Finite Group Databasebeta

← #13140698

Subgroups — #13140698

Order 1728.

362 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721854C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic
3C9×#835C211yesmaximal, characteristic
4C3×#63395C211yesmaximal, characteristic
5#11720277C211yesmaximal, characteristic
6#11720271C211yesmaximal, characteristic
7#11720269C211yesmaximal, characteristic
8#10591538—31nomaximal
9#10588472—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×#6189—21no—
13#88534—21no—
14#88534—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64153—31no—
22#63919—31no—
23C3×#835—31no—
24C3×#835D311yescharacteristic
25#63567—31no—
26#63567—31no—
27#63553—31no—
28#63506—31no—
29#63506—61no—
30#63450—31no—
31C9×#103—31no—
32#63395D311yescharacteristic
33#63376—31no—
34C8×Dic9—31no—
35#63368—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47#6189—42no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4792—91no—
51#4792—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×#634—31no—
54C2×#627—61no—
55#3750—61no—
56#3750—61no—
57C4×C3⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62#3724—61no—
63C2×Q8×C9—21no—
64C2×Q8×C9C2×D311yescharacteristic
65C2×Dic18—31no—
66C2×Dic18—61no—
67C2×Dic18—61no—
68C2×C8×C9—31no—
69C9×C4⋊C4—31no—
70C4⋊Dic9C2×D311yescharacteristic
71Dic9⋊C4—61no—
72Dic9⋊C4—61no—
73Dic9⋊C4—61no—
74Dic9⋊C4—62no—
75Dic9⋊C4—62no—
76C4×Dic9—31no—
77C4×Dic9—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—41no—
84C3⋊Dic9SD1612yes—
85C3⋊Dic9Q1612yes—
86C9×Dic3—41no—
87C3×Dic9—41no—
88#944—91no—
89#944—91no—
90#944—181no—
91#888—91no—
92#888—91no—
93C3×#103—31no—
94C3×#103—91no—
95#835—31no—
96#835D911yescharacteristic
97#816—91no—
98#816—91no—
99C8×Dic3—91no—
100C8×Dic3—91no—
101#810—91no—
102#810—91no—
103C32×Q8—21no—
104C32×Q8—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110#634—61no—
111#634—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114#627—122no—
115C3×C3⋊C8—21no—
116C3×C3⋊C8—61no—
117Q8×C9—21no—
118Q8×C9—21no—
119Q8×C9—41no—
120Q8×C9—41no—
121C2×C4×C9—22no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9—31no—
128C2×Dic9—31no—
129C2×Dic9—61no—
130C2×Dic9—61no—
131C2×Dic9—62no—
132C2×Dic9—61no—
133Dic18—61no—
134Dic18—61no—
135Dic18—121no—
136Dic18—121no—
137Dic18—122no—
138C8×C9—61no—
139C9⋊C8—21no—
140#487—271noSylow
141C2×C3×C9#12411yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9C2×Q1611yescharacteristic
144C2×C3×Q8—21no—
145C2×C3×Q8C2×D911yescharacteristic
146C2×C3×Q8D3211yescharacteristic
147C2×Dic6—91no—
148C2×Dic6—91no—
149C2×Dic6—181no—
150C2×Dic6—181no—
151C2×Dic6—181no—
152C2×C3×C8—31no—
153C2×C3×C8—91no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—91no—
156C4⋊Dic3—31no—
157C4⋊Dic3C2×D911yescharacteristic
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—182no—
165Dic3⋊C4—182no—
166C4×Dic3—91no—
167C4×Dic3—91no—
168C4×Dic3—181no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32D3×D411yescharacteristic
172C32×C4—41no—
173C32×C4#23511yescharacteristic
174C32×C4#23711yescharacteristic
175C3⋊Dic3—32no—
176C3⋊Dic3—32no—
177C3⋊Dic3—121no—
178C3×Dic3—41no—
179C3×Dic3—121no—
180C22×C9—21no—
181C22×C9D3×D411yescharacteristic
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—42no—
185C4×C9—41no—
186C4×C9—121no—
187C4×C9#23511yescharacteristic
188C4×C9#23711yescharacteristic
189Dic9—32no—
190Dic9—32no—
191Dic9—41no—
192Dic9—62no—
193Dic9—62no—
194Dic9—122no—
195Dic9—121no—
196C4⋊Q8—271no—
197C4⋊Q8—271no—
198#103—31no—
199#103—91no—
200#103—271no—
201#103—271no—
202C4×C8—271no—
203C3×C9#48711yescharacteristic, complemented, Sylow
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8—21no—
207C3×Q8—21no—
208C3×Q8—41no—
209C3×Q8—41no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—61no—
215C2×C3×C4—181no—
216C2×C3×C4C22×D911yescharacteristic
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—61no—
220C2×Dic3—91no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C2×Dic3—182no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229Dic6—181no—
230Dic6—181no—
231Dic6—181no—
232Dic6—181no—
233Dic6—361no—
234Dic6—361no—
235Dic6—362no—
236Dic6—362no—
237C3×C8—61no—
238C3×C8—181no—
239C3⋊C8—21no—
240C3⋊C8—61no—
241C2×C32#88511yescharacteristic
242C2×C32D3×SD1611yescharacteristic
243C2×C32D3×Q1611yescharacteristic
244C2×C9—21no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9#88511yescharacteristic
248C2×C9D3×SD1611yescharacteristic
249C2×C9D3×Q1611yescharacteristic
250C2×Q8—271no—
251C2×Q8—541no—
252C2×Q8D3×D911yescharacteristic
253C2×C8—31no—
254C2×C8—91no—
255C4⋊C4—31no—
256C4⋊C4—91no—
257C4⋊C4—541no—
258C4⋊C4—541no—
259C4⋊C4—541no—
260C42—271no—
261C22×C3—21no—
262C22×C3D4×D911yescharacteristic
263C22×C3#381111yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—42no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—121no—
270C3×C4—361no—
271C3×C4#369711yescharacteristic
272C3×C4#369911yescharacteristic
273C3×C4#380411yescharacteristic
274C3×C4#380411yescharacteristic
275Dic3—41no—
276Dic3—92no—
277Dic3—92no—
278Dic3—92no—
279Dic3—92no—
280Dic3—121no—
281Dic3—182no—
282Dic3—182no—
283Dic3—362no—
284Dic3—361no—
285Dic3—361no—
286C32#479211yescharacteristic
287C9—21no—
288C9#479211yescharacteristic, complemented
289Q8—21no—
290Q8—21no—
291Q8—541no—
292Q8—541no—
293Q8—1082no—
294C2×C4—21no—
295C2×C4—61no—
296C2×C4—181no—
297C2×C4—271no—
298C2×C4—271no—
299C2×C4—541no—
300C2×C4C2×D3×D911yescharacteristic
301C8—61no—
302C8—181no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3#6344711yescharacteristic
307C2×C3SD16×D911yescharacteristic
308C2×C3Q16×D911yescharacteristic
309C2×C3#6394011yescharacteristic
310C2×C3#6394411yescharacteristic
311C2×C3#6395311yescharacteristic
312C22#8866811yescharacteristic
313C4—41no—
314C4—121no—
315C4—272no—
316C4—272no—
317C4—361no—
318C4—1081no—
319C4#8865811yescharacteristic
320C4#8866211yescharacteristic
321C3—21no—
322C3#1058847211yescharacteristic, complemented
323C3#1059153811yescharacteristic
324C2#1172121211yescharacteristic
325C2#1172122011yescharacteristic
326C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#487
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
203. C3×C9#487semidirect
288. C9#4792semidirect
322. C3#10588472semidirect

Maximal subgroups

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

subgroupindex
#117218542
#117209712
C9×#8352
C3×#633952
#117202772
#117202712
#117202692
#105915383
#105884723