Finite Group Databasebeta

← C3×#13403 #10632010

Subgroups — C3×#13403 #10632010

Order 768.

384 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#473C211yesmaximal, characteristic
2C3×#2155C211yesmaximal, characteristic, complemented
3C2×C3×#362C211yesmaximal, characteristic, complemented
4C3×#2141C211yesmaximal, characteristic, complemented
5C3×#2115C211yesmaximal, characteristic
6C3×#1814C211yesmaximal, characteristic
7C3×#1388C211yesmaximal, characteristic, complemented
8#13403C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C22×C3×Q16C2211yescharacteristic
11C2×C3×#131C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#473—22no—
15C3×#473—22no—
16C2×C3×#108C2211yescharacteristic
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#421—21no—
20C2×C3×#101—21no—
21C3×#365—22no—
22C3×#362—22no—
23C3×#358—22no—
24C3×#349—22no—
25C3×#344—21no—
26C2×#473C2×C311yescharacteristic
27#2155C2×C311yescharacteristic, complemented
28C2×#362C2×C311yescharacteristic, complemented
29#2141C2×C311yescharacteristic, complemented
30#2115C2×C311yescharacteristic
31#1814C2×C311yescharacteristic
32#1388C2×C311yescharacteristic, complemented
33C2×C3×#41D411yescharacteristic
34C22×C3×Q8—21no—
35C3×#137—24no—
36C3×#137—42no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—41no—
43C2×C3×Q16D412yes—
44C2×C3×SD16—41no—
45C3×#131—21no—
46C3×#131—21no—
47C3×#131—21no—
48C3×#131—21no—
49C2×C3×M16—21no—
50C2×C3×M16—21no—
51C2×C3×M16—21no—
52C2×C3×M16D411yescharacteristic
53C22×C3×C8—21no—
54C22×C3×C8C2311yescharacteristic
55C3×#117—21no—
56C3×M32—21no—
57C3×M32—21no—
58C3×M32—21no—
59C3×M32D412yes—
60C2×C3×C16—21no—
61C2×C3×C16—21no—
62C3×#108—21no—
63C3×#108—21no—
64C3×#108—21no—
65C3×#108D412yes—
66C3×#107—21no—
67C3×#106—21no—
68C3×#104—42no—
69C3×#101—24no—
70C2×#137—21no—
71C22×Q16C22×C311yescharacteristic
72C2×#131C22×C311yescharacteristic
73C2×M32C22×C311yescharacteristic
74C2×M32C22×C311yescharacteristic
75#473—22no—
76#473—22no—
77C2×#108C22×C311yescharacteristic
78C2×#108C22×C311yescharacteristic
79#427C22×C311yescharacteristic
80#421—21no—
81C2×#101—21no—
82#365—22no—
83#362—22no—
84#358—22no—
85#349—22no—
86#344—21no—
87C3×#41—21no—
88C3×#41—21no—
89C3×#41—21no—
90C3×#41—21no—
91C2×C3×Q8—22no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8SD1611yescharacteristic
98C2×C3×D4—21no—
99C2×C3×D4SD1611yescharacteristic
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C3×Q16—22no—
104C3×Q16—22no—
105C3×Q16—44no—
106C3×Q16—41no—
107C3×Q16—41no—
108C3×SD16—44no—
109C3×M16—21no—
110C3×M16—22no—
111C3×M16—21no—
112C3×M16—41no—
113C3×M16—41no—
114C3×M16—41no—
115C3×M16—41no—
116C3×M16—41no—
117C2×C3×C8—21no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8C2×D411yescharacteristic
123C2×C3×C8C2×D411yescharacteristic
124C2×C3×C8C2×D411yescharacteristic
125C2×C3×C8#4111yescharacteristic
126C2×C3×C8#4111yescharacteristic
127C2×C3×C8#4111yescharacteristic
128C3×C4⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42—41no—
131C3×C16—22no—
132C3×C16—41no—
133C2×#41C3×D411yescharacteristic
134C22×Q8—21no—
135#137—24no—
136#137—42no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—41no—
143C2×Q16C3×D412yes—
144C2×SD16—41no—
145#131—21no—
146#131—21no—
147#131—21no—
148#131—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16C3×D411yescharacteristic
153C22×C8—21no—
154C22×C8C23×C311yescharacteristic
155#117—21no—
156M32—21no—
157M32—21no—
158M32—21no—
159M32C3×D412yes—
160C2×C16—21no—
161C2×C16—21no—
162#108—21no—
163#108—21no—
164#108—21no—
165#108C3×D412yes—
166#107—21no—
167#106—21no—
168#104—42no—
169#101—24no—
170C23×C3—21no—
171C23×C3#13511yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—42no—
175C3×Q8—42no—
176C3×Q8—81no—
177C3×Q8—81no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—41no—
181C3×D4—41no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4C2×SD1611yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C3×C8—41no—
196C3×C8—42no—
197C3×C8—81no—
198C3×C8#12111yescharacteristic
199C3×C8#12111yescharacteristic
200C3×C8#12311yescharacteristic
201C3×C8#12411yescharacteristic
202#41—21no—
203#41—21no—
204#41—21no—
205#41—21no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8C3×SD1611yescharacteristic
213C2×D4—21no—
214C2×D4C3×SD1611yescharacteristic
215C22×C4—21no—
216C22×C4—41no—
217C22×C4C2×C3×D411yescharacteristic
218Q16—22no—
219Q16—22no—
220Q16—44no—
221Q16—41no—
222Q16—41no—
223SD16—44no—
224M16—21no—
225M16—22no—
226M16—21no—
227M16—41no—
228M16—41no—
229M16—41no—
230M16—41no—
231M16—41no—
232C2×C8—21no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8C2×C3×D411yescharacteristic
238C2×C8C2×C3×D411yescharacteristic
239C2×C8C2×C3×D411yescharacteristic
240C2×C8C3×#4111yescharacteristic
241C2×C8C3×#4111yescharacteristic
242C2×C8C3×#4111yescharacteristic
243C4⋊C4—41no—
244C22⋊C4—41no—
245C42—41no—
246C16—22no—
247C16—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—81no—
251C22×C3#44911yescharacteristic
252C22×C3#46511yescharacteristic
253C22×C3#48211yescharacteristic
254C3×C4—41no—
255C3×C4—82no—
256C3×C4—161no—
257C3×C4#39411yescharacteristic
258C3×C4#45011yescharacteristic
259C3×C4#46911yescharacteristic
260C3×C4#48111yescharacteristic
261C23—21no—
262C23C3×#13511yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—42no—
266Q8—42no—
267Q8—81no—
268Q8—81no—
269D4—21no—
270D4—21no—
271D4—41no—
272D4—41no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4C3×#12011yescharacteristic
281C2×C4C3×#12111yescharacteristic
282C2×C4C3×#12311yescharacteristic
283C2×C4C2×C3×SD1611yescharacteristic
284C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
285C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
286C8—41no—
287C8—42no—
288C8—81no—
289C8C3×#12111yescharacteristic
290C8C3×#12111yescharacteristic
291C8C3×#12311yescharacteristic
292C8C3×#12411yescharacteristic
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—81no—
297C2×C3#203411yescharacteristic
298C22—41no—
299C22—41no—
300C22—81no—
301C22C3×#44911yescharacteristic
302C22C3×#46511yescharacteristic
303C22C3×#48211yescharacteristic
304C4—41no—
305C4—82no—
306C4—161no—
307C4C3×#39411yescharacteristic
308C4C3×#45011yescharacteristic
309C4C3×#46911yescharacteristic
310C4C3×#48111yescharacteristic
311C3#1340311yescharacteristic, direct product factor, Sylow
312C2—21no—
313C2—21no—
314C2—21no—
315C2—81no—
316C2C3×#203411yescharacteristic
subgroupG/H
center, socleC2×C3#2034
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#41D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2155C2semidirect
3. C2×C3×#362C2semidirect
4. C3×#2141C2semidirect
7. C3×#1388C2semidirect
8. #13403C3direct (N × H)
27. #2155C2×C3semidirect
28. C2×#362C2×C3semidirect
29. #2141C2×C3semidirect
32. #1388C2×C3semidirect
311. C3#13403direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4732
C3×#21552
C2×C3×#3622
C3×#21412
C3×#21152
C3×#18142
C3×#13882
#134033