Finite Group Databasebeta

← C3×#29643 #10648250

Subgroups — C3×#29643 #10648250

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic
3C2×C3×#483C211yesmaximal, characteristic
4C3×#2855C211yesmaximal, characteristic
5C3×#2059C214yesmaximal, complemented
6C3×#2026C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic
8C3×#1823C211yesmaximal, characteristic
9C3×#1817C211yesmaximal, characteristic
10#29643C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#125—24no—
12C2×C3×#125—24no—
13C2×C3×#125—22no—
14C2×C3×#125C2214yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4—21no—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C3×#484—24no—
22C3×#483—24no—
23C2×C3×#107C2212yes—
24C2×C3×#106C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#391C2212yes—
30C3×#389C2212yes—
31C3×#387—22no—
32C3×#385—21no—
33C3×#381—21no—
34C3×#380C2211yescharacteristic
35C3×#340C2214yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C22×#125C2×C311yescharacteristic
39C2×#484C2×C311yescharacteristic
40C2×#483C2×C311yescharacteristic
41#2855C2×C311yescharacteristic
42#2059C2×C314yescomplemented
43#2026C2×C314yescomplemented
44#1824C2×C311yescharacteristic
45#1823C2×C311yescharacteristic
46#1817C2×C311yescharacteristic
47C22×C3×Q8C2311yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4—21no—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×#125—216no—
56C3×#125—28no—
57C3×#125—28no—
58C3×C22⋊Q8—22no—
59C3×C22⋊Q8—22no—
60C2×C3×C4⋊C4—28no—
61C2×C3×C4⋊C4—28no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D414yes—
71C2×C3×C4⋊C4D414yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D412yes—
81C2×C3×C42C2312yes—
82C3×#107—24no—
83C3×#106—24no—
84C3×#103—24no—
85C3×#103—24no—
86C3×C22⋊C8—22no—
87C3×C22⋊C8—22no—
88C3×#95—21no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—22no—
95C2×#125C22×C314yes—
96C2×C22⋊Q8C22×C311yescharacteristic
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C311yescharacteristic
101C22×C42C22×C311yescharacteristic
102#484—24no—
103#483—24no—
104C2×#107C22×C312yes—
105C2×#106C22×C312yes—
106C2×#103C22×C312yes—
107C2×#103C22×C312yes—
108C2×C22⋊C8C22×C311yescharacteristic
109C2×C22⋊C8C22×C311yescharacteristic
110#391C22×C312yes—
111#389C22×C312yes—
112#387—22no—
113#385—21no—
114#381—21no—
115#380C22×C311yescharacteristic
116#340C22×C314yes—
117#340C22×C314yes—
118#336C22×C314yes—
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—24no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4—44no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2411yescharacteristic
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—24no—
148C2×C3×C8—24no—
149C3×C4⋊C4—216no—
150C3×C4⋊C4—216no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—44no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4C2×D414yes—
161C3×C4⋊C4C2×D414yes—
162C3×C22⋊C4—42no—
163C3×C22⋊C4—42no—
164C3×C42—24no—
165C3×C42—22no—
166C3×C42—22no—
167C22×Q8C23×C311yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C3×D411yescharacteristic
171C23×C4C3×D411yescharacteristic
172C23×C4C23×C311yescharacteristic
173C22×C8C23×C312yes—
174C22×C8C23×C312yes—
175#125—216no—
176#125—28no—
177#125—28no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C3×D414yes—
191C2×C4⋊C4C3×D414yes—
192C2×C4⋊C4C23×C311yescharacteristic
193C2×C4⋊C4C23×C312yes—
194C2×C4⋊C4C23×C312yes—
195C2×C4⋊C4C23×C312yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C3×D412yes—
201C2×C42C23×C312yes—
202#107—24no—
203#106—24no—
204#103—24no—
205#103—24no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95—22no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3#12012yes—
215C23×C3#13512yes—
216C23×C3#13712yes—
217C23×C3C22×D411yescharacteristic
218C3×Q8—42no—
219C3×Q8—42no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—44no—
232C2×C3×C4—48no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—48no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4#12012yes—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C42−1+414yes—
249C3×C8—42no—
250C3×C8—42no—
251C24C2×C3×D411yescharacteristic
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C2×C3×D412yes—
272C22×C4C2×C3×D412yes—
273C22×C4C2×C3×D411yescharacteristic
274C22×C4C2×C3×D412yes—
275C22×C4C2×C3×D412yes—
276C22×C4C24×C311yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C4⋊C4—216no—
282C4⋊C4—216no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—24no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4C2×C3×D414yes—
293C4⋊C4C2×C3×D414yes—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C42—24no—
297C42—22no—
298C42—22no—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—24no—
302C22×C3—22no—
303C22×C3#45214yes—
304C22×C3C2×#12011yescharacteristic
305C22×C3#53612yes—
306C22×C3C2×#13511yescharacteristic
307C22×C3C2×#13711yescharacteristic
308C22×C3#57512yes—
309C3×C4—22no—
310C3×C4—42no—
311C3×C4—44no—
312C3×C4—42no—
313C3×C4—44no—
314C3×C4—81no—
315C3×C4—82no—
316C3×C4—81no—
317C3×C4#53612yes—
318C3×C4#53612yes—
319C23—22no—
320C23—22no—
321C23C3×#12012yes—
322C23C3×#13512yes—
323C23C3×#13712yes—
324C23C22×C3×D411yescharacteristic
325Q8—42no—
326Q8—42no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4C3×#12012yes—
353C2×C4C22×C3×D411yescharacteristic
354C2×C4C22×C3×D411yescharacteristic
355C2×C4C3×2−1+414yes—
356C8—42no—
357C8—42no—
358C2×C3—24no—
359C2×C3#261411yescharacteristic
360C2×C3C2×#45211yescharacteristic
361C2×C3#300511yescharacteristic
362C2×C3#323212yes—
363C2×C3#323612yes—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22C3×#45214yes—
369C22C2×C3×#12011yescharacteristic
370C22C3×#53612yes—
371C22C2×C3×#13511yescharacteristic
372C22C2×C3×#13711yescharacteristic
373C22C3×#57512yes—
374C4—22no—
375C4—42no—
376C4—44no—
377C4—42no—
378C4—44no—
379C4—82no—
380C4—81no—
381C4—81no—
382C4C3×#53612yes—
383C4C3×#53612yes—
384C3#2964311yescharacteristic, direct product factor, Sylow
385C2—24no—
386C2C3×#261411yescharacteristic
387C2C2×C3×#45211yescharacteristic
388C2C3×#300511yescharacteristic
389C2C3×#323212yes—
390C2C3×#323612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#2059C2semidirect
6. C3×#2026C2semidirect
10. #29643C3direct (N × H)
42. #2059C2×C3semidirect
43. #2026C2×C3semidirect
384. C3#29643direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×#4842
C2×C3×#4832
C3×#28552
C3×#20592
C3×#20262
C3×#18242
C3×#18232
C3×#18172
#296433