Finite Group Databasebeta

← C3×#10254 #10628861

Subgroups — C3×#10254 #10628861

Order 768.

634 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C3×#3245C211yesmaximal, characteristic, complemented
2C3×#3036C212yesmaximal, complemented
3C3×#1954C211yesmaximal, characteristic, complemented
4C3×#1492C212yesmaximal, complemented
5C3×#1292C211yesmaximal, characteristic, complemented
6#10254C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×AGL(1,ℤ/8)—22no—
8C3×#532C2212yes—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#474—22no—
11C3×#466—22no—
12C3×#463—24no—
13C3×#461—22no—
14C3×#459—22no—
15C3×#459—22no—
16C3×#423C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C3×#417—22no—
19C2×C3×C22⋊C8—22no—
20C2×C3×C4×C8C2211yescharacteristic
21C3×D4⋊C8—24no—
22#3245C2×C311yescharacteristic, complemented
23#3036C2×C312yescomplemented
24#1954C2×C311yescharacteristic, complemented
25#1492C2×C312yescomplemented
26#1292C2×C311yescharacteristic, complemented
27C2×C3×#41—22no—
28C22×C3×D4D411yescharacteristic
29C23×C3×C4—21no—
30C3×AGL(1,ℤ/8)—44no—
31C3×AGL(1,ℤ/8)—44no—
32C2×C3×SD16—24no—
33C2×C3×D8—24no—
34C2×C3×D8—22no—
35C2×C3×M16—22no—
36C22×C3×C8—21no—
37C22×C3×C8D412yes—
38C3×C4⋊Q8D412yes—
39C3×#127D412yes—
40C3×#124—22no—
41C3×#121—22no—
42C3×#121—22no—
43C3×C4×D4—22no—
44C3×C4×D4—22no—
45C3×C4×D4—22no—
46C3×C4×D4—22no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C3×#107—22no—
51C3×C4⋊C8—22no—
52C3×C4⋊C8—22no—
53C3×C4⋊C8D414yes—
54C3×#103—24no—
55C3×#102—24no—
56C3×#102—24no—
57C3×C22⋊C8—44no—
58C3×C4×C8—22no—
59C2×AGL(1,ℤ/8)—22no—
60#532C22×C312yes—
61C2×C4×D4C22×C311yescharacteristic
62#474—22no—
63#466—22no—
64#463—24no—
65#461—22no—
66#459—22no—
67#459—22no—
68#423C22×C312yes—
69C2×C4⋊C8C22×C311yescharacteristic
70#417—22no—
71C2×C22⋊C8—22no—
72C2×C4×C8C22×C311yescharacteristic
73D4⋊C8—24no—
74C24×C3—21no—
75C3×#41—42no—
76C3×#41—42no—
77C3×#41—42no—
78C3×#41—42no—
79C2×C3×Q8—22no—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—21no—
85C2×C3×D4—22no—
86C2×C3×D4—42no—
87C2×C3×D4D812yescomplemented
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4—41no—
92C22×C3×C4—42no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C3×SD16—48no—
98C3×D8—48no—
99C3×D8—82no—
100C3×M16—42no—
101C3×M16—42no—
102C2×C3×C8—22no—
103C2×C3×C8—22no—
104C2×C3×C8—24no—
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C2×C3×C8—42no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C42C2×D412yes—
115C3×C42C2×D412yes—
116C2×#41—22no—
117C22×D4C3×D411yescharacteristic
118C23×C4—21no—
119AGL(1,ℤ/8)—44no—
120AGL(1,ℤ/8)—44no—
121C2×SD16—24no—
122C2×D8—24no—
123C2×D8—22no—
124C2×M16—22no—
125C22×C8—21no—
126C22×C8C3×D412yes—
127C4⋊Q8C3×D412yes—
128#127C3×D412yes—
129#124—22no—
130#121—22no—
131#121—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C2×C4⋊C4C3×D411yescharacteristic
137C2×C22⋊C4—21no—
138C2×C42C23×C311yescharacteristic
139#107—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8C3×D414yes—
143#103—24no—
144#102—24no—
145#102—24no—
146C22⋊C8—44no—
147C4×C8—22no—
148C23×C3—22no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3—42no—
154C23×C3#12111yescharacteristic
155C3×Q8—42no—
156C3×Q8—42no—
157C3×D4—24no—
158C3×D4—44no—
159C3×D4—42no—
160C3×D4—42no—
161C3×D4—42no—
162C3×D4—42no—
163C3×D4—42no—
164C3×D4—82no—
165C3×D4—82no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—82no—
173C2×C3×C4—82no—
174C2×C3×C4—82no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4#12012yes—
178C2×C3×C4#12011yescharacteristic
179C2×C3×C4#12112yes—
180C2×C3×C4#12711yescharacteristic
181C2×C3×C4C2×D811yescharacteristic
182C2×C3×C4#13511yescharacteristic
183C2×C3×C4AGL(1,ℤ/8)12yes—
184C3×C8—42no—
185C3×C8—44no—
186C3×C8—42no—
187C24—21no—
188#41—42no—
189#41—42no—
190#41—42no—
191#41—42no—
192C2×Q8—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—42no—
200C2×D4C3×D812yescomplemented
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4C2×C3×D411yescharacteristic
208C22×C4C2×C3×D411yescharacteristic
209C22×C4C3×#4111yescharacteristic
210SD16—48no—
211D8—48no—
212D8—82no—
213M16—42no—
214M16—42no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C42C2×C3×D412yes—
228C42C2×C3×D412yes—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—44no—
232C22×C3—42no—
233C22×C3—82no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3—82no—
237C22×C3—82no—
238C22×C3—82no—
239C22×C3#39211yescharacteristic
240C22×C3#46611yescharacteristic
241C22×C3#46911yescharacteristic
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—82no—
245C3×C4—81no—
246C3×C4#44712yes—
247C3×C4#44912yes—
248C3×C4#45311yescharacteristic
249C3×C4#45411yescharacteristic
250C23—22no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23C3×#12111yescharacteristic
257Q8—42no—
258Q8—42no—
259D4—24no—
260D4—44no—
261D4—42no—
262D4—42no—
263D4—42no—
264D4—42no—
265D4—42no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4C3×#12012yes—
280C2×C4C3×#12011yescharacteristic
281C2×C4C3×#12112yes—
282C2×C4C3×#12711yescharacteristic
283C2×C4C2×C3×D811yescharacteristic
284C2×C4C3×#13511yescharacteristic
285C2×C4C3×AGL(1,ℤ/8)12yes—
286C8—42no—
287C8—44no—
288C8—42no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—82no—
292C2×C3—162no—
293C2×C3#199911yescharacteristic
294C2×C3#200611yescharacteristic
295C2×C3#201711yescharacteristic
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22C3×#39211yescharacteristic
307C22C3×#46611yescharacteristic
308C22C3×#46911yescharacteristic
309C4—21no—
310C4—22no—
311C4—82no—
312C4—81no—
313C4C3×#44712yes—
314C4C3×#44912yes—
315C4C3×#45311yescharacteristic
316C4C3×#45411yescharacteristic
317C3#1025411yescharacteristic, direct product factor, Sylow
318C2—21no—
319C2—21no—
320C2—82no—
321C2—162no—
322C2C3×#199911yescharacteristic
323C2C3×#200611yescharacteristic
324C2C3×#201711yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3245C2semidirect
2. C3×#3036C2semidirect
3. C3×#1954C2semidirect
4. C3×#1492C2semidirect
5. C3×#1292C2semidirect
6. #10254C3direct (N × H)
22. #3245C2×C3semidirect
23. #3036C2×C3semidirect
24. #1954C2×C3semidirect
25. #1492C2×C3semidirect
26. #1292C2×C3semidirect
87. C2×C3×D4D8semidirect
200. C2×D4C3×D8semidirect
317. C3#10254direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32452
C3×#30362
C3×#19542
C3×#14922
C3×#12922
#102543