Finite Group Databasebeta

← #13143831

Subgroups — #13143831

Order 1728.

950 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 189 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×C4×C3⋊D9C211yesmaximal, characteristic
2C2×#88523C212yesmaximal, complemented
3C3×C9×C22⋊C8C211yesmaximal, characteristic, complemented
4#11720571C211yesmaximal, characteristic, complemented
5#11720542C212yesmaximal, complemented
6#10592425—31nomaximal
7#10588408—33nomaximal
8C23×C3⋊D9C411yescharacteristic
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3⋊Dic9C411yescharacteristic
11C2×C4×C3⋊D9—24no—
12C2×C4×C3⋊D9C411yescharacteristic
13C2×C4×C3⋊D9C411yescharacteristic
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C3×C8×C9C2212yescomplemented
17C2×C3⋊(C9⋊C8)C2212yescomplemented
18#88523—24no—
19C22×C4×C3⋊D3—31no—
20C2×#3743—32no—
21C22×C4×D9—33no—
22C32×C22⋊C8D311yescharacteristic
23#63657—31no—
24#63647—32no—
25C2×#3663—36no—
26C9×C22⋊C8D313yescomplemented
27#63389—33no—
28(C2×D9)⋊C8—36no—
29C23×C3×C9C2×C411yescharacteristic
30C22×C3⋊D9—22no—
31C22×C3⋊D9—22no—
32C22×C3⋊D9—22no—
33C22×C3⋊D9C2×C411yescharacteristic
34C22×C3⋊D9C2×C411yescharacteristic
35C2×C3×C4×C9—22no—
36C2×C3×C4×C9C2×C411yescharacteristic
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—22no—
39C2×C3⋊Dic9C2×C411yescharacteristic
40C2×C3⋊Dic9C2×C411yescharacteristic
41C4×C3⋊D9—22no—
42C4×C3⋊D9—22no—
43C4×C3⋊D9—22no—
44C4×C3⋊D9D414yes—
45C3×C8×C9—22no—
46C3⋊(C9⋊C8)—22no—
47#4726—93no—
48#4726—91no—
49C23×C3⋊D3—31no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C3⋊Dic3—31no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—64no—
57C23×D9—33no—
58C2×C32×C8C2×D312yes—
59C2×C3⋊(C3⋊C8)—32no—
60#3743—64no—
61C22×C4×C9C2×D313yes—
62C22×Dic9—33no—
63C2×C4×D9—33no—
64C2×C4×D9—33no—
65C2×C4×D9—33no—
66C2×C4×D9—33no—
67C2×C4×D9—612no—
68C2×C8×C9C2×D316yescomplemented
69C2×C9⋊C8—36no—
70#3663—612no—
71C22×C3×C9—21no—
72C22×C3×C9—21no—
73C22×C3×C9M1612yescomplemented
74C22×C3×C9C22×C411yescharacteristic
75C2×C3⋊D9—28no—
76C2×C3⋊D9—21no—
77C2×C3⋊D9—21no—
78C2×C3⋊D9—21no—
79C2×C3⋊D9—21no—
80C2×C3⋊D9C22⋊C412yes—
81C2×C3⋊D9M1612yescomplemented
82C3×C4×C9—21no—
83C3×C4×C9C2×D412yes—
84C3⋊Dic9—21no—
85C3⋊Dic9C22⋊C412yes—
86C22×C4×D3—93no—
87C22×C4×D3—91no—
88C2×ΓL(1,𝔽25)—96no—
89C2×ΓL(1,𝔽25)—92no—
90C3×C22⋊C8D913yescomplemented
91C3×C22⋊C8C3⋊D311yescharacteristic
92#830—93no—
93#830—91no—
94(C2×D3)⋊C8—96no—
95(C2×D3)⋊C8—92no—
96C23×C32C4×D311yescharacteristic
97C22×C3⋊D3—31no—
98C22×C3⋊D3—31no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—62no—
101C22×C3⋊D3—62no—
102C2×C32×C4—22no—
103C2×C32×C4C4×D311yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—62no—
108C4×C3⋊D3—34no—
109C4×C3⋊D3—62no—
110C4×C3⋊D3—62no—
111C4×C3⋊D3—62no—
112C23×C9C4×D313yes—
113C22×D9—33no—
114C22×D9—33no—
115C22×D9—66no—
116C22×D9—66no—
117C22×D9—66no—
118C32×C8—22no—
119C3⋊(C3⋊C8)—62no—
120C2×C4×C9—26no—
121C2×C4×C9C4×D313yes—
122C2×C4×C9C22×D313yes—
123C2×Dic9—33no—
124C2×Dic9—33no—
125C2×Dic9—66no—
126C4×D9—312no—
127C4×D9—66no—
128C4×D9—66no—
129C4×D9—66no—
130C8×C9—26no—
131C9⋊C8—66no—
132#407—271noSylow
133C2×C3×C9—22no—
134C2×C3×C9C2×C22⋊C411yescharacteristic
135C2×C3×C9C2×M1611yescharacteristic
136C2×C3×C9C2×M1611yescharacteristic
137C3⋊D9—22no—
138C3⋊D9—22no—
139C23×D3—93no—
140C23×D3—91no—
141C22×C3×C4C2×D913yes—
142C22×C3×C4C2×C3⋊D311yescharacteristic
143C22×Dic3—93no—
144C22×Dic3—91no—
145C2×C4×D3—93no—
146C2×C4×D3—91no—
147C2×C4×D3—93no—
148C2×C4×D3—93no—
149C2×C4×D3—93no—
150C2×C4×D3—91no—
151C2×C4×D3—91no—
152C2×C4×D3—91no—
153C2×C4×D3—184no—
154C2×C4×D3—1812no—
155C2×C3×C8C2×D916yescomplemented
156C2×C3×C8C2×C3⋊D312yes—
157C2×C3⋊C8—96no—
158C2×C3⋊C8—92no—
159ΓL(1,𝔽25)—184no—
160ΓL(1,𝔽25)—1812no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32ΓL(1,𝔽25)12yes—
164C22×C32C2×C4×D311yescharacteristic
165C2×C3⋊D3—32no—
166C2×C3⋊D3—32no—
167C2×C3⋊D3—68no—
168C2×C3⋊D3—61no—
169C2×C3⋊D3—61no—
170C2×C3⋊D3—61no—
171C2×C3⋊D3—61no—
172C32×C4—21no—
173C32×C4D3×D412yes—
174C3⋊Dic3—32no—
175C3⋊Dic3—61no—
176C22×C9—23no—
177C22×C9—23no—
178C22×C9ΓL(1,𝔽25)16yescomplemented
179C22×C9C2×C4×D313yes—
180C2×D9—36no—
181C2×D9—36no—
182C2×D9—624no—
183C2×D9—63no—
184C2×D9—63no—
185C2×D9—63no—
186C2×D9—63no—
187C4×C9—23no—
188C4×C9D3×D416yes—
189Dic9—36no—
190Dic9—63no—
191C23×C4—271no—
192C2×M16—272no—
193C22⋊C8—271no—
194C22⋊C8—272no—
195C22⋊C8C3⋊D911yescharacteristic, complemented
196C3×C9#40711yescharacteristic, complemented, Sylow
197C23×C3C4×D913yes—
198C23×C3C4×C3⋊D311yescharacteristic
199C22×D3—93no—
200C22×D3—93no—
201C22×D3—91no—
202C22×D3—91no—
203C22×D3—182no—
204C22×D3—182no—
205C22×D3—186no—
206C22×D3—186no—
207C22×D3—186no—
208C22×D3—182no—
209C2×C3×C4—26no—
210C2×C3×C4—22no—
211C2×C3×C4C4×D913yes—
212C2×C3×C4C22×D913yes—
213C2×C3×C4C4×C3⋊D311yescharacteristic
214C2×C3×C4C22×C3⋊D311yescharacteristic
215C2×Dic3—93no—
216C2×Dic3—91no—
217C2×Dic3—93no—
218C2×Dic3—91no—
219C2×Dic3—182no—
220C2×Dic3—186no—
221C4×D3—912no—
222C4×D3—94no—
223C4×D3—182no—
224C4×D3—186no—
225C4×D3—182no—
226C4×D3—186no—
227C4×D3—186no—
228C4×D3—182no—
229C3×C8—26no—
230C3×C8—22no—
231C3⋊C8—186no—
232C3⋊C8—182no—
233C2×C32—22no—
234C2×C32D3×C22⋊C411yescharacteristic
235C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
236C2×C32D3×M1611yescharacteristic
237C3⋊D3—62no—
238C3⋊D3—62no—
239C2×C9—26no—
240C2×C9D3×C22⋊C413yes—
241C2×C9C2×ΓL(1,𝔽25)13yes—
242C2×C9D3×M1613yes—
243D9—66no—
244D9—66no—
245C24—271no—
246C22×C4—271no—
247C22×C4—271no—
248C22×C4—271no—
249C22×C4—271no—
250C22×C4—271no—
251C22×C4—544no—
252C22×C4C2×C3⋊D911yescharacteristic
253M16—544no—
254C2×C8—272no—
255C2×C8C2×C3⋊D912yescomplemented
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—23no—
259C22×C3—23no—
260C22×C3#366316yescomplemented
261C22×C3C2×C4×D913yes—
262C22×C3#374312yes—
263C22×C3C2×C4×C3⋊D311yescharacteristic
264C2×D3—92no—
265C2×D3—92no—
266C2×D3—96no—
267C2×D3—96no—
268C2×D3—188no—
269C2×D3—1824no—
270C2×D3—181no—
271C2×D3—181no—
272C2×D3—181no—
273C2×D3—181no—
274C2×D3—183no—
275C2×D3—183no—
276C2×D3—183no—
277C2×D3—183no—
278C3×C4—21no—
279C3×C4—23no—
280C3×C4D4×D916yes—
281C3×C4D4×C3⋊D312yes—
282Dic3—92no—
283Dic3—96no—
284Dic3—181no—
285Dic3—183no—
286C32#472611yescharacteristic
287C9#472613yescomplemented
288C23—271no—
289C23—271no—
290C23—542no—
291C23—542no—
292C23—542no—
293C23C4×C3⋊D911yescharacteristic
294C2×C4—22no—
295C2×C4—271no—
296C2×C4—274no—
297C2×C4—271no—
298C2×C4—542no—
299C2×C4—542no—
300C2×C4—542no—
301C2×C4—542no—
302C2×C4C4×C3⋊D911yescharacteristic
303C2×C4C22×C3⋊D911yescharacteristic
304C8—22no—
305C8—542no—
306C2×C3—22no—
307C2×C3—26no—
308C2×C3C22⋊C4×D913yes—
309C2×C3C2×#366313yes—
310C2×C3M16×D913yes—
311C2×C3C22⋊C4×C3⋊D311yescharacteristic
312C2×C3C2×#374311yescharacteristic
313C2×C3M16×C3⋊D311yescharacteristic
314D3—182no—
315D3—182no—
316D3—186no—
317D3—186no—
318C22—21no—
319C22—21no—
320C22—272no—
321C22—272no—
322C22—548no—
323C22—541no—
324C22—541no—
325C22—541no—
326C22—541no—
327C22#8852312yescomplemented
328C22C2×C4×C3⋊D911yescharacteristic
329C4—21no—
330C4—272no—
331C4—541no—
332C4D4×C3⋊D912yes—
333C3#1058840813yescomplemented
334C3#1059242511yescharacteristic
335C2—22no—
336C2—542no—
337C2—542no—
338C2C22⋊C4×C3⋊D911yescharacteristic
339C2C2×#8852311yescharacteristic
340C2M16×C3⋊D911yescharacteristic
subgroupG/H
centerC2×C4C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×C22⋊C8C2
socleC22×C32C2×C4×D3
prime radicalC23×C3⋊D9C4
cosocleC2×C3×C4×C9C23
hypercenterC22⋊C8C3⋊D9
hypocenterC3×C9#407
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#88523C2semidirect
3. C3×C9×C22⋊C8C2semidirect
4. #11720571C2semidirect
5. #11720542C2semidirect
16. C2×C3×C8×C9C22semidirect
17. C2×C3⋊(C9⋊C8)C22semidirect
26. C9×C22⋊C8D3semidirect
68. C2×C8×C9C2×D3semidirect
73. C22×C3×C9M16semidirect
81. C2×C3⋊D9M16semidirect
90. C3×C22⋊C8D9semidirect
155. C2×C3×C8C2×D9semidirect
178. C22×C9ΓL(1,𝔽25)semidirect
195. C22⋊C8C3⋊D9semidirect
196. C3×C9#407semidirect
255. C2×C8C2×C3⋊D9semidirect
260. C22×C3#3663semidirect
287. C9#4726semidirect
327. C22#88523semidirect
333. C3#10588408semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×#885232
C3×C9×C22⋊C82
#117205712
#117205422
#105924253
#105884083