Finite Group Databasebeta

← C3×#13402 #10632009

Subgroups — C3×#13402 #10632009

Order 768.

404 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#472C211yesmaximal, characteristic
2C3×#2155C211yesmaximal, characteristic, complemented
3C2×C3×#361C211yesmaximal, characteristic, complemented
4C3×#2140C211yesmaximal, characteristic, complemented
5C3×#2115C211yesmaximal, characteristic, complemented
6C3×#1814C211yesmaximal, characteristic, complemented
7C3×#1388C211yesmaximal, characteristic, complemented
8#13402C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C22×C3×D8C2211yescharacteristic
11C2×C3×#131C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#472—22no—
15C3×#472—22no—
16C2×C3×#108C2211yescharacteristic
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#421—21no—
20C2×C3×#100—21no—
21C3×#365—22no—
22C3×#361—22no—
23C3×#357—22no—
24C3×#349—22no—
25C3×#344—21no—
26C2×#472C2×C311yescharacteristic
27#2155C2×C311yescharacteristic, complemented
28C2×#361C2×C311yescharacteristic, complemented
29#2140C2×C311yescharacteristic, complemented
30#2115C2×C311yescharacteristic, complemented
31#1814C2×C311yescharacteristic, complemented
32#1388C2×C311yescharacteristic, complemented
33C2×C3×#41D411yescharacteristic
34C22×C3×D4—21no—
35C3×AGL(1,ℤ/8)—24no—
36C3×AGL(1,ℤ/8)—42no—
37C2×C3×SD16—41no—
38C2×C3×D8—21no—
39C2×C3×D8—21no—
40C2×C3×D8—21no—
41C2×C3×D8—21no—
42C2×C3×D8—21no—
43C2×C3×D8—41no—
44C2×C3×D8D412yes—
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×#100—24no—
70C2×AGL(1,ℤ/8)—21no—
71C22×D8C22×C311yescharacteristic
72C2×#131C22×C311yescharacteristic
73C2×M32C22×C311yescharacteristic
74C2×M32C22×C311yescharacteristic
75#472—22no—
76#472—22no—
77C2×#108C22×C311yescharacteristic
78C2×#108C22×C311yescharacteristic
79#427C22×C311yescharacteristic
80#421—21no—
81C2×#100—21no—
82#365—22no—
83#361—22no—
84#357—22no—
85#349—22no—
86#344—21no—
87C24×C3—41no—
88C3×#41—21no—
89C3×#41—21no—
90C3×#41—21no—
91C3×#41—21no—
92C2×C3×Q8SD1611yescharacteristic
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4SD1611yescharacteristic
101C22×C3×C4—21no—
102C22×C3×C4C2×D411yescharacteristic
103C3×SD16—44no—
104C3×D8—22no—
105C3×D8—22no—
106C3×D8—44no—
107C3×D8—41no—
108C3×D8—41no—
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×D4—21no—
135AGL(1,ℤ/8)—24no—
136AGL(1,ℤ/8)—42no—
137C2×SD16—41no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—41no—
144C2×D8C3×D412yes—
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#100—24no—
170C23×C3—21no—
171C23×C3—42no—
172C23×C3—81no—
173C23×C3—81no—
174C23×C3—162no—
175C23×C3#13511yescharacteristic
176C3×Q8—21no—
177C3×Q8—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—41no—
181C3×D4—41no—
182C3×D4—42no—
183C3×D4—42no—
184C3×D4—81no—
185C3×D4—81no—
186C2×C3×C4—21no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4C2×SD1611yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C3×C8—41no—
197C3×C8—42no—
198C3×C8—81no—
199C3×C8#12111yescharacteristic
200C3×C8#12111yescharacteristic
201C3×C8#12311yescharacteristic
202C3×C8#12411yescharacteristic
203C24—41no—
204#41—21no—
205#41—21no—
206#41—21no—
207#41—21no—
208C2×Q8C3×SD1611yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4C3×SD1611yescharacteristic
217C22×C4—21no—
218C22×C4C2×C3×D411yescharacteristic
219SD16—44no—
220D8—22no—
221D8—22no—
222D8—44no—
223D8—41no—
224D8—41no—
225M16—21no—
226M16—22no—
227M16—21no—
228M16—41no—
229M16—41no—
230M16—41no—
231M16—41no—
232M16—41no—
233C2×C8—21no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8C2×C3×D411yescharacteristic
239C2×C8C2×C3×D411yescharacteristic
240C2×C8C2×C3×D411yescharacteristic
241C2×C8C3×#4111yescharacteristic
242C2×C8C3×#4111yescharacteristic
243C2×C8C3×#4111yescharacteristic
244C4⋊C4—41no—
245C22⋊C4—41no—
246C42—41no—
247C16—22no—
248C16—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—81no—
252C22×C3—82no—
253C22×C3—162no—
254C22×C3—162no—
255C22×C3—162no—
256C22×C3#44911yescharacteristic
257C22×C3#46511yescharacteristic
258C22×C3#48211yescharacteristic
259C3×C4—41no—
260C3×C4—161no—
261C3×C4#39411yescharacteristic
262C3×C4#45011yescharacteristic
263C3×C4#46911yescharacteristic
264C3×C4#48111yescharacteristic
265C23—21no—
266C23—42no—
267C23—81no—
268C23—81no—
269C23—162no—
270C23C3×#13511yescharacteristic
271Q8—21no—
272Q8—21no—
273D4—21no—
274D4—21no—
275D4—41no—
276D4—41no—
277D4—42no—
278D4—42no—
279D4—81no—
280D4—81no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4C3×#12011yescharacteristic
286C2×C4C3×#12111yescharacteristic
287C2×C4C3×#12311yescharacteristic
288C2×C4C2×C3×SD1611yescharacteristic
289C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
290C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
291C8—41no—
292C8—42no—
293C8—81no—
294C8C3×#12111yescharacteristic
295C8C3×#12111yescharacteristic
296C8C3×#12311yescharacteristic
297C8C3×#12411yescharacteristic
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—81no—
302C2×C3—162no—
303C2×C3#203411yescharacteristic
304C22—41no—
305C22—41no—
306C22—81no—
307C22—82no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22C3×#44911yescharacteristic
312C22C3×#46511yescharacteristic
313C22C3×#48211yescharacteristic
314C4—41no—
315C4—161no—
316C4C3×#39411yescharacteristic
317C4C3×#45011yescharacteristic
318C4C3×#46911yescharacteristic
319C4C3×#48111yescharacteristic
320C3#1340211yescharacteristic, direct product factor, Sylow
321C2—21no—
322C2—21no—
323C2—21no—
324C2—81no—
325C2—162no—
326C2C3×#203411yescharacteristic
subgroupG/H
center, socleC2×C3#2034
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#472C2
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×#361C2semidirect
4. C3×#2140C2semidirect
5. C3×#2115C2semidirect
6. C3×#1814C2semidirect
7. C3×#1388C2semidirect
8. #13402C3direct (N × H)
27. #2155C2×C3semidirect
28. C2×#361C2×C3semidirect
29. #2140C2×C3semidirect
30. #2115C2×C3semidirect
31. #1814C2×C3semidirect
32. #1388C2×C3semidirect
320. C3#13402direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4722
C3×#21552
C2×C3×#3612
C3×#21402
C3×#21152
C3×#18142
C3×#13882
#134023