Finite Group Databasebeta

← #13138639

Subgroups — #13138639

Order 1728.

504 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 95 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×D3×Dic18C211yesmaximal, characteristic
2C2×#88521C211yesmaximal, characteristic, complemented
3#11720961C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×(C2×D3)⋊C8C211yesmaximal, characteristic
6C3×#63376C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10590952—31nomaximal
9#10588405—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×D3×Dic9—21no—
15C2×#6189—21no—
16D3×Dic18—22no—
17D3×Dic18—22no—
18C2×C3×C8×C9C2211yescharacteristic
19#88521—22no—
20C3×C4⋊Dic9C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22(C2×D3)⋊Dic9—21no—
23Dic9⋊Dic3—21no—
24#88418—22no—
25C2×D3×Dic6—31no—
26C2×#3746—31no—
27#63857—31no—
28C2×#3719—31no—
29C22×Dic18—31no—
30C3×(C2×D3)⋊C8D311yescharacteristic
31C3×#816—31no—
32#63567—31no—
33#63486—31no—
34C2×Dic36—31no—
35C2×Dic36—31no—
36C2×Dic36—61no—
37C9×C22⋊C8—31no—
38#63376—31no—
39#63376D311yescharacteristic, complemented
40C22×D3×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—21no—
44#6226—21no—
45#6226—21no—
46#6226—21no—
47#6226D412yes—
48C2×C9×Dic3D411yescharacteristic
49C2×C3×Dic9—21no—
50C2×C3×Dic9—21no—
51C4×D3×C9—21no—
52C4×D3×C9—21no—
53C3×Dic18—21no—
54C3×Dic18D412yes—
55D3×Dic9—24no—
56#6189—24no—
57C3×C8×C9—21no—
58C9×C3⋊C8—21no—
59#4809—91no—
60#4739—91no—
61C2×#634—31no—
62C2×#634—31no—
63C2×C3×C4×D3C2×D311yescharacteristic
64C2×C3×Dic6—31no—
65C2×#627—61no—
66C2×D3×Dic3—61no—
67D3×Dic6—62no—
68D3×Dic6—62no—
69C2×C32×C8C2×D311yescharacteristic
70#3746—62no—
71C3×C4⋊Dic3—31no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75#3719—62no—
76C22×C4×C9—31no—
77C22×Dic9—61no—
78C2×Dic18—31no—
79C2×Dic18—31no—
80C2×Dic18—62no—
81C2×Dic18—61no—
82C2×Dic18—62no—
83C2×Dic18—61no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C2×C8×C9—21no—
87C2×C8×C9—31no—
88C2×C8×C9C2×D311yescharacteristic
89#3676—61no—
90C4⋊Dic9C2×D311yescharacteristic
91Dic9⋊C4—61no—
92Dic36—62no—
93Dic36—64no—
94Dic36—62no—
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—21no—
97C2×D3×C9—21no—
98C2×D3×C9Q1612yescomplemented
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—22no—
102C3⋊Dic9—41no—
103C9×Dic3—21no—
104C3×Dic9—22no—
105C3×Dic9—41no—
106C2×D3×Q8—91no—
107C22×Dic6—91no—
108C2×#216—91no—
109#924—91no—
110C2×Dic12—91no—
111C2×Dic12—91no—
112C2×Dic12—91no—
113C2×Dic12—181no—
114#897—91no—
115C3×#103—91no—
116C3×C22⋊C8—31no—
117(C2×D3)⋊C8D911yescharacteristic
118#816—31no—
119#816—91no—
120#810—91no—
121C22×C3×D3D1211yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—61no—
125#634—32no—
126#634—61no—
127#634—61no—
128#634—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3D1211yescharacteristic
132C3×C4×D3—21no—
133C3×C4×D3—21no—
134C3×Dic6—32no—
135C3×Dic6—61no—
136#627—64no—
137D3×Dic3—64no—
138C23×C9—31no—
139C32×C8—21no—
140C3×C3⋊C8—21no—
141C2×C4×C9—21no—
142C2×C4×C9—31no—
143C2×C4×C9—61no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146C2×Dic9—21no—
147C2×Dic9—21no—
148C2×Dic9—64no—
149C2×Dic9—61no—
150C2×Dic9—61no—
151C2×Dic9—121no—
152C2×Dic9—121no—
153Dic18—21no—
154Dic18—32no—
155Dic18—62no—
156Dic18—64no—
157Dic18—61no—
158Dic18—62no—
159Dic18—61no—
160Dic18—61no—
161Dic18—121no—
162Dic18—121no—
163Dic18D1212yes—
164C8×C9—22no—
165C8×C9—21no—
166C8×C9—61no—
167#451—271noSylow
168C2×C3×C9#12011yescharacteristic
169C2×C3×C9C2×Q1611yescharacteristic
170C2×C3×C9#13711yescharacteristic
171D3×C9—22no—
172C2×C3×Q8—91no—
173C22×C3×C4—31no—
174C22×Dic3—181no—
175D3×Q8—182no—
176D3×Q8—182no—
177C2×C4×D3—181no—
178C2×C4×D3C2×D911yescharacteristic
179C2×Dic6—31no—
180C2×Dic6—91no—
181C2×Dic6—91no—
182C2×Dic6—91no—
183C2×Dic6—91no—
184C2×Dic6—181no—
185C2×Dic6—181no—
186C2×Dic6—182no—
187C2×Dic6—182no—
188C2×Dic6—181no—
189C2×Dic6—181no—
190C2×C3×C8—21no—
191C2×C3×C8—31no—
192C2×C3×C8C2×D911yescharacteristic
193C2×C3×C8D3211yescharacteristic
194C3×C4⋊C4—91no—
195#217—181no—
196#216—182no—
197(C2×D3)⋊C4—181no—
198C4⋊Dic3—31no—
199C4⋊Dic3—181no—
200Dic3⋊C4—181no—
201C2×C3⋊C8C2×D911yescharacteristic
202Dic12—182no—
203Dic12—182no—
204Dic12—184no—
205Dic12—182no—
206C22×C32C2×D1211yescharacteristic
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C2×C3×D3Dic1212yes—
210C32×C4D3×D411yescharacteristic
211C32×C4D3×D411yescharacteristic
212C3⋊Dic3—62no—
213C3⋊Dic3—121no—
214C3×Dic3—21no—
215C3×Dic3—62no—
216C3×Dic3—121no—
217C22×C9—21no—
218C22×C9—32no—
219C22×C9—61no—
220C22×C9—61no—
221C22×C9D3×D411yescharacteristic
222C4×C9—21no—
223C4×C9—21no—
224C4×C9—61no—
225C4×C9C2×D1211yescharacteristic
226C4×C9D3×D411yescharacteristic
227Dic9—22no—
228Dic9—41no—
229Dic9—62no—
230Dic9—122no—
231Dic9—122no—
232Dic9—121no—
233C22×Q8—271no—
234C2×Q16—271no—
235C2×Q16—271no—
236C22⋊Q8—271no—
237#103—91no—
238#103—271no—
239C22⋊C8—31no—
240C3×C9#45111yescharacteristic, complemented, Sylow
241C23×C3—31no—
242C22×D3D3611yescharacteristic
243C3×Q8—92no—
244C3×Q8—181no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260C2×Dic3—361no—
261C2×Dic3—361no—
262C2×Dic3D3611yescharacteristic
263C4×D3—21no—
264C4×D3—21no—
265C4×D3—184no—
266Dic6—32no—
267Dic6—61no—
268Dic6—92no—
269Dic6—92no—
270Dic6—182no—
271Dic6—184no—
272Dic6—184no—
273Dic6—181no—
274Dic6—181no—
275Dic6—182no—
276Dic6—181no—
277Dic6—181no—
278Dic6—181no—
279Dic6—181no—
280Dic6—361no—
281Dic6—361no—
282C3×C8—22no—
283C3×C8—21no—
284C3×C8—21no—
285C3×C8—61no—
286C3⋊C8—21no—
287C2×C32#88211yescharacteristic
288C2×C32C2×Dic1211yescharacteristic
289C2×C32#90911yescharacteristic
290C3×D3—22no—
291C2×C9—21no—
292C2×C9—21no—
293C2×C9—21no—
294C2×C9—62no—
295C2×C9#88211yescharacteristic
296C2×C9#91511yescharacteristic
297C2×C9D3×Q1611yescharacteristic
298C2×Q8—91no—
299C2×Q8—271no—
300C2×Q8—271no—
301C2×Q8—541no—
302C2×Q8—542no—
303C2×Q8—542no—
304C22×C4—31no—
305C22×C4—541no—
306Q16—542no—
307Q16—542no—
308C2×C8—31no—
309C2×C8D3×D911yescharacteristic
310C4⋊C4—91no—
311C4⋊C4—541no—
312C22⋊C4—541no—
313C22×C3—21no—
314C22×C3—32no—
315C22×C3—61no—
316C22×C3—61no—
317C22×C3C2×D3611yescharacteristic
318C22×C3D3×D1211yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3Dic3612yescomplemented
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—61no—
325C3×C4—182no—
326C3×C4—361no—
327C3×C4D4×D911yescharacteristic
328C3×C4D4×D911yescharacteristic
329C3×C4D3×D1211yescharacteristic
330C3×C4#381111yescharacteristic
331Dic3—21no—
332Dic3—62no—
333Dic3—121no—
334Dic3—182no—
335Dic3—182no—
336Dic3—362no—
337Dic3—362no—
338Dic3—361no—
339Dic3—361no—
340C32#473911yescharacteristic
341C9—21no—
342C9#480911yescharacteristic, complemented
343C23—31no—
344Q8—92no—
345Q8—181no—
346Q8—272no—
347Q8—544no—
348Q8—541no—
349Q8—541no—
350Q8—541no—
351C2×C4—31no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—544no—
357C2×C4—541no—
358C2×C4—541no—
359C2×C4C2×D3×D911yescharacteristic
360C8—21no—
361C8—61no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—62no—
366C2×C3#6344411yescharacteristic
367C2×C3C2×Dic3611yescharacteristic
368C2×C3#6347111yescharacteristic
369C2×C3D3×Dic1211yescharacteristic
370C2×C3#6380011yescharacteristic
371C2×C3#6392311yescharacteristic
372D3—22no—
373C22—32no—
374C22—61no—
375C22—61no—
376C22D3×D3611yescharacteristic
377C4—61no—
378C4—182no—
379C4—361no—
380C4—542no—
381C4—1081no—
382C4D9×D1211yescharacteristic
383C4#8866811yescharacteristic
384C3—21no—
385C3#1058840511yescharacteristic, complemented
386C3#1059095211yescharacteristic
387C2—62no—
388C2D3×Dic3611yescharacteristic
389C2#1172079611yescharacteristic
390C2#1172105111yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC22×C3×D3D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88521C2semidirect
4. C2×#88418C2semidirect
6. C3×#63376C2semidirect
7. #11720277C2semidirect
39. #63376D3semidirect
98. C2×D3×C9Q16semidirect
240. C3×C9#451semidirect
321. C2×D3Dic36semidirect
342. C9#4809semidirect
385. C3#10588405semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic182
C2×#885212
#117209612
C2×#884182
C9×(C2×D3)⋊C82
C3×#633762
#117202772
#105909523
#105884053