Finite Group Databasebeta

← C3×#13056 #10631663

Subgroups — C3×#13056 #10631663

Order 768.

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#456C211yesmaximal, characteristic
2C3×#2381C211yesmaximal, characteristic, complemented
3C2×C3×#356C211yesmaximal, characteristic, complemented
4C3×#1793C212yesmaximal
5C3×#1394C212yesmaximal, complemented
6#13056C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×2−1+4C2211yescharacteristic
8C2×C3×#137—21no—
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#456—22no—
11C3×#456C414yescomplemented
12C2×C3×#104—21no—
13C2×C3×#104—22no—
14C3×#417—22no—
15C2×C3×#101C2212yes—
16C2×C3×#101C2211yescharacteristic
17C3×#391—21no—
18C3×#391C2212yes—
19C3×#387—22no—
20C3×#387—22no—
21C3×#383—21no—
22C3×#356—22no—
23C2×#456C2×C311yescharacteristic
24#2381C2×C311yescharacteristic, complemented
25C2×#356C2×C311yescharacteristic, complemented
26#1793C2×C312yes—
27#1394C2×C312yescomplemented
28C3×2−1+4—21no—
29C3×2−1+4—22no—
30C3×2−1+4—42no—
31C3×2−1+4C2×C412yes—
32C2×C3×#41—21no—
33C2×C3×#41—22no—
34C2×C3×#41—41no—
35C22×C3×Q8—41no—
36C22×C3×Q8D411yescharacteristic
37C22×C3×Q8C2311yescharacteristic
38C3×#137—24no—
39C3×#137—42no—
40C2×C3×Q16—41no—
41C2×C3×SD16—41no—
42C2×C3×M16—21no—
43C2×C3×M16—22no—
44C3×C4⋊Q8—22no—
45C3×C4⋊Q8C2×C412yes—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—22no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C42—21no—
51C2×C3×C42D411yescharacteristic
52C2×C3×C42D412yes—
53C3×#104—24no—
54C3×#104—44no—
55C3×#103—42no—
56C3×#102—42no—
57C3×#101—21no—
58C3×#101—22no—
59C3×#101—22no—
60C3×#101C2×C412yes—
61C3×#95—21no—
62C3×#95—22no—
63C3×#95—21no—
64C3×#95—22no—
65C3×#95—22no—
66C3×#95—22no—
67C2×2−1+4C22×C311yescharacteristic
68C2×#137—21no—
69C2×C4⋊Q8C22×C311yescharacteristic
70#456—22no—
71#456C3×C414yescomplemented
72C2×#104—21no—
73C2×#104—22no—
74#417—22no—
75C2×#101C22×C312yes—
76C2×#101C22×C311yescharacteristic
77#391—21no—
78#391C22×C312yes—
79#387—22no—
80#387—22no—
81#383—21no—
82#356—22no—
83C3×#41—24no—
84C3×#41—22no—
85C3×#41—22no—
86C3×#41—24no—
87C3×#41—42no—
88C3×#41—41no—
89C3×#41—42no—
90C3×#41—42no—
91C3×#41—44no—
92C3×#41—42no—
93C3×#41—41no—
94C2×C3×Q8—21no—
95C2×C3×Q8—22no—
96C2×C3×Q8—21no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—42no—
102C2×C3×Q8—42no—
103C2×C3×Q8—44no—
104C2×C3×Q8C22⋊C412yes—
105C2×C3×Q8C22×C411yescharacteristic
106C2×C3×Q8C2×D411yescharacteristic
107C2×C3×Q8#4112yes—
108C2×C3×D4—22no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—42no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D412yes—
119C3×Q16—44no—
120C3×SD16—44no—
121C3×M16—22no—
122C3×M16—41no—
123C3×M16—42no—
124C3×M16—42no—
125C2×C3×C8—41no—
126C2×C3×C8—42no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—42no—
130C3×C4⋊C4—81no—
131C3×C42—21no—
132C3×C42—42no—
133C3×C42—81no—
134C3×C42C22⋊C412yes—
1352−1+4—21no—
1362−1+4—22no—
1372−1+4—42no—
1382−1+4C2×C3×C412yes—
139C2×#41—21no—
140C2×#41—22no—
141C2×#41—41no—
142C22×Q8—41no—
143C22×Q8C3×D411yescharacteristic
144C22×Q8C23×C311yescharacteristic
145#137—24no—
146#137—42no—
147C2×Q16—41no—
148C2×SD16—41no—
149C2×M16—21no—
150C2×M16—22no—
151C4⋊Q8—22no—
152C4⋊Q8C2×C3×C412yes—
153C4⋊Q8C3×D412yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C3×D412yes—
157C2×C42—21no—
158C2×C42C3×D411yescharacteristic
159C2×C42C3×D412yes—
160#104—24no—
161#104—44no—
162#103—42no—
163#102—42no—
164#101—21no—
165#101—22no—
166#101—22no—
167#101C2×C3×C412yes—
168#95—21no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95—22no—
173#95—22no—
174C23×C3—41no—
175C23×C3#12011yescharacteristic
176C3×Q8—22no—
177C3×Q8—24no—
178C3×Q8—41no—
179C3×Q8—41no—
180C3×Q8—42no—
181C3×Q8—42no—
182C3×Q8—42no—
183C3×Q8—41no—
184C3×Q8—81no—
185C3×D4—22no—
186C3×D4—24no—
187C3×D4—42no—
188C3×D4—41no—
189C3×D4—42no—
190C3×D4—42no—
191C2×C3×C4—22no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—42no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D412yes—
211C2×C3×C4#12112yes—
212C2×C3×C4#12311yescharacteristic
213C3×C8—42no—
214C3×C8—82no—
215#41—24no—
216#41—22no—
217#41—22no—
218#41—24no—
219#41—42no—
220#41—41no—
221#41—42no—
222#41—42no—
223#41—44no—
224#41—42no—
225#41—41no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—21no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8C3×C22⋊C412yes—
237C2×Q8C22×C3×C411yescharacteristic
238C2×Q8C2×C3×D411yescharacteristic
239C2×Q8C3×#4112yes—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4C2×C3×D411yescharacteristic
250C22×C4C2×C3×D412yes—
251Q16—44no—
252SD16—44no—
253M16—22no—
254M16—41no—
255M16—42no—
256M16—42no—
257C2×C8—41no—
258C2×C8—42no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—42no—
262C4⋊C4—81no—
263C42—21no—
264C42—42no—
265C42—81no—
266C42C3×C22⋊C412yes—
267C22×C3—22no—
268C22×C3—42no—
269C22×C3—82no—
270C22×C3#38611yescharacteristic
271C22×C3B(4,𝔽2)11yescharacteristic
272C22×C3B(4,𝔽2)11yescharacteristic
273C3×C4—21no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—42no—
280C3×C4—82no—
281C3×C4—82no—
282C23—41no—
283C23C3×#12011yescharacteristic
284Q8—22no—
285Q8—24no—
286Q8—41no—
287Q8—41no—
288Q8—42no—
289Q8—42no—
290Q8—42no—
291Q8—41no—
292Q8—81no—
293D4—22no—
294D4—24no—
295D4—42no—
296D4—41no—
297D4—42no—
298D4—42no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4C2×C3×C22⋊C411yescharacteristic
318C2×C4C3×C4×D412yes—
319C2×C4C3×#12112yes—
320C2×C4C3×#12311yescharacteristic
321C8—42no—
322C8—82no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—82no—
326C2×C3#188911yescharacteristic
327C2×C3#220312yes—
328C22—22no—
329C22—42no—
330C22—82no—
331C22C3×#38611yescharacteristic
332C22C3×B(4,𝔽2)11yescharacteristic
333C22C3×B(4,𝔽2)11yescharacteristic
334C4—21no—
335C4—22no—
336C4—22no—
337C4—21no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—82no—
343C3#1305611yescharacteristic, direct product factor, Sylow
344C2—21no—
345C2—21no—
346C2—82no—
347C2C3×#188911yescharacteristic
348C2C3×#220312yes—
subgroupG/H
center, socleC22×C3B(4,𝔽2)
commutatorC2×Q8C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC2×C3×2−1+4C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2381C2semidirect
3. C2×C3×#356C2semidirect
5. C3×#1394C2semidirect
6. #13056C3direct (N × H)
11. C3×#456C4semidirect
24. #2381C2×C3semidirect
25. C2×#356C2×C3semidirect
27. #1394C2×C3semidirect
71. #456C3×C4semidirect
343. C3#13056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4562
C3×#23812
C2×C3×#3562
C3×#17932
C3×#13942
#130563