Finite Group Databasebeta

← #13140736

Subgroups — #13140736

Order 1728.

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

normal subgroups only

This group has 402 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×#88547C211yesmaximal, characteristic, complemented
2C3×#63508C211yesmaximal, characteristic
3#11720945C211yesmaximal, characteristic
4C2×D9×C3⋊C8C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic, complemented
6#11720277C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10591596—31nomaximal
9#10588493—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13#88547—22no—
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21Dic3⋊Dic9—21no—
22D9×C3⋊C8—22no—
23C2×#3756—31no—
24C3×#946—31no—
25#63857—31no—
26C2×D3×C3⋊C8—31no—
27C3×#835D311yescharacteristic
28#63576—31no—
29#63567—31no—
30#63508D311yescharacteristic
31C2×#3674—31no—
32C2×#3674—61no—
33C2×C8×D9—31no—
34#63459—31no—
35C9×#103—31no—
36#63376—31no—
37#63366—31no—
38C22×C3×D9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44#6226—21no—
45#6226—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9D411yescharacteristic
49C3×C4×D9—21no—
50C3×C4×D9—21no—
51C3⋊(C9⋊C8)D412yes—
52C9×C3⋊C8—21no—
53#5051—91no—
54#4813—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×#634—31no—
57C2×C3×C4×D3—31no—
58#3756—62no—
59C2×C3⋊(C3⋊C8)—31no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C3×Dic3⋊C4—61no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67D3×C3⋊C8—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C2×D311yescharacteristic
70C2×C4×D9C2×D311yescharacteristic
71C2×Dic18—31no—
72C2×Dic18—61no—
73C2×C8×C9—31no—
74C9×C4⋊C4—31no—
75#3674—62no—
76#3674—64no—
77(C2×D9)⋊C4—21no—
78(C2×D9)⋊C4—61no—
79C4⋊Dic9—61no—
80C4⋊Dic9C2×D311yescharacteristic
81Dic9⋊C4—21no—
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84C8×D9—62no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C2×C3×D9Q1612yescomplemented
89C3×C4×C9—41no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3⋊Dic9—41no—
93C9×Dic3—41no—
94C3×Dic9—21no—
95C3×Dic9—41no—
96C3×C22⋊Q8—91no—
97#946—31no—
98C2×#216—91no—
99C2×#216—91no—
100C2×#216—181no—
101#924—91no—
102C22×C3⋊C8—91no—
103C2×D3×C8—91no—
104#897—91no—
105C3×#103—31no—
106#835D911yescharacteristic, complemented
107#816—91no—
108#810—91no—
109#807—91no—
110#807—91no—
111C22×C3×D3—31no—
112C32×Q8—21no—
113C32×Q8—21no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—61no—
117#634—61no—
118#634—61no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—61no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C22×D9#6711yescharacteristic
125C3⋊(C3⋊C8)—32no—
126C3×C3⋊C8—21no—
127Q8×C9—22no—
128Q8×C9—21no—
129Q8×C9—21no—
130Q8×C9—41no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—41no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136C2×Dic9—21no—
137C2×Dic9—61no—
138C2×Dic9—121no—
139C2×Dic9#6711yescharacteristic
140C4×D9—21no—
141C4×D9—21no—
142Dic18—62no—
143Dic18—61no—
144Dic18—61no—
145Dic18—121no—
146C8×C9—61no—
147C9⋊C8—32no—
148C9⋊C8—62no—
149#467—271noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9C2×Q1611yescharacteristic
152C2×C3×C9#13511yescharacteristic
153C3×D9—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8C2×D911yescharacteristic
156C2×C3×Q8D3211yescharacteristic
157C22×C3×C4—91no—
158C2×C4×D3—31no—
159C2×Dic6—91no—
160C2×Dic6—91no—
161C2×Dic6—181no—
162C2×C3×C8—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167#217—181no—
168#216—182no—
169#216—182no—
170#216—184no—
171(C2×D3)⋊C4—61no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—181no—
175C4⋊Dic3C2×D911yescharacteristic
176Dic3⋊C4—61no—
177Dic3⋊C4—181no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—181no—
181C2×C3⋊C8—182no—
182C2×C3⋊C8C2×D911yescharacteristic
183D3×C8—182no—
184C22×C32D3×D411yescharacteristic
185C2×C3×D3—32no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C32×C4—41no—
189C32×C4#23511yescharacteristic
190C32×C4D3×D411yescharacteristic
191C3⋊Dic3—121no—
192C3×Dic3—41no—
193C3×Dic3—61no—
194C3×Dic3—121no—
195C22×C9—21no—
196C22×C9C2×#6711yescharacteristic
197C2×D9—21no—
198C2×D9—21no—
199C2×D9#21612yescomplemented
200C4×C9—21no—
201C4×C9—21no—
202C4×C9—42no—
203C4×C9—41no—
204C4×C9—121no—
205C4×C9D3×D411yescharacteristic
206C4×C9#23711yescharacteristic
207Dic9—21no—
208Dic9—41no—
209Dic9—122no—
210Dic9—121no—
211C2×Q16—271no—
212C22×C8—271no—
213C22⋊Q8—91no—
214C22⋊Q8—271no—
215#107—271no—
216#103—31no—
217#103—271no—
218C3×C9#46711yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C3×Q8—22no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—41no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—61no—
232C2×C3×C4—91no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4C22×D911yescharacteristic
237C2×C3×C4C2×D3211yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—31no—
240C2×Dic3—61no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—361no—
244C4×D3—61no—
245C4×D3—61no—
246Dic6—182no—
247Dic6—181no—
248Dic6—181no—
249Dic6—181no—
250Dic6—181no—
251Dic6—361no—
252C3×C8—61no—
253C3⋊C8—21no—
254C3⋊C8—92no—
255C3⋊C8—92no—
256C3⋊C8—182no—
257C2×C32#88411yescharacteristic
258C2×C32#91611yescharacteristic
259C2×C32D3×Q1611yescharacteristic
260C3×D3—62no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—21no—
264C2×C9#93911yescharacteristic
265C2×C9C2×#21611yescharacteristic
266C2×C9#95011yescharacteristic
267D9—22no—
268C2×Q8—271no—
269C2×Q8D3×D911yescharacteristic
270C22×C4—91no—
271Q16—542no—
272C2×C8—31no—
273C2×C8—271no—
274C2×C8—542no—
275C4⋊C4—31no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—541no—
279C22⋊C4—181no—
280C22⋊C4—541no—
281C22×C3—21no—
282C22×C3—92no—
283C22×C3—181no—
284C22×C3—181no—
285C22×C3D4×D911yescharacteristic
286C22×C3D3×#6711yescharacteristic
287C2×D3—32no—
288C2×D3—61no—
289C2×D3—61no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—121no—
296C3×C4—181no—
297C3×C4—361no—
298C3×C4#369711yescharacteristic
299C3×C4D4×D911yescharacteristic
300C3×C4#380411yescharacteristic
301C3×C4#381111yescharacteristic
302Dic3—41no—
303Dic3—61no—
304Dic3—121no—
305Dic3—362no—
306Dic3—361no—
307Dic3—361no—
308C32#481311yescharacteristic
309C9—21no—
310C9#505111yescharacteristic, complemented
311C23—91no—
312Q8—21no—
313Q8—21no—
314Q8—541no—
315Q8—541no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—91no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—541no—
323C2×C4C2×D3×D911yescharacteristic
324C8—61no—
325C8—272no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—182no—
330C2×C3#6344611yescharacteristic
331C2×C3#6347811yescharacteristic
332C2×C3Q16×D911yescharacteristic
333C2×C3D3×#21611yescharacteristic
334C2×C3#6394711yescharacteristic
335C2×C3#6397111yescharacteristic
336D3—62no—
337C22—92no—
338C22—181no—
339C22—181no—
340C22D9×#6711yescharacteristic
341C4—41no—
342C4—121no—
343C4—181no—
344C4—361no—
345C4—1081no—
346C4#8865811yescharacteristic
347C4#8866811yescharacteristic
348C3—21no—
349C3#1058849311yescharacteristic, complemented
350C3#1059159611yescharacteristic
351C2—182no—
352C2#1172121511yescharacteristic
353C2D9×#21611yescharacteristic
354C2#1172127111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#467
LCSG ▷ C2×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
1. C2×#88547C2semidirect
5. C9×#835C2semidirect
6. #11720277C2semidirect
7. #11720263C2semidirect
88. C2×C3×D9Q16semidirect
106. #835D9semidirect
199. C2×D9#216semidirect
218. C3×C9#467semidirect
310. C9#5051semidirect
349. C3#10588493semidirect

Maximal subgroups

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

subgroupindex
C2×#885472
C3×#635082
#117209452
C2×D9×C3⋊C82
C9×#8352
#117202772
#117202632
#105915963
#105884933