Finite Group Databasebeta

← C2×#11720923 #13150141

Subgroups — C2×#11720923 #13150141

Order 1728.

1202 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic
6C2×#88445C211yesmaximal, characteristic
7C2×#88442C211yesmaximal, characteristic, complemented
8#11720923C218yesmaximal, direct product factor
9C2×#63865—31nomaximal
10C2×#63458—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C2×C4×C3⋊D9—24no—
15C2×C4×C3⋊D9C2214yes—
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6189—24no—
21C2×#6189C2214yescomplemented
22C9×Dic3⋊C4C2214yescomplemented
23C3×Dic9⋊C4C2214yescomplemented
24#88445C2214yes—
25#88445C2214yes—
26#88442C2214yescomplemented
27C22×C4×C3⋊D3—31no—
28C22×#627—31no—
29C2×C3×Dic3⋊C4—31no—
30C2×C3×Dic3⋊C4D311yescharacteristic
31C2×#3724—31no—
32C2×#3722—31no—
33C2×#3722—31no—
34#63865—38no—
35C22×C4×D9—31no—
36C22×C4×D9—61no—
37C22×Dic18—31no—
38C2×C9×C4⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4—31no—
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⋊Dic9D414yes—
52C2×C3⋊Dic9C2312yes—
53C4×C3⋊D9—28no—
54C2×C9×Dic3—22no—
55C2×C9×Dic3—22no—
56C2×C9×Dic3C2312yes—
57C2×C9×Dic3C2312yes—
58C2×C3×Dic9—22no—
59C2×C3×Dic9—22no—
60C2×C3×Dic9C2312yes—
61C2×C3×Dic9C2312yes—
62#6189—28no—
63C2×#896—91no—
64C2×#896—91no—
65C23×C3⋊D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C2×C4×C3⋊D3—34no—
69C2×C4×C3⋊D3—64no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C2×D311yescharacteristic
74C2×#627—34no—
75C2×#627—64no—
76C23×D9—31no—
77C23×D9—61no—
78C3×Dic3⋊C4—34no—
79C3×Dic3⋊C4C2×D314yes—
80#3724—34no—
81#3722—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×C4×D9—64no—
94C2×C4×D9—68no—
95C2×Dic18—34no—
96C2×Dic18—64no—
97C9×C4⋊C4—34no—
98(C2×D9)⋊C4—34no—
99(C2×D9)⋊C4—34no—
100Dic9⋊C4—34no—
101Dic9⋊C4C2×D314yescomplemented
102C22×C3×C9C2×D412yes—
103C22×C3×C9C2×Q812yes—
104C22×C3×C9#4112yes—
105C22×C3×C9C2411yescharacteristic
106C2×C3⋊D9—28no—
107C2×C3⋊D9—22no—
108C2×C3⋊D9—22no—
109C2×C3⋊D9C2×Q814yescomplemented
110C3×C4×C9—22no—
111C3⋊Dic9C2×D414yes—
112C9×Dic3—22no—
113C9×Dic3—22no—
114C3×Dic9—22no—
115C3×Dic9—22no—
116C22×C4×D3—91no—
117C22×C4×D3—91no—
118C22×C4×D3—181no—
119C22×Dic6—91no—
120C22×Dic6—91no—
121C2×C3×C4⋊C4—31no—
122C2×C3×C4⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—91no—
126C2×(C2×D3)⋊C4—91no—
127C2×Dic3⋊C4—31no—
128C2×Dic3⋊C4—91no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4D911yescharacteristic, complemented
131#896—98no—
132#896—98no—
133C23×C32C22×D311yescharacteristic
134C22×C3⋊D3—34no—
135C22×C3⋊D3—32no—
136C22×C3⋊D3—62no—
137C22×C3⋊D3—62no—
138C2×C32×C4—22no—
139C2×C32×C4C22×D312yes—
140C2×C3⋊Dic3—34no—
141C2×C3⋊Dic3—32no—
142C4×C3⋊D3—68no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3C22×D312yes—
150C2×C3×Dic3C22×D312yes—
151#627—68no—
152C23×C9—21no—
153C23×C9C22×D311yescharacteristic
154C22×D9—34no—
155C22×D9—32no—
156C22×D9—62no—
157C22×D9—64no—
158C22×D9—64no—
159C22×D9—64no—
160C22×D9—62no—
161C22×D9—62no—
162C2×C4×C9—24no—
163C2×C4×C9—22no—
164C2×C4×C9—22no—
165C2×C4×C9—32no—
166C2×C4×C9—32no—
167C2×C4×C9—62no—
168C2×C4×C9—62no—
169C2×C4×C9C22×D312yes—
170C2×Dic9—22no—
171C2×Dic9—22no—
172C2×Dic9—34no—
173C2×Dic9—32no—
174C2×Dic9—64no—
175C2×Dic9—62no—
176C2×Dic9C22×D312yes—
177C2×Dic9C22×D312yes—
178C4×D9—68no—
179C4×D9—616no—
180Dic18—68no—
181C2×C22⋊Q8—271noSylow
182C2×C3×C9C22⋊Q814yescomplemented
183C2×C3×C9C22×D411yescharacteristic
184C2×C3×C9C22×Q811yescharacteristic
185C2×C3×C9C2×#4111yescharacteristic
186C3⋊D9—24no—
187C23×D3—91no—
188C23×D3—91no—
189C23×D3—181no—
190C22×C3×C4—21no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4C2×D911yescharacteristic
196C22×C3×C4D3211yescharacteristic
197C22×Dic3—31no—
198C22×Dic3—31no—
199C22×Dic3—91no—
200C22×Dic3—91no—
201C22×Dic3—181no—
202C22×Dic3C2×D911yescharacteristic
203C22×Dic3C2×D911yescharacteristic
204C2×C4×D3—94no—
205C2×C4×D3—94no—
206C2×C4×D3—184no—
207C2×C4×D3—184no—
208C2×C4×D3—184no—
209C2×C4×D3—188no—
210C2×Dic6—94no—
211C2×Dic6—94no—
212C2×Dic6—184no—
213C2×Dic6—184no—
214C3×C4⋊C4—34no—
215C3×C4⋊C4—94no—
216(C2×D3)⋊C4—94no—
217(C2×D3)⋊C4—94no—
218(C2×D3)⋊C4—94no—
219(C2×D3)⋊C4—94no—
220Dic3⋊C4—34no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223Dic3⋊C4C2×D914yescomplemented
224C22×C32#23512yes—
225C22×C32D3×D412yes—
226C22×C32D3×Q812yes—
227C22×C32C23×D311yescharacteristic
228C2×C3⋊D3—34no—
229C2×C3⋊D3—68no—
230C2×C3⋊D3—62no—
231C2×C3⋊D3—62no—
232C32×C4—22no—
233C3⋊Dic3—34no—
234C3×Dic3—22no—
235C3×Dic3—22no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C22×C9—21no—
239C22×C9—22no—
240C22×C9—22no—
241C22×C9—22no—
242C22×C9#23512yes—
243C22×C9D3×D412yes—
244C22×C9D3×Q812yes—
245C22×C9C23×D311yescharacteristic
246C2×D9—34no—
247C2×D9—64no—
248C2×D9—64no—
249C2×D9—64no—
250C2×D9—616no—
251C2×D9—68no—
252C2×D9—62no—
253C2×D9—62no—
254C4×C9—24no—
255C4×C9—22no—
256C4×C9—62no—
257C4×C9—62no—
258Dic9—22no—
259Dic9—22no—
260Dic9—34no—
261Dic9—64no—
262C22×Q8—271no—
263C23×C4—271no—
264C22⋊Q8—278no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4—271no—
268C2×C22⋊C4—271no—
269C2×C22⋊C4—271no—
270C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
271C23×C3—21no—
272C23×C3C22×D911yescharacteristic
273C23×C3C2×D3211yescharacteristic
274C22×D3—92no—
275C22×D3—94no—
276C22×D3—92no—
277C22×D3—94no—
278C22×D3—182no—
279C22×D3—184no—
280C22×D3—184no—
281C22×D3—184no—
282C22×D3—182no—
283C22×D3—182no—
284C22×D3—182no—
285C22×D3—182no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—182no—
297C2×C3×C4—182no—
298C2×C3×C4C22×D912yes—
299C2×C3×C4C2×D3212yes—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—92no—
307C2×Dic3—94no—
308C2×Dic3—92no—
309C2×Dic3—94no—
310C2×Dic3—182no—
311C2×Dic3—184no—
312C2×Dic3C22×D912yes—
313C2×Dic3C22×D912yes—
314C4×D3—188no—
315C4×D3—188no—
316C4×D3—1816no—
317Dic6—188no—
318Dic6—188no—
319C2×C32#89614yes—
320C2×C32C2×#23511yescharacteristic
321C2×C32C2×D3×D411yescharacteristic
322C2×C32C2×D3×Q811yescharacteristic
323C3⋊D3—64no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—21no—
327C2×C9—24no—
328C2×C9#89614yescomplemented
329C2×C9C2×#23511yescharacteristic
330C2×C9C2×D3×D411yescharacteristic
331C2×C9C2×D3×Q811yescharacteristic
332D9—68no—
333D9—64no—
334C24—271no—
335C2×Q8—274no—
336C2×Q8—544no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—91no—
340C22×C4—91no—
341C22×C4—271no—
342C22×C4—274no—
343C22×C4—544no—
344C22×C4D3×D911yescharacteristic
345C4⋊C4—34no—
346C4⋊C4—94no—
347C4⋊C4—274no—
348C22⋊C4—274no—
349C22⋊C4—274no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—21no—
353C22×C3—22no—
354C22×C3#369712yes—
355C22×C3D4×D912yes—
356C22×C3Q8×D912yes—
357C22×C3C23×D911yescharacteristic
358C22×C3#379712yes—
359C22×C3#379812yes—
360C22×C3#381112yes—
361C22×C3C22×D3211yescharacteristic
362C2×D3—94no—
363C2×D3—94no—
364C2×D3—184no—
365C2×D3—184no—
366C2×D3—1816no—
367C2×D3—184no—
368C2×D3—182no—
369C2×D3—182no—
370C2×D3—188no—
371C2×D3—182no—
372C2×D3—182no—
373C2×D3—188no—
374C3×C4—24no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—62no—
378C3×C4—62no—
379C3×C4—182no—
380C3×C4—182no—
381Dic3—22no—
382Dic3—22no—
383Dic3—62no—
384Dic3—62no—
385Dic3—94no—
386Dic3—94no—
387Dic3—184no—
388C32C2×#89611yescharacteristic
389C9—21no—
390C9C2×#89611yescharacteristic, complemented
391C23—274no—
392C23—272no—
393C23—542no—
394C23—542no—
395C23C2×D3×D911yescharacteristic
396Q8—548no—
397C2×C4—22no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—92no—
403C2×C4—92no—
404C2×C4—182no—
405C2×C4—182no—
406C2×C4—272no—
407C2×C4—274no—
408C2×C4—548no—
409C2×C4C2×D3×D912yes—
410C2×C3—21no—
411C2×C3—24no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3#6345814yescomplemented
415C2×C3C2×#369711yescharacteristic
416C2×C3C2×D4×D911yescharacteristic
417C2×C3C2×Q8×D911yescharacteristic
418C2×C3#6386514yes—
419C2×C3C2×#379711yescharacteristic
420C2×C3C2×#379811yescharacteristic
421C2×C3C2×#381111yescharacteristic
422D3—188no—
423D3—184no—
424D3—184no—
425C22—274no—
426C22—542no—
427C22—548no—
428C22—542no—
429C22(C3⋊D9)⋊Q812yes—
430C22#8864012yes—
431C22#8866812yes—
432C22C22×D3×D911yescharacteristic
433C4—22no—
434C4—62no—
435C4—62no—
436C4—182no—
437C4—182no—
438C4—274no—
439C3—21no—
440C3C2×#6345811yescharacteristic, complemented
441C3C2×#6386511yescharacteristic
442C2—544no—
443C2#1172092314yesdirect product factor
444C2C2×(C3⋊D9)⋊Q811yescharacteristic
445C2C2×#8864011yescharacteristic
446C2C2×#8866811yescharacteristic
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
3. C2×C9×Dic3⋊C4C2semidirect
4. C2×C3×Dic9⋊C4C2semidirect
7. C2×#88442C2semidirect
8. #11720923C2direct (N × H)
21. C2×#6189C22semidirect
22. C9×Dic3⋊C4C22semidirect
23. C3×Dic9⋊C4C22semidirect
26. #88442C22semidirect
42. C2×Dic9⋊C4D3semidirect
47. C22×C3⋊D9Q8semidirect
101. Dic9⋊C4C2×D3semidirect
109. C2×C3⋊D9C2×Q8semidirect
130. C2×Dic3⋊C4D9semidirect
182. C2×C3×C9C22⋊Q8semidirect
223. Dic3⋊C4C2×D9semidirect
270. C3×C9C2×C22⋊Q8semidirect
328. C2×C9#896semidirect
390. C9C2×#896semidirect
414. C2×C3#63458semidirect
440. C3C2×#63458semidirect
443. C2#11720923direct (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×#61892
C2×C9×Dic3⋊C42
C2×C3×Dic9⋊C42
C2×#884452
C2×#884422
#117209232
C2×#638653
C2×#634583