Finite Group Databasebeta

← C2×C3×#1740 #10638960

Subgroups — C2×C3×#1740 #10638960

Order 768.

2350 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#117C212yesmaximal
2C22×C3×#117C211yesmaximal, characteristic
3C3×#1740C218yesmaximal, direct product factor
4C2×C3×#342C214yesmaximal
5C2×#1740C311yesmaximal, characteristic, direct product factor, Sylow
6C24×C3×C4C2211yescharacteristic
7C2×C3×#117—28no—
8C2×C3×#117—22no—
9C2×C3×#117—24no—
10C2×C3×#117—22no—
11C2×C3×#117—24no—
12C2×C3×#117C4116yescomplemented
13C2×C3×#117C414yescomplemented
14C2×C3×#117C2218yes—
15C2×C3×#117C2214yes—
16C22×C3×C4⋊C4—22no—
17C22×C3×C4⋊C4C412yescomplemented
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2214yes—
20C22×C3×C42—22no—
21C22×C3×C42C412yescomplemented
22C3×#342C22116yes—
23C22×#117C2×C312yes—
24C22×#117C2×C311yescharacteristic
25#1740C2×C318yesdirect product factor
26C2×#342C2×C314yes—
27C25×C3C2311yescharacteristic
28C23×C3×C4—24no—
29C23×C3×C4C2×C414yes—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4C2×C418yes—
32C23×C3×C4C2×C412yes—
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D412yes—
35C23×C3×C4D414yes—
36C23×C3×C4D414yes—
37C23×C3×C4D412yes—
38C23×C3×C4Q814yes—
39C23×C3×C4C2312yes—
40C3×#117—232no—
41C3×#117—28no—
42C3×#117—28no—
43C3×#117—216no—
44C3×#117—28no—
45C3×#117—28no—
46C3×#117C2×C4132yescomplemented
47C2×C3×C4⋊C4—216no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4—28no—
51C2×C3×C4⋊C4C2×C418yescomplemented
52C2×C3×C22⋊C4—24no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C22⋊C4—28no—
56C2×C3×C22⋊C4C2×C414yes—
57C2×C3×C22⋊C4C2×C4116yes—
58C2×C3×C22⋊C4C2314yes—
59C2×C3×C22⋊C4C2318yes—
60C2×C3×C42—216no—
61C2×C3×C42—22no—
62C2×C3×C42—24no—
63C2×C3×C42—22no—
64C2×C3×C42—28no—
65C2×C3×C42C2×C418yescomplemented
66C24×C4C22×C311yescharacteristic
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—22no—
70C2×#117—28no—
71C2×#117—24no—
72C2×#117C3×C4116yescomplemented
73C2×#117C3×C414yescomplemented
74C2×#117C22×C318yes—
75C2×#117C22×C314yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C3×C412yescomplemented
78C22×C22⋊C4C22×C312yes—
79C22×C22⋊C4C22×C314yes—
80C22×C42—22no—
81C22×C42C3×C412yescomplemented
82#342C22×C3116yes—
83C24×C3—44no—
84C24×C3C22×C411yescharacteristic
85C24×C3C22×C414yes—
86C24×C3C22×C411yescharacteristic
87C24×C3C2×D412yes—
88C24×C3C2×D412yes—
89C24×C3C2×D412yes—
90C24×C3C2×Q812yes—
91C24×C3C2411yescharacteristic
92C22×C3×C4—24no—
93C22×C3×C4—24no—
94C22×C3×C4—216no—
95C22×C3×C4—28no—
96C22×C3×C4—416no—
97C22×C3×C4—48no—
98C22×C3×C4C4214yes—
99C22×C3×C4C4218yes—
100C22×C3×C4C4214yes—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C412yes—
103C22×C3×C4C22⋊C418yes—
104C22×C3×C4C22⋊C418yes—
105C22×C3×C4C22⋊C412yes—
106C22×C3×C4C22⋊C412yes—
107C22×C3×C4C22⋊C414yes—
108C22×C3×C4C22⋊C4116yes—
109C22×C3×C4C22⋊C414yes—
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C4⋊C414yes—
112C22×C3×C4C4⋊C418yes—
113C22×C3×C4C4⋊C418yes—
114C22×C3×C4C4⋊C4116yes—
115C22×C3×C4C4⋊C418yes—
116C22×C3×C4C4⋊C414yes—
117C22×C3×C4C22×C418yes—
118C22×C3×C4C22×C412yes—
119C22×C3×C4C22×C412yes—
120C22×C3×C4C22×C418yes—
121C22×C3×C4C2×D414yes—
122C22×C3×C4C2×D414yes—
123C22×C3×C4C2×D414yes—
124C22×C3×C4C2×Q814yes—
125C3×C4⋊C4—232no—
126C3×C4⋊C4—216no—
127C3×C22⋊C4—28no—
128C3×C22⋊C4—216no—
129C3×C22⋊C4—28no—
130C3×C22⋊C4—28no—
131C3×C22⋊C4C22×C4116yes—
132C3×C42—232no—
133C3×C42—28no—
134C3×C42—28no—
135C25C23×C311yescharacteristic
136C23×C4—24no—
137C23×C4C2×C3×C414yes—
138C23×C4C2×C3×C411yescharacteristic
139C23×C4C2×C3×C418yes—
140C23×C4C2×C3×C412yes—
141C23×C4C2×C3×C411yescharacteristic
142C23×C4C3×D412yes—
143C23×C4C3×D414yes—
144C23×C4C3×D414yes—
145C23×C4C3×D412yes—
146C23×C4C3×Q814yes—
147C23×C4C23×C312yes—
148#117—232no—
149#117—28no—
150#117—28no—
151#117—216no—
152#117—28no—
153#117—28no—
154#117C2×C3×C4132yescomplemented
155C2×C4⋊C4—216no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4C2×C3×C418yescomplemented
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4C2×C3×C414yes—
165C2×C22⋊C4C2×C3×C4116yes—
166C2×C22⋊C4C23×C314yes—
167C2×C22⋊C4C23×C318yes—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—216no—
172C2×C42—28no—
173C2×C42C2×C3×C418yescomplemented
174C23×C3—24no—
175C23×C3—28no—
176C23×C3—42no—
177C23×C3—48no—
178C23×C3—44no—
179C23×C3—44no—
180C23×C3—44no—
181C23×C3—42no—
182C23×C3C2×C4212yes—
183C23×C3C2×C4212yes—
184C23×C3C2×C22⋊C412yes—
185C23×C3C2×C22⋊C411yescharacteristic
186C23×C3C2×C22⋊C414yes—
187C23×C3C2×C22⋊C414yes—
188C23×C3C2×C22⋊C411yescharacteristic
189C23×C3C2×C4⋊C412yes—
190C23×C3C2×C4⋊C414yes—
191C23×C3C2×C4⋊C412yes—
192C23×C3C2×C4⋊C414yes—
193C23×C3C23×C411yescharacteristic
194C23×C3C23×C412yes—
195C23×C3C22×D412yes—
196C23×C3C22×D411yescharacteristic
197C23×C3C22×Q811yescharacteristic
198C2×C3×C4—28no—
199C2×C3×C4—216no—
200C2×C3×C4—416no—
201C2×C3×C4—432no—
202C2×C3×C4—48no—
203C2×C3×C4—48no—
204C2×C3×C4—48no—
205C2×C3×C4#9518yes—
206C2×C3×C4#95116yes—
207C2×C3×C4#9518yes—
208C2×C3×C4#95116yes—
209C2×C3×C4#9518yes—
210C2×C3×C4#9514yes—
211C2×C3×C4#9514yes—
212C2×C3×C4C2×C4214yes—
213C2×C3×C4C2×C4214yes—
214C2×C3×C4C2×C22⋊C418yes—
215C2×C3×C4C2×C22⋊C412yes—
216C2×C3×C4C2×C22⋊C412yes—
217C2×C3×C4C2×C22⋊C418yes—
218C2×C3×C4C2×C22⋊C414yes—
219C2×C3×C4C2×C4⋊C414yes—
220C2×C3×C4C2×C4⋊C414yes—
221C2×C3×C4C2×C4⋊C418yes—
222C2×C3×C4C2×C4⋊C418yes—
223C24—44no—
224C24C22×C3×C411yescharacteristic
225C24C22×C3×C414yes—
226C24C22×C3×C411yescharacteristic
227C24C2×C3×D412yes—
228C24C2×C3×D412yes—
229C24C2×C3×D412yes—
230C24C2×C3×Q812yes—
231C24C24×C311yescharacteristic
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—216no—
235C22×C4—28no—
236C22×C4—416no—
237C22×C4—48no—
238C22×C4C3×C4214yes—
239C22×C4C3×C4214yes—
240C22×C4C3×C4218yes—
241C22×C4C3×C22⋊C412yes—
242C22×C4C3×C22⋊C418yes—
243C22×C4C3×C22⋊C411yescharacteristic
244C22×C4C3×C22⋊C411yescharacteristic
245C22×C4C3×C22⋊C418yes—
246C22×C4C3×C22⋊C412yes—
247C22×C4C3×C22⋊C412yes—
248C22×C4C3×C22⋊C414yes—
249C22×C4C3×C22⋊C414yes—
250C22×C4C3×C22⋊C4116yes—
251C22×C4C3×C4⋊C414yes—
252C22×C4C3×C4⋊C418yes—
253C22×C4C3×C4⋊C418yes—
254C22×C4C3×C4⋊C414yes—
255C22×C4C3×C4⋊C418yes—
256C22×C4C3×C4⋊C4116yes—
257C22×C4C22×C3×C418yes—
258C22×C4C22×C3×C412yes—
259C22×C4C22×C3×C412yes—
260C22×C4C22×C3×C418yes—
261C22×C4C2×C3×D414yes—
262C22×C4C2×C3×D414yes—
263C22×C4C2×C3×D414yes—
264C22×C4C2×C3×Q814yes—
265C4⋊C4—232no—
266C4⋊C4—216no—
267C22⋊C4—28no—
268C22⋊C4—216no—
269C22⋊C4—28no—
270C22⋊C4—28no—
271C22⋊C4C22×C3×C4116yes—
272C42—232no—
273C42—28no—
274C42—28no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—24no—
278C22×C3—24no—
279C22×C3—28no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3—28no—
283C22×C3—42no—
284C22×C3—44no—
285C22×C3—42no—
286C22×C3—44no—
287C22×C3—42no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3C2×#9512yes—
291C22×C3C2×#9514yes—
292C22×C3C2×#9512yes—
293C22×C3#41014yes—
294C22×C3C22×C4211yescharacteristic
295C22×C3C22×C22⋊C412yes—
296C22×C3C22×C22⋊C411yescharacteristic
297C22×C3C22×C4⋊C411yescharacteristic
298C22×C3C22×C4⋊C412yes—
299C3×C4—416no—
300C3×C4—48no—
301C3×C4C2×#9514yes—
302C3×C4C2×#9518yes—
303C3×C4C2×#9514yes—
304C23—24no—
305C23—28no—
306C23—48no—
307C23—42no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23C2×C3×C4212yes—
313C23C2×C3×C4212yes—
314C23C2×C3×C22⋊C412yes—
315C23C2×C3×C22⋊C414yes—
316C23C2×C3×C22⋊C411yescharacteristic
317C23C2×C3×C22⋊C414yes—
318C23C2×C3×C22⋊C411yescharacteristic
319C23C2×C3×C4⋊C412yes—
320C23C2×C3×C4⋊C414yes—
321C23C2×C3×C4⋊C414yes—
322C23C2×C3×C4⋊C412yes—
323C23C23×C3×C411yescharacteristic
324C23C23×C3×C412yes—
325C23C22×C3×D412yes—
326C23C22×C3×D411yescharacteristic
327C23C22×C3×Q811yescharacteristic
328C2×C4—28no—
329C2×C4—216no—
330C2×C4—432no—
331C2×C4—416no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4C3×#9518yes—
336C2×C4C3×#95116yes—
337C2×C4C3×#9518yes—
338C2×C4C3×#95116yes—
339C2×C4C3×#9514yes—
340C2×C4C3×#9518yes—
341C2×C4C3×#9514yes—
342C2×C4C2×C3×C4214yes—
343C2×C4C2×C3×C4214yes—
344C2×C4C2×C3×C22⋊C418yes—
345C2×C4C2×C3×C22⋊C412yes—
346C2×C4C2×C3×C22⋊C412yes—
347C2×C4C2×C3×C22⋊C418yes—
348C2×C4C2×C3×C22⋊C414yes—
349C2×C4C2×C3×C4⋊C414yes—
350C2×C4C2×C3×C4⋊C414yes—
351C2×C4C2×C3×C4⋊C418yes—
352C2×C4C2×C3×C4⋊C418yes—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—24no—
358C2×C3—21no—
359C2×C3#174014yesdirect product factor
360C2×C3C22×#9511yescharacteristic
361C2×C3C2×#41012yes—
362C22—28no—
363C22—24no—
364C22—24no—
365C22—28no—
366C22—22no—
367C22—22no—
368C22—24no—
369C22—24no—
370C22—42no—
371C22—44no—
372C22—42no—
373C22—44no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22C2×C3×#9512yes—
378C22C2×C3×#9514yes—
379C22C2×C3×#9512yes—
380C22C3×#41014yes—
381C22C22×C3×C4211yescharacteristic
382C22C22×C3×C22⋊C412yes—
383C22C22×C3×C22⋊C411yescharacteristic
384C22C22×C3×C4⋊C411yescharacteristic
385C22C22×C3×C4⋊C412yes—
386C4—416no—
387C4—48no—
388C4C2×C3×#9514yes—
389C4C2×C3×#9514yes—
390C4C2×C3×#9518yes—
391C3C2×#174011yescharacteristic, direct product factor, Sylow
392C2—22no—
393C2—24no—
394C2—22no—
395C2—21no—
396C2—21no—
397C2—22no—
398C2C3×#174014yesdirect product factor
399C2C22×C3×#9511yescharacteristic
400C2C2×C3×#41012yes—
subgroupG/H
centerC22×C3×C4C22⋊C4
commutatorC22C22×C3×C42
Frattini, cosocleC24C24×C3
socleC23×C3C2×C22⋊C4
prime radicalC25×C3C23
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1740C2direct (N × H)
5. C2×#1740C3direct (N × H)
12. C2×C3×#117C4semidirect
13. C2×C3×#117C4semidirect
17. C22×C3×C4⋊C4C4semidirect
21. C22×C3×C42C4semidirect
25. #1740C2×C3direct (N × H)
46. C3×#117C2×C4semidirect
51. C2×C3×C4⋊C4C2×C4semidirect
65. C2×C3×C42C2×C4semidirect
72. C2×#117C3×C4semidirect
73. C2×#117C3×C4semidirect
77. C22×C4⋊C4C3×C4semidirect
81. C22×C42C3×C4semidirect
154. #117C2×C3×C4semidirect
159. C2×C4⋊C4C2×C3×C4semidirect
173. C2×C42C2×C3×C4semidirect
359. C2×C3#1740direct (N × H)
391. C3C2×#1740direct (N × H)
398. C2C3×#1740direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C3×#17402
C2×C3×#3422
C2×#17403