Finite Group Databasebeta

← #13140362

Subgroups — #13140362

Order 1728.

440 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88543C211yesmaximal, characteristic, complemented
2C3×#63500C211yesmaximal, characteristic
3#11720945C211yesmaximal, characteristic, complemented
4C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6#11720277C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10591460—31nomaximal
9#10588458—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×#611—21no—
13C2×C3×C4×D9C2211yescharacteristic
14#88543—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×#3676—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21Dic3⋊Dic9—21no—
22D9×C3⋊C8—22no—
23C2×#3754—31no—
24C3×#938—31no—
25#63857—31no—
26C2×D3×C3⋊C8—31no—
27C3×#832D311yescharacteristic
28#63574—31no—
29#63567—31no—
30#63500D311yescharacteristic
31C2×#3672—31no—
32C2×#3672—61no—
33C2×C8×D9—31no—
34#63459—31no—
35C9×#102—31no—
36#63376—31no—
37#63367—31no—
38C23×C3×C9—21no—
39C22×C3×D9D411yescharacteristic
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44#6226—21no—
45#6226—21no—
46C3×#611—42no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C2×C3×Dic9D411yescharacteristic
50C3×C4×D9—21no—
51C3×C4×D9—21no—
52C3⋊(C9⋊C8)D412yescomplemented
53C9×C3⋊C8—21no—
54#5041—91no—
55#4778—91no—
56C2×C32×D4C2×D311yescharacteristic
57C2×#634—31no—
58C2×C3×#67—61no—
59C2×C3×C4×D3—31no—
60#3754—62no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×#217—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65C2×C3×C3⋊C8C2×D311yescharacteristic
66Dic3⋊Dic3—61no—
67(C2×D3)⋊Dic3—61no—
68D3×C3⋊C8—62no—
69C2×D4×C9—21no—
70C2×D4×C9C2×D311yescharacteristic
71C2×#611—21no—
72C2×C4×D9C2×D311yescharacteristic
73C2×Dic18—31no—
74C2×Dic18—61no—
75C2×C8×C9—31no—
76C9×C4⋊C4—31no—
77#3676—21no—
78#3672—62no—
79#3672—64no—
80(C2×D9)⋊C4—61no—
81C4⋊Dic9—61no—
82C4⋊Dic9C2×D311yescharacteristic
83C2×C9⋊C8—31no—
84C2×C9⋊C8—61no—
85C8×D9—62no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9SD1612yescomplemented
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C3⋊Dic9—41no—
96C9×Dic3—41no—
97C3×Dic9—21no—
98C3×Dic9—41no—
99C3×#121—91no—
100#938—31no—
101C2×#214—91no—
102C2×#214—91no—
103C2×#214—181no—
104#924—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—91no—
107#897—91no—
108C3×#102—31no—
109#832D911yescharacteristic, complemented
110#816—91no—
111#810—91no—
112#808—91no—
113#808—91no—
114C23×C32—21no—
115C22×C3×D3—31no—
116C32×D4—21no—
117C32×D4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—61no—
120#634—61no—
121#634—61no—
122C3×#67—122no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C23×C9—21no—
129C23×C9—41no—
130C22×D9#6711yescharacteristic
131C3⋊(C3⋊C8)—32no—
132C3×C3⋊C8—21no—
133D4×C9—22no—
134D4×C9—21no—
135D4×C9—21no—
136D4×C9—41no—
137C2×C4×C9—21no—
138C2×C4×C9—61no—
139C2×C4×C9C22×D311yescharacteristic
140#611—42no—
141C2×Dic9—21no—
142C2×Dic9—61no—
143C2×Dic9—121no—
144C2×Dic9#6711yescharacteristic
145C4×D9—21no—
146C4×D9—21no—
147Dic18—62no—
148Dic18—61no—
149Dic18—61no—
150Dic18—121no—
151C8×C9—61no—
152C9⋊C8—32no—
153C9⋊C8—62no—
154#465—271noSylow
155C2×C3×C9—81no—
156C2×C3×C9#12111yescharacteristic
157C2×C3×C9C2×SD1611yescharacteristic
158C2×C3×C9#13511yescharacteristic
159C3×D9—22no—
160C2×C3×D4—21no—
161C2×C3×D4—181no—
162C2×C3×D4C2×D911yescharacteristic
163C2×C3×D4D3211yescharacteristic
164C22×C3×C4—91no—
165C2×#67—61no—
166C2×C4×D3—31no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×Dic6—181no—
170C2×C3×C8—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—91no—
173C3×C22⋊C4—181no—
174#217—61no—
175#217—181no—
176#214—182no—
177#214—182no—
178#214—184no—
179(C2×D3)⋊C4—181no—
180C4⋊Dic3—31no—
181C4⋊Dic3—181no—
182C4⋊Dic3C2×D911yescharacteristic
183Dic3⋊C4—181no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—181no—
187C2×C3⋊C8—182no—
188C2×C3⋊C8C2×D911yescharacteristic
189D3×C8—182no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32D3×D411yescharacteristic
194C2×C3×D3—32no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C32×C4#23511yescharacteristic
198C32×C4D3×D411yescharacteristic
199C3⋊Dic3—121no—
200C3×Dic3—41no—
201C3×Dic3—61no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9—42no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9—41no—
208C22×C9—81no—
209C22×C9—81no—
210C22×C9C2×#6711yescharacteristic
211C2×D9—21no—
212C2×D9—21no—
213C2×D9#21412yescomplemented
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—121no—
217C4×C9D3×D411yescharacteristic
218C4×C9#23711yescharacteristic
219Dic9—21no—
220Dic9—41no—
221Dic9—122no—
222Dic9—121no—
223C2×SD16—271no—
224C22×C8—271no—
225C22⋊Q8—271no—
226#121—91no—
227#106—271no—
228#103—271no—
229#102—31no—
230C3×C9#46511yescharacteristic, complemented, Sylow
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—41no—
234C23×C3—91no—
235C22×D3—31no—
236C3×D4—22no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—41no—
242C3×D4—362no—
243C2×C3×C4—21no—
244C2×C3×C4—61no—
245C2×C3×C4—91no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C22×D911yescharacteristic
250C2×C3×C4C2×D3211yescharacteristic
251#67—122no—
252C2×Dic3—21no—
253C2×Dic3—31no—
254C2×Dic3—61no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—361no—
258C4×D3—61no—
259C4×D3—61no—
260Dic6—182no—
261Dic6—181no—
262Dic6—181no—
263Dic6—181no—
264Dic6—181no—
265Dic6—361no—
266C3×C8—61no—
267C3⋊C8—21no—
268C3⋊C8—92no—
269C3⋊C8—92no—
270C3⋊C8—182no—
271C2×C32—81no—
272C2×C32#88411yescharacteristic
273C2×C32#91211yescharacteristic
274C2×C32D3×SD1611yescharacteristic
275C3×D3—62no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—82no—
280C2×C9—81no—
281C2×C9C2×#21411yescharacteristic
282C2×C9#93911yescharacteristic
283C2×C9#95011yescharacteristic
284D9—22no—
285C2×Q8—271no—
286C2×D4—181no—
287C2×D4D3×D911yescharacteristic
288C22×C4—91no—
289SD16—542no—
290C2×C8—31no—
291C2×C8—271no—
292C2×C8—542no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—541no—
296C22⋊C4—181no—
297C22⋊C4—541no—
298C22×C3—21no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—92no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3D4×D911yescharacteristic
312C22×C3D3×#6711yescharacteristic
313C2×D3—32no—
314C2×D3—61no—
315C2×D3—61no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—121no—
319C3×C4—181no—
320C3×C4—361no—
321C3×C4#369711yescharacteristic
322C3×C4D4×D911yescharacteristic
323C3×C4#380411yescharacteristic
324C3×C4#381111yescharacteristic
325Dic3—41no—
326Dic3—61no—
327Dic3—121no—
328Dic3—362no—
329Dic3—361no—
330Dic3—361no—
331C32#477811yescharacteristic
332C9—21no—
333C9#504111yescharacteristic, complemented
334C23—21no—
335C23—91no—
336Q8—541no—
337Q8—541no—
338D4—21no—
339D4—21no—
340D4—362no—
341C2×C4—61no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—541no—
347C2×C4C2×D3×D911yescharacteristic
348C8—61no—
349C8—272no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—82no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—182no—
357C2×C3#6344611yescharacteristic
358C2×C3#6347411yescharacteristic
359C2×C3SD16×D911yescharacteristic
360C2×C3D3×#21411yescharacteristic
361C2×C3#6393311yescharacteristic
362C2×C3#6397111yescharacteristic
363D3—62no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—92no—
368C22—181no—
369C22—181no—
370C22D9×#6711yescharacteristic
371C4—121no—
372C4—181no—
373C4—361no—
374C4—1081no—
375C4#8865811yescharacteristic
376C4#8866811yescharacteristic
377C3—21no—
378C3#1058845811yescharacteristic, complemented
379C3#1059146011yescharacteristic
380C2—81no—
381C2—182no—
382C2#1172117711yescharacteristic
383C2D9×#21411yescharacteristic
384C2#1172127111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC3×#63500C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#465
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
1. C2×#88543C2semidirect
3. #11720945C2semidirect
4. C2×D9×C3⋊C8C2semidirect
5. C9×#832C2semidirect
6. #11720277C2semidirect
7. #11720265C2semidirect
52. C3⋊(C9⋊C8)D4semidirect
92. C2×C3×D9SD16semidirect
109. #832D9semidirect
213. C2×D9#214semidirect
230. C3×C9#465semidirect
333. C9#5041semidirect
378. C3#10588458semidirect

Maximal subgroups

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

subgroupindex
C2×#885432
C3×#635002
#117209452
C2×D9×C3⋊C82
C9×#8322
#117202772
#117202652
#105914603
#105884583