Finite Group Databasebeta

← C2×#11720994 #13150211

Subgroups — C2×#11720994 #13150211

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3C2×#88536C211yesmaximal, characteristic
4C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
5C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
6C2×#88445C211yesmaximal, characteristic
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720994C218yesmaximal, direct product factor
9C2×#63895—31nomaximal
10C2×#63458—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)⋊C4—24no—
19C2×(C3⋊D9)⋊C4C2214yes—
20C2×#6189—24no—
21C2×#6189C2214yescomplemented
22#88536C2214yes—
23C9×C4⋊Dic3C2214yescomplemented
24C3×Dic9⋊C4C2214yescomplemented
25#88445C2214yes—
26Dic9⋊Dic3C2214yescomplemented
27C22×#627—31no—
28C22×#624—31no—
29C2×#3752—31no—
30C2×C3×C4⋊Dic3D311yescharacteristic
31C2×C3×Dic3⋊C4—31no—
32C2×Dic3⋊Dic3—31no—
33C2×#3722—31no—
34#63895—38no—
35C22×C4×D9—31no—
36C22×Dic18—31no—
37C2×C9×C4⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4—61no—
41C2×Dic9⋊C4—31no—
42C2×Dic9⋊C4D311yescharacteristic, complemented
43#63458—38no—
44C23×C3×C9C2311yescharacteristic
45C22×C3⋊D9—22no—
46C22×C3⋊D9—22no—
47C22×C3⋊D9Q814yescomplemented
48C22×C3⋊D9C2312yes—
49C2×C3×C4×C9—22no—
50C2×C3×C4×C9C2312yes—
51C2×C3⋊Dic9—22no—
52C2×C3⋊Dic9C2312yes—
53C2×C9×Dic3—22no—
54C2×C9×Dic3—22no—
55C2×C9×Dic3C2312yes—
56C2×C9×Dic3C2312yes—
57C2×C3×Dic9—22no—
58C2×C3×Dic9D414yes—
59C2×C3×Dic9C2312yes—
60C2×C3×Dic9C2312yes—
61(C3⋊D9)⋊C4—28no—
62#6189—28no—
63C2×#897—91no—
64C2×#896—91no—
65C23×C3⋊D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3C2×D311yescharacteristic
71C22×C3×Dic3C2×D311yescharacteristic
72C2×#627—34no—
73C2×#627—64no—
74C2×#624—34no—
75C2×#624—64no—
76C23×D9—31no—
77C23×D9—61no—
78#3752—34no—
79C3×C4⋊Dic3C2×D314yes—
80C3×Dic3⋊C4—34no—
81Dic3⋊Dic3—34no—
82#3722—34no—
83C22×C4×C9—21no—
84C22×C4×C9—31no—
85C22×C4×C9—31no—
86C22×C4×C9C2×D311yescharacteristic
87C22×Dic9—31no—
88C22×Dic9—61no—
89C22×Dic9C2×D311yescharacteristic
90C22×Dic9C2×D311yescharacteristic
91C2×C4×D9—34no—
92C2×C4×D9—64no—
93C2×Dic18—34no—
94C2×Dic18—64no—
95C9×C4⋊C4—34no—
96(C2×D9)⋊C4—34no—
97(C2×D9)⋊C4—34no—
98(C2×D9)⋊C4—64no—
99Dic9⋊C4—34no—
100Dic9⋊C4C2×D314yescomplemented
101C22×C3×C9C2×D412yes—
102C22×C3×C9C2×Q812yes—
103C22×C3×C9#4112yes—
104C22×C3×C9C2411yescharacteristic
105C2×C3⋊D9—28no—
106C2×C3⋊D9—22no—
107C2×C3⋊D9—22no—
108C2×C3⋊D9C2×Q814yescomplemented
109C3×C4×C9—22no—
110C3⋊Dic9—22no—
111C9×Dic3—22no—
112C9×Dic3—22no—
113C3×Dic9—22no—
114C3×Dic9C2×D414yes—
115C22×C4×D3—91no—
116C22×C4×D3—91no—
117C22×Dic6—91no—
118C22×Dic6—91no—
119C2×C3×C4⋊C4—31no—
120C2×C3×C4⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—181no—
126C2×C4⋊Dic3—91no—
127C2×C4⋊Dic3D911yescharacteristic, complemented
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—91no—
130#897—98no—
131#896—98no—
132C23×C32C22×D311yescharacteristic
133C22×C3⋊D3—34no—
134C22×C3⋊D3—32no—
135C22×C3⋊D3—62no—
136C22×C3⋊D3—62no—
137C2×C32×C4—22no—
138C2×C32×C4C22×D312yes—
139C2×C3⋊Dic3—32no—
140C2×C3⋊Dic3—62no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3C22×D312yes—
148C2×C3×Dic3C22×D312yes—
149#627—68no—
150#624—68no—
151C23×C9—21no—
152C23×C9C22×D311yescharacteristic
153C22×D9—34no—
154C22×D9—32no—
155C22×D9—62no—
156C22×D9—64no—
157C22×D9—62no—
158C22×D9—62no—
159C22×D9—122no—
160C22×D9—122no—
161C2×C4×C9—22no—
162C2×C4×C9—22no—
163C2×C4×C9—32no—
164C2×C4×C9—32no—
165C2×C4×C9—42no—
166C2×C4×C9—62no—
167C2×C4×C9—62no—
168C2×C4×C9C22×D312yes—
169C2×Dic9—22no—
170C2×Dic9—32no—
171C2×Dic9—62no—
172C2×Dic9—62no—
173C2×Dic9—122no—
174C2×Dic9D1214yes—
175C2×Dic9C22×D312yes—
176C2×Dic9C22×D312yes—
177C4×D9—68no—
178Dic18—68no—
179C2×C22⋊Q8—271noSylow
180C2×C3×C9C22⋊Q814yescomplemented
181C2×C3×C9C22×D411yescharacteristic
182C2×C3×C9C22×Q811yescharacteristic
183C2×C3×C9C2×#4111yescharacteristic
184C3⋊D9—24no—
185C23×D3—91no—
186C23×D3—91no—
187C23×D3—181no—
188C22×C3×C4—21no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4C2×D911yescharacteristic
194C22×C3×C4D3211yescharacteristic
195C22×Dic3—31no—
196C22×Dic3—31no—
197C22×Dic3—91no—
198C22×Dic3—91no—
199C22×Dic3—181no—
200C22×Dic3C2×D911yescharacteristic
201C22×Dic3C2×D911yescharacteristic
202C2×C4×D3—94no—
203C2×C4×D3—94no—
204C2×C4×D3—184no—
205C2×C4×D3—184no—
206C2×Dic6—94no—
207C2×Dic6—94no—
208C2×Dic6—184no—
209C2×Dic6—184no—
210C3×C4⋊C4—34no—
211C3×C4⋊C4—94no—
212(C2×D3)⋊C4—94no—
213(C2×D3)⋊C4—94no—
214(C2×D3)⋊C4—94no—
215(C2×D3)⋊C4—94no—
216(C2×D3)⋊C4—184no—
217C4⋊Dic3—94no—
218C4⋊Dic3C2×D914yescomplemented
219Dic3⋊C4—34no—
220Dic3⋊C4—94no—
221C22×C32#23512yes—
222C22×C32D3×D412yes—
223C22×C32D3×Q812yes—
224C22×C32C23×D311yescharacteristic
225C2×C3⋊D3—34no—
226C2×C3⋊D3—68no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—62no—
229C32×C4—22no—
230C3⋊Dic3—62no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—34no—
234C3×Dic3—62no—
235C22×C9—21no—
236C22×C9—22no—
237C22×C9—22no—
238C22×C9—22no—
239C22×C9C2×D1212yes—
240C22×C9#23712yes—
241C22×C9D3×Q812yes—
242C22×C9C23×D311yescharacteristic
243C2×D9—34no—
244C2×D9—64no—
245C2×D9—68no—
246C2×D9—62no—
247C2×D9—62no—
248C2×D9—122no—
249C2×D9—122no—
250C2×D9—128no—
251C4×C9—22no—
252C4×C9—42no—
253C4×C9—62no—
254C4×C9—62no—
255Dic9—22no—
256Dic9—62no—
257Dic9—122no—
258Dic9C2×D1214yes—
259C22×Q8—271no—
260C23×C4—271no—
261C22⋊Q8—278no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—91no—
264C2×C4⋊C4—271no—
265C2×C22⋊C4—271no—
266C2×C22⋊C4—271no—
267C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
268C23×C3—21no—
269C23×C3C22×D911yescharacteristic
270C23×C3C2×D3211yescharacteristic
271C22×D3—92no—
272C22×D3—94no—
273C22×D3—92no—
274C22×D3—94no—
275C22×D3—182no—
276C22×D3—184no—
277C22×D3—182no—
278C22×D3—182no—
279C22×D3—182no—
280C22×D3—182no—
281C22×D3—362no—
282C22×D3—362no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—32no—
287C2×C3×C4—32no—
288C2×C3×C4—42no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—94no—
292C2×C3×C4—92no—
293C2×C3×C4—92no—
294C2×C3×C4—182no—
295C2×C3×C4C22×D912yes—
296C2×C3×C4C2×D3212yes—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—32no—
300C2×Dic3—34no—
301C2×Dic3—32no—
302C2×Dic3—62no—
303C2×Dic3—92no—
304C2×Dic3—92no—
305C2×Dic3—182no—
306C2×Dic3—182no—
307C2×Dic3—182no—
308C2×Dic3—362no—
309C2×Dic3C22×D912yes—
310C2×Dic3C22×D912yes—
311C4×D3—188no—
312C4×D3—188no—
313Dic6—188no—
314Dic6—188no—
315C2×C32#89614yes—
316C2×C32C2×#23511yescharacteristic
317C2×C32C2×D3×D411yescharacteristic
318C2×C32C2×D3×Q811yescharacteristic
319C3⋊D3—64no—
320C2×C9—21no—
321C2×C9—21no—
322C2×C9—21no—
323C2×C9—24no—
324C2×C9#89714yescomplemented
325C2×C9C22×D1211yescharacteristic
326C2×C9C2×#23711yescharacteristic
327C2×C9C2×D3×Q811yescharacteristic
328D9—64no—
329D9—124no—
330C24—271no—
331C2×Q8—274no—
332C2×Q8—544no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—91no—
336C22×C4—91no—
337C22×C4—274no—
338C22×C4—271no—
339C22×C4—544no—
340C22×C4D3×D911yescharacteristic
341C4⋊C4—34no—
342C4⋊C4—94no—
343C4⋊C4—274no—
344C22⋊C4—274no—
345C22⋊C4—274no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3#369712yes—
351C22×C3D4×D912yes—
352C22×C3Q8×D912yes—
353C22×C3C23×D911yescharacteristic
354C22×C3#379712yes—
355C22×C3D3×D1212yes—
356C22×C3#380412yes—
357C22×C3C22×D3211yescharacteristic
358C2×D3—94no—
359C2×D3—94no—
360C2×D3—184no—
361C2×D3—182no—
362C2×D3—182no—
363C2×D3—188no—
364C2×D3—182no—
365C2×D3—182no—
366C2×D3—188no—
367C2×D3—362no—
368C2×D3—368no—
369C2×D3—362no—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—42no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—94no—
376C3×C4—182no—
377Dic3—22no—
378Dic3—22no—
379Dic3—34no—
380Dic3—62no—
381Dic3—182no—
382Dic3—182no—
383Dic3—362no—
384C32C2×#89611yescharacteristic
385C9—21no—
386C9C2×#89711yescharacteristic, complemented
387C23—274no—
388C23—272no—
389C23—542no—
390C23—542no—
391C23C2×D3×D911yescharacteristic
392Q8—548no—
393C2×C4—22no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—92no—
399C2×C4—94no—
400C2×C4—92no—
401C2×C4—182no—
402C2×C4—272no—
403C2×C4—548no—
404C2×C4—542no—
405C2×C4C2×D3×D912yes—
406C2×C3—21no—
407C2×C3—24no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3#6345814yescomplemented
411C2×C3C2×#369711yescharacteristic
412C2×C3C2×D4×D911yescharacteristic
413C2×C3C2×Q8×D911yescharacteristic
414C2×C3#6389514yes—
415C2×C3C2×#379711yescharacteristic
416C2×C3C2×D3×D1211yescharacteristic
417C2×C3C2×#380411yescharacteristic
418D3—184no—
419D3—184no—
420D3—364no—
421C22—274no—
422C22—542no—
423C22—548no—
424C22—542no—
425C22(C3⋊D9)⋊Q812yes—
426C22D9×D1212yes—
427C22#8865812yes—
428C22C22×D3×D911yescharacteristic
429C4—22no—
430C4—62no—
431C4—62no—
432C4—94no—
433C4—182no—
434C4—542no—
435C3—21no—
436C3C2×#6345811yescharacteristic, complemented
437C3C2×#6389511yescharacteristic
438C2—544no—
439C2#1172099414yesdirect product factor
440C2C2×(C3⋊D9)⋊Q811yescharacteristic
441C2C2×D9×D1211yescharacteristic
442C2C2×#8865811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6189C2semidirect
4. C2×C9×C4⋊Dic3C2semidirect
5. C2×C3×Dic9⋊C4C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720994C2direct (N × H)
21. C2×#6189C22semidirect
23. C9×C4⋊Dic3C22semidirect
24. C3×Dic9⋊C4C22semidirect
26. Dic9⋊Dic3C22semidirect
42. C2×Dic9⋊C4D3semidirect
47. C22×C3⋊D9Q8semidirect
100. Dic9⋊C4C2×D3semidirect
108. C2×C3⋊D9C2×Q8semidirect
127. C2×C4⋊Dic3D9semidirect
180. C2×C3×C9C22⋊Q8semidirect
218. C4⋊Dic3C2×D9semidirect
267. C3×C9C2×C22⋊Q8semidirect
324. C2×C9#897semidirect
386. C9C2×#897semidirect
410. C2×C3#63458semidirect
436. C3C2×#63458semidirect
439. C2#11720994direct (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
C22×#61892
C2×#885362
C2×C9×C4⋊Dic32
C2×C3×Dic9⋊C42
C2×#884452
C2×Dic9⋊Dic32
#117209942
C2×#638953
C2×#634583