Finite Group Databasebeta

← C2×#11720921 #13150061

Subgroups — C2×#11720921 #13150061

Order 1728.

870 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic
2C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
4C2×#88445C211yesmaximal, characteristic
5C2×#88445C211yesmaximal, characteristic
6C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11720921C218yesmaximal, direct product factor
9C2×#63848—31nomaximal
10C2×#63460—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18#88536C2214yes—
19C9×Dic3⋊C4C2214yescomplemented
20C3×C4⋊Dic9C2214yescomplemented
21#88445C2214yes—
22#88445C2214yes—
23Dic9⋊Dic3C2214yescomplemented
24Dic3×Dic9C2214yescomplemented
25C2×#3752—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×Dic3⋊C4D311yescharacteristic
28C2×Dic3⋊Dic3—31no—
29C2×#3722—31no—
30C2×#3722—31no—
31C2×Dic32—31no—
32#63848—38no—
33C2×C9×C4⋊C4—31no—
34C2×(C2×D9)⋊C4—31no—
35C2×(C2×D9)⋊C4—31no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—61no—
38C2×C4⋊Dic9D311yescharacteristic, complemented
39C2×Dic9⋊C4—31no—
40C2×C4×Dic9—31no—
41#63460—38no—
42C23×C3×C9C2311yescharacteristic
43C22×C3⋊D9—22no—
44C22×C3⋊D9—22no—
45C22×C3⋊D9—22no—
46C22×C3⋊D9C2312yes—
47C2×C3×C4×C9—22no—
48C2×C3×C4×C9C2312yes—
49C2×C3⋊Dic9—22no—
50C2×C3⋊Dic9C2312yes—
51C2×C9×Dic3—22no—
52C2×C9×Dic3—22no—
53C2×C9×Dic3C2312yes—
54C2×C9×Dic3C2312yes—
55C2×C3×Dic9—22no—
56C2×C3×Dic9—22no—
57C2×C3×Dic9C2312yes—
58C2×C3×Dic9C2312yes—
59C2×#898—91no—
60C2×#898—91no—
61C23×C3⋊D3—31no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C3⋊Dic3—31no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3C2×D311yescharacteristic
67C22×C3×Dic3C2×D311yescharacteristic
68C23×D9—31no—
69C23×D9—61no—
70#3752—34no—
71C3×C4⋊Dic3—34no—
72C3×Dic3⋊C4C2×D314yes—
73Dic3⋊Dic3—34no—
74#3722—34no—
75#3722—34no—
76Dic32—34no—
77C22×C4×C9—21no—
78C22×C4×C9—31no—
79C22×C4×C9—31no—
80C22×C4×C9C2×D311yescharacteristic
81C22×Dic9—31no—
82C22×Dic9—61no—
83C22×Dic9C2×D311yescharacteristic
84C22×Dic9C2×D311yescharacteristic
85C9×C4⋊C4—34no—
86(C2×D9)⋊C4—34no—
87(C2×D9)⋊C4—34no—
88(C2×D9)⋊C4—34no—
89(C2×D9)⋊C4—64no—
90C4⋊Dic9C2×D314yescomplemented
91Dic9⋊C4—34no—
92C4×Dic9—34no—
93C22×C3×C9#4112yes—
94C22×C3×C9#4112yes—
95C22×C3×C9#4112yes—
96C22×C3×C9C2411yescharacteristic
97C2×C3⋊D9—22no—
98C2×C3⋊D9—22no—
99C2×C3⋊D9—22no—
100C2×C3⋊D9—44no—
101C3×C4×C9—22no—
102C3⋊Dic9—22no—
103C9×Dic3—22no—
104C9×Dic3—22no—
105C3×Dic9—22no—
106C3×Dic9—22no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—91no—
109C2×(C2×D3)⋊C4—91no—
110C2×(C2×D3)⋊C4—91no—
111C2×(C2×D3)⋊C4—91no—
112C2×(C2×D3)⋊C4—91no—
113C2×(C2×D3)⋊C4—91no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—181no—
116C2×C4⋊Dic3—31no—
117C2×C4⋊Dic3—91no—
118C2×Dic3⋊C4—91no—
119C2×Dic3⋊C4D911yescharacteristic, complemented
120C2×C4×Dic3—91no—
121C2×C4×Dic3—91no—
122#898—98no—
123#898—98no—
124C23×C32C22×D311yescharacteristic
125C22×C3⋊D3—32no—
126C22×C3⋊D3—62no—
127C22×C3⋊D3—62no—
128C22×C3⋊D3—62no—
129C2×C32×C4—22no—
130C2×C32×C4C22×D312yes—
131C2×C3⋊Dic3—32no—
132C2×C3⋊Dic3—62no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3C22×D312yes—
140C2×C3×Dic3C22×D312yes—
141C23×C9—21no—
142C23×C9C22×D311yescharacteristic
143C22×D9—32no—
144C22×D9—62no—
145C22×D9—64no—
146C22×D9—62no—
147C22×D9—62no—
148C22×D9—62no—
149C22×D9—122no—
150C22×D9—122no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—32no—
154C2×C4×C9—32no—
155C2×C4×C9—42no—
156C2×C4×C9—62no—
157C2×C4×C9—62no—
158C2×C4×C9C22×D312yes—
159C2×Dic9—22no—
160C2×Dic9—22no—
161C2×Dic9—32no—
162C2×Dic9—62no—
163C2×Dic9—62no—
164C2×Dic9—122no—
165C2×Dic9C22×D312yes—
166C2×Dic9C22×D312yes—
167C2×#126—271noSylow
168C2×C3×C9#12614yescomplemented
169C2×C3×C9C2×#4111yescharacteristic
170C2×C3×C9C2×#4111yescharacteristic
171C2×C3×C9C2×#4111yescharacteristic
172C3⋊D9—42no—
173C23×D3—91no—
174C23×D3—91no—
175C23×D3—181no—
176C22×C3×C4—21no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4C2×D911yescharacteristic
182C22×C3×C4D3211yescharacteristic
183C22×Dic3—31no—
184C22×Dic3—31no—
185C22×Dic3—91no—
186C22×Dic3—91no—
187C22×Dic3—181no—
188C22×Dic3C2×D911yescharacteristic
189C22×Dic3C2×D911yescharacteristic
190C3×C4⋊C4—34no—
191C3×C4⋊C4—94no—
192(C2×D3)⋊C4—94no—
193(C2×D3)⋊C4—94no—
194(C2×D3)⋊C4—94no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—94no—
197(C2×D3)⋊C4—94no—
198(C2×D3)⋊C4—184no—
199C4⋊Dic3—34no—
200C4⋊Dic3—94no—
201Dic3⋊C4—94no—
202Dic3⋊C4C2×D914yescomplemented
203C4×Dic3—94no—
204C4×Dic3—94no—
205C22×C32#23512yes—
206C22×C32#23712yes—
207C22×C32#23912yes—
208C22×C32C23×D311yescharacteristic
209C2×C3⋊D3—62no—
210C2×C3⋊D3—62no—
211C2×C3⋊D3—62no—
212C2×C3⋊D3—124no—
213C32×C4—22no—
214C3⋊Dic3—62no—
215C3×Dic3—22no—
216C3×Dic3—22no—
217C3×Dic3—62no—
218C3×Dic3—62no—
219C22×C9—21no—
220C22×C9—22no—
221C22×C9—22no—
222C22×C9—22no—
223C22×C9#23512yes—
224C22×C9#23712yes—
225C22×C9#23912yes—
226C22×C9C23×D311yescharacteristic
227C2×D9—64no—
228C2×D9—62no—
229C2×D9—62no—
230C2×D9—62no—
231C2×D9—122no—
232C2×D9—122no—
233C2×D9—128no—
234C2×D9—124no—
235C4×C9—22no—
236C4×C9—42no—
237C4×C9—62no—
238C4×C9—62no—
239Dic9—22no—
240Dic9—22no—
241Dic9—62no—
242Dic9—122no—
243#126—278no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—91no—
246C2×C4⋊C4—271no—
247C2×C22⋊C4—271no—
248C2×C22⋊C4—271no—
249C2×C22⋊C4—271no—
250C2×C42—271no—
251C3×C9C2×#12611yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3C22×D911yescharacteristic
254C23×C3C2×D3211yescharacteristic
255C22×D3—92no—
256C22×D3—92no—
257C22×D3—182no—
258C22×D3—184no—
259C22×D3—182no—
260C22×D3—182no—
261C22×D3—182no—
262C22×D3—182no—
263C22×D3—182no—
264C22×D3—182no—
265C22×D3—362no—
266C22×D3—362no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—32no—
271C2×C3×C4—32no—
272C2×C3×C4—42no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—92no—
276C2×C3×C4—92no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279C2×C3×C4C22×D912yes—
280C2×C3×C4C2×D3212yes—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—92no—
288C2×Dic3—92no—
289C2×Dic3—182no—
290C2×Dic3—182no—
291C2×Dic3—182no—
292C2×Dic3—362no—
293C2×Dic3C22×D912yes—
294C2×Dic3C22×D912yes—
295C2×C32#89814yes—
296C2×C32C2×#23511yescharacteristic
297C2×C32C2×#23711yescharacteristic
298C2×C32C2×#23911yescharacteristic
299C3⋊D3—122no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—24no—
304C2×C9#89814yescomplemented
305C2×C9C2×#23511yescharacteristic
306C2×C9C2×#23711yescharacteristic
307C2×C9C2×#23911yescharacteristic
308D9—124no—
309D9—122no—
310C24—271no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—271no—
316C22×C4D3×D911yescharacteristic
317C4⋊C4—34no—
318C4⋊C4—94no—
319C4⋊C4—274no—
320C22⋊C4—274no—
321C22⋊C4—274no—
322C22⋊C4—274no—
323C42—274no—
324C22×C3—22no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3#369712yes—
329C22×C3#369912yes—
330C22×C3#370112yes—
331C22×C3C23×D911yescharacteristic
332C22×C3#379612yes—
333C22×C3#379912yes—
334C22×C3#380412yes—
335C22×C3C22×D3211yescharacteristic
336C2×D3—184no—
337C2×D3—182no—
338C2×D3—182no—
339C2×D3—182no—
340C2×D3—182no—
341C2×D3—182no—
342C2×D3—182no—
343C2×D3—362no—
344C2×D3—362no—
345C2×D3—368no—
346C2×D3—364no—
347C2×D3—364no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—42no—
351C3×C4—62no—
352C3×C4—62no—
353C3×C4—182no—
354C3×C4—182no—
355Dic3—22no—
356Dic3—22no—
357Dic3—62no—
358Dic3—62no—
359Dic3—182no—
360Dic3—182no—
361Dic3—362no—
362C32C2×#89811yescharacteristic
363C9—21no—
364C9C2×#89811yescharacteristic, complemented
365C23—272no—
366C23—542no—
367C23—542no—
368C23—542no—
369C23C2×D3×D911yescharacteristic
370C2×C4—22no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—92no—
376C2×C4—92no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4—272no—
380C2×C4—542no—
381C2×C4C2×D3×D912yes—
382C2×C3—21no—
383C2×C3—24no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3#6346014yescomplemented
387C2×C3C2×#369711yescharacteristic
388C2×C3C2×#369911yescharacteristic
389C2×C3C2×#370111yescharacteristic
390C2×C3#6384814yes—
391C2×C3C2×#379611yescharacteristic
392C2×C3C2×#379911yescharacteristic
393C2×C3C2×#380411yescharacteristic
394D3—364no—
395D3—362no—
396D3—362no—
397C22—542no—
398C22—542no—
399C22—542no—
400C22—1084no—
401C22#8863412yes—
402C22#8864212yes—
403C22#8866212yes—
404C22C22×D3×D911yescharacteristic
405C4—22no—
406C4—62no—
407C4—62no—
408C4—182no—
409C4—182no—
410C4—542no—
411C3—21no—
412C3C2×#6346011yescharacteristic, complemented
413C3C2×#6384811yescharacteristic
414C2—1082no—
415C2#1172092114yesdirect product factor
416C2C2×#8863411yescharacteristic
417C2C2×#8864211yescharacteristic
418C2C2×#8866211yescharacteristic
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×#126
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. C2×C9×Dic3⋊C4C2semidirect
3. C2×C3×C4⋊Dic9C2semidirect
6. C2×Dic9⋊Dic3C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11720921C2direct (N × H)
19. C9×Dic3⋊C4C22semidirect
20. C3×C4⋊Dic9C22semidirect
23. Dic9⋊Dic3C22semidirect
24. Dic3×Dic9C22semidirect
38. C2×C4⋊Dic9D3semidirect
90. C4⋊Dic9C2×D3semidirect
119. C2×Dic3⋊C4D9semidirect
168. C2×C3×C9#126semidirect
202. Dic3⋊C4C2×D9semidirect
251. C3×C9C2×#126semidirect
304. C2×C9#898semidirect
364. C9C2×#898semidirect
386. C2×C3#63460semidirect
412. C3C2×#63460semidirect
415. C2#11720921direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#885362
C2×C9×Dic3⋊C42
C2×C3×C4⋊Dic92
C2×#884452
C2×Dic9⋊Dic32
C2×Dic3×Dic92
#117209212
C2×#638483
C2×#634603