Finite Group Databasebeta

← C2×#11720939 #13150135

Subgroups — C2×#11720939 #13150135

Order 1728.

1058 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×C4×D9C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3C2×#88534C211yesmaximal, characteristic, complemented
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720939C218yesmaximal, direct product factor
9C2×#63851—31nomaximal
10C2×#63458—31nomaximal
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C3×C4×D9—24no—
19C2×C3×C4×D9C2214yes—
20C2×#6189—24no—
21C2×#6189C2214yescomplemented
22#88534C2214yescomplemented
23C9×Dic3⋊C4C2214yescomplemented
24(C2×D9)⋊Dic3C2214yes—
25(C2×D9)⋊Dic3C2214yes—
26Dic9⋊Dic3C2214yescomplemented
27C22×C3×C4×D3—31no—
28C22×#627—31no—
29C2×#3750—31no—
30C2×C3×Dic3⋊C4D311yescharacteristic
31C2×Dic3⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C2×(C2×D3)⋊Dic3—31no—
34#63851—38no—
35C22×C4×D9D311yescharacteristic
36C22×Dic18—31no—
37C2×C9×C4⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×Dic9⋊C4—31no—
41C2×Dic9⋊C4—31no—
42C2×Dic9⋊C4—61no—
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⋊Dic9—22no—
53C2×C3⋊Dic9C2312yes—
54C2×C3⋊Dic9C2312yes—
55C2×C9×Dic3—22no—
56C2×C9×Dic3—22no—
57C2×C9×Dic3C2312yes—
58C2×C9×Dic3C2312yes—
59C2×C3×Dic9D414yes—
60C2×C3×Dic9C2312yes—
61C3×C4×D9—28no—
62#6189—28no—
63C2×#924—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×C3×C4×D3—34no—
73C2×C3×C4×D3—64no—
74C2×#627—34no—
75C2×#627—64no—
76C23×D9C2×D311yescharacteristic
77#3750—34no—
78C3×Dic3⋊C4C2×D314yes—
79Dic3⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81(C2×D3)⋊Dic3—34no—
82C22×C4×C9—21no—
83C22×C4×C9—31no—
84C22×C4×C9—31no—
85C22×C4×C9C2×D311yescharacteristic
86C22×Dic9—31no—
87C22×Dic9—31no—
88C22×Dic9—61no—
89C22×Dic9—61no—
90C22×Dic9C2×D311yescharacteristic
91C2×C4×D9—24no—
92C2×C4×D9C2×D314yes—
93C2×Dic18—34no—
94C2×Dic18—64no—
95C9×C4⋊C4—34no—
96(C2×D9)⋊C4—34no—
97(C2×D9)⋊C4—34no—
98Dic9⋊C4—34no—
99Dic9⋊C4—34no—
100Dic9⋊C4—64no—
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—
111C3⋊Dic9—22no—
112C9×Dic3—22no—
113C9×Dic3—22no—
114C3×Dic9C2×D414yes—
115C23×C3×C4—91no—
116C22×C4×D3—31no—
117C22×Dic6—91no—
118C22×Dic6—91no—
119C2×C3×C4⋊C4—31no—
120C2×#217—91no—
121C2×#217—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×C4⋊Dic3—91no—
125#924—98no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C2×Dic3⋊C4—181no—
130C2×Dic3⋊C4D911yescharacteristic, complemented
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—32no—
141C2×C3⋊Dic3—62no—
142C2×C3⋊Dic3—62no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3C22×D312yes—
148C2×C3×Dic3C22×D312yes—
149C3×C4×D3—68no—
150#627—68no—
151C23×C9—21no—
152C23×C9C22×D311yescharacteristic
153C22×D9—22no—
154C22×D9—22no—
155C22×D9Dic614yescomplemented
156C22×D9C22×D312yes—
157C2×C4×C9—22no—
158C2×C4×C9—22no—
159C2×C4×C9—32no—
160C2×C4×C9—32no—
161C2×C4×C9—42no—
162C2×C4×C9—62no—
163C2×C4×C9—62no—
164C2×C4×C9C22×D312yes—
165C2×Dic9—32no—
166C2×Dic9—32no—
167C2×Dic9—62no—
168C2×Dic9—62no—
169C2×Dic9—64no—
170C2×Dic9—62no—
171C2×Dic9—62no—
172C2×Dic9—122no—
173C2×Dic9#6714yes—
174C2×Dic9C22×D312yes—
175C4×D9—28no—
176Dic18—68no—
177C2×C22⋊Q8—271noSylow
178C2×C3×C9C22⋊Q814yescomplemented
179C2×C3×C9C22×D411yescharacteristic
180C2×C3×C9C22×Q811yescharacteristic
181C2×C3×C9C2×#4111yescharacteristic
182C3×D9—24no—
183C24×C3—91no—
184C23×D3—31no—
185C22×C3×C4—21no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—91no—
189C22×C3×C4—94no—
190C22×C3×C4—184no—
191C22×C3×C4C2×D911yescharacteristic
192C22×C3×C4D3211yescharacteristic
193C22×Dic3—31no—
194C22×Dic3—91no—
195C22×Dic3—91no—
196C22×Dic3—91no—
197C22×Dic3—91no—
198C22×Dic3—181no—
199C22×Dic3—181no—
200C22×Dic3C2×D911yescharacteristic
201C22×Dic3C2×D911yescharacteristic
202C2×C4×D3—34no—
203C2×C4×D3—64no—
204C2×Dic6—94no—
205C2×Dic6—94no—
206C2×Dic6—184no—
207C2×Dic6—184no—
208C3×C4⋊C4—34no—
209#217—94no—
210#217—94no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—94no—
213C4⋊Dic3—94no—
214Dic3⋊C4—94no—
215Dic3⋊C4—94no—
216Dic3⋊C4—94no—
217Dic3⋊C4—184no—
218Dic3⋊C4C2×D914yescomplemented
219C22×C32#23512yes—
220C22×C32D3×D412yes—
221C22×C32D3×Q812yes—
222C22×C32C23×D311yescharacteristic
223C2×C3×D3—34no—
224C2×C3×D3—68no—
225C2×C3×D3—62no—
226C2×C3×D3—62no—
227C32×C4—22no—
228C3⋊Dic3—62no—
229C3⋊Dic3—62no—
230C3×Dic3—22no—
231C3×Dic3—22no—
232C3×Dic3—34no—
233C22×C9—21no—
234C22×C9—22no—
235C22×C9—22no—
236C22×C9—22no—
237C22×C9C2×Dic612yes—
238C22×C9#23512yes—
239C22×C9C2×#6712yes—
240C22×C9C23×D311yescharacteristic
241C2×D9—28no—
242C2×D9—22no—
243C2×D9—22no—
244C2×D9C2×Dic614yescomplemented
245C4×C9—22no—
246C4×C9—42no—
247C4×C9—62no—
248C4×C9—62no—
249Dic9—64no—
250Dic9—62no—
251Dic9—62no—
252Dic9—122no—
253Dic9C2×#6714yes—
254C22×Q8—271no—
255C23×C4—91no—
256C22⋊Q8—278no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—271no—
259C2×C4⋊C4—271no—
260C2×C22⋊C4—271no—
261C2×C22⋊C4—271no—
262C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
263C23×C3—21no—
264C23×C3—94no—
265C23×C3—92no—
266C23×C3—182no—
267C23×C3—182no—
268C23×C3C22×D911yescharacteristic
269C23×C3C2×D3211yescharacteristic
270C22×D3—32no—
271C22×D3—34no—
272C22×D3—62no—
273C22×D3—62no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—32no—
278C2×C3×C4—32no—
279C2×C3×C4—42no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—94no—
283C2×C3×C4—92no—
284C2×C3×C4—188no—
285C2×C3×C4C22×D912yes—
286C2×C3×C4C2×D3212yes—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—32no—
290C2×Dic3—34no—
291C2×Dic3—92no—
292C2×Dic3—92no—
293C2×Dic3—92no—
294C2×Dic3—92no—
295C2×Dic3—182no—
296C2×Dic3—182no—
297C2×Dic3—184no—
298C2×Dic3—182no—
299C2×Dic3—182no—
300C2×Dic3—182no—
301C2×Dic3—182no—
302C2×Dic3—362no—
303C2×Dic3C22×D912yes—
304C2×Dic3C22×D912yes—
305C4×D3—68no—
306Dic6—188no—
307Dic6—188no—
308C2×C32#89614yes—
309C2×C32C2×#23511yescharacteristic
310C2×C32C2×D3×D411yescharacteristic
311C2×C32C2×D3×Q811yescharacteristic
312C3×D3—64no—
313C2×C9—21no—
314C2×C9—21no—
315C2×C9—21no—
316C2×C9—24no—
317C2×C9#92414yescomplemented
318C2×C9C22×Dic611yescharacteristic
319C2×C9C2×#23511yescharacteristic
320C2×C9C22×#6711yescharacteristic
321D9—24no—
322C24—91no—
323C2×Q8—274no—
324C2×Q8—544no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—91no—
328C22×C4—94no—
329C22×C4—184no—
330C22×C4—271no—
331C22×C4—271no—
332C22×C4D3×D911yescharacteristic
333C4⋊C4—34no—
334C4⋊C4—274no—
335C4⋊C4—274no—
336C22⋊C4—274no—
337C22⋊C4—274no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—22no—
342C22×C3—94no—
343C22×C3—182no—
344C22×C3—182no—
345C22×C3—188no—
346C22×C3#369712yes—
347C22×C3D4×D912yes—
348C22×C3Q8×D912yes—
349C22×C3C23×D911yescharacteristic
350C22×C3D3×Dic612yes—
351C22×C3#379812yes—
352C22×C3D3×#6712yes—
353C22×C3C22×D3211yescharacteristic
354C2×D3—34no—
355C2×D3—62no—
356C2×D3—62no—
357C2×D3—68no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—42no—
361C3×C4—62no—
362C3×C4—62no—
363C3×C4—94no—
364Dic3—22no—
365Dic3—22no—
366Dic3—34no—
367Dic3—184no—
368Dic3—182no—
369Dic3—182no—
370Dic3—182no—
371Dic3—182no—
372Dic3—362no—
373C32C2×#89611yescharacteristic
374C9—21no—
375C9C2×#92411yescharacteristic, complemented
376C23—94no—
377C23—92no—
378C23—182no—
379C23—182no—
380C23C2×D3×D911yescharacteristic
381Q8—548no—
382C2×C4—22no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—92no—
388C2×C4—94no—
389C2×C4—188no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—542no—
393C2×C4—542no—
394C2×C4C2×D3×D912yes—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—24no—
399C2×C3—184no—
400C2×C3#6345814yescomplemented
401C2×C3C2×#369711yescharacteristic
402C2×C3C2×D4×D911yescharacteristic
403C2×C3C2×Q8×D911yescharacteristic
404C2×C3#6385114yes—
405C2×C3C2×D3×Dic611yescharacteristic
406C2×C3C2×#379811yescharacteristic
407C2×C3C2×D3×#6711yescharacteristic
408D3—64no—
409C22—94no—
410C22—182no—
411C22—182no—
412C22—188no—
413C22D9×Dic612yes—
414C22#8864012yes—
415C22D9×#6712yes—
416C22C22×D3×D911yescharacteristic
417C4—22no—
418C4—62no—
419C4—62no—
420C4—94no—
421C4—542no—
422C4—542no—
423C3—21no—
424C3C2×#6345811yescharacteristic, complemented
425C3C2×#6385111yescharacteristic
426C2—184no—
427C2#1172093914yesdirect product factor
428C2C2×D9×Dic611yescharacteristic
429C2C2×#8864011yescharacteristic
430C2C2×D9×#6711yescharacteristic
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×#88534C2semidirect
4. C2×C9×Dic3⋊C4C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720939C2direct (N × H)
21. C2×#6189C22semidirect
22. #88534C22semidirect
23. C9×Dic3⋊C4C22semidirect
26. Dic9⋊Dic3C22semidirect
47. C22×C3×D9Q8semidirect
108. C2×C3×D9C2×Q8semidirect
130. C2×Dic3⋊C4D9semidirect
155. C22×D9Dic6semidirect
178. C2×C3×C9C22⋊Q8semidirect
218. Dic3⋊C4C2×D9semidirect
244. C2×D9C2×Dic6semidirect
262. C3×C9C2×C22⋊Q8semidirect
317. C2×C9#924semidirect
375. C9C2×#924semidirect
400. C2×C3#63458semidirect
424. C3C2×#63458semidirect
427. C2#11720939direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C22×#61892
C2×#885342
C2×C9×Dic3⋊C42
C2×(C2×D9)⋊Dic32
C2×Dic9⋊Dic32
#117209392
C2×#638513
C2×#634583