Finite Group Databasebeta

← C2×#11723679 #13178503

Subgroups — C2×#11723679 #13178503

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C212yesmaximal
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×#88815C211yesmaximal, characteristic
4C2×#88815C211yesmaximal, characteristic
5C2×#88815C212yesmaximal
6#11723679C218yesmaximal, direct product factor
7C2×#63870—31nomaximal
8C2×#63867—32nomaximal
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2212yes—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C3×#3750C2218yes—
15C3×C4×C3⋊Dic3C2214yes—
16#88815C2214yes—
17#88815C2214yes—
18#88815C2218yes—
19C2×#3750D312yes—
20C2×C4×C3⋊Dic3D311yescharacteristic
21C2×C3×Dic3⋊C4—32no—
22C2×C3×Dic3⋊C4—32no—
23C2×C3×Dic3⋊C4—62no—
24C2×C3×C4×Dic3—32no—
25C2×C3×C4×Dic3—61no—
26C2×#3724—32no—
27C2×#3724—31no—
28C2×#3724—31no—
29C2×Dic3⋊Dic3—32no—
30C2×Dic3⋊Dic3—32no—
31C2×Dic3⋊Dic3—32no—
32C2×Dic3⋊Dic3—32no—
33#63870—38no—
34#63867—316no—
35C23×C33C2311yescharacteristic
36C2×C33×C4—22no—
37C2×C33×C4C2312yes—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3—24no—
41C2×C3×C3⋊Dic3Q814yes—
42C2×C3×C3⋊Dic3C2312yes—
43C2×C3×C3⋊Dic3C2314yes—
44C2×C3×C3⋊Dic3C2314yes—
45C2×C3×C3⋊Dic3C2312yes—
46C2×#889—92no—
47C2×#870—91no—
48C22×C32×C4—21no—
49C22×C32×C4—22no—
50C22×C32×C4—41no—
51C22×C32×C4C2×D312yes—
52C22×C32×C4C2×D311yescharacteristic
53C22×C3⋊Dic3C2×D311yescharacteristic
54C22×C3⋊Dic3C2×D312yes—
55C22×C3⋊Dic3C2×D312yes—
56C22×C3⋊Dic3C2×D311yescharacteristic
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—62no—
66C22×C3×Dic3—62no—
67#3750C2×D318yes—
68C4×C3⋊Dic3C2×D314yes—
69C3×Dic3⋊C4—38no—
70C3×Dic3⋊C4—38no—
71C3×Dic3⋊C4—68no—
72C3×C4×Dic3—38no—
73C3×C4×Dic3—64no—
74#3724—38no—
75#3724—34no—
76#3724—34no—
77Dic3⋊Dic3—38no—
78Dic3⋊Dic3—38no—
79Dic3⋊Dic3—38no—
80Dic3⋊Dic3—38no—
81C22×C33C2×Q812yes—
82C22×C33#4112yes—
83C22×C33#4112yes—
84C22×C33C2411yescharacteristic
85C33×C4—22no—
86C3×C3⋊Dic3—22no—
87C3×C3⋊Dic3—24no—
88C3×C3⋊Dic3—24no—
89C3×C3⋊Dic3C2×Q814yes—
90C2×C3×C4⋊C4—92no—
91C2×C3×C42—91no—
92C2×C4⋊Dic3—91no—
93C2×C4⋊Dic3—91no—
94C2×C4⋊Dic3—92no—
95C2×Dic3⋊C4—32no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4—62no—
98C2×Dic3⋊C4—92no—
99C2×Dic3⋊C4—92no—
100C2×Dic3⋊C4—92no—
101C2×Dic3⋊C4—92no—
102C2×C4×Dic3—32no—
103C2×C4×Dic3—61no—
104#889—916no—
105#870—98no—
106C23×C32—22no—
107C23×C32—21no—
108C23×C32—41no—
109C23×C32C22×D311yescharacteristic
110C23×C32C22×D312yes—
111C2×C32×C4—22no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—24no—
116C2×C32×C4—42no—
117C2×C32×C4—44no—
118C2×C32×C4—44no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D312yes—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—24no—
123C2×C3⋊Dic3—24no—
124C2×C3⋊Dic3Dic614yes—
125C2×C3⋊Dic3C22×D312yes—
126C2×C3⋊Dic3C22×D314yes—
127C2×C3⋊Dic3C22×D314yes—
128C2×C3⋊Dic3C22×D312yes—
129C2×C3×Dic3—38no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—68no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—124no—
149C2×#125—271noSylow
150C2×C33#12514yescomplemented
151C2×C33C22×Q811yescharacteristic
152C2×C33C2×#4111yescharacteristic
153C2×C33C2×#4111yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—41no—
157C22×C3×C4—91no—
158C22×C3×C4—91no—
159C22×C3×C4—92no—
160C22×C3×C4—92no—
161C22×C3×C4D3212yes—
162C22×C3×C4D3211yescharacteristic
163C22×Dic3—32no—
164C22×Dic3—32no—
165C22×Dic3—32no—
166C22×Dic3—32no—
167C22×Dic3—32no—
168C22×Dic3—32no—
169C22×Dic3—61no—
170C22×Dic3—61no—
171C22×Dic3—62no—
172C22×Dic3—62no—
173C3×C4⋊C4—98no—
174C3×C42—94no—
175C4⋊Dic3—94no—
176C4⋊Dic3—94no—
177C4⋊Dic3—98no—
178Dic3⋊C4—38no—
179Dic3⋊C4—38no—
180Dic3⋊C4—68no—
181Dic3⋊C4—98no—
182Dic3⋊C4—98no—
183Dic3⋊C4—98no—
184Dic3⋊C4—98no—
185C4×Dic3—38no—
186C4×Dic3—64no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—24no—
190C22×C32—22no—
191C22×C32—24no—
192C22×C32—22no—
193C22×C32—24no—
194C22×C32—22no—
195C22×C32—42no—
196C22×C32—42no—
197C22×C32—42no—
198C22×C32—41no—
199C22×C32C2×Dic612yes—
200C22×C32#23512yes—
201C22×C32#23514yes—
202C22×C32#23512yes—
203C22×C32#23714yes—
204C22×C32D3×Q814yes—
205C22×C32C23×D311yescharacteristic
206C22×C32C23×D312yes—
207C32×C4—22no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—44no—
211C32×C4—44no—
212C3⋊Dic3—22no—
213C3⋊Dic3—24no—
214C3⋊Dic3—24no—
215C3⋊Dic3C2×Dic614yes—
216C3×Dic3—38no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—68no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C3×Dic3—64no—
223C3×Dic3—64no—
224C3×Dic3—64no—
225C3×Dic3—124no—
226#125—278no—
227C2×C4⋊C4—92no—
228C2×C4⋊C4—271no—
229C2×C4⋊C4—271no—
230C2×C4⋊C4—272no—
231C2×C42—91no—
232C33C2×#12511yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—22no—
235C23×C3—41no—
236C23×C3C2×D3212yes—
237C23×C3C2×D3211yescharacteristic
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—44no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4—92no—
247C2×C3×C4—94no—
248C2×C3×C4—92no—
249C2×C3×C4—94no—
250C2×C3×C4—94no—
251C2×C3×C4—184no—
252C2×C3×C4—182no—
253C2×C3×C4—184no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C2×D3212yes—
256C2×Dic3—34no—
257C2×Dic3—38no—
258C2×Dic3—34no—
259C2×Dic3—34no—
260C2×Dic3—34no—
261C2×Dic3—34no—
262C2×Dic3—34no—
263C2×Dic3—64no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268C2×Dic3—64no—
269C2×Dic3—68no—
270C2×Dic3—64no—
271C2×Dic3—64no—
272C2×Dic3—64no—
273C2×Dic3—64no—
274C2×Dic3—64no—
275C2×Dic3—124no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—22no—
281C2×C32—21no—
282C2×C32—28no—
283C2×C32—24no—
284C2×C32—44no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32#87014yescomplemented
289C2×C32#88918yescomplemented
290C2×C32C22×Dic611yescharacteristic
291C2×C32C2×#23511yescharacteristic
292C2×C32C2×#23512yes—
293C2×C32C2×#23511yescharacteristic
294C2×C32C2×#23712yes—
295C2×C32C2×D3×Q812yes—
296C22×C4—91no—
297C22×C4—91no—
298C22×C4—92no—
299C22×C4—92no—
300C22×C4#111311yescharacteristic
301C4⋊C4—98no—
302C4⋊C4—274no—
303C4⋊C4—274no—
304C4⋊C4—278no—
305C42—94no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—24no—
310C22×C3—22no—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—22no—
314C22×C3—41no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3D3×Dic614yes—
319C22×C3#379712yes—
320C22×C3#379814yes—
321C22×C3#379812yes—
322C22×C3#380414yes—
323C22×C3#380512yes—
324C22×C3C22×D3212yes—
325C22×C3C22×D3211yescharacteristic
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—22no—
329C3×C4—44no—
330C3×C4—44no—
331C3×C4—94no—
332C3×C4—184no—
333C3×C4—182no—
334C3×C4—184no—
335Dic3—38no—
336Dic3—64no—
337Dic3—64no—
338Dic3—68no—
339Dic3—64no—
340Dic3—64no—
341Dic3—64no—
342Dic3—64no—
343Dic3—64no—
344Dic3—124no—
345C32—22no—
346C32—21no—
347C32—41no—
348C32C2×#87011yescharacteristic, complemented
349C32C2×#88912yescomplemented
350C23C2×#111311yescharacteristic
351C2×C4—22no—
352C2×C4—92no—
353C2×C4—94no—
354C2×C4—92no—
355C2×C4—94no—
356C2×C4—94no—
357C2×C4—182no—
358C2×C4—184no—
359C2×C4—184no—
360C2×C4C2×#111312yes—
361C2×C3—21no—
362C2×C3—22no—
363C2×C3—21no—
364C2×C3—22no—
365C2×C3—22no—
366C2×C3—21no—
367C2×C3—24no—
368C2×C3—28no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—44no—
373C2×C3#6386718yescomplemented
374C2×C3#6387014yescomplemented
375C2×C3C2×D3×Dic612yes—
376C2×C3C2×#379711yescharacteristic
377C2×C3C2×#379812yes—
378C2×C3C2×#379811yescharacteristic
379C2×C3C2×#380412yes—
380C2×C3C2×#380511yescharacteristic
381C22#8904012yes—
382C22#8904212yes—
383C22#8904612yes—
384C22C22×#111311yescharacteristic
385C4—22no—
386C4—94no—
387C4—182no—
388C4—184no—
389C4—184no—
390C3—21no—
391C3—22no—
392C3—41no—
393C3C2×#6386712yescomplemented
394C3C2×#6387011yescharacteristic, complemented
395C2#1172367914yesdirect product factor
396C2C2×#8904011yescharacteristic
397C2C2×#8904211yescharacteristic
398C2C2×#8904611yescharacteristic
subgroupG/H
centerC23C2×#1113
commutator, cosocleC22×C33C24
FrattiniC22C22×#1113
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C4#1113
hypocenterC33C2×#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11723679C2direct (N × H)
150. C2×C33#125semidirect
232. C33C2×#125semidirect
288. C2×C32#870semidirect
289. C2×C32#889semidirect
348. C32C2×#870semidirect
349. C32C2×#889semidirect
373. C2×C3#63867semidirect
374. C2×C3#63870semidirect
393. C3C2×#63867semidirect
394. C3C2×#63870semidirect
395. C2#11723679direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
C2×#888152
#117236792
C2×#638703
C2×#638673