Finite Group Databasebeta

← #13147632

Subgroups — #13147632

Order 1728.

796 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 161 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#11721709C211yesmaximal, characteristic, complemented
2C9×#874C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721045C212yesmaximal, complemented
5#11721025C212yesmaximal, complemented
6#11721017C214yesmaximal, complemented
7#11721015C214yesmaximal, complemented
8#10593950—31nomaximal
9#10589821—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×D12C2212yes—
12C2×C9×D12C2214yes—
13C2×#6195C2214yes—
14C2×D3×Dic9C2214yes—
15C3×C42×C9C2211yescharacteristic
16#88536C2212yes—
17#88534C2212yes—
18C4×C3⋊Dic9C2211yescharacteristic
19C3×C4⋊Dic9C2212yes—
20C3×Dic9⋊C4C2214yes—
21(C2×D3)⋊Dic9C2214yes—
22(C2×D3)⋊Dic9C2214yes—
23#64081—31no—
24C3×#874D311yescharacteristic
25C3×#870—31no—
26#63916—32no—
27#63898—32no—
28#63886—34no—
29#63884—34no—
30C9×#127—31no—
31#63501—34no—
32#63500—32no—
33#63497—34no—
34D4×Dic9—32no—
35#63434—31no—
36#63434—61no—
37#63432D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C22×D3×C9C2314yes—
40C2×C3×C4×C9C2312yes—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9C2312yes—
43C2×C3⋊Dic9C2311yescharacteristic
44C4×C3⋊D9—22no—
45C2×C3×Dic9C2314yes—
46C9×D12—24no—
47C9×D12—44no—
48#6195—44no—
49D3×Dic9—44no—
50#5719—91no—
51#5697—91no—
52C2×C4×C3⋊D3—31no—
53C2×C3×D12C2×D312yes—
54C2×C3×D12C2×D314yes—
55C2×#626—34no—
56C2×D3×Dic3—34no—
57C32×C42C2×D311yescharacteristic
58#3752—32no—
59#3750—32no—
60C4×C3⋊Dic3—31no—
61C3×C4⋊Dic3—32no—
62C3×Dic3⋊C4—34no—
63(C2×D3)⋊Dic3—34no—
64(C2×D3)⋊Dic3—34no—
65C2×D4×C9—32no—
66C2×D4×C9—34no—
67C2×#611—34no—
68C22×Dic9—34no—
69C2×C4×D9—31no—
70C2×C4×D9—61no—
71C42×C9—21no—
72C42×C9C2×D311yescharacteristic
73#3676—34no—
74#3676—34no—
75(C2×D9)⋊C4—32no—
76(C2×D9)⋊C4—62no—
77C4⋊Dic9C2×D312yes—
78Dic9⋊C4—32no—
79Dic9⋊C4—62no—
80Dic9⋊C4C2×D314yes—
81C4×Dic9—31no—
82C4×Dic9—61no—
83C22×C3×C9C2411yescharacteristic
84C2×C3⋊D9—21no—
85C2×C3⋊D9—22no—
86C2×D3×C9—24no—
87C2×D3×C9—24no—
88C2×D3×C9—24no—
89C3×C4×C9—22no—
90C3×C4×C9#4112yes—
91C3⋊Dic9—21no—
92C3⋊Dic9—22no—
93C3×Dic9—24no—
94C3×#127—31no—
95C3×#125—91no—
96#939—94no—
97#938—92no—
98#935—94no—
99D4×Dic3—92no—
100#895—94no—
101#895—92no—
102#894—94no—
103#893—92no—
104#874D911yescharacteristic, complemented
105#872—91no—
106#872—91no—
107#872—181no—
108#870—31no—
109C22×C3⋊D3—31no—
110C22×C3×D3C22×D314yes—
111C2×C32×C4C22×D312yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—32no—
115C4×C3⋊D3—62no—
116C2×C3×Dic3—34no—
117C3×D12—24no—
118C3×D12—44no—
119#626—124no—
120D3×Dic3—124no—
121C23×C9—34no—
122C22×D9—31no—
123C22×D9—61no—
124D4×C9—64no—
125D4×C9—124no—
126C2×C4×C9—21no—
127C2×C4×C9—22no—
128C2×C4×C9C22×D312yes—
129C2×C4×C9C22×D311yescharacteristic
130#611—124no—
131C2×Dic9—32no—
132C2×Dic9—31no—
133C2×Dic9—61no—
134C2×Dic9—62no—
135C2×Dic9—124no—
136C2×Dic9C22×D314yes—
137C4×D9—64no—
138C4×D9—62no—
139#540—271noSylow
140C2×C3×C9C2×#4111yescharacteristic
141C2×C3×C9H(5,𝔽2)12yes—
142C3⋊D9—41no—
143D3×C9—44no—
144C2×C3×D4—32no—
145C2×C3×D4—34no—
146C2×#67—94no—
147C22×Dic3—94no—
148C2×D12—94no—
149C2×D12C2×D912yes—
150C2×D12C2×D914yes—
151C2×C4×D3—91no—
152C2×C4×D3—91no—
153C2×C4×D3—94no—
154C2×C4×D3—181no—
155C3×C4⋊C4—92no—
156C3×C4⋊C4—94no—
157C3×C42—21no—
158C3×C42C2×D911yescharacteristic
159C3×C42D3211yescharacteristic
160#217—94no—
161#217—94no—
162(C2×D3)⋊C4—94no—
163(C2×D3)⋊C4—94no—
164(C2×D3)⋊C4—92no—
165(C2×D3)⋊C4—92no—
166(C2×D3)⋊C4—182no—
167C4⋊Dic3—32no—
168Dic3⋊C4—34no—
169Dic3⋊C4—92no—
170Dic3⋊C4—92no—
171Dic3⋊C4—182no—
172C4×Dic3—91no—
173C4×Dic3—91no—
174C4×Dic3—181no—
175C22×C32C23×D311yescharacteristic
176C2×C3⋊D3—61no—
177C2×C3⋊D3—62no—
178C2×C3×D3—24no—
179C2×C3×D3—24no—
180C2×C3×D3—24no—
181C32×C4—22no—
182C32×C4#23712yes—
183C3⋊Dic3—61no—
184C3⋊Dic3—62no—
185C3×Dic3—64no—
186C22×C9—21no—
187C22×C9—64no—
188C22×C9—64no—
189C22×C9—64no—
190C22×C9C23×D311yescharacteristic
191C2×D9—62no—
192C2×D9—61no—
193C2×D9—62no—
194C2×D9—122no—
195C4×C9—22no—
196C4×C9—22no—
197C4×C9—42no—
198C4×C9#23912yes—
199Dic9—24no—
200Dic9—62no—
201Dic9—61no—
202Dic9—62no—
203Dic9—122no—
204#127—31no—
205#125—91no—
206#123—274no—
207#121—272no—
208#121—274no—
209C4×D4—272no—
210#117—271no—
211C3×C9#54011yescharacteristic, complemented, Sylow
212C23×C3—34no—
213C22×D3—91no—
214C22×D3—91no—
215C22×D3—181no—
216C22×D3C22×D914yes—
217C3×D4—64no—
218C3×D4—124no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—94no—
222C2×C3×C4C22×D912yes—
223C2×C3×C4C22×D911yescharacteristic
224C2×C3×C4C2×D3212yes—
225C2×C3×C4C2×D3211yescharacteristic
226#67—364no—
227C2×Dic3—34no—
228C2×Dic3—92no—
229C2×Dic3—91no—
230C2×Dic3—92no—
231C2×Dic3—91no—
232C2×Dic3—181no—
233C2×Dic3—182no—
234C2×Dic3—364no—
235D12—24no—
236D12—44no—
237D12—364no—
238C4×D3—184no—
239C4×D3—182no—
240C4×D3—182no—
241C4×D3—364no—
242C2×C32C2×#23711yescharacteristic
243C2×C32#100412yes—
244C3⋊D3—121no—
245C3×D3—44no—
246C2×C9—21no—
247C2×C9—22no—
248C2×C9—124no—
249C2×C9C2×#23911yescharacteristic
250C2×C9#100912yes—
251D9—122no—
252D9—121no—
253C2×D4—32no—
254C2×D4—34no—
255C2×D4—274no—
256C22×C4—274no—
257C22×C4—271no—
258C4⋊C4—92no—
259C4⋊C4—94no—
260C4⋊C4—272no—
261C22⋊C4—274no—
262C22⋊C4—274no—
263C22⋊C4—272no—
264C42—271no—
265C42D3×D911yescharacteristic
266C22×C3—21no—
267C22×C3—64no—
268C22×C3—64no—
269C22×C3—64no—
270C22×C3C23×D911yescharacteristic
271C22×C3C22×D3211yescharacteristic
272C2×D3—24no—
273C2×D3—24no—
274C2×D3—24no—
275C2×D3—182no—
276C2×D3—181no—
277C2×D3—182no—
278C2×D3—181no—
279C2×D3—182no—
280C2×D3—362no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—22no—
284C3×C4—42no—
285C3×C4—184no—
286C3×C4#369912yes—
287C3×C4#379612yes—
288Dic3—64no—
289Dic3—182no—
290Dic3—181no—
291Dic3—182no—
292Dic3—181no—
293Dic3—182no—
294Dic3—362no—
295C32#571911yescharacteristic
296C9—21no—
297C9#569711yescharacteristic, complemented
298C23—34no—
299C23—271no—
300D4—64no—
301D4—124no—
302D4—1084no—
303C2×C4—94no—
304C2×C4—272no—
305C2×C4—271no—
306C2×C4—542no—
307C2×C4—1084no—
308C2×C4C2×D3×D912yes—
309C2×C4C2×D3×D911yescharacteristic
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—124no—
313C2×C3C2×#369911yescharacteristic
314C2×C3#6371012yes—
315C2×C3C2×#379611yescharacteristic
316C2×C3#6430712yes—
317D3—44no—
318D3—362no—
319D3—361no—
320D3—361no—
321C22—64no—
322C22—64no—
323C22—64no—
324C22—541no—
325C22—542no—
326C22C22×D3×D911yescharacteristic
327C4—22no—
328C4—184no—
329C4—541no—
330C4—542no—
331C4#8863912yes—
332C3—21no—
333C3#1058982111yescharacteristic, complemented
334C3#1059395011yescharacteristic
335C2—124no—
336C2—1081no—
337C2C2×#8863911yescharacteristic
338C2#1172247612yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#540
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721709C2semidirect
2. C9×#874C2semidirect
3. C3×#63432C2semidirect
4. #11721045C2semidirect
5. #11721025C2semidirect
6. #11721017C2semidirect
7. #11721015C2semidirect
37. #63432D3semidirect
104. #874D9semidirect
211. C3×C9#540semidirect
297. C9#5697semidirect
333. C3#10589821semidirect

Maximal subgroups

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

subgroupindex
#117217092
C9×#8742
C3×#634322
#117210452
#117210252
#117210172
#117210152
#105939503
#105898213