—92no—
233C3×Q8—184no—
234C3×Q8—181no—
235C3×Q8C4×D912yes—
236C3×Q8D3×Dic312yes—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—92no—
244C2×C3×C4—92no—
245C2×C3×C4—184no—
246C2×C3×C4—181no—
247C2×C3×C4C22×D911yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249C2×Dic3—21no—
250C2×Dic3—31no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—361no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—64no—
258Dic6—32no—
259Dic6—64no—
260Dic6—61no—
261C3×C8—61no—
262C3⋊C8—21no—
263C3⋊C8—181no—
264C3⋊C8—181no—
265C3⋊C8—361no—
266C2×C32D3×C22⋊C411yescharacteristic
267C2×C32#91511yescharacteristic
268C2×C32#91811yescharacteristic
269C3×D3—62no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9#94211yescharacteristic
274C2×C9#94211yescharacteristic
275C2×C9C2×#21711yescharacteristic
276D9—22no—
277C2×Q8—94no—
278C2×Q8—91no—
279C2×Q8—181no—
280C2×Q8—182no—
281C2×Q8D3×D911yescharacteristic
282C22×C4—91no—
283C22×C4—181no—
284M16—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4—31no—
288C4⋊C4—271no—
289C22⋊C4—541no—
290C42—541no—
291C22×C3—21no—
292C22×C3—92no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3D4×D911yescharacteristic
296C22×C3D3×#6711yescharacteristic
297C2×D3—32no—
298C2×D3—61no—
299C2×D3—61no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—92no—
306C3×C4—121no—
307C3×C4—182no—
308C3×C4C2×C4×D911yescharacteristic
309C3×C4D4×D911yescharacteristic
310C3×C4C2×D3×Dic311yescharacteristic
311C3×C4D3×#6711yescharacteristic
312Dic3—32no—
313Dic3—41no—
314Dic3—62no—
315Dic3—361no—
316Dic3—361no—
317Dic3—721no—
318C32#480211yescharacteristic
319C9—21no—
320C9#522511yescharacteristic, complemented
321C23—91no—
322Q8—21no—
323Q8—92no—
324Q8—184no—
325Q8—181no—
326Q8Dic3×D912yes—
327C2×C4—21no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—184no—
333C2×C4—181no—
334C2×C4—541no—
335C2×C4C2×D3×D911yescharacteristic
336C8—61no—
337C8—541no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—182no—
342C2×C3C22⋊C4×D911yescharacteristic
343C2×C3#6347711yescharacteristic
344C2×C3#6348011yescharacteristic
345C2×C3#6394611yescharacteristic
346C2×C3#6394811yescharacteristic
347C2×C3D3×#21711yescharacteristic
348D3—62no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22D9×#6711yescharacteristic
353C4—22no—
354C4—92no—
355C4—121no—
356C4—182no—
357C4—1081no—
358C4C2×Dic3×D911yescharacteristic
359C4D9×#6711yescharacteristic
360C3—21no—
361C3#1058848211yescharacteristic, complemented
362C3#1059155711yescharacteristic
363C2—182no—
364C2#1172121411yescharacteristic
365C2#1172122211yescharacteristic
366C2D9×#21711yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#419
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
4. #11720582C2semidirect
5. C9×#835C2semidirect
6. #11720276C2semidirect
7. #11720268C2semidirect
12. C3×Q8×D9C4semidirect
75. Q8×D9Dic3semidirect
112. #835D9semidirect
225. C3×C9#419semidirect
320. C9#5225semidirect
361. C3#10588482semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
#117209072
C2×#884132
#117205822
C9×#8352
#117202762
#117202682
#105915573
#105884823