Finite Group Databasebeta

← #13139518

Subgroups — #13139518

Order 1728.

600 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#6226C211yesmaximal, characteristic
2C3×#63486C211yesmaximal, characteristic
3C2×#88418C212yesmaximal, complemented
4C9×#830C211yesmaximal, characteristic
5#11720277C212yesmaximal, complemented
6#10591190—31nomaximal
7#10588405—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3⋊Dic9—21no—
10C2×#6226—21no—
11C2×#6226—24no—
12C2×#6226C2212yes—
13C2×C3×Dic18C2211yescharacteristic
14C3×#3676—21no—
15C3×C4⋊Dic9C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C2×C9×C3⋊C8C2212yes—
18#88418—24no—
19C22×#634—31no—
20C3×#924—31no—
21C2×#3719—32no—
22C22×Dic18—31no—
23C22×Dic18—61no—
24C3×#830D311yescharacteristic
25#63567—32no—
26#63486D311yescharacteristic
27C2×Dic36—32no—
28C9×C22⋊C8—31no—
29#63376—32no—
30C23×C3×C9D411yescharacteristic
31C2×C3×C4×C9—22no—
32C2×C3×C4×C9D411yescharacteristic
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3⋊Dic9—24no—
35C2×C3⋊Dic9—22no—
36#6226—24no—
37#6226—22no—
38#6226D414yes—
39C2×C3×Dic9—21no—
40C2×C3×Dic9—21no—
41C3×Dic18—22no—
42C9×C3⋊C8—22no—
43#5050—91no—
44#4739—91no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C3⋊Dic3—61no—
47C2×#634—32no—
48C2×#634—64no—
49C2×#634—61no—
50C2×C3×Dic6—31no—
51C3×#217—61no—
52C3×C4⋊Dic3—31no—
53C3×Dic3⋊C4—61no—
54C2×C3×C3⋊C8C2×D312yes—
55#3719—64no—
56C22×C4×C9—21no—
57C22×C4×C9C2×D311yescharacteristic
58C22×Dic9—62no—
59C22×Dic9—61no—
60C2×Dic18—32no—
61C2×Dic18—64no—
62C2×Dic18—62no—
63C2×Dic18—68no—
64C2×Dic18—62no—
65C2×Dic18—61no—
66C2×Dic18C2×D311yescharacteristic
67C2×C8×C9—32no—
68#3676—21no—
69C4⋊Dic9C2×D311yescharacteristic
70Dic9⋊C4—21no—
71Dic36—64no—
72C22×C3×C9—21no—
73C22×C3×C9—21no—
74C22×C3×C9Q1612yescomplemented
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—21no—
77C3×C4×C9C2×D412yes—
78C3⋊Dic9—24no—
79C3×Dic9—41no—
80C3×Dic9—41no—
81C22×Dic6—91no—
82C22×Dic6—91no—
83C22×Dic6—181no—
84C3×C22⋊Q8—91no—
85C2×#216—92no—
86#924—31no—
87C2×Dic12—92no—
88C3×C22⋊C8—31no—
89#830D911yescharacteristic
90#816—92no—
91#810—92no—
92C23×C32D1211yescharacteristic
93C2×C32×C4—22no—
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3—64no—
97C2×C3⋊Dic3—62no—
98#634—34no—
99#634—64no—
100#634—62no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C3×Dic6—62no—
104C23×C9—21no—
105C23×C9#6711yescharacteristic
106C3×C3⋊C8—22no—
107C2×C4×C9—24no—
108C2×C4×C9—22no—
109C2×C4×C9—21no—
110C2×C4×C9—21no—
111C2×C4×C9#6711yescharacteristic
112C2×C4×C9C22×D311yescharacteristic
113C2×Dic9—21no—
114C2×Dic9—21no—
115C2×Dic9—68no—
116C2×Dic9—64no—
117C2×Dic9—64no—
118C2×Dic9—62no—
119Dic18—22no—
120Dic18—34no—
121Dic18—64no—
122Dic18—68no—
123Dic18—64no—
124Dic18—64no—
125Dic18—62no—
126C8×C9—62no—
127#451—271noSylow
128C2×C3×C9—22no—
129C2×C3×C9#12011yescharacteristic
130C2×C3×C9C2×Q1611yescharacteristic
131C2×C3×C9#13711yescharacteristic
132C2×C3×Q8—91no—
133C22×C3×C4—21no—
134C22×C3×C4C2×D911yescharacteristic
135C22×C3×C4D3211yescharacteristic
136C22×Dic3—182no—
137C22×Dic3—181no—
138C22×Dic3—181no—
139C2×Dic6—31no—
140C2×Dic6—92no—
141C2×Dic6—92no—
142C2×Dic6—182no—
143C2×Dic6—184no—
144C2×Dic6—184no—
145C2×Dic6—188no—
146C2×Dic6—182no—
147C2×Dic6—181no—
148C2×Dic6—181no—
149C2×C3×C8—32no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C22⋊C4—181no—
153#217—61no—
154#216—184no—
155C4⋊Dic3—31no—
156Dic3⋊C4—61no—
157C2×C3⋊C8C2×D912yes—
158Dic12—184no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32Dic1212yes—
162C22×C32C2×D1211yescharacteristic
163C32×C4—21no—
164C32×C4D3×D412yes—
165C3⋊Dic3—64no—
166C3×Dic3—121no—
167C3×Dic3—121no—
168C22×C9—22no—
169C22×C9—22no—
170C22×C9—21no—
171C22×C9—21no—
172C22×C9—22no—
173C22×C9—21no—
174C22×C9#21612yescomplemented
175C22×C9C2×#6711yescharacteristic
176C4×C9—22no—
177C4×C9—21no—
178C4×C9—22no—
179C4×C9D3×D412yes—
180Dic9—41no—
181Dic9—41no—
182Dic9—68no—
183Dic9—64no—
184C22×Q8—271no—
185C2×Q16—272no—
186C22⋊Q8—91no—
187#103—272no—
188C22⋊C8—31no—
189C3×C9#45111yescharacteristic, complemented, Sylow
190C23×C3—21no—
191C23×C3D3611yescharacteristic
192C23×C3#62611yescharacteristic
193C3×Q8—182no—
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4D3611yescharacteristic
202C2×C3×C4C22×D911yescharacteristic
203C2×C3×C4#62611yescharacteristic
204C2×C3×C4C2×D3211yescharacteristic
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—184no—
208C2×Dic3—188no—
209C2×Dic3—184no—
210C2×Dic3—184no—
211C2×Dic3—182no—
212C2×Dic3—182no—
213Dic6—62no—
214Dic6—94no—
215Dic6—94no—
216Dic6—184no—
217Dic6—184no—
218Dic6—188no—
219Dic6—184no—
220Dic6—184no—
221Dic6—182no—
222Dic6—182no—
223C3×C8—62no—
224C3⋊C8—22no—
225C2×C32—22no—
226C2×C32#88211yescharacteristic
227C2×C32C2×Dic1211yescharacteristic
228C2×C32#90911yescharacteristic
229C2×C9—24no—
230C2×C9—22no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9#93711yescharacteristic
235C2×C9C2×#21611yescharacteristic
236C2×C9#95111yescharacteristic
237C2×Q8—91no—
238C2×Q8—272no—
239C2×Q8—544no—
240C2×Q8—541no—
241C22×C4—541no—
242C22×C4D3×D911yescharacteristic
243Q16—544no—
244C2×C8—32no—
245C4⋊C4—91no—
246C4⋊C4—181no—
247C22⋊C4—181no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—22no—
256C22×C3Dic3612yescomplemented
257C22×C3C2×D3611yescharacteristic
258C22×C3#371912yes—
259C22×C3C2×#62611yescharacteristic
260C3×C4—22no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—361no—
265C3×C4—361no—
266C3×C4D4×D912yes—
267C3×C4#381112yes—
268Dic3—121no—
269Dic3—121no—
270Dic3—188no—
271Dic3—184no—
272Dic3—184no—
273C32#473911yescharacteristic
274C9—21no—
275C9#505011yescharacteristic, complemented
276C23#619211yescharacteristic
277Q8—182no—
278Q8—274no—
279Q8—544no—
280Q8—542no—
281C2×C4—22no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4—544no—
285C2×C4—542no—
286C2×C4#619211yescharacteristic
287C2×C4C2×D3×D911yescharacteristic
288C8—62no—
289C2×C3—24no—
290C2×C3—22no—
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3#6344411yescharacteristic
296C2×C3C2×Dic3611yescharacteristic
297C2×C3#6347111yescharacteristic
298C2×C3#6383111yescharacteristic
299C2×C3C2×#371911yescharacteristic
300C2×C3#6398111yescharacteristic
301C22—21no—
302C22—21no—
303C22#8841812yescomplemented
304C22C2×#619211yescharacteristic
305C4—21no—
306C4—361no—
307C4—361no—
308C4—544no—
309C4#8866812yes—
310C3—21no—
311C3#1058840511yescharacteristic, complemented
312C3#1059119011yescharacteristic
313C2—22no—
314C2#1172086911yescharacteristic
315C2C2×#8841811yescharacteristic
316C2#1172127811yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
prime radicalC23×C32D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88418C2semidirect
5. #11720277C2semidirect
74. C22×C3×C9Q16semidirect
174. C22×C9#216semidirect
189. C3×C9#451semidirect
256. C22×C3Dic36semidirect
275. C9#5050semidirect
303. C22#88418semidirect
311. C3#10588405semidirect

Maximal subgroups

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

subgroupindex
C22×#62262
C3×#634862
C2×#884182
C9×#8302
#117202772
#105911903
#105884053