Finite Group Databasebeta

← C2×#11722722 #13166571

Subgroups — C2×#11722722 #13166571

Order 1728.

1240 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊#157C211yesmaximal, characteristic
2C23×C3×#157C211yesmaximal, characteristic
3C22×Dic3×C3⋊D3C211yesmaximal, characteristic
4#11722722C214yesmaximal, direct product factor
5C2×#63781—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157—22no—
8C22×C3⋊#157—22no—
9C22×C3⋊#157C414yescomplemented
10C22×C3⋊#157C2212yes—
11C22×C3×#157—22no—
12C22×C3×#157—22no—
13C22×C3×#157C414yescomplemented
14C22×C3×#157C2212yes—
15C22×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
16C22×C32×Dic3C411yescharacteristic, complemented
17C2×Dic3×C3⋊D3—22no—
18C2×Dic3×C3⋊D3—22no—
19C2×Dic3×C3⋊D3C412yescomplemented
20C2×Dic3×C3⋊D3C2212yes—
21C23×#157—31no—
22C23×#157D311yescharacteristic
23C22×C4×C3⋊D3—31no—
24C22×D3×Dic3—62no—
25#63781—34no—
26C23×C33C2×C411yescharacteristic
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C22×C3×C3⋊D3C2×C412yes—
29C22×C3×C3⋊D3C2×C412yes—
30C22×C3×C3⋊D3D412yes—
31C22×C3×C3⋊D3D412yes—
32C22×C3×C3⋊D3D412yes—
33C22×C3×C3⋊D3Q812yes—
34C22×C3×C3⋊D3C2311yescharacteristic
35C2×C3⋊#157—28no—
36C2×C3⋊#157—22no—
37C2×C3⋊#157—22no—
38C2×C3⋊#157C2×C414yescomplemented
39C2×C3×#157—28no—
40C2×C3×#157—22no—
41C2×C3×#157—22no—
42C2×C3×#157C2×C414yescomplemented
43C2×C3⋊(C3⋊Dic3)—22no—
44C2×C3⋊(C3⋊Dic3)C2×C412yescomplemented
45C2×C32×Dic3—22no—
46C2×C32×Dic3C2×C412yescomplemented
47Dic3×C3⋊D3—24no—
48Dic3×C3⋊D3—22no—
49Dic3×C3⋊D3—22no—
50C2×C3⋊#95—91nomaximal
51C23×C3⋊D3C2×D311yescharacteristic
52C23×C3×D3—62no—
53C22×#157—22no—
54C22×#157—22no—
55C22×#157—34no—
56C22×#157—32no—
57C22×#157—62no—
58C22×#157—62no—
59C22×#157Dic314yescomplemented
60C22×#157C2×D312yes—
61C22×C32×C4—31no—
62C22×C3⋊Dic3—31no—
63C22×C3⋊Dic3—62no—
64C22×C3⋊Dic3—122no—
65C2×C4×C3⋊D3—32no—
66C2×C4×C3⋊D3—32no—
67C2×C4×C3⋊D3—62no—
68C2×C4×C3⋊D3—62no—
69C22×C3×Dic3—22no—
70C2×D3×Dic3—64no—
71C2×D3×Dic3—64no—
72C2×D3×Dic3—68no—
73C2×D3×Dic3—68no—
74C22×C33C4212yes—
75C22×C33C22⋊C412yes—
76C22×C33C4⋊C412yes—
77C22×C33C22×C411yescharacteristic
78C2×C3×C3⋊D3C4212yes—
79C2×C3×C3⋊D3C22⋊C412yes—
80C2×C3×C3⋊D3C22⋊C414yes—
81C2×C3×C3⋊D3C22⋊C414yes—
82C2×C3×C3⋊D3C4⋊C412yes—
83C2×C3×C3⋊D3C4⋊C414yes—
84C2×C3×C3⋊D3C4⋊C414yes—
85C2×C3×C3⋊D3C22×C411yescharacteristic
86C2×C3×C3⋊D3C22×C411yescharacteristic
87C2×C3×C3⋊D3C2×D411yescharacteristic
88C2×C3×C3⋊D3C2×D411yescharacteristic
89C2×C3×C3⋊D3C2×D411yescharacteristic
90C2×C3×C3⋊D3C2×Q811yescharacteristic
91C3⋊#157—24no—
92C3×#157—24no—
93C3⋊(C3⋊Dic3)—22no—
94C32×Dic3—22no—
95C23×C3×C4—91no—
96C23×Dic3—91no—
97C23×Dic3—91no—
98C22×C4×D3—182no—
99C3⋊#95—94no—
100C23×C32—22no—
101C23×C32—42no—
102C23×C32C4×D311yescharacteristic
103C22×C3⋊D3Dic612yes—
104C22×C3⋊D3C4×D311yescharacteristic
105C22×C3⋊D3C4×D312yes—
106C22×C3⋊D3D1212yes—
107C22×C3⋊D3C2×Dic312yes—
108C22×C3⋊D3#6712yes—
109C22×C3⋊D3#6712yes—
110C22×C3⋊D3C22×D311yescharacteristic
111C22×C3×D3—64no—
112C22×C3×D3—64no—
113C22×C3×D3—62no—
114C22×C3×D3—64no—
115C22×C3×D3—64no—
116C22×C3×D3—64no—
117C22×C3×D3—64no—
118C22×C3×D3—62no—
119C2×#157—28no—
120C2×#157—22no—
121C2×#157—22no—
122C2×#157—34no—
123C2×#157—68no—
124C2×#157—62no—
125C2×#157—62no—
126C2×#157C2×Dic314yescomplemented
127C2×C32×C4—32no—
128C2×C32×C4—62no—
129C2×C3⋊Dic3—32no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—62no—
132C2×C3⋊Dic3—68no—
133C2×C3⋊Dic3—128no—
134C2×C3⋊Dic3—124no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—62no—
137C4×C3⋊D3—62no—
138C2×C3×Dic3—28no—
139C2×C3×Dic3—24no—
140D3×Dic3—616no—
141D3×Dic3—68no—
142D3×Dic3—68no—
143C2×#95—271noSylow
144C2×C33#9514yescomplemented
145C2×C33C2×C4211yescharacteristic
146C2×C33C2×C22⋊C411yescharacteristic
147C2×C33C2×C4⋊C411yescharacteristic
148C3×C3⋊D3#9514yescomplemented
149C3×C3⋊D3C2×C22⋊C411yescharacteristic
150C3×C3⋊D3C2×C22⋊C411yescharacteristic
151C3×C3⋊D3C2×C4⋊C411yescharacteristic
152C3×C3⋊D3C2×C4⋊C411yescharacteristic
153C24×C3—91no—
154C23×D3—62no—
155C22×C3×C4—62no—
156C22×C3×C4—94no—
157C22×C3×C4—92no—
158C22×C3×C4—182no—
159C22×C3×C4—182no—
160C22×Dic3—94no—
161C22×Dic3—92no—
162C22×Dic3—91no—
163C22×Dic3—92no—
164C22×Dic3—92no—
165C22×Dic3—182no—
166C22×Dic3—182no—
167C22×Dic3—182no—
168C22×Dic3—182no—
169C22×Dic3—182no—
170C22×Dic3—362no—
171C22×Dic3#15711yescharacteristic, complemented
172C2×C4×D3—184no—
173C2×C4×D3—184no—
174C2×C4×D3—188no—
175C2×C4×D3—188no—
176C22×C32—22no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—24no—
180C22×C32—44no—
181C22×C32—44no—
182C22×C32—44no—
183C22×C32—42no—
184C22×C32C4×Dic312yes—
185C22×C32Dic3⋊C412yes—
186C22×C32(C2×D3)⋊C412yes—
187C22×C32C2×C4×D311yescharacteristic
188C2×C3⋊D3C4×Dic312yes—
189C2×C3⋊D3Dic3⋊C412yes—
190C2×C3⋊D3Dic3⋊C414yes—
191C2×C3⋊D3C4⋊Dic314yes—
192C2×C3⋊D3(C2×D3)⋊C412yes—
193C2×C3⋊D3(C2×D3)⋊C414yes—
194C2×C3⋊D3#21714yes—
195C2×C3⋊D3C2×Dic611yescharacteristic
196C2×C3⋊D3C2×C4×D311yescharacteristic
197C2×C3⋊D3C2×D1211yescharacteristic
198C2×C3⋊D3C22×Dic311yescharacteristic
199C2×C3⋊D3C2×#6711yescharacteristic
200C2×C3⋊D3C2×#6711yescharacteristic
201C2×C3×D3—64no—
202C2×C3×D3—64no—
203C2×C3×D3—68no—
204C2×C3×D3—68no—
205C2×C3×D3—64no—
206C2×C3×D3—68no—
207C2×C3×D3—68no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214#157—24no—
215#157—64no—
216C32×C4—62no—
217C3⋊Dic3—62no—
218C3⋊Dic3—68no—
219C3⋊Dic3—128no—
220C3×Dic3—28no—
221C23×C4—91no—
222C23×C4—271no—
223C23×C4—271no—
224#95—274no—
225C33C2×#9511yescharacteristic, complemented, Sylow
226C23×C3—22no—
227C23×C3—42no—
228C23×C3—91no—
229C23×C3—91no—
230C23×C3—92no—
231C23×C3—92no—
232C23×C3—92no—
233C23×C3—92no—
234C23×C3—92no—
235C23×C3—92no—
236C23×C3C2×#15711yescharacteristic
237C22×D3—64no—
238C22×D3—62no—
239C22×D3—64no—
240C22×D3—64no—
241C22×D3—64no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—62no—
245C2×C3×C4—64no—
246C2×C3×C4—68no—
247C2×C3×C4—94no—
248C2×C3×C4—182no—
249C2×C3×C4—188no—
250C2×C3×C4—182no—
251C2×Dic3—22no—
252C2×Dic3—94no—
253C2×Dic3—92no—
254C2×Dic3—184no—
255C2×Dic3—182no—
256C2×Dic3—182no—
257C2×Dic3—184no—
258C2×Dic3—188no—
259C2×Dic3—182no—
260C2×Dic3—188no—
261C2×Dic3—182no—
262C2×Dic3—182no—
263C2×Dic3—364no—
264C2×Dic3—368no—
265C2×Dic3C2×#15712yescomplemented
266C4×D3—1816no—
267C4×D3—188no—
268C4×D3—188no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—28no—
272C2×C32—22no—
273C2×C32—48no—
274C2×C32—42no—
275C2×C32—42no—
276C2×C32—42no—
277C2×C32C3⋊#9514yescomplemented
278C2×C32C2×C4×Dic311yescharacteristic
279C2×C32C2×Dic3⋊C411yescharacteristic
280C2×C32C2×(C2×D3)⋊C411yescharacteristic
281C3⋊D3C3⋊#9514yescomplemented
282C3⋊D3C2×Dic3⋊C411yescharacteristic
283C3⋊D3C2×C4⋊Dic311yescharacteristic
284C3⋊D3C2×(C2×D3)⋊C411yescharacteristic
285C3⋊D3C2×#21711yescharacteristic
286C3×D3—68no—
287C3×D3—62no—
288C3×D3—62no—
289C3×D3—62no—
290C3×D3—62no—
291C24—91no—
292C22×C4—31no—
293C22×C4—94no—
294C22×C4—92no—
295C22×C4—182no—
296C22×C4—182no—
297C22×C4—272no—
298C22×C4—272no—
299C22×C4—272no—
300C22×C4—274no—
301C22×C4—271no—
302C22×C4—542no—
303C22×C4—542no—
304C22×C4—542no—
305C22×C4—542no—
306C22×C3—22no—
307C22×C3—24no—
308C22×C3—24no—
309C22×C3—24no—
310C22×C3—42no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—91no—
315C22×C3—91no—
316C22×C3—92no—
317C22×C3—91no—
318C22×C3—91no—
319C22×C3—92no—
320C22×C3—94no—
321C22×C3—94no—
322C22×C3—94no—
323C22×C3—94no—
324C22×C3—91no—
325C22×C3—91no—
326C22×C3—92no—
327C22×C3C4×#15712yes—
328C22×C3C4⋊#15712yes—
329C22×C3C22⋊#15712yes—
330C22×C3C22×#15711yescharacteristic
331C2×D3—64no—
332C2×D3—62no—
333C2×D3—62no—
334C2×D3—64no—
335C2×D3—68no—
336C2×D3—64no—
337C2×D3—68no—
338C2×D3—68no—
339C2×D3—68no—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—62no—
343C2×D3—62no—
344C3×C4—68no—
345C3×C4—184no—
346Dic3—22no—
347Dic3—188no—
348Dic3—184no—
349Dic3—182no—
350Dic3—368no—
351C32—22no—
352C32—42no—
353C32C2×C3⋊#9511yescharacteristic, complemented
354C23—91no—
355C23—91no—
356C23—92no—
357C23—92no—
358C23—92no—
359C23—92no—
360C23—92no—
361C23—92no—
362C23D3×#15711yescharacteristic
363C2×C4—32no—
364C2×C4—62no—
365C2×C4—94no—
366C2×C4—188no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—274no—
370C2×C4—272no—
371C2×C4—542no—
372C2×C4—542no—
373C2×C4—544no—
374C2×C4—548no—
375C2×C4—542no—
376C2×C4—542no—
377C2×C4—542no—
378C2×C3—28no—
379C2×C3—22no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—48no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—42no—
386C2×C3—91no—
387C2×C3—91no—
388C2×C3—91no—
389C2×C3—91no—
390C2×C3—94no—
391C2×C3#6378114yescomplemented
392C2×C3C2×C4×#15711yescharacteristic
393C2×C3C2×C4⋊#15711yescharacteristic
394C2×C3C2×C22⋊#15711yescharacteristic
395D3—68no—
396D3—62no—
397D3—62no—
398D3—62no—
399D3—62no—
400C22—94no—
401C22—94no—
402C22—94no—
403C22—94no—
404C22—91no—
405C22—91no—
406C22—92no—
407C22—91no—
408C22—91no—
409C22—92no—
410C22—91no—
411C22—91no—
412C22—92no—
413C22Dic3×#15712yes—
414C22(C2×D3)⋊#15712yes—
415C22Dic3⋊#15712yes—
416C22C2×D3×#15711yescharacteristic
417C4—62no—
418C4—184no—
419C4—544no—
420C4—542no—
421C3—22no—
422C3—42no—
423C3C2×#6378111yescharacteristic, complemented
424C2—91no—
425C2—91no—
426C2—94no—
427C2—91no—
428C2—91no—
429C2#1172272214yesdirect product factor
430C2C2×Dic3×#15711yescharacteristic
431C2C2×(C2×D3)⋊#15711yescharacteristic
432C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC23D3×#157
commutatorC2×C33C2×C42
FrattiniC22C2×D3×#157
Fitting, socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33C2×#95
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722722C2direct (N × H)
9. C22×C3⋊#157C4semidirect
13. C22×C3×#157C4semidirect
15. C22×C3⋊(C3⋊Dic3)C4semidirect
16. C22×C32×Dic3C4semidirect
19. C2×Dic3×C3⋊D3C4semidirect
38. C2×C3⋊#157C2×C4semidirect
42. C2×C3×#157C2×C4semidirect
44. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
46. C2×C32×Dic3C2×C4semidirect
59. C22×#157Dic3semidirect
126. C2×#157C2×Dic3semidirect
144. C2×C33#95semidirect
148. C3×C3⋊D3#95semidirect
171. C22×Dic3#157semidirect
225. C33C2×#95semidirect
265. C2×Dic3C2×#157semidirect
277. C2×C32C3⋊#95semidirect
281. C3⋊D3C3⋊#95semidirect
353. C32C2×C3⋊#95semidirect
391. C2×C3#63781semidirect
423. C3C2×#63781semidirect
429. C2#11722722direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C23×C3×#1572
C22×Dic3×C3⋊D32
#117227222
C2×#637813
C2×C3⋊#959