Finite Group Databasebeta

← C2×#11720899 #13150171

Subgroups — C2×#11720899 #13150171

Order 1728.

1122 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
2C2×#88536C211yesmaximal, characteristic
3C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic
6C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11720899C218yesmaximal, direct product factor
9C2×#63836—31nomaximal
10C2×#63454—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C418yescomplemented
19C2×(C3⋊D9)⋊C4C2214yes—
20#88536C2214yes—
21C9×Dic3⋊C4C2214yescomplemented
22C3×C4×Dic9C2214yescomplemented
23#88445C2214yes—
24Dic3⋊Dic9C2214yescomplemented
25Dic3×Dic9C2214yescomplemented
26C22×#624—31no—
27C2×#3752—31no—
28C2×C3×Dic3⋊C4D311yescharacteristic
29C2×C3×C4×Dic3—31no—
30C2×Dic3⋊Dic3—31no—
31C2×#3722—31no—
32C2×Dic32—31no—
33#63836—38no—
34C22×C4×D9—31no—
35C2×C9×C4⋊C4—31no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—31no—
38C2×(C2×D9)⋊C4—61no—
39C2×C4⋊Dic9—31no—
40C2×C4×Dic9—31no—
41C2×C4×Dic9D311yescharacteristic, complemented
42#63454—38no—
43C23×C3×C9C2311yescharacteristic
44C22×C3⋊D9—22no—
45C22×C3⋊D9—22no—
46C22×C3⋊D9C2×C414yes—
47C22×C3⋊D9C2312yes—
48C2×C3×C4×C9—22no—
49C2×C3×C4×C9C2312yes—
50C2×C3⋊Dic9—22no—
51C2×C3⋊Dic9C2312yes—
52C2×C9×Dic3—22no—
53C2×C9×Dic3C2×C414yes—
54C2×C9×Dic3C2312yes—
55C2×C9×Dic3C2312yes—
56C2×C3×Dic9—22no—
57C2×C3×Dic9C2×C414yes—
58C2×C3×Dic9C2312yes—
59C2×C3×Dic9C2312yes—
60(C3⋊D9)⋊C4C2×C4116yescomplemented
61C2×#892—91no—
62C2×#872—91no—
63C23×C3⋊D3—31no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C3⋊Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3C2×D311yescharacteristic
69C22×C3×Dic3C2×D311yescharacteristic
70C2×#624—38no—
71C2×#624—34no—
72C23×D9—31no—
73C23×D9—61no—
74#3752—34no—
75C3×Dic3⋊C4C2×D314yes—
76C3×C4×Dic3—34no—
77Dic3⋊Dic3—34no—
78#3722—34no—
79Dic32—34no—
80C22×C4×C9—21no—
81C22×C4×C9—31no—
82C22×C4×C9—31no—
83C22×C4×C9C2×D311yescharacteristic
84C22×Dic9—31no—
85C22×Dic9—61no—
86C22×Dic9C2×D311yescharacteristic
87C22×Dic9C2×D311yescharacteristic
88C2×C4×D9—38no—
89C2×C4×D9—34no—
90C9×C4⋊C4—34no—
91(C2×D9)⋊C4—34no—
92(C2×D9)⋊C4—34no—
93(C2×D9)⋊C4—64no—
94C4⋊Dic9—34no—
95C4×Dic9—34no—
96C4×Dic9C2×D314yescomplemented
97C22×C3×C9C22×C412yes—
98C22×C3×C9#4112yes—
99C22×C3×C9#4112yes—
100C22×C3×C9C2411yescharacteristic
101C2×C3⋊D9—28no—
102C2×C3⋊D9—22no—
103C2×C3⋊D9—22no—
104C2×C3⋊D9C22×C414yes—
105C3×C4×C9—22no—
106C3⋊Dic9—22no—
107C9×Dic3—22no—
108C9×Dic3C22×C414yes—
109C3×Dic9—22no—
110C3×Dic9C22×C414yes—
111C22×C4×D3—91no—
112C22×C4×D3—91no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C42—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—181no—
120C2×C4⋊Dic3—91no—
121C2×Dic3⋊C4—91no—
122C2×Dic3⋊C4D911yescharacteristic, complemented
123C2×C4×Dic3—31no—
124C2×C4×Dic3—91no—
125C2×C4×Dic3—91no—
126#892—98no—
127#872—98no—
128C23×C32C22×D311yescharacteristic
129C22×C3⋊D3—34no—
130C22×C3⋊D3—32no—
131C22×C3⋊D3—62no—
132C22×C3⋊D3—62no—
133C2×C32×C4—22no—
134C2×C32×C4C22×D312yes—
135C2×C3⋊Dic3—32no—
136C2×C3⋊Dic3—62no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3C4×D314yes—
143C2×C3×Dic3C22×D312yes—
144C2×C3×Dic3C22×D312yes—
145#624—316no—
146C23×C9—21no—
147C23×C9C22×D311yescharacteristic
148C22×D9—34no—
149C22×D9—32no—
150C22×D9—62no—
151C22×D9—64no—
152C22×D9—62no—
153C22×D9—62no—
154C22×D9—122no—
155C22×D9—122no—
156C2×C4×C9—22no—
157C2×C4×C9—22no—
158C2×C4×C9—34no—
159C2×C4×C9—32no—
160C2×C4×C9—32no—
161C2×C4×C9—42no—
162C2×C4×C9—62no—
163C2×C4×C9C22×D312yes—
164C2×Dic9—22no—
165C2×Dic9—32no—
166C2×Dic9—62no—
167C2×Dic9—62no—
168C2×Dic9—122no—
169C2×Dic9C4×D314yes—
170C2×Dic9C22×D312yes—
171C2×Dic9C22×D312yes—
172C4×D9—316no—
173C2×#117—271noSylow
174C2×C3×C9#11714yescomplemented
175C2×C3×C9C23×C411yescharacteristic
176C2×C3×C9C2×#4111yescharacteristic
177C2×C3×C9C2×#4111yescharacteristic
178C3⋊D9—24no—
179C23×D3—91no—
180C23×D3—91no—
181C23×D3—181no—
182C22×C3×C4—21no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—91no—
186C22×C3×C4—91no—
187C22×C3×C4C2×D911yescharacteristic
188C22×C3×C4D3211yescharacteristic
189C22×Dic3—31no—
190C22×Dic3—31no—
191C22×Dic3—91no—
192C22×Dic3—91no—
193C22×Dic3—181no—
194C22×Dic3C2×D911yescharacteristic
195C22×Dic3C2×D911yescharacteristic
196C2×C4×D3—94no—
197C2×C4×D3—98no—
198C2×C4×D3—98no—
199C2×C4×D3—94no—
200C3×C4⋊C4—34no—
201C3×C42—94no—
202(C2×D3)⋊C4—94no—
203(C2×D3)⋊C4—94no—
204(C2×D3)⋊C4—94no—
205(C2×D3)⋊C4—94no—
206(C2×D3)⋊C4—184no—
207C4⋊Dic3—94no—
208Dic3⋊C4—94no—
209Dic3⋊C4C2×D914yescomplemented
210C4×Dic3—34no—
211C4×Dic3—94no—
212C4×Dic3—94no—
213C22×C32C2×C4×D312yes—
214C22×C32#23712yes—
215C22×C32#23912yes—
216C22×C32C23×D311yescharacteristic
217C2×C3⋊D3—34no—
218C2×C3⋊D3—68no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—62no—
221C32×C4—22no—
222C3⋊Dic3—62no—
223C3×Dic3—22no—
224C3×Dic3—34no—
225C3×Dic3—62no—
226C3×Dic3C2×C4×D314yes—
227C22×C9—21no—
228C22×C9—22no—
229C22×C9—22no—
230C22×C9—22no—
231C22×C9C2×C4×D312yes—
232C22×C9#23512yes—
233C22×C9#23512yes—
234C22×C9C23×D311yescharacteristic
235C2×D9—34no—
236C2×D9—64no—
237C2×D9—68no—
238C2×D9—62no—
239C2×D9—62no—
240C2×D9—122no—
241C2×D9—122no—
242C2×D9—128no—
243C4×C9—22no—
244C4×C9—34no—
245C4×C9—42no—
246C4×C9—62no—
247Dic9—22no—
248Dic9—62no—
249Dic9—122no—
250Dic9C2×C4×D314yes—
251C23×C4—271no—
252#117—278no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—271no—
255C2×C22⋊C4—271no—
256C2×C22⋊C4—271no—
257C2×C42—91no—
258C2×C42—271no—
259C3×C9C2×#11711yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3C22×D911yescharacteristic
262C23×C3C2×D3211yescharacteristic
263C22×D3—92no—
264C22×D3—94no—
265C22×D3—92no—
266C22×D3—94no—
267C22×D3—182no—
268C22×D3—184no—
269C22×D3—182no—
270C22×D3—182no—
271C22×D3—182no—
272C22×D3—182no—
273C22×D3—362no—
274C22×D3—362no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—32no—
279C2×C3×C4—34no—
280C2×C3×C4—32no—
281C2×C3×C4—42no—
282C2×C3×C4—62no—
283C2×C3×C4—94no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—182no—
287C2×C3×C4C22×D912yes—
288C2×C3×C4C2×D3212yes—
289C2×Dic3—22no—
290C2×Dic3—32no—
291C2×Dic3—34no—
292C2×Dic3—32no—
293C2×Dic3—62no—
294C2×Dic3—92no—
295C2×Dic3—92no—
296C2×Dic3—182no—
297C2×Dic3—182no—
298C2×Dic3—182no—
299C2×Dic3—362no—
300C2×Dic3C4×D914yes—
301C2×Dic3C22×D912yes—
302C2×Dic3C22×D912yes—
303C4×D3—916no—
304C4×D3—916no—
305C2×C32#89214yes—
306C2×C32C22×C4×D311yescharacteristic
307C2×C32C2×#23711yescharacteristic
308C2×C32C2×#23911yescharacteristic
309C3⋊D3—64no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9—21no—
313C2×C9—24no—
314C2×C9#87214yescomplemented
315C2×C9C22×C4×D311yescharacteristic
316C2×C9C2×#23511yescharacteristic
317C2×C9C2×#23511yescharacteristic
318D9—64no—
319D9—124no—
320C24—271no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4—91no—
324C22×C4—91no—
325C22×C4—274no—
326C22×C4—278no—
327C22×C4—271no—
328C22×C4D3×D911yescharacteristic
329C4⋊C4—34no—
330C4⋊C4—274no—
331C22⋊C4—274no—
332C22⋊C4—274no—
333C42—94no—
334C42—274no—
335C22×C3—22no—
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3C2×C4×D912yes—
340C22×C3#369912yes—
341C22×C3#370112yes—
342C22×C3C23×D911yescharacteristic
343C22×C3#379912yes—
344C22×C3C4×D3212yes—
345C22×C3#380412yes—
346C22×C3C22×D3211yescharacteristic
347C2×D3—94no—
348C2×D3—94no—
349C2×D3—184no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—188no—
353C2×D3—182no—
354C2×D3—188no—
355C2×D3—182no—
356C2×D3—362no—
357C2×D3—368no—
358C2×D3—362no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—34no—
362C3×C4—42no—
363C3×C4—62no—
364C3×C4—94no—
365C3×C4—182no—
366Dic3—22no—
367Dic3—34no—
368Dic3—62no—
369Dic3—182no—
370Dic3—182no—
371Dic3—362no—
372Dic3C2×C4×D914yes—
373C32C2×#89211yescharacteristic
374C9—21no—
375C9C2×#87211yescharacteristic, complemented
376C23—272no—
377C23—274no—
378C23—542no—
379C23—542no—
380C23C2×D3×D911yescharacteristic
381C2×C4—22no—
382C2×C4—32no—
383C2×C4—34no—
384C2×C4—32no—
385C2×C4—62no—
386C2×C4—92no—
387C2×C4—94no—
388C2×C4—92no—
389C2×C4—182no—
390C2×C4—2716no—
391C2×C4—272no—
392C2×C4—542no—
393C2×C4C2×D3×D912yes—
394C2×C3—21no—
395C2×C3—24no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3#6345414yescomplemented
399C2×C3C22×C4×D911yescharacteristic
400C2×C3C2×#369911yescharacteristic
401C2×C3C2×#370111yescharacteristic
402C2×C3#6383614yes—
403C2×C3C2×#379911yescharacteristic
404C2×C3C2×C4×D3211yescharacteristic
405C2×C3C2×#380411yescharacteristic
406D3—184no—
407D3—184no—
408D3—364no—
409C22—274no—
410C22—542no—
411C22—548no—
412C22—542no—
413C22#8863512yes—
414C22C4×D3×D912yes—
415C22#8866212yes—
416C22C22×D3×D911yescharacteristic
417C4—22no—
418C4—34no—
419C4—62no—
420C4—94no—
421C4—182no—
422C4—542no—
423C3—21no—
424C3C2×#6345411yescharacteristic, complemented
425C3C2×#6383611yescharacteristic
426C2—544no—
427C2#1172089914yesdirect product factor
428C2C2×#8863511yescharacteristic
429C2C2×C4×D3×D911yescharacteristic
430C2C2×#8866211yescharacteristic
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×C9C2×#117
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×Dic3⋊C4C2semidirect
4. C2×C3×C4×Dic9C2semidirect
6. C2×Dic3⋊Dic9C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11720899C2direct (N × H)
18. C2×(C3⋊D9)⋊C4C4semidirect
21. C9×Dic3⋊C4C22semidirect
22. C3×C4×Dic9C22semidirect
24. Dic3⋊Dic9C22semidirect
25. Dic3×Dic9C22semidirect
41. C2×C4×Dic9D3semidirect
60. (C3⋊D9)⋊C4C2×C4semidirect
96. C4×Dic9C2×D3semidirect
122. C2×Dic3⋊C4D9semidirect
174. C2×C3×C9#117semidirect
209. Dic3⋊C4C2×D9semidirect
259. C3×C9C2×#117semidirect
314. C2×C9#872semidirect
375. C9C2×#872semidirect
398. C2×C3#63454semidirect
424. C3C2×#63454semidirect
427. C2#11720899direct (N × H)

Maximal subgroups

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

subgroupindex
C22×(C3⋊D9)⋊C42
C2×#885362
C2×C9×Dic3⋊C42
C2×C3×C4×Dic92
C2×#884452
C2×Dic3⋊Dic92
C2×Dic3×Dic92
#117208992
C2×#638363
C2×#634543