Finite Group Databasebeta

← C2×C3×#1886 #10639557

Subgroups — C2×C3×#1886 #10639557

Order 768.

1750 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#519C211yesmaximal, characteristic
3C2×C3×#419C212yesmaximal
4C22×C3×#101C211yesmaximal, characteristic
5C3×#1886C218yesmaximal, direct product factor
6C2×C3×#339C212yesmaximal
7C2×#1886C311yesmaximal, characteristic, direct product factor, Sylow
8C23×C3×Q8C2211yescharacteristic
9C23×C3×Q8C2211yescharacteristic
10C22×C3×M16C2212yes—
11C2×C3×C4⋊Q8—24no—
12C2×C3×C4⋊Q8—22no—
13C2×C3×C4⋊Q8C418yescomplemented
14C2×C3×C4⋊Q8C2214yes—
15C3×#519—22no—
16C3×#519—24no—
17C3×#519C2214yes—
18C2×C3×C4×Q8—24no—
19C2×C3×#117—22no—
20C2×C3×#117C2212yes—
21C22×C3×C4⋊C4—22no—
22C22×C3×C42C2211yescharacteristic
23C3×#419—24no—
24C3×#419C2218yes—
25C2×C3×#103—24no—
26C2×C3×#101—24no—
27C2×C3×#101C2214yes—
28C3×#339C2218yes—
29C22×C4⋊Q8C2×C311yescharacteristic
30C2×#519C2×C311yescharacteristic
31C2×#419C2×C312yes—
32C22×#101C2×C311yescharacteristic
33#1886C2×C318yesdirect product factor
34C2×#339C2×C312yes—
35C22×C3×Q8—22no—
36C22×C3×Q8—22no—
37C22×C3×Q8—22no—
38C22×C3×Q8—21no—
39C22×C3×Q8—24no—
40C22×C3×Q8—22no—
41C22×C3×Q8—22no—
42C22×C3×Q8—24no—
43C22×C3×Q8C2×C414yes—
44C22×C3×Q8C2×C414yes—
45C22×C3×Q8D414yes—
46C22×C3×Q8D412yes—
47C22×C3×Q8C2312yes—
48C22×C3×Q8C2312yes—
49C23×C3×C4—21no—
50C23×C3×C4—21no—
51C23×C3×C4D412yes—
52C23×C3×C4C2311yescharacteristic
53C2×C3×M16—22no—
54C2×C3×M16—24no—
55C2×C3×M16—24no—
56C2×C3×M16C2314yes—
57C22×C3×C8—22no—
58C3×C4⋊Q8—216no—
59C3×C4⋊Q8—28no—
60C3×C4⋊Q8C2×C4116yescomplemented
61C3×C4×Q8—216no—
62C3×C4×Q8—216no—
63C3×#117—28no—
64C3×#117—28no—
65C3×#117—22no—
66C3×#117—22no—
67C3×#117D414yes—
68C3×#117C2314yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—216no—
71C2×C3×C4⋊C4—28no—
72C2×C3×C4⋊C4D414yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—24no—
75C2×C3×C42—21no—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42C2×C414yes—
79C2×C3×C42C2312yes—
80C3×#103—216no—
81C3×#101—28no—
82C23×Q8C22×C311yescharacteristic
83C23×Q8C22×C311yescharacteristic
84C22×M16C22×C312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C3×C418yescomplemented
88C2×C4⋊Q8C22×C314yes—
89#519—22no—
90#519—24no—
91#519C22×C314yes—
92C2×C4×Q8—24no—
93C2×#117—22no—
94C2×#117C22×C312yes—
95C22×C4⋊C4—22no—
96C22×C42C22×C311yescharacteristic
97#419—24no—
98#419C22×C318yes—
99C2×#103—24no—
100C2×#101—24no—
101C2×#101C22×C314yes—
102#339C22×C318yes—
103C24×C3C2×D411yescharacteristic
104C2×C3×Q8—24no—
105C2×C3×Q8—28no—
106C2×C3×Q8—28no—
107C2×C3×Q8—24no—
108C2×C3×Q8—28no—
109C2×C3×Q8—28no—
110C2×C3×Q8—24no—
111C2×C3×Q8—24no—
112C2×C3×Q8—22no—
113C2×C3×Q8—24no—
114C2×C3×Q8—28no—
115C2×C3×Q8—28no—
116C2×C3×Q8—24no—
117C2×C3×Q8—24no—
118C2×C3×Q8—42no—
119C2×C3×Q8—44no—
120C2×C3×Q8—44no—
121C2×C3×Q8—42no—
122C2×C3×Q8C22⋊C418yes—
123C2×C3×Q8C22×C414yes—
124C2×C3×Q8C22×C414yes—
125C2×C3×Q8C2×D414yes—
126C22×C3×C4—28no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—42no—
138C22×C3×C4C22⋊C418yes—
139C22×C3×C4C22×C412yes—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4#4114yes—
146C22×C3×C4C2411yescharacteristic
147C3×M16—24no—
148C3×M16—24no—
149C3×M16—44no—
150C2×C3×C8—24no—
151C2×C3×C8—44no—
152C3×C4⋊C4—216no—
153C3×C4⋊C4—232no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4C2×D418yes—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C42—216no—
159C3×C42—24no—
160C3×C42—24no—
161C3×C42—22no—
162C3×C42—24no—
163C3×C42C22×C414yes—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—21no—
168C22×Q8—24no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8C2×C3×C414yes—
173C22×Q8C2×C3×C414yes—
174C22×Q8C3×D414yes—
175C22×Q8C3×D412yes—
176C22×Q8C23×C312yes—
177C22×Q8C23×C312yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C3×D412yes—
181C23×C4C23×C311yescharacteristic
182C2×M16—22no—
183C2×M16—24no—
184C2×M16—24no—
185C2×M16C23×C314yes—
186C22×C8—22no—
187C4⋊Q8—216no—
188C4⋊Q8—28no—
189C4⋊Q8C2×C3×C4116yescomplemented
190C4×Q8—216no—
191C4×Q8—216no—
192#117—28no—
193#117—28no—
194#117—22no—
195#117—22no—
196#117C3×D414yes—
197#117C23×C314yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—216no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4C3×D414yes—
202C2×C22⋊C4—22no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42C2×C3×C414yes—
208C2×C42C23×C312yes—
209#103—216no—
210#101—28no—
211C23×C3—24no—
212C23×C3C2×C22⋊C412yes—
213C23×C3#12011yescharacteristic
214C23×C3#12011yescharacteristic
215C23×C3#12312yes—
216C23×C3C22×D411yescharacteristic
217C3×Q8—28no—
218C3×Q8—28no—
219C3×Q8—216no—
220C3×Q8—28no—
221C3×Q8—44no—
222C3×Q8—48no—
223C2×C3×C4—216no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—28no—
229C2×C3×C4—28no—
230C2×C3×C4—24no—
231C2×C3×C4—28no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—44no—
235C2×C3×C4—44no—
236C2×C3×C4—48no—
237C2×C3×C4C2×C22⋊C418yes—
238C2×C3×C4C2×C22⋊C412yes—
239C2×C3×C4C4×D414yes—
240C2×C3×C4C4×D414yes—
241C2×C3×C4#12012yes—
242C2×C3×C4#12114yes—
243C2×C3×C4#12114yes—
244C2×C3×C4#12312yes—
245C2×C3×C4C23×C411yescharacteristic
246C2×C3×C4C22×D412yes—
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C2×#4112yes—
249C3×C8—44no—
250C24C2×C3×D411yescharacteristic
251C2×Q8—24no—
252C2×Q8—28no—
253C2×Q8—28no—
254C2×Q8—24no—
255C2×Q8—28no—
256C2×Q8—28no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—28no—
262C2×Q8—28no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—42no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8C3×C22⋊C418yes—
270C2×Q8C22×C3×C414yes—
271C2×Q8C22×C3×C414yes—
272C2×Q8C2×C3×D414yes—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4C3×C22⋊C418yes—
286C22×C4C22×C3×C412yes—
287C22×C4C2×C3×D414yes—
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D412yes—
290C22×C4C2×C3×D412yes—
291C22×C4C2×C3×D412yes—
292C22×C4C3×#4114yes—
293C22×C4C24×C311yescharacteristic
294M16—24no—
295M16—24no—
296M16—44no—
297C2×C8—24no—
298C2×C8—44no—
299C4⋊C4—216no—
300C4⋊C4—232no—
301C4⋊C4—24no—
302C4⋊C4C2×C3×D418yes—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C42—216no—
306C42—24no—
307C42—24no—
308C42—22no—
309C42—24no—
310C42C22×C3×C414yes—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3#38612yes—
316C22×C3#38612yes—
317C22×C3#45614yes—
318C22×C3C22×C22⋊C411yescharacteristic
319C22×C3C2×#12011yescharacteristic
320C22×C3C2×#12311yescharacteristic
321C3×C4—28no—
322C3×C4—28no—
323C3×C4—48no—
324C3×C4—44no—
325C3×C4#38614yes—
326C3×C4C2×C4×D412yes—
327C3×C4C2×#12112yes—
328C23—24no—
329C23C2×C3×C22⋊C412yes—
330C23C3×#12011yescharacteristic
331C23C3×#12011yescharacteristic
332C23C3×#12312yes—
333C23C22×C3×D411yescharacteristic
334Q8—216no—
335Q8—28no—
336Q8—28no—
337Q8—28no—
338Q8—48no—
339Q8—44no—
340C2×C4—28no—
341C2×C4—216no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4C2×C3×C22⋊C418yes—
355C2×C4C2×C3×C22⋊C412yes—
356C2×C4C3×C4×D414yes—
357C2×C4C3×C4×D414yes—
358C2×C4C3×#12012yes—
359C2×C4C3×#12114yes—
360C2×C4C3×#12114yes—
361C2×C4C3×#12312yes—
362C2×C4C23×C3×C411yescharacteristic
363C2×C4C22×C3×D412yes—
364C2×C4C22×C3×D411yescharacteristic
365C2×C4C2×C3×#4112yes—
366C8—44no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—22no—
370C2×C3#188614yesdirect product factor
371C2×C3C2×#38611yescharacteristic
372C2×C3C2×#45612yes—
373C22—22no—
374C22—22no—
375C22—24no—
376C22—24no—
377C22C3×#38612yes—
378C22C3×#38612yes—
379C22C3×#45614yes—
380C22C22×C3×C22⋊C411yescharacteristic
381C22C2×C3×#12011yescharacteristic
382C22C2×C3×#12311yescharacteristic
383C4—28no—
384C4—28no—
385C4—48no—
386C4—44no—
387C4C3×#38614yes—
388C4C2×C3×C4×D412yes—
389C4C2×C3×#12112yes—
390C3C2×#188611yescharacteristic, direct product factor, Sylow
391C2—21no—
392C2—22no—
393C2—21no—
394C2C3×#188614yesdirect product factor
395C2C2×C3×#38611yescharacteristic
396C2C2×C3×#45612yes—
subgroupG/H
center, socleC23×C3#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#1886C2direct (N × H)
7. C2×#1886C3direct (N × H)
13. C2×C3×C4⋊Q8C4semidirect
33. #1886C2×C3direct (N × H)
60. C3×C4⋊Q8C2×C4semidirect
87. C2×C4⋊Q8C3×C4semidirect
189. C4⋊Q8C2×C3×C4semidirect
370. C2×C3#1886direct (N × H)
390. C3C2×#1886direct (N × H)
394. C2C3×#1886direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#5192
C2×C3×#4192
C22×C3×#1012
C3×#18862
C2×C3×#3392
C2×#18863