Finite Group Databasebeta

← C2×#11721000 #13150233

Subgroups — C2×#11721000 #13150233

Order 1728.

1490 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C212yesmaximal
3C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
5C2×#88442C212yesmaximal, complemented
6#11721000C218yesmaximal, direct product factor
7C2×#63900—31nomaximal
8C2×C4⋊C4×D9—31nomaximal
9C23×C3⋊D9C2211yescharacteristic
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C2×C4×C3⋊D9C418yescomplemented
13C2×C4×C3⋊D9C2214yes—
14C22×C9×Dic3C2212yes—
15C22×C3×Dic9C2212yes—
16C2×(C3⋊D9)⋊C4—28no—
17C2×(C3⋊D9)⋊C4C2218yes—
18C9×C4⋊Dic3C2214yescomplemented
19C3×C4⋊Dic9C2214yescomplemented
20#88442C2218yescomplemented
21C22×C4×C3⋊D3—31no—
22C22×#624—32no—
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4⋊Dic3D311yescharacteristic
25C2×#3724—32no—
26#63900—38no—
27C22×C4×D9—31no—
28C22×C4×D9—32no—
29C22×C4×D9—61no—
30C2×C9×C4⋊C4—31no—
31C2×C4⋊Dic9D311yescharacteristic, complemented
32C2×Dic9⋊C4—32no—
33C4⋊C4×D9—38no—
34C23×C3×C9C2311yescharacteristic
35C22×C3⋊D9C2×C414yes—
36C22×C3⋊D9D414yes—
37C22×C3⋊D9Q814yes—
38C22×C3⋊D9C2312yes—
39C2×C3×C4×C9C2×C414yescomplemented
40C2×C3×C4×C9C2312yes—
41C2×C3⋊Dic9C2×C414yescomplemented
42C2×C3⋊Dic9C2312yes—
43C4×C3⋊D9C2×C4116yescomplemented
44C2×C9×Dic3—24no—
45C2×C9×Dic3C2314yes—
46C2×C3×Dic9—24no—
47C2×C3×Dic9C2314yes—
48(C3⋊D9)⋊C4—216no—
49C2×D3×C4⋊C4—91no—
50C2×D3×C4⋊C4—91no—
51C23×C3⋊D3—31no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C3⋊Dic3—31no—
54C2×C4×C3⋊D3—38no—
55C2×C4×C3⋊D3—34no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3C2×D312yes—
58C2×#624—38no—
59C2×#624—68no—
60C23×D9—31no—
61C23×D9—61no—
62C3×C4⋊Dic3—34no—
63C3×C4⋊Dic3C2×D314yes—
64#3724—38no—
65C22×C4×C9—21no—
66C22×C4×C9—32no—
67C22×C4×C9C2×D311yescharacteristic
68C22×Dic9—31no—
69C22×Dic9—61no—
70C22×Dic9C2×D312yes—
71C2×C4×D9—38no—
72C2×C4×D9—34no—
73C2×C4×D9—38no—
74C2×C4×D9—68no—
75C2×C4×D9—68no—
76C2×C4×D9—64no—
77C9×C4⋊C4—34no—
78C4⋊Dic9C2×D314yescomplemented
79Dic9⋊C4—38no—
80C22×C3×C9C22×C412yes—
81C22×C3×C9C2×D412yes—
82C22×C3×C9C2×Q812yes—
83C22×C3×C9C2411yescharacteristic
84C2×C3⋊D9C4⋊C4116yescomplemented
85C2×C3⋊D9C22×C414yes—
86C2×C3⋊D9C2×D414yes—
87C2×C3⋊D9C2×Q814yes—
88C3×C4×C9C22×C414yescomplemented
89C3⋊Dic9C22×C414yescomplemented
90C9×Dic3—24no—
91C3×Dic9—24no—
92C22×C4×D3—91no—
93C22×C4×D3—91no—
94C22×C4×D3—92no—
95C22×C4×D3—92no—
96C22×C4×D3—181no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊C4—91no—
99C2×C4⋊Dic3—31no—
100C2×C4⋊Dic3D911yescharacteristic, complemented
101C2×Dic3⋊C4—92no—
102C2×Dic3⋊C4—92no—
103D3×C4⋊C4—98no—
104D3×C4⋊C4—98no—
105C23×C32C22×D311yescharacteristic
106C22×C3⋊D3—34no—
107C22×C3⋊D3—34no—
108C22×C3⋊D3—34no—
109C22×C3⋊D3—32no—
110C2×C32×C4C4×D314yes—
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—32no—
114C4×C3⋊D3—316no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3C22×D314yes—
119#624—616no—
120C23×C9—21no—
121C23×C9C22×D311yescharacteristic
122C22×D9—34no—
123C22×D9—34no—
124C22×D9—34no—
125C22×D9—32no—
126C22×D9—64no—
127C22×D9—62no—
128C22×D9—64no—
129C22×D9—64no—
130C2×C4×C9—24no—
131C2×C4×C9—22no—
132C2×C4×C9—34no—
133C2×C4×C9—64no—
134C2×C4×C9C4×D314yescomplemented
135C2×C4×C9C22×D312yes—
136C2×Dic9—24no—
137C2×Dic9—34no—
138C2×Dic9—32no—
139C2×Dic9—64no—
140C2×Dic9—62no—
141C2×Dic9C22×D314yes—
142C4×D9—316no—
143C4×D9—616no—
144C4×D9—616no—
145C22×C4⋊C4—271noSylow
146C2×C3×C9C2×C4⋊C414yescomplemented
147C2×C3×C9C23×C411yescharacteristic
148C2×C3×C9C22×D411yescharacteristic
149C2×C3×C9C22×Q811yescharacteristic
150C3⋊D9C2×C4⋊C418yescomplemented
151C23×D3—91no—
152C23×D3—91no—
153C23×D3—181no—
154C22×C3×C4—21no—
155C22×C3×C4—32no—
156C22×C3×C4—92no—
157C22×C3×C4C2×D911yescharacteristic
158C22×C3×C4D3211yescharacteristic
159C22×Dic3—32no—
160C22×Dic3—91no—
161C22×Dic3—91no—
162C22×Dic3—181no—
163C22×Dic3C2×D912yes—
164C2×C4×D3—98no—
165C2×C4×D3—94no—
166C2×C4×D3—98no—
167C2×C4×D3—94no—
168C2×C4×D3—98no—
169C2×C4×D3—98no—
170C2×C4×D3—188no—
171C2×C4×D3—184no—
172C2×C4×D3—188no—
173C2×C4×D3—188no—
174C3×C4⋊C4—34no—
175C3×C4⋊C4—94no—
176C4⋊Dic3—34no—
177C4⋊Dic3C2×D914yescomplemented
178Dic3⋊C4—98no—
179Dic3⋊C4—98no—
180C22×C32C2×C4×D312yes—
181C22×C32D3×D412yes—
182C22×C32D3×Q812yes—
183C22×C32C23×D311yescharacteristic
184C2×C3⋊D3—316no—
185C2×C3⋊D3—34no—
186C2×C3⋊D3—34no—
187C2×C3⋊D3—34no—
188C32×C4C2×C4×D314yes—
189C3⋊Dic3—34no—
190C3×Dic3—24no—
191C3×Dic3—64no—
192C22×C9—21no—
193C22×C9—22no—
194C22×C9—22no—
195C22×C9—22no—
196C22×C9C2×C4×D312yes—
197C22×C9D3×D412yes—
198C22×C9D3×Q812yes—
199C22×C9C23×D311yescharacteristic
200C2×D9—316no—
201C2×D9—34no—
202C2×D9—34no—
203C2×D9—34no—
204C2×D9—616no—
205C2×D9—64no—
206C2×D9—64no—
207C2×D9—64no—
208C4×C9—24no—
209C4×C9—64no—
210C4×C9C2×C4×D314yescomplemented
211Dic9—24no—
212Dic9—34no—
213Dic9—64no—
214C23×C4—272no—
215C23×C4—271no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—91no—
218C2×C4⋊C4—278no—
219C2×C4⋊C4—272no—
220C3×C9C22×C4⋊C411yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3C22×D911yescharacteristic
223C23×C3C2×D3211yescharacteristic
224C22×D3—92no—
225C22×D3—94no—
226C22×D3—94no—
227C22×D3—94no—
228C22×D3—92no—
229C22×D3—94no—
230C22×D3—94no—
231C22×D3—94no—
232C22×D3—182no—
233C22×D3—184no—
234C22×D3—184no—
235C22×D3—184no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—34no—
239C2×C3×C4—64no—
240C2×C3×C4—94no—
241C2×C3×C4—184no—
242C2×C3×C4C4×D914yescomplemented
243C2×C3×C4C22×D912yes—
244C2×C3×C4#62414yes—
245C2×C3×C4C2×D3212yes—
246C2×Dic3—24no—
247C2×Dic3—34no—
248C2×Dic3—64no—
249C2×Dic3—92no—
250C2×Dic3—94no—
251C2×Dic3—92no—
252C2×Dic3—94no—
253C2×Dic3—182no—
254C2×Dic3—184no—
255C2×Dic3C22×D914yes—
256C4×D3—916no—
257C4×D3—916no—
258C4×D3—1816no—
259C4×D3—1816no—
260C4×D3—1816no—
261C2×C32D3×C4⋊C414yes—
262C2×C32C22×C4×D311yescharacteristic
263C2×C32C2×D3×D411yescharacteristic
264C2×C32C2×D3×Q811yescharacteristic
265C3⋊D3—38no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9—24no—
270C2×C9D3×C4⋊C414yescomplemented
271C2×C9C22×C4×D311yescharacteristic
272C2×C9C2×D3×D411yescharacteristic
273C2×C9C2×D3×Q811yescharacteristic
274D9—38no—
275D9—68no—
276C24—271no—
277C22×C4—32no—
278C22×C4—92no—
279C22×C4—278no—
280C22×C4—271no—
281C22×C4—278no—
282C22×C4—274no—
283C22×C4—548no—
284C22×C4D3×D911yescharacteristic
285C4⋊C4—34no—
286C4⋊C4—94no—
287C4⋊C4—278no—
288C22×C3—22no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3C2×C4×D912yes—
293C22×C3D4×D912yes—
294C22×C3Q8×D912yes—
295C22×C3C23×D911yescharacteristic
296C22×C3#379712yes—
297C22×C3#380212yes—
298C22×C3C2×#62412yes—
299C22×C3C22×D3211yescharacteristic
300C2×D3—94no—
301C2×D3—94no—
302C2×D3—94no—
303C2×D3—916no—
304C2×D3—94no—
305C2×D3—94no—
306C2×D3—916no—
307C2×D3—94no—
308C2×D3—184no—
309C2×D3—1816no—
310C2×D3—184no—
311C2×D3—184no—
312C3×C4—24no—
313C3×C4—64no—
314C3×C4—184no—
315C3×C4C2×C4×D914yescomplemented
316C3×C4C2×#62414yes—
317Dic3—24no—
318Dic3—64no—
319Dic3—94no—
320Dic3—94no—
321Dic3—184no—
322C32C2×D3×C4⋊C411yescharacteristic
323C9—21no—
324C9C2×D3×C4⋊C411yescharacteristic, complemented
325C23—274no—
326C23—272no—
327C23—274no—
328C23—274no—
329C23C2×D3×D911yescharacteristic
330C2×C4—34no—
331C2×C4—64no—
332C2×C4—94no—
333C2×C4—184no—
334C2×C4—272no—
335C2×C4—274no—
336C2×C4—2716no—
337C2×C4—5416no—
338C2×C4(C3⋊D9)⋊C414yescomplemented
339C2×C4C2×D3×D912yes—
340C2×C3—21no—
341C2×C3—24no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3C4⋊C4×D914yescomplemented
345C2×C3C22×C4×D911yescharacteristic
346C2×C3C2×D4×D911yescharacteristic
347C2×C3C2×Q8×D911yescharacteristic
348C2×C3#6390014yes—
349C2×C3C2×#379711yescharacteristic
350C2×C3C2×#380211yescharacteristic
351C2×C3C22×#62411yescharacteristic
352D3—98no—
353D3—98no—
354D3—188no—
355C22—274no—
356C22—2716no—
357C22—274no—
358C22—274no—
359C22(C3⋊D9)⋊Q812yes—
360C22#8864612yes—
361C22C2×(C3⋊D9)⋊C412yes—
362C22C22×D3×D911yescharacteristic
363C4—64no—
364C4—184no—
365C4—274no—
366C4C2×(C3⋊D9)⋊C414yescomplemented
367C3—21no—
368C3C2×C4⋊C4×D911yescharacteristic, complemented
369C3C2×#6390011yescharacteristic
370C2—278no—
371C2#1172100014yesdirect product factor
372C2C2×(C3⋊D9)⋊Q811yescharacteristic
373C2C2×#8864611yescharacteristic
374C2C22×(C3⋊D9)⋊C411yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C22×C4⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×C9×C4⋊Dic3C2semidirect
4. C2×C3×C4⋊Dic9C2semidirect
5. C2×#88442C2semidirect
6. #11721000C2direct (N × H)
12. C2×C4×C3⋊D9C4semidirect
18. C9×C4⋊Dic3C22semidirect
19. C3×C4⋊Dic9C22semidirect
20. #88442C22semidirect
31. C2×C4⋊Dic9D3semidirect
39. C2×C3×C4×C9C2×C4semidirect
41. C2×C3⋊Dic9C2×C4semidirect
43. C4×C3⋊D9C2×C4semidirect
78. C4⋊Dic9C2×D3semidirect
84. C2×C3⋊D9C4⋊C4semidirect
88. C3×C4×C9C22×C4semidirect
89. C3⋊Dic9C22×C4semidirect
100. C2×C4⋊Dic3D9semidirect
134. C2×C4×C9C4×D3semidirect
146. C2×C3×C9C2×C4⋊C4semidirect
150. C3⋊D9C2×C4⋊C4semidirect
177. C4⋊Dic3C2×D9semidirect
210. C4×C9C2×C4×D3semidirect
220. C3×C9C22×C4⋊C4semidirect
242. C2×C3×C4C4×D9semidirect
270. C2×C9D3×C4⋊C4semidirect
315. C3×C4C2×C4×D9semidirect
324. C9C2×D3×C4⋊C4semidirect
338. C2×C4(C3⋊D9)⋊C4semidirect
344. C2×C3C4⋊C4×D9semidirect
366. C4C2×(C3⋊D9)⋊C4semidirect
368. C3C2×C4⋊C4×D9semidirect
371. C2#11721000direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C22×(C3⋊D9)⋊C42
C2×C9×C4⋊Dic32
C2×C3×C4⋊Dic92
C2×#884422
#117210002
C2×#639003
C2×C4⋊C4×D93