Finite Group Databasebeta

← C2×#11723407 #13176815

Subgroups — C2×#11723407 #13176815

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88857C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C2×#88803C211yesmaximal, characteristic
6C2×#88803C211yesmaximal, characteristic
7C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#11723407C218yesmaximal, direct product factor
9C2×#64107—31nomaximal
10C2×#63858—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18#88857C2214yescomplemented
19C3×#3752C2214yes—
20C32×C4⋊Dic3C2214yescomplemented
21Dic3⋊(C3⋊Dic3)C2214yescomplemented
22#88803C2214yes—
23#88803C2214yes—
24Dic3×C3⋊Dic3C2214yescomplemented
25C2×C32×C4⋊C4—31no—
26C2×#3752—31no—
27C2×#3752—31no—
28C2×#3752D311yescharacteristic
29C2×C4⋊(C3⋊Dic3)—31no—
30C2×#3750—31no—
31C2×#3750—34no—
32C2×#3750—64no—
33C2×C4×C3⋊Dic3—31no—
34#64107—38no—
35C2×C3×(C2×D3)⋊C4—34no—
36C2×C3×C4⋊Dic3D314yescomplemented
37C2×Dic3⋊Dic3—34no—
38C2×(C2×D3)⋊Dic3—34no—
39C2×(C2×D3)⋊Dic3—34no—
40C2×Dic32—34no—
41#63858—332no—
42C23×C33C2311yescharacteristic
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3—22no—
45C22×C3×C3⋊D3—22no—
46C22×C3×C3⋊D3C2312yes—
47C2×C33×C4—22no—
48C2×C33×C4C2312yes—
49C2×C3⋊(C3⋊Dic3)—22no—
50C2×C3⋊(C3⋊Dic3)—22no—
51C2×C3⋊(C3⋊Dic3)C2312yes—
52C2×C3⋊(C3⋊Dic3)C2312yes—
53C2×C3×C3⋊Dic3—22no—
54C2×C3×C3⋊Dic3C2312yes—
55C2×C32×Dic3—22no—
56C2×C32×Dic3—22no—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C2×#898—94no—
60C2×#879—91no—
61C23×C3⋊D3C2×D311yescharacteristic
62C23×C3×D3—34no—
63C22×C32×C4—24no—
64C22×C32×C4—31no—
65C22×C32×C4—31no—
66C22×C32×C4C2×D314yes—
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3—31no—
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—34no—
71C22×C3⋊Dic3—34no—
72C22×C3⋊Dic3—64no—
73C22×C3⋊Dic3—64no—
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3×Dic3—34no—
76C22×C3×Dic3C2×D314yes—
77C22×C3×Dic3C2×D314yes—
78C32×C4⋊C4—34no—
79#3752—34no—
80#3752—34no—
81#3752C2×D314yes—
82C4⋊(C3⋊Dic3)—34no—
83#3750—34no—
84#3750—316no—
85#3750—616no—
86C4×C3⋊Dic3—34no—
87C3×(C2×D3)⋊C4—316no—
88C3×C4⋊Dic3C2×D3116yescomplemented
89Dic3⋊Dic3—316no—
90(C2×D3)⋊Dic3—316no—
91(C2×D3)⋊Dic3—316no—
92Dic32—316no—
93C22×C33#4112yes—
94C22×C33#4112yes—
95C22×C33#4112yes—
96C22×C33C2411yescharacteristic
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—44no—
101C33×C4—22no—
102C3⋊(C3⋊Dic3)—22no—
103C3⋊(C3⋊Dic3)—22no—
104C3×C3⋊Dic3—22no—
105C32×Dic3—22no—
106C32×Dic3—22no—
107C2×C3×C4⋊C4—34no—
108C2×C3×C22⋊C4—91no—
109C2×#217—91no—
110C2×#217—91no—
111C2×(C2×D3)⋊C4—34no—
112C2×(C2×D3)⋊C4—94no—
113C2×(C2×D3)⋊C4—94no—
114C2×C4⋊Dic3—94no—
115C2×C4⋊Dic3C3⋊D311yescharacteristic, complemented
116C2×Dic3⋊C4—94no—
117C2×Dic3⋊C4—91no—
118C2×Dic3⋊C4—91no—
119C2×Dic3⋊C4—184no—
120C2×C4×Dic3—94no—
121C2×C4×Dic3—91no—
122#898—932no—
123#879—98no—
124C23×C32—24no—
125C23×C32C22×D314yes—
126C23×C32C22×D311yescharacteristic
127C22×C3⋊D3—22no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3C22×D312yes—
131C22×C3×D3—38no—
132C22×C3×D3—68no—
133C22×C3×D3—68no—
134C22×C3×D3—68no—
135C2×C32×C4—22no—
136C2×C32×C4—28no—
137C2×C32×C4—28no—
138C2×C32×C4—32no—
139C2×C32×C4—32no—
140C2×C32×C4—48no—
141C2×C32×C4—62no—
142C2×C32×C4—62no—
143C2×C32×C4C22×D318yes—
144C2×C32×C4C22×D312yes—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—38no—
149C2×C3⋊Dic3—38no—
150C2×C3⋊Dic3—68no—
151C2×C3⋊Dic3—68no—
152C2×C3⋊Dic3—616no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—68no—
156C2×C3⋊Dic3—68no—
157C2×C3⋊Dic3—128no—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3×Dic3—28no—
160C2×C3×Dic3—28no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—68no—
163C2×C3×Dic3C22×D318yes—
164C2×C3×Dic3C22×D318yes—
165C2×#126—271noSylow
166C2×C33#12614yescomplemented
167C2×C33C2×#4111yescharacteristic
168C2×C33C2×#4111yescharacteristic
169C2×C33C2×#4111yescharacteristic
170C3×C3⋊D3—42no—
171C24×C3—91no—
172C23×D3—34no—
173C22×C3×C4—24no—
174C22×C3×C4—34no—
175C22×C3×C4—34no—
176C22×C3×C4—91no—
177C22×C3×C4D3214yes—
178C22×C3×C4C2×C3⋊D311yescharacteristic
179C22×Dic3—34no—
180C22×Dic3—94no—
181C22×Dic3—94no—
182C22×Dic3—91no—
183C22×Dic3—91no—
184C22×Dic3—184no—
185C22×Dic3—184no—
186C22×Dic3C2×C3⋊D311yescharacteristic
187C22×Dic3C2×C3⋊D311yescharacteristic
188C3×C4⋊C4—316no—
189C3×C22⋊C4—94no—
190#217—94no—
191#217—94no—
192(C2×D3)⋊C4—316no—
193(C2×D3)⋊C4—916no—
194(C2×D3)⋊C4—916no—
195C4⋊Dic3—916no—
196C4⋊Dic3C2×C3⋊D314yescomplemented
197Dic3⋊C4—916no—
198Dic3⋊C4—94no—
199Dic3⋊C4—94no—
200Dic3⋊C4—1816no—
201C4×Dic3—916no—
202C4×Dic3—94no—
203C22×C32—24no—
204C22×C32—28no—
205C22×C32—28no—
206C22×C32—28no—
207C22×C32#23518yes—
208C22×C32#23512yes—
209C22×C32#23718yes—
210C22×C32#23712yes—
211C22×C32#23712yes—
212C22×C32#23918yes—
213C22×C32C23×D314yes—
214C22×C32C23×D311yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—22no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—44no—
219C2×C3×D3—68no—
220C2×C3×D3—68no—
221C2×C3×D3—68no—
222C2×C3×D3—1216no—
223C32×C4—22no—
224C32×C4—28no—
225C32×C4—48no—
226C32×C4—62no—
227C32×C4—62no—
228C3⋊Dic3—22no—
229C3⋊Dic3—616no—
230C3⋊Dic3—62no—
231C3⋊Dic3—62no—
232C3⋊Dic3—68no—
233C3⋊Dic3—68no—
234C3⋊Dic3—128no—
235C3×Dic3—28no—
236C3×Dic3—28no—
237C3×Dic3—68no—
238#126—278no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—271no—
241C2×C4⋊C4—271no—
242C2×C22⋊C4—91no—
243C2×C22⋊C4—271no—
244C2×C22⋊C4—271no—
245C2×C42—271no—
246C33C2×#12611yescharacteristic, complemented, Sylow
247C23×C3—24no—
248C23×C3—92no—
249C23×C3—182no—
250C23×C3—182no—
251C23×C3—182no—
252C23×C3C2×D3214yes—
253C23×C3C22×C3⋊D311yescharacteristic
254C22×D3—38no—
255C22×D3—68no—
256C22×D3—68no—
257C22×D3—68no—
258C2×C3×C4—28no—
259C2×C3×C4—22no—
260C2×C3×C4—28no—
261C2×C3×C4—38no—
262C2×C3×C4—38no—
263C2×C3×C4—48no—
264C2×C3×C4—68no—
265C2×C3×C4—68no—
266C2×C3×C4—92no—
267C2×C3×C4—182no—
268C2×C3×C4C2×D3218yes—
269C2×C3×C4C22×C3⋊D312yes—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—38no—
273C2×Dic3—68no—
274C2×Dic3—98no—
275C2×Dic3—98no—
276C2×Dic3—92no—
277C2×Dic3—92no—
278C2×Dic3—188no—
279C2×Dic3—188no—
280C2×Dic3—1816no—
281C2×Dic3—188no—
282C2×Dic3—188no—
283C2×Dic3—182no—
284C2×Dic3—182no—
285C2×Dic3—368no—
286C2×Dic3C22×C3⋊D312yes—
287C2×Dic3C22×C3⋊D312yes—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—216no—
292C2×C32#87914yescomplemented
293C2×C32#898116yescomplemented
294C2×C32C2×#23514yes—
295C2×C32C2×#23511yescharacteristic
296C2×C32C2×#23714yes—
297C2×C32C2×#23711yescharacteristic
298C2×C32C2×#23711yescharacteristic
299C2×C32C2×#23914yes—
300C3⋊D3—42no—
301C3×D3—128no—
302C24—91no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—91no—
306C22×C4—271no—
307C22×C4—271no—
308C22×C4D3×C3⋊D311yescharacteristic
309C4⋊C4—34no—
310C4⋊C4—274no—
311C4⋊C4—274no—
312C22⋊C4—94no—
313C22⋊C4—274no—
314C22⋊C4—274no—
315C42—274no—
316C22×C3—24no—
317C22×C3—28no—
318C22×C3—28no—
319C22×C3—28no—
320C22×C3—182no—
321C22×C3—182no—
322C22×C3—182no—
323C22×C3—364no—
324C22×C3#379518yes—
325C22×C3#379618yes—
326C22×C3#380418yes—
327C22×C3#382812yes—
328C22×C3#383012yes—
329C22×C3#383212yes—
330C22×C3C22×D3214yes—
331C22×C3C23×C3⋊D311yescharacteristic
332C2×D3—68no—
333C2×D3—68no—
334C2×D3—68no—
335C2×D3—1216no—
336C3×C4—28no—
337C3×C4—22no—
338C3×C4—48no—
339C3×C4—68no—
340C3×C4—68no—
341C3×C4—182no—
342Dic3—22no—
343Dic3—22no—
344Dic3—68no—
345Dic3—1816no—
346Dic3—188no—
347Dic3—188no—
348Dic3—182no—
349Dic3—182no—
350Dic3—368no—
351C32—24no—
352C32C2×#87911yescharacteristic, complemented
353C32C2×#89814yescomplemented
354C23—92no—
355C23—182no—
356C23—182no—
357C23—182no—
358C23C2×D3×C3⋊D311yescharacteristic
359C2×C4—22no—
360C2×C4—32no—
361C2×C4—32no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—92no—
365C2×C4—182no—
366C2×C4—272no—
367C2×C4—272no—
368C2×C4—542no—
369C2×C4—542no—
370C2×C4C2×D3×C3⋊D312yes—
371C2×C3—24no—
372C2×C3—24no—
373C2×C3—24no—
374C2×C3—216no—
375C2×C3—362no—
376C2×C3#63858116yescomplemented
377C2×C3#6410714yescomplemented
378C2×C3C2×#379514yes—
379C2×C3C2×#379614yes—
380C2×C3C2×#380414yes—
381C2×C3C2×#382811yescharacteristic
382C2×C3C2×#383011yescharacteristic
383C2×C3C2×#383211yescharacteristic
384D3—128no—
385C22—182no—
386C22—182no—
387C22—182no—
388C22—364no—
389C22#8901412yes—
390C22#8901712yes—
391C22#8903112yes—
392C22C22×D3×C3⋊D311yescharacteristic
393C4—22no—
394C4—62no—
395C4—62no—
396C4—182no—
397C4—542no—
398C4—542no—
399C3—24no—
400C3C2×#6385814yescomplemented
401C3C2×#6410711yescharacteristic, complemented
402C2—362no—
403C2#1172340714yesdirect product factor
404C2C2×#8901411yescharacteristic
405C2C2×#8901711yescharacteristic
406C2C2×#8903111yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88857C2semidirect
3. C2×C32×C4⋊Dic3C2semidirect
4. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. C2×Dic3×C3⋊Dic3C2semidirect
8. #11723407C2direct (N × H)
18. #88857C22semidirect
20. C32×C4⋊Dic3C22semidirect
21. Dic3⋊(C3⋊Dic3)C22semidirect
24. Dic3×C3⋊Dic3C22semidirect
36. C2×C3×C4⋊Dic3D3semidirect
88. C3×C4⋊Dic3C2×D3semidirect
115. C2×C4⋊Dic3C3⋊D3semidirect
166. C2×C33#126semidirect
196. C4⋊Dic3C2×C3⋊D3semidirect
246. C33C2×#126semidirect
292. C2×C32#879semidirect
293. C2×C32#898semidirect
352. C32C2×#879semidirect
353. C32C2×#898semidirect
376. C2×C3#63858semidirect
377. C2×C3#64107semidirect
400. C3C2×#63858semidirect
401. C3C2×#64107semidirect
403. C2#11723407direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888572
C2×C3×#37522
C2×C32×C4⋊Dic32
C2×Dic3⋊(C3⋊Dic3)2
C2×#888032
C2×Dic3×C3⋊Dic32
#117234072
C2×#641073
C2×#638583