Finite Group Databasebeta

← #13137626

Subgroups — #13137626

Order 1728.

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

normal subgroups only

This group has 416 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
1#11721753C211yesmaximal, characteristic
2#11720936C211yesmaximal, characteristic
3C2×#88418C211yesmaximal, characteristic, complemented
4C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic
5C3×#63368C211yesmaximal, characteristic, complemented
6C9×#807C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590525—31nomaximal
9#10588567—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14#88536—21no—
15C4⋊(C3⋊Dic9)—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic3⋊Dic9—21no—
21#88418—22no—
22(C3⋊D9)⋊C8—22no—
23#64106—31no—
24#63871—31no—
25C2×#3719—31no—
26C2×#3710—31no—
27C3×#810—31no—
28C3×#807D311yescharacteristic
29#63567—31no—
30C2×C8×D9—31no—
31C2×Dic36—31no—
32#63459—31no—
33#63459—31no—
34#63459—61no—
35C9×#107—31no—
36#63368—31no—
37#63368D311yescharacteristic, complemented
38C22×C3⋊D9D411yescharacteristic
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9D411yescharacteristic
43C4×C3⋊D9—21no—
44C4×C3⋊D9—21no—
45#6226—21no—
46#6226—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C9×C3⋊C8D412yes—
52C3×C9⋊C8—21no—
53#4887—91no—
54#4813—91no—
55C2×C4×C3⋊D3—31no—
56C2×#634—31no—
57C2×C3×Dic6—31no—
58C32×C4⋊C4C2×D311yescharacteristic
59#3752—61no—
60C4⋊(C3⋊Dic3)—61no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64Dic3⋊Dic3—61no—
65#3722—61no—
66#3719—62no—
67#3710—62no—
68C2×C4×D9—31no—
69C2×C4×D9—61no—
70C2×Dic18—31no—
71C2×Dic18—61no—
72C2×Dic18C2×D311yescharacteristic
73C2×C8×C9—31no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4—31no—
76C9×C4⋊C4C2×D311yescharacteristic
77(C2×D9)⋊C4—61no—
78(C2×D9)⋊C4—61no—
79(C2×D9)⋊C4—62no—
80C4⋊Dic9—61no—
81C4⋊Dic9—61no—
82C4⋊Dic9—62no—
83C2×C9⋊C8C2×D311yescharacteristic
84C8×D9—62no—
85Dic36—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—21no—
88C2×C3⋊D9—21no—
89C2×C3⋊D9Q1612yescomplemented
90C3×C4×C9—41no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—21no—
94C3⋊Dic9—41no—
95C9×Dic3—41no—
96C3×Dic9—41no—
97#946—91no—
98C2×#216—91no—
99C2×Dic12—91no—
100C2×D3×C8—91no—
101C2×D3×C8—91no—
102#897—91no—
103#897—91no—
104#897—91no—
105#897—181no—
106C3×#107—31no—
107C3×#103—91no—
108#816—91no—
109#810—31no—
110#810—91no—
111#807D911yescharacteristic, complemented
112C22×C3⋊D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117C4×C3⋊D3—61no—
118C4×C3⋊D3—61no—
119#634—61no—
120#634—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—61no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125C22×D9—31no—
126C22×D9—61no—
127C3×C3⋊C8—61no—
128C3×C3⋊C8D1212yes—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9—31no—
136C2×Dic9—61no—
137C2×Dic9—62no—
138C2×Dic9—61no—
139C4×D9—61no—
140C4×D9—61no—
141C4×D9—121no—
142C4×D9—121no—
143Dic18—21no—
144Dic18—21no—
145Dic18—61no—
146Dic18—61no—
147Dic18—121no—
148Dic18—121no—
149C8×C9—32no—
150C9⋊C8—21no—
151#467—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9C2×Q1611yescharacteristic
154C2×C3×C9#13511yescharacteristic
155C3⋊D9—22no—
156C2×C3×Q8—91no—
157C2×C4×D3—91no—
158C2×C4×D3—91no—
159C2×C4×D3—181no—
160C2×Dic6—31no—
161C2×Dic6—91no—
162C2×Dic6—91no—
163C2×Dic6—181no—
164C2×C3×C8—31no—
165C2×C3×C8—91no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4C2×D911yescharacteristic
169C3×C4⋊C4D3211yescharacteristic
170#216—182no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—182no—
176C4⋊Dic3—181no—
177C4⋊Dic3—181no—
178C4⋊Dic3—181no—
179C4⋊Dic3—182no—
180C4⋊Dic3C2×D911yescharacteristic
181Dic3⋊C4—181no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8C2×D911yescharacteristic
184Dic12—182no—
185D3×C8—182no—
186D3×C8—182no—
187C22×C32D3×D411yescharacteristic
188C2×C3⋊D3—32no—
189C2×C3⋊D3—61no—
190C2×C3⋊D3—61no—
191C32×C4—41no—
192C32×C4C2×D1211yescharacteristic
193C32×C4#23911yescharacteristic
194C3⋊Dic3—61no—
195C3⋊Dic3—121no—
196C3×Dic3—41no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9D3×D411yescharacteristic
200C2×D9—32no—
201C2×D9—62no—
202C2×D9—61no—
203C2×D9—61no—
204C2×D9—121no—
205C2×D9—121no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—42no—
209C4×C9—41no—
210C4×C9—121no—
211C4×C9#23511yescharacteristic
212C4×C9D3×D411yescharacteristic
213Dic9—41no—
214Dic9—61no—
215Dic9—121no—
216Dic9—122no—
217Dic9—121no—
218C2×Q16—271no—
219C22×C8—271no—
220C22⋊Q8—271no—
221C22⋊Q8—271no—
222#107—31no—
223#103—91no—
224#103—271no—
225C3×C9#46711yescharacteristic, complemented, Sylow
226C22×D3—91no—
227C22×D3—91no—
228C22×D3—181no—
229C3×Q8—181no—
230C3×Q8—181no—
231C2×C3×C4—22no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—61no—
236C2×C3×C4—181no—
237C2×C3×C4C22×D911yescharacteristic
238C2×C3×C4C2×D3211yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C2×Dic3—181no—
244C2×Dic3—182no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C4×D3—181no—
248C4×D3—181no—
249C4×D3—181no—
250C4×D3—181no—
251C4×D3—361no—
252C4×D3—361no—
253Dic6—61no—
254Dic6—61no—
255Dic6—181no—
256Dic6—181no—
257Dic6—181no—
258Dic6—181no—
259Dic6—361no—
260Dic6—361no—
261C3×C8—32no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8D3612yes—
265C2×C32#89511yescharacteristic
266C2×C32#91211yescharacteristic
267C2×C32D3×Q1611yescharacteristic
268C3⋊D3—62no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9#88411yescharacteristic
273C2×C9#91611yescharacteristic
274C2×C9D3×Q1611yescharacteristic
275D9—62no—
276D9—122no—
277C2×Q8—91no—
278C2×Q8—271no—
279C22×C4—271no—
280Q16—542no—
281C2×C8—31no—
282C2×C8—91no—
283C2×C8—542no—
284C4⋊C4—31no—
285C4⋊C4—541no—
286C4⋊C4—541no—
287C4⋊C4D3×D911yescharacteristic
288C22⋊C4—541no—
289C22⋊C4—541no—
290C22×C3—21no—
291C22×C3D4×D911yescharacteristic
292C22×C3#381111yescharacteristic
293C2×D3—92no—
294C2×D3—92no—
295C2×D3—182no—
296C2×D3—181no—
297C2×D3—181no—
298C2×D3—181no—
299C2×D3—181no—
300C2×D3—361no—
301C2×D3—361no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—121no—
308C3×C4—361no—
309C3×C4C2×D3611yescharacteristic
310C3×C4#370111yescharacteristic
311C3×C4#379911yescharacteristic
312C3×C4D3×D1211yescharacteristic
313Dic3—41no—
314Dic3—121no—
315Dic3—181no—
316Dic3—181no—
317Dic3—361no—
318Dic3—362no—
319Dic3—361no—
320Dic3—361no—
321C32#488711yescharacteristic
322C9—21no—
323C9#481311yescharacteristic, complemented
324C23—271no—
325Q8—181no—
326Q8—181no—
327Q8—541no—
328Q8—541no—
329C2×C4—21no—
330C2×C4—61no—
331C2×C4—181no—
332C2×C4—271no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4—541no—
336C2×C4C2×D3×D911yescharacteristic
337C8—32no—
338C8—181no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3#6345711yescharacteristic
343C2×C3#6347411yescharacteristic
344C2×C3Q16×D911yescharacteristic
345C2×C3#6391011yescharacteristic
346C2×C3#6393611yescharacteristic
347C2×C3#6394411yescharacteristic
348D3—182no—
349D3—182no—
350D3—362no—
351C22—272no—
352C22—541no—
353C22—541no—
354C22#8866811yescharacteristic
355C4—41no—
356C4—121no—
357C4—361no—
358C4—541no—
359C4—1081no—
360C4#8863511yescharacteristic
361C4D3×D3611yescharacteristic
362C3—21no—
363C3#1058856711yescharacteristic, complemented
364C3#1059052511yescharacteristic
365C2—542no—
366C2#1172100511yescharacteristic
367C2#1172117811yescharacteristic
368C2#1172122011yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#467
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#88418C2semidirect
5. C3×#63368C2semidirect
6. C9×#807C2semidirect
7. #11720269C2semidirect
37. #63368D3semidirect
89. C2×C3⋊D9Q16semidirect
111. #807D9semidirect
225. C3×C9#467semidirect
323. C9#4813semidirect
363. C3#10588567semidirect

Maximal subgroups

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

subgroupindex
#117217532
#117209362
C2×#884182
C2×(C3⋊D9)⋊C82
C3×#633682
C9×#8072
#117202692
#105905253
#105885673