Finite Group Databasebeta

← #13147083

Subgroups — #13147083

Order 1728.

970 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C23×C3×C22⋊C9C211yesmaximal, characteristic
2C2×C3⋊((C22⋊C9)⋊C4)C212yesmaximal, complemented
3#10592946—31nomaximal
4#10589534—33nomaximal
5C22×C3×C22⋊C9—22no—
6C22×C3×C22⋊C9C412yescomplemented
7C22×C3×C22⋊C9C2211yescharacteristic
8C3⋊((C22⋊C9)⋊C4)—22no—
9#88551—41nomaximal
10C25×C32D311yescharacteristic
11C23×C22⋊C9D313yes—
12C2×#3757—32no—
13C2×#3757—31no—
14C2×#3757—31no—
15C2×#3757—32no—
16C2×(C22⋊C9)⋊C4—36no—
17C23×C3×C9—41no—
18C2×C3×C22⋊C9—22no—
19C2×C3×C22⋊C9C2×C411yescharacteristic
20C2×C3×C22⋊C9D412yes—
21C2×C3⋊Dic9—42no—
22#5247—93no—
23#5247—91no—
24C24×C32—22no—
25C24×C32—31no—
26C24×C32—31no—
27C24×C32—32no—
28C24×C32—62no—
29C24×C32Dic312yes—
30C24×C32C2×D311yescharacteristic
31C22×C3⋊Dic3—32no—
32C22×C3⋊Dic3—32no—
33C22×C22⋊C9—26no—
34C22×C22⋊C9Dic316yescomplemented
35C22×C22⋊C9C2×D313yes—
36#3757—62no—
37#3757—64no—
38#3757—62no—
39#3757—61no—
40#3757—62no—
41#3757—61no—
42(C22⋊C9)⋊C4—66no—
43#3676—123no—
44C22×C3×C9—41no—
45C22×C3×C9—42no—
46C22×C3×C9—82no—
47C3×C22⋊C9C22⋊C411yescharacteristic, complemented
48C3⋊Dic9—82no—
49C25×C3D913yes—
50C25×C3C3⋊D311yescharacteristic
51C2×#217—96no—
52C2×#217—92no—
53C2×#217—93no—
54C2×#217—91no—
55C2×#217—93no—
56C2×#217—91no—
57C2×#217—96no—
58C2×#217—92no—
59C23×C32—22no—
60C23×C32—31no—
61C23×C32—32no—
62C23×C32—31no—
63C23×C32—32no—
64C23×C32—34no—
65C23×C32—32no—
66C23×C32—32no—
67C23×C32—31no—
68C23×C32—62no—
69C23×C32—62no—
70C23×C32—62no—
71C23×C32—61no—
72C23×C32—64no—
73C23×C32—62no—
74C23×C32—64no—
75C23×C32C2×Dic311yescharacteristic
76C23×C32#6712yes—
77C23×C32S411yescharacteristic
78C2×C3⋊Dic3—62no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—62no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—62no—
83C2×C3⋊Dic3—62no—
84C23×C9—43no—
85C2×C22⋊C9—26no—
86C2×C22⋊C9C2×Dic313yes—
87C2×C22⋊C9#6716yes—
88C2×Dic9—126no—
89#379—271noSylow
90C2×C3×C9—41no—
91C2×C3×C9—42no—
92C2×C3×C9—82no—
93C24×C3—26no—
94C24×C3—22no—
95C24×C3—33no—
96C24×C3—31no—
97C24×C3—33no—
98C24×C3—31no—
99C24×C3—36no—
100C24×C3—32no—
101C24×C3—66no—
102C24×C3—62no—
103C24×C3Dic916yescomplemented
104C24×C3C2×D913yes—
105C24×C3C3⋊Dic312yes—
106C24×C3C2×C3⋊D311yescharacteristic
107C22×Dic3—92no—
108C22×Dic3—96no—
109C22×Dic3—92no—
110C22×Dic3—96no—
111#217—186no—
112#217—182no—
113#217—184no—
114#217—1812no—
115#217—186no—
116#217—182no—
117#217—181no—
118#217—183no—
119#217—186no—
120#217—183no—
121#217—182no—
122#217—181no—
123C22×C32—22no—
124C22×C32—31no—
125C22×C32—32no—
126C22×C32—31no—
127C22×C32—31no—
128C22×C32—32no—
129C22×C32—32no—
130C22×C32—34no—
131C22×C32—32no—
132C22×C32—62no—
133C22×C32—64no—
134C22×C32—62no—
135C22×C32—61no—
136C22×C32—62no—
137C22×C32—64no—
138C22×C32—62no—
139C22×C32#21711yescharacteristic
140C22×C32SL(2,ℤ/4)12yes—
141C22×C32C2×S411yescharacteristic
142C3⋊Dic3—122no—
143C3⋊Dic3—122no—
144C22×C9—43no—
145C22×C9—46no—
146C22×C9—86no—
147C22⋊C9#21713yescomplemented
148Dic9—246no—
149C25C3⋊D911yescharacteristic
150C2×C22⋊C4—272no—
151C2×C22⋊C4—271no—
152C2×C22⋊C4—271no—
153C2×C22⋊C4—272no—
154C3×C9—41noSylow
155C23×C3—26no—
156C23×C3—22no—
157C23×C3—31no—
158C23×C3—33no—
159C23×C3—36no—
160C23×C3—32no—
161C23×C3—36no—
162C23×C3—33no—
163C23×C3—32no—
164C23×C3—31no—
165C23×C3—34no—
166C23×C3—312no—
167C23×C3—31no—
168C23×C3—36no—
169C23×C3—36no—
170C23×C3—33no—
171C23×C3—32no—
172C23×C3—32no—
173C23×C3—62no—
174C23×C3—66no—
175C23×C3—62no—
176C23×C3—66no—
177C23×C3—66no—
178C23×C3—62no—
179C23×C3—61no—
180C23×C3—63no—
181C23×C3—612no—
182C23×C3—64no—
183C23×C3—62no—
184C23×C3—66no—
185C23×C3—612no—
186C23×C3—64no—
187C23×C3C2×Dic913yes—
188C23×C3#61116yes—
189C23×C3(C22⋊C9)⋊C213yes—
190C23×C3C2×C3⋊Dic311yescharacteristic
191C23×C3#63812yes—
192C23×C3C3⋊S411yescharacteristic
193C2×Dic3—182no—
194C2×Dic3—182no—
195C2×Dic3—182no—
196C2×Dic3—186no—
197C2×Dic3—186no—
198C2×Dic3—186no—
199C2×Dic3—182no—
200C2×Dic3—182no—
201C2×Dic3—186no—
202C2×Dic3—186no—
203C2×Dic3—182no—
204C2×Dic3—186no—
205C2×C32—22no—
206C2×C32—32no—
207C2×C32—31no—
208C2×C32—31no—
209C2×C32—62no—
210C2×C32C2×SL(2,ℤ/4)11yescharacteristic
211C2×C32GL(2,ℤ/4)12yes—
212C2×C9—43no—
213C2×C9—46no—
214C2×C9—86no—
215C24—22no—
216C24—31no—
217C24—31no—
218C24—32no—
219C24—62no—
220C24C3⋊Dic912yescomplemented
221C24C2×C3⋊D911yescharacteristic
222C22×C4—272no—
223C22×C4—272no—
224C22⋊C4—542no—
225C22⋊C4—544no—
226C22⋊C4—542no—
227C22⋊C4—541no—
228C22⋊C4—542no—
229C22⋊C4—541no—
230C22×C3—22no—
231C22×C3—26no—
232C22×C3—33no—
233C22×C3—36no—
234C22×C3—31no—
235C22×C3—32no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—32no—
239C22×C3—32no—
240C22×C3—33no—
241C22×C3—33no—
242C22×C3—36no—
243C22×C3—36no—
244C22×C3—312no—
245C22×C3—34no—
246C22×C3—32no—
247C22×C3—36no—
248C22×C3—66no—
249C22×C3—62no—
250C22×C3—612no—
251C22×C3—66no—
252C22×C3—64no—
253C22×C3—62no—
254C22×C3—63no—
255C22×C3—61no—
256C22×C3—66no—
257C22×C3—62no—
258C22×C3—612no—
259C22×C3—64no—
260C22×C3—62no—
261C22×C3—66no—
262C22×C3#367613yescomplemented
263C22×C3(C22⋊C9)⋊C416yescomplemented
264C22×C3#375711yescharacteristic
265C22×C3C2×(C22⋊C9)⋊C213yes—
266C22×C3C3⋊SL(2,ℤ/4)12yes—
267C22×C3C2×C3⋊S411yescharacteristic
268Dic3—366no—
269Dic3—362no—
270Dic3—362no—
271Dic3—366no—
272C32#543211yescharacteristic
273C9—43no—
274C23—22no—
275C23—31no—
276C23—32no—
277C23—32no—
278C23—31no—
279C23—34no—
280C23—32no—
281C23—32no—
282C23—31no—
283C23—62no—
284C23—62no—
285C23—62no—
286C23—61no—
287C23—64no—
288C23—62no—
289C23—64no—
290C23C2×C3⋊Dic911yescharacteristic
291C23#623312yes—
292C23C3⋊((C22⋊C9)⋊C2)11yescharacteristic
293C2×C4—542no—
294C2×C4—542no—
295C2×C4—542no—
296C2×C4—542no—
297C2×C4—542no—
298C2×C4—542no—
299C2×C3—26no—
300C2×C3—22no—
301C2×C3—36no—
302C2×C3—33no—
303C2×C3—33no—
304C2×C3—32no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—66no—
308C2×C3—62no—
309C2×C3C2×(C22⋊C9)⋊C413yes—
310C2×C3#6369416yes—
311C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
312C2×C3#6426912yes—
313C22—22no—
314C22—31no—
315C22—32no—
316C22—31no—
317C22—31no—
318C22—32no—
319C22—32no—
320C22—34no—
321C22—32no—
322C22—62no—
323C22—64no—
324C22—62no—
325C22—61no—
326C22—62no—
327C22—64no—
328C22—62no—
329C22#8855111yescharacteristic, complemented
330C22C3⋊((C22⋊C9)⋊C4)12yescomplemented
331C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
332C4—1082no—
333C4—1082no—
334C3#1058953413yescomplemented
335C3#1059366611yescharacteristic
336C2—22no—
337C2—32no—
338C2—31no—
339C2—31no—
340C2—62no—
341C2C2×C3⋊((C22⋊C9)⋊C4)11yescharacteristic
342C2#1172239612yes—
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutatorC2×C3×C22⋊C9C2×C4
FrattiniC22×C3C2×C3⋊S4
Fitting, prime radicalC25×C32D3
socleC24×C32C2×D3
cosocleC22×C3×C22⋊C9C22
hypercenterC23C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9C22⋊C4
LCSG ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊((C22⋊C9)⋊C4)C2semidirect
6. C22×C3×C22⋊C9C4semidirect
34. C22×C22⋊C9Dic3semidirect
47. C3×C22⋊C9C22⋊C4semidirect
103. C24×C3Dic9semidirect
147. C22⋊C9#217semidirect
220. C24C3⋊Dic9semidirect
262. C22×C3#3676semidirect
263. C22×C3(C22⋊C9)⋊C4semidirect
329. C22#88551semidirect
330. C22C3⋊((C22⋊C9)⋊C4)semidirect
334. C3#10589534semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C2×C3⋊((C22⋊C9)⋊C4)2
#105929463
#105895343
#885514