Finite Group Databasebeta

← C2×#11723596 #13178034

Subgroups — C2×#11723596 #13178034

Order 1728.

2004 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×#88864C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×#88807C211yesmaximal, characteristic, complemented
5C2×#88806C211yesmaximal, characteristic, complemented
6C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7#11723596C218yesmaximal, direct product factor
8C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×#64088—31nomaximal
10C2×#63956—34nomaximal
11C24×C33C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18#88864C2214yes—
19C3×#3757C2214yes—
20C32×#217C2214yes—
21#88807C2214yescomplemented
22#88806C2214yescomplemented
23Dic3⋊(C3⋊Dic3)C2214yescomplemented
24Dic3×C3⋊Dic3C2214yescomplemented
25C2×C32×C22⋊C4—31no—
26C2×#3757—31no—
27C2×#3757—34no—
28C2×#3757—64no—
29C2×#3757D311yescharacteristic
30C2×C4⋊(C3⋊Dic3)—31no—
31C2×#3750—31no—
32C2×#3750—31no—
33C2×C4×C3⋊Dic3—31no—
34#64088—38no—
35C2×C3×#217—34no—
36C2×C3×#217D314yes—
37C2×#3724—34no—
38C2×Dic3⋊Dic3—34no—
39C2×Dic3⋊Dic3—34no—
40#63956—332no—
41C2×Dic32—34no—
42C23×C33—22no—
43C23×C33—22no—
44C23×C33—22no—
45C23×C33C2312yes—
46C23×C33C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—22no—
48C2×C3⋊(C3⋊Dic3)—22no—
49C2×C3⋊(C3⋊Dic3)C2312yes—
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2312yes—
55C2×C32×Dic3—22no—
56C2×C32×Dic3—22no—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C2×#879—94no—
60C2×#879—91no—
61C24×C32—24no—
62C24×C32C2×D314yes—
63C24×C32C2×D311yescharacteristic
64C22×C32×C4—31no—
65C22×C32×C4—31no—
66C22×C3⋊Dic3—31no—
67C22×C3⋊Dic3—31no—
68C22×C3⋊Dic3—34no—
69C22×C3⋊Dic3—34no—
70C22×C3⋊Dic3—64no—
71C22×C3⋊Dic3—64no—
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3×Dic3—34no—
75C22×C3×Dic3—34no—
76C22×C3×Dic3C2×D314yes—
77C22×C3×Dic3C2×D314yes—
78C32×C22⋊C4—34no—
79#3757—34no—
80#3757—316no—
81#3757—616no—
82#3757C2×D314yes—
83C4⋊(C3⋊Dic3)—34no—
84#3750—34no—
85#3750—34no—
86C4×C3⋊Dic3—34no—
87C3×#217—316no—
88C3×#217C2×D3116yes—
89#3724—316no—
90Dic3⋊Dic3—316no—
91Dic3⋊Dic3—316no—
92Dic32—316no—
93C22×C33—22no—
94C22×C33—22no—
95C22×C33—22no—
96C22×C33—44no—
97C22×C33#4112yes—
98C22×C33#4112yes—
99C22×C33#4112yes—
100C22×C33C2411yescharacteristic
101C3⋊(C3⋊Dic3)—22no—
102C3⋊(C3⋊Dic3)—22no—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3—22no—
105C32×Dic3—22no—
106C32×Dic3—22no—
107C2×C3×C22⋊C4—34no—
108C2×C3×C22⋊C4—91no—
109C2×#217—34no—
110C2×#217—94no—
111C2×#217—91no—
112C2×#217—184no—
113C2×#217C3⋊D311yescharacteristic
114C2×C4⋊Dic3—94no—
115C2×C4⋊Dic3—91no—
116C2×Dic3⋊C4—94no—
117C2×Dic3⋊C4—94no—
118C2×Dic3⋊C4—91no—
119C2×Dic3⋊C4—91no—
120C2×C4×Dic3—94no—
121C2×C4×Dic3—91no—
122#879—932no—
123#879—98no—
124C23×C32—216no—
125C23×C32—22no—
126C23×C32—28no—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32—24no—
130C23×C32—28no—
131C23×C32—28no—
132C23×C32—28no—
133C23×C32—48no—
134C23×C32—48no—
135C23×C32C22×D318yes—
136C23×C32C22×D314yes—
137C23×C32C22×D312yes—
138C23×C32C22×D311yescharacteristic
139C2×C32×C4—32no—
140C2×C32×C4—32no—
141C2×C32×C4—62no—
142C2×C32×C4—62no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—38no—
148C2×C3⋊Dic3—38no—
149C2×C3⋊Dic3—68no—
150C2×C3⋊Dic3—68no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—68no—
154C2×C3⋊Dic3—68no—
155C2×C3⋊Dic3—128no—
156C2×C3⋊Dic3—128no—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3×Dic3—28no—
160C2×C3×Dic3—28no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—38no—
163C2×C3×Dic3—68no—
164C2×C3×Dic3—68no—
165C2×C3×Dic3C22×D318yes—
166C2×C3×Dic3C22×D318yes—
167C2×#126—271noSylow
168C2×C33—42no—
169C2×C33#12614yescomplemented
170C2×C33C2×#4111yescharacteristic
171C2×C33C2×#4111yescharacteristic
172C2×C33C2×#4111yescharacteristic
173C24×C3—24no—
174C24×C3D3214yes—
175C24×C3C2×C3⋊D311yescharacteristic
176C22×C3×C4—34no—
177C22×C3×C4—34no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×Dic3—34no—
181C22×Dic3—34no—
182C22×Dic3—94no—
183C22×Dic3—94no—
184C22×Dic3—91no—
185C22×Dic3—91no—
186C22×Dic3—184no—
187C22×Dic3—184no—
188C22×Dic3C2×C3⋊D311yescharacteristic
189C22×Dic3C2×C3⋊D311yescharacteristic
190C3×C22⋊C4—316no—
191C3×C22⋊C4—94no—
192#217—316no—
193#217—916no—
194#217—94no—
195#217—1816no—
196#217C2×C3⋊D314yes—
197C4⋊Dic3—916no—
198C4⋊Dic3—94no—
199Dic3⋊C4—916no—
200Dic3⋊C4—916no—
201Dic3⋊C4—94no—
202Dic3⋊C4—94no—
203C4×Dic3—916no—
204C4×Dic3—94no—
205C22×C32—216no—
206C22×C32—22no—
207C22×C32—28no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—24no—
211C22×C32—28no—
212C22×C32—28no—
213C22×C32—28no—
214C22×C32—28no—
215C22×C32—28no—
216C22×C32—48no—
217C22×C32—48no—
218C22×C32—432no—
219C22×C32—416no—
220C22×C32—44no—
221C22×C32#23518yes—
222C22×C32#23512yes—
223C22×C32#23718yes—
224C22×C32#23718yes—
225C22×C32#23712yes—
226C22×C32#23712yes—
227C22×C32C23×D314yes—
228C22×C32C23×D311yescharacteristic
229C32×C4—62no—
230C32×C4—62no—
231C3⋊Dic3—22no—
232C3⋊Dic3—22no—
233C3⋊Dic3—62no—
234C3⋊Dic3—62no—
235C3⋊Dic3—68no—
236C3⋊Dic3—68no—
237C3⋊Dic3—128no—
238C3⋊Dic3—128no—
239C3×Dic3—28no—
240C3×Dic3—28no—
241C3×Dic3—68no—
242C3×Dic3—68no—
243#126—278no—
244C2×C4⋊C4—271no—
245C2×C4⋊C4—271no—
246C2×C4⋊C4—271no—
247C2×C22⋊C4—31no—
248C2×C22⋊C4—91no—
249C2×C22⋊C4—271no—
250C2×C42—271no—
251C33C2×#12611yescharacteristic, complemented, Sylow
252C23×C3—216no—
253C23×C3—28no—
254C23×C3—28no—
255C23×C3—28no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—28no—
260C23×C3—24no—
261C23×C3—48no—
262C23×C3—48no—
263C23×C3C2×D3218yes—
264C23×C3C2×D3214yes—
265C23×C3C22×C3⋊D312yes—
266C23×C3C22×C3⋊D311yescharacteristic
267C2×C3×C4—38no—
268C2×C3×C4—38no—
269C2×C3×C4—68no—
270C2×C3×C4—68no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—182no—
274C2×C3×C4—182no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—38no—
278C2×Dic3—38no—
279C2×Dic3—68no—
280C2×Dic3—68no—
281C2×Dic3—98no—
282C2×Dic3—98no—
283C2×Dic3—92no—
284C2×Dic3—92no—
285C2×Dic3—188no—
286C2×Dic3—188no—
287C2×Dic3—188no—
288C2×Dic3—188no—
289C2×Dic3—182no—
290C2×Dic3—182no—
291C2×Dic3—368no—
292C2×Dic3—368no—
293C2×Dic3C22×C3⋊D312yes—
294C2×Dic3C22×C3⋊D312yes—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—216no—
299C2×C32—416no—
300C2×C32—48no—
301C2×C32—42no—
302C2×C32#879116yescomplemented
303C2×C32#87914yescomplemented
304C2×C32C2×#23514yes—
305C2×C32C2×#23511yescharacteristic
306C2×C32C2×#23714yes—
307C2×C32C2×#23714yes—
308C2×C32C2×#23711yescharacteristic
309C2×C32C2×#23711yescharacteristic
310C24D3×C3⋊D311yescharacteristic
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—271no—
316C22×C4—271no—
317C4⋊C4—274no—
318C4⋊C4—274no—
319C4⋊C4—274no—
320C22⋊C4—34no—
321C22⋊C4—94no—
322C22⋊C4—274no—
323C42—274no—
324C22×C3—216no—
325C22×C3—28no—
326C22×C3—28no—
327C22×C3—28no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—24no—
332C22×C3—28no—
333C22×C3—28no—
334C22×C3—28no—
335C22×C3—48no—
336C22×C3—48no—
337C22×C3—432no—
338C22×C3—44no—
339C22×C3—416no—
340C22×C3#380418yes—
341C22×C3#380418yes—
342C22×C3#380518yes—
343C22×C3#382812yes—
344C22×C3#383012yes—
345C22×C3#383012yes—
346C22×C3C22×D3214yes—
347C22×C3C23×C3⋊D311yescharacteristic
348C3×C4—68no—
349C3×C4—68no—
350C3×C4—182no—
351C3×C4—182no—
352Dic3—22no—
353Dic3—22no—
354Dic3—68no—
355Dic3—68no—
356Dic3—188no—
357Dic3—188no—
358Dic3—182no—
359Dic3—182no—
360Dic3—368no—
361Dic3—368no—
362C32—24no—
363C32C2×#87914yescomplemented
364C32C2×#87911yescharacteristic, complemented
365C23—22no—
366C23—22no—
367C23—22no—
368C23C2×D3×C3⋊D312yes—
369C23C2×D3×C3⋊D311yescharacteristic
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—92no—
375C2×C4—92no—
376C2×C4—182no—
377C2×C4—182no—
378C2×C4—272no—
379C2×C4—272no—
380C2×C4—542no—
381C2×C4—542no—
382C2×C3—24no—
383C2×C3—24no—
384C2×C3—24no—
385C2×C3—216no—
386C2×C3—416no—
387C2×C3—42no—
388C2×C3—48no—
389C2×C3#63956116yescomplemented
390C2×C3#6408814yescomplemented
391C2×C3C2×#380414yes—
392C2×C3C2×#380414yes—
393C2×C3C2×#380514yes—
394C2×C3C2×#382811yescharacteristic
395C2×C3C2×#383011yescharacteristic
396C2×C3C2×#383011yescharacteristic
397C22—22no—
398C22—22no—
399C22—22no—
400C22—44no—
401C22#8902812yes—
402C22#8902912yes—
403C22#8903112yes—
404C22C22×D3×C3⋊D311yescharacteristic
405C4—62no—
406C4—62no—
407C4—182no—
408C4—182no—
409C4—542no—
410C4—542no—
411C3—24no—
412C3C2×#6395614yescomplemented
413C3C2×#6408811yescharacteristic, complemented
414C2—42no—
415C2#1172359614yesdirect product factor
416C2C2×#8902811yescharacteristic
417C2C2×#8902911yescharacteristic
418C2C2×#8903111yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
Fitting, prime radicalC24×C33C22
socleC23×C33C23
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88807C2semidirect
5. C2×#88806C2semidirect
6. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. #11723596C2direct (N × H)
8. C2×Dic3×C3⋊Dic3C2semidirect
21. #88807C22semidirect
22. #88806C22semidirect
23. Dic3⋊(C3⋊Dic3)C22semidirect
24. Dic3×C3⋊Dic3C22semidirect
169. C2×C33#126semidirect
251. C33C2×#126semidirect
302. C2×C32#879semidirect
303. C2×C32#879semidirect
363. C32C2×#879semidirect
364. C32C2×#879semidirect
389. C2×C3#63956semidirect
390. C2×C3#64088semidirect
412. C3C2×#63956semidirect
413. C3C2×#64088semidirect
415. C2#11723596direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888642
C2×C3×#37572
C2×C32×#2172
C2×#888072
C2×#888062
C2×Dic3⋊(C3⋊Dic3)2
#117235962
C2×Dic3×C3⋊Dic32
C2×#640883
C2×#639563