Finite Group Databasebeta

← C2×#11722074 #13161807

Subgroups — C2×#11722074 #13161807

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C3×#3748C212yesmaximal
3#11722074C214yesmaximal, direct product factor
4C2×#63561—31nomaximal
5C2×#63560—32nomaximal
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C412yes—
8C2×C3×#634—22no—
9C2×C3×#634—22no—
10C2×C3×#634C412yes—
11C2×C3×#634C2212yes—
12C3×#3748—22no—
13C3×#3748C2214yes—
14C2×C3×C3⋊(C3⋊C8)—22no—
15C22×#634D311yescharacteristic
16C22×C3×Dic6—32no—
17C22×C3×Dic6—61no—
18C2×#3748D312yes—
19C2×C3×#208—32no—
20C2×C3×#208—32no—
21C2×C3×#208—62no—
22#63561—34no—
23#63560—38no—
24C23×C33C2×C411yescharacteristic
25C2×C33×C4C2×C411yescharacteristic
26C2×C33×C4D412yes—
27C2×C33×C4D412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2×C414yes—
32C3×#634—24no—
33C3×#634—24no—
34C3×C3⋊(C3⋊C8)—24no—
35C2×#836—91no—
36C2×#824—92no—
37C22×C32×C4—22no—
38C22×C32×C4—21no—
39C22×C32×C4—41no—
40C22×C32×C4C2×D312yes—
41C22×C32×C4C2×D311yescharacteristic
42C22×C3⋊Dic3Dic312yes—
43C2×#634—22no—
44C2×#634—22no—
45C2×#634Dic312yes—
46C2×#634C2×D312yes—
47C22×C3×Dic3—34no—
48C22×C3×Dic3—62no—
49C2×C3×Dic6—34no—
50C2×C3×Dic6—34no—
51C2×C3×Dic6—62no—
52C2×C3×Dic6—62no—
53C2×C3×Dic6—64no—
54C2×C3×Dic6—64no—
55C2×C3×Dic6—62no—
56C2×C3×Dic6—62no—
57C2×C3×Dic6—62no—
58C2×C3×Dic6—62no—
59#3748—22no—
60#3748C2×D314yes—
61C2×C3⋊(C3⋊C8)—22no—
62C3×#208—34no—
63C3×#208—34no—
64C3×#208—64no—
65C3×#208—64no—
66C3×#208—62no—
67C3×#208—62no—
68C2×C3×C3⋊C8—64no—
69C2×C3×C3⋊C8—62no—
70C2×C3×C3⋊C8—62no—
71C22×C33—22no—
72C22×C33C22⋊C411yescharacteristic
73C22×C33C22⋊C411yescharacteristic
74C22×C33C22×C411yescharacteristic
75C33×C4C22⋊C412yes—
76C33×C4C2×D412yes—
77C3×C3⋊Dic3—24no—
78C22×C3×Q8—91no—
79C22×Dic6—32no—
80C22×Dic6—61no—
81C2×C3×M16—92no—
82C2×#208—32no—
83C2×#208—32no—
84C2×#208—62no—
85#836—94no—
86#824—98no—
87C23×C32—22no—
88C23×C32—21no—
89C23×C32—41no—
90C23×C32C4×D312yes—
91C23×C32C2×Dic311yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—42no—
104C2×C32×C4—42no—
105C2×C32×C4—41no—
106C2×C32×C4C4×D312yes—
107C2×C32×C4D1212yes—
108C2×C32×C4D1212yes—
109C2×C32×C4C2×Dic311yescharacteristic
110C2×C32×C4#6712yes—
111C2×C32×C4#6712yes—
112C2×C32×C4#6712yes—
113C2×C32×C4#6712yes—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3C2×Dic314yes—
119#634—24no—
120#634—24no—
121C2×C3×Dic3—38no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×Dic6—68no—
128C3×Dic6—68no—
129C3×Dic6—64no—
130C3×Dic6—64no—
131C3×Dic6—64no—
132C3×Dic6—64no—
133C3⋊(C3⋊C8)—24no—
134C3×C3⋊C8—68no—
135C3×C3⋊C8—64no—
136C3×C3⋊C8—64no—
137C2×#101—271noSylow
138C2×C33—21no—
139C2×C33—21no—
140C2×C33#10112yescomplemented
141C2×C33C2×C22⋊C411yescharacteristic
142C2×C3×Q8—92no—
143C2×C3×Q8—92no—
144C2×C3×Q8—182no—
145C2×C3×Q8—182no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—41no—
149C22×C3×C4—92no—
150C22×C3×C4D3212yes—
151C22×C3×C4D3211yescharacteristic
152C22×Dic3—34no—
153C22×Dic3—62no—
154C2×Dic6—34no—
155C2×Dic6—34no—
156C2×Dic6—62no—
157C2×Dic6—62no—
158C2×Dic6—64no—
159C2×Dic6—64no—
160C2×Dic6—62no—
161C2×Dic6—62no—
162C2×Dic6—62no—
163C2×Dic6—62no—
164C3×M16—94no—
165C3×M16—182no—
166C2×C3×C8—182no—
167#208—34no—
168#208—34no—
169#208—64no—
170#208—64no—
171#208—62no—
172#208—62no—
173C2×C3⋊C8—64no—
174C2×C3⋊C8—62no—
175C2×C3⋊C8—62no—
176C22×C32—22no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—44no—
189C22×C32—44no—
190C22×C32(C2×D3)⋊C412yes—
191C22×C32(C2×D3)⋊C412yes—
192C22×C32#21711yescharacteristic
193C22×C32#21711yescharacteristic
194C22×C32C2×C4×D312yes—
195C22×C32C22×Dic311yescharacteristic
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4—42no—
203C32×C4—42no—
204C32×C4(C2×D3)⋊C412yes—
205C32×C4(C2×D3)⋊C412yes—
206C32×C4#21712yes—
207C32×C4C2×D1212yes—
208C32×C4C2×#6712yes—
209C32×C4C2×#6712yes—
210C3⋊Dic3—24no—
211C3×Dic3—68no—
212C3×Dic3—68no—
213C22×Q8—91no—
214C2×M16—92no—
215#101—274no—
216C33C2×#10111yescharacteristic, complemented, Sylow
217C23×C3—21no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3D3×Dic312yes—
221C23×C3#62411yescharacteristic
222C3×Q8—184no—
223C3×Q8—184no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—94no—
239C2×C3×C4—182no—
240C2×C3×C4—182no—
241C2×C3×C4D3×Dic312yes—
242C2×C3×C4#62411yescharacteristic
243C2×C3×C4#62512yes—
244C2×C3×C4#62512yes—
245C2×C3×C4#62612yes—
246C2×C3×C4#62612yes—
247C2×C3×C4#62612yes—
248C2×C3×C4#62612yes—
249C2×C3×C4C2×D3212yes—
250C2×C3×C4C2×D3211yescharacteristic
251C2×Dic3—38no—
252C2×Dic3—64no—
253C2×Dic3—64no—
254C2×Dic3—64no—
255C2×Dic3—64no—
256C2×Dic3—64no—
257Dic6—68no—
258Dic6—68no—
259Dic6—64no—
260Dic6—64no—
261Dic6—64no—
262Dic6—64no—
263C3×C8—184no—
264C3⋊C8—68no—
265C3⋊C8—64no—
266C3⋊C8—64no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—22no—
272C2×C32—22no—
273C2×C32—24no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32—42no—
278C2×C32—41no—
279C2×C32—42no—
280C2×C32—42no—
281C2×C32—42no—
282C2×C32—42no—
283C2×C32#82414yescomplemented
284C2×C32#83612yescomplemented
285C2×C32C2×(C2×D3)⋊C412yes—
286C2×C32C2×#21711yescharacteristic
287C2×Q8—92no—
288C2×Q8—92no—
289C2×Q8—182no—
290C2×Q8—182no—
291C22×C4—92no—
292C22×C4#111311yescharacteristic
293M16—94no—
294M16—182no—
295C2×C8—182no—
296C22×C3—24no—
297C22×C3—24no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—21no—
302C22×C3—22no—
303C22×C3—21no—
304C22×C3—22no—
305C22×C3—44no—
306C22×C3—44no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3(C2×D3)⋊Dic312yes—
311C22×C3(C2×D3)⋊Dic312yes—
312C22×C3#372211yescharacteristic
313C22×C3#372211yescharacteristic
314C22×C3C2×D3×Dic312yes—
315C22×C3C2×#62411yescharacteristic
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—42no—
324C3×C4—184no—
325C3×C4(C2×D3)⋊Dic312yes—
326C3×C4(C2×D3)⋊Dic312yes—
327C3×C4#372212yes—
328C3×C4C2×#62512yes—
329C3×C4C2×#62612yes—
330C3×C4C2×#62612yes—
331Dic3—68no—
332Dic3—68no—
333C32—22no—
334C32—21no—
335C32—41no—
336C32C2×#82412yescomplemented
337C32C2×#83611yescharacteristic, complemented
338C23#629411yescharacteristic
339Q8—184no—
340Q8—184no—
341C2×C4—94no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4#629411yescharacteristic
345C2×C4#629512yes—
346C2×C4#629512yes—
347C2×C4C2×#111311yescharacteristic
348C8—184no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—24no—
357C2×C3—21no—
358C2×C3—22no—
359C2×C3—42no—
360C2×C3—42no—
361C2×C3—42no—
362C2×C3—42no—
363C2×C3—42no—
364C2×C3—41no—
365C2×C3#6356014yescomplemented
366C2×C3#6356112yescomplemented
367C2×C3C2×(C2×D3)⋊Dic312yes—
368C2×C3C2×#372211yescharacteristic
369C22—22no—
370C22#8881411yescharacteristic
371C22#8881411yescharacteristic
372C22C2×#629411yescharacteristic
373C4—184no—
374C4#8881412yes—
375C4C2×#629512yes—
376C3—21no—
377C3—22no—
378C3—41no—
379C3C2×#6356012yescomplemented
380C3C2×#6356111yescharacteristic, complemented
381C2—21no—
382C2—21no—
383C2#1172207412yesdirect product factor
384C2C2×#8881411yescharacteristic
subgroupG/H
centerC22#88814
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11722074C2direct (N × H)
140. C2×C33#101semidirect
216. C33C2×#101semidirect
283. C2×C32#824semidirect
284. C2×C32#836semidirect
336. C32C2×#824semidirect
337. C32C2×#836semidirect
365. C2×C3#63560semidirect
366. C2×C3#63561semidirect
379. C3C2×#63560semidirect
380. C3C2×#63561semidirect
383. C2#11722074direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C3×#37482
#117220742
C2×#635613
C2×#635603