Finite Group Databasebeta

← #13140469

Subgroups — #13140469

Order 1728.

454 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 93 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
1D4×C3⋊Dic9C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic
3C2×C3×#3672C211yesmaximal, characteristic
4#11720930C211yesmaximal, characteristic, complemented
5#11720298C211yesmaximal, characteristic, complemented
6#11720276C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10591497—31nomaximal
9#10588842—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14#88551—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C4×C3⋊Dic9C2211yescharacteristic
17C9×#214—22no—
18C3×#3672—22no—
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88442—21no—
22#88442—21no—
23D4×C3⋊Dic3—31no—
24C2×C3×#214—31no—
25C2×C3×#214D311yescharacteristic
26#63873—31no—
27#63572—31no—
28#63566—31no—
29#63566—31no—
30C2×C9×SD16—31no—
31#63506—31no—
32D4×Dic9—31no—
33D4×Dic9—61no—
34C2×#3672D311yescharacteristic
35#63395—31no—
36#63376—31no—
37Dic9⋊C8—31no—
38C23×C3×C9—21no—
39C3×D4×C9—21no—
40C3×D4×C9D412yes—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—42no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C9×Dic6—21no—
49C9×Dic6—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C9×C3⋊C8—21no—
53C3×C9⋊C8—21no—
54#5162—91no—
55#5162—91no—
56C2×C32×D4C2×D311yescharacteristic
57C22×C3⋊Dic3—61no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6C2×D311yescharacteristic
60#3757—61no—
61C4⋊(C3⋊Dic3)—31no—
62C4×C3⋊Dic3—31no—
63C3×#214—22no—
64C3×#214—62no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67#3724—61no—
68#3724—61no—
69C2×Q8×C9—31no—
70C2×D4×C9—21no—
71C2×D4×C9C2×D311yescharacteristic
72C22×Dic9—61no—
73C22×Dic9—62no—
74C2×Dic18C2×D311yescharacteristic
75C9×SD16—62no—
76C2×C8×C9—31no—
77#3676—61no—
78#3676—62no—
79#3672—22no—
80C4⋊Dic9—31no—
81C4⋊Dic9—61no—
82Dic9⋊C4—61no—
83Dic9⋊C4—61no—
84C4×Dic9—31no—
85C4×Dic9—61no—
86C2×C9⋊C8C2×D311yescharacteristic
87C22×C3×C9—22no—
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9C2×D411yescharacteristic
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—21no—
94C3⋊Dic9—41no—
95C3⋊Dic9SD1612yescomplemented
96C9×Dic3—41no—
97C3×Dic9—41no—
98C2×C3×SD16—31no—
99C2×C3×SD16—91no—
100#944—91no—
101#944—91no—
102D4×Dic3—91no—
103D4×Dic3—91no—
104D4×Dic3—181no—
105C2×#214—31no—
106C2×#214D911yescharacteristic
107#835—91no—
108#835—91no—
109#816—91no—
110#816—91no—
111Dic3⋊C8—91no—
112Dic3⋊C8—91no—
113C23×C32—21no—
114C32×D4—21no—
115C32×D4#6712yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—122no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—61no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C23×C9—22no—
128C23×C9—21no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8—61no—
131Q8×C9—61no—
132Q8×C9—61no—
133D4×C9—22no—
134D4×C9—21no—
135D4×C9—22no—
136D4×C9#6712yes—
137C2×C4×C9—21no—
138C2×C4×C9—61no—
139C2×C4×C9C22×D311yescharacteristic
140C2×Dic9—21no—
141C2×Dic9—31no—
142C2×Dic9—31no—
143C2×Dic9—61no—
144C2×Dic9—61no—
145C2×Dic9—62no—
146C2×Dic9—61no—
147C2×Dic9—122no—
148C2×Dic9—124no—
149Dic18—21no—
150Dic18—21no—
151C8×C9—61no—
152C9⋊C8—21no—
153#461—271noSylow
154C2×C3×C9—42no—
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9C2×SD1611yescharacteristic
157C2×C3×C9#13711yescharacteristic
158C2×C3×Q8—31no—
159C2×C3×Q8—91no—
160C2×C3×D4—21no—
161C2×C3×D4C2×D911yescharacteristic
162C2×C3×D4D3211yescharacteristic
163C22×Dic3—181no—
164C22×Dic3—181no—
165C22×Dic3—182no—
166C2×Dic6—31no—
167C2×Dic6C2×D911yescharacteristic
168C3×SD16—62no—
169C3×SD16—182no—
170C2×C3×C8—31no—
171C2×C3×C8—91no—
172#217—181no—
173#217—181no—
174#217—182no—
175#214—22no—
176#214—62no—
177C4⋊Dic3—91no—
178C4⋊Dic3—91no—
179C4⋊Dic3—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184C4×Dic3—91no—
185C4×Dic3—91no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8C2×D911yescharacteristic
189C22×C32—22no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32D3×D411yescharacteristic
193C32×C4#23711yescharacteristic
194C32×C4C2×#6711yescharacteristic
195C3⋊Dic3—32no—
196C3⋊Dic3—61no—
197C3⋊Dic3—121no—
198C3×Dic3—41no—
199C3×Dic3—121no—
200C22×C9—24no—
201C22×C9—22no—
202C22×C9—21no—
203C22×C9—42no—
204C22×C9—42no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9D3×D411yescharacteristic
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—121no—
211C4×C9#23711yescharacteristic
212C4×C9C2×#6711yescharacteristic
213Dic9—32no—
214Dic9—41no—
215Dic9—62no—
216Dic9—61no—
217Dic9—121no—
218Dic9—122no—
219Dic9—121no—
220C2×SD16—31no—
221C2×SD16—91no—
222C4⋊Q8—271no—
223C4×D4—271no—
224C4⋊C8—271no—
225#103—271no—
226#103—271no—
227C3×C9#46111yescharacteristic, complemented, Sylow
228C23×C3—22no—
229C23×C3—21no—
230C23×C3—21no—
231C3×Q8—61no—
232C3×Q8—61no—
233C3×Q8—181no—
234C3×Q8—181no—
235C3×D4—22no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—22no—
239C3×D4#61112yes—
240C3×D4#62512yes—
241C2×C3×C4—21no—
242C2×C3×C4—61no—
243C2×C3×C4—181no—
244C2×C3×C4C22×D911yescharacteristic
245C2×C3×C4C2×D3211yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—182no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—362no—
258C2×Dic3—362no—
259C2×Dic3—364no—
260Dic6—21no—
261Dic6—21no—
262Dic6—61no—
263Dic6—61no—
264C3×C8—61no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—61no—
268C2×C32—42no—
269C2×C32D3×SD1611yescharacteristic
270C2×C32#91511yescharacteristic
271C2×C32#93911yescharacteristic
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—44no—
276C2×C9—42no—
277C2×C9D3×SD1611yescharacteristic
278C2×C9#91511yescharacteristic
279C2×C9#93911yescharacteristic
280C2×Q8—31no—
281C2×Q8—91no—
282C2×D4D3×D911yescharacteristic
283C22×C4—541no—
284SD16—62no—
285SD16—182no—
286C2×C8—31no—
287C2×C8—91no—
288C4⋊C4—271no—
289C4⋊C4—541no—
290C4⋊C4—541no—
291C22⋊C4—541no—
292C42—271no—
293C22×C3—24no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—21no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3D4×D911yescharacteristic
304C22×C3#381111yescharacteristic
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—121no—
308C3×C4—361no—
309C3×C4#369911yescharacteristic
310C3×C4C2×#61111yescharacteristic
311C3×C4#380511yescharacteristic
312C3×C4C2×#62511yescharacteristic
313Dic3—41no—
314Dic3—92no—
315Dic3—92no—
316Dic3—121no—
317Dic3—182no—
318Dic3—181no—
319Dic3—181no—
320Dic3—361no—
321Dic3—362no—
322Dic3—361no—
323Dic3—361no—
324C32#516211yescharacteristic
325C9—21no—
326C9#516211yescharacteristic, complemented
327C23—21no—
328Q8—61no—
329Q8—61no—
330Q8—181no—
331Q8—181no—
332D4—21no—
333D4#619412yes—
334C2×C4—61no—
335C2×C4—181no—
336C2×C4—271no—
337C2×C4—271no—
338C2×C4—541no—
339C2×C4—1082no—
340C2×C4C2×D3×D911yescharacteristic
341C8—61no—
342C8—181no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—44no—
347C2×C3—42no—
348C2×C3—42no—
349C2×C3SD16×D911yescharacteristic
350C2×C3#6347711yescharacteristic
351C2×C3#6350111yescharacteristic
352C2×C3#6393011yescharacteristic
353C2×C3#6393111yescharacteristic
354C2×C3#6398011yescharacteristic
355C22—22no—
356C22—41no—
357C22—41no—
358C22#8866811yescharacteristic
359C4—121no—
360C4—272no—
361C4—361no—
362C4—541no—
363C4—1081no—
364C4#8866311yescharacteristic
365C4C2×#619411yescharacteristic
366C3—21no—
367C3#1058884211yescharacteristic, complemented
368C3#1059149711yescharacteristic
369C2—42no—
370C2#1172118211yescharacteristic
371C2#1172118311yescharacteristic
372C2#1172129011yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#461
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11720930C2semidirect
5. #11720298C2semidirect
6. #11720276C2semidirect
7. #11720273C2semidirect
95. C3⋊Dic9SD16semidirect
227. C3×C9#461semidirect
326. C9#5162semidirect
367. C3#10588842semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C2×C9×#2142
C2×C3×#36722
#117209302
#117202982
#117202762
#117202732
#105914973
#105888423