Finite Group Databasebeta

← C3×#69121 #11798306

Subgroups — C3×#69121 #11798306

Order 1152.

461 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 98 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×#5748C211yesmaximal, characteristic
2C3×#5137C211yesmaximal, characteristic, complemented
3C3×#4909C212yesmaximal, complemented
4C32×#349C211yesmaximal, characteristic, complemented
5C3×#4518C212yesmaximal, complemented
6C3×#2191—31nomaximal
7#69121C311yesmaximal, characteristic, direct product factor
8C2×C3×#235—21no—
9C2×C32×M16C2211yescharacteristic
10C3×#924—21no—
11C3×#909—22no—
12C3×#908—22no—
13C2×C3×Dic12C2211yescharacteristic, complemented
14C3×#904—21no—
15C3×#904C2212yes—
16C32×M32C2212yescomplemented
17C3×#817C2211yescharacteristic, complemented
18C3×#816—21no—
19C3×Dic24—22no—
20C3×#800—22no—
21C3×#575—31no—
22#5748C2×C311yescharacteristic
23C3×#510—32no—
24C3×#470—31no—
25#5137C2×C311yescharacteristic, complemented
26#4909C2×C312yescomplemented
27C3×#360—32no—
28C3×#349—21no—
29C3×#349D311yescharacteristic, complemented
30C3×#349C2×C311yescharacteristic, complemented
31#4518C2×C312yescomplemented
32C22×C32×C4D411yescharacteristic
33C2×C3×#67—41no—
34C3×#235—22no—
35C3×#235—22no—
36C3×#235—41no—
37C3×#235—41no—
38C2×C3×D12—21no—
39C2×C3×C4×D3—41no—
40C2×C3×Dic6—21no—
41C2×C3×Dic6—21no—
42C32×M16—22no—
43C2×C32×C8D411yescharacteristic
44C2×C32×C8C2311yescharacteristic
45C3×#217—41no—
46C3×C4⋊Dic3—21no—
47C3×Dic3⋊C4—41no—
48C3×Dic12—21no—
49C3×Dic12—22no—
50C3×Dic12D412yes—
51C3×D24—21no—
52C3×D24D412yes—
53C3×#204—22no—
54C3×#204—22no—
55C32×C16—22no—
56#2191—31noSylow
57C2×C3×#41—61no—
58C2×#235—21no—
59C3×#137—62no—
60C3×AGL(1,ℤ/8)—62no—
61C3×#135—32no—
62C3×#135—61no—
63C2×C3×Q16—31no—
64C2×C3×M16—21no—
65C2×C3×M16C2×D311yescharacteristic
66C2×C3×M16C22×C311yescharacteristic
67C3×C22⋊Q8—61no—
68#924—21no—
69#909—22no—
70#908—22no—
71C2×Dic12C22×C311yescharacteristic, complemented
72#904—21no—
73#904C22×C312yes—
74C3×Q32—62no—
75C3×SD32—62no—
76C3×M32—22no—
77C3×M32C2×D312yescomplemented
78C3×M32C22×C312yescomplemented
79C3×#106—31no—
80C3×#103—61no—
81#817C22×C311yescharacteristic, complemented
82#816—21no—
83Dic24—22no—
84#800—22no—
85C23×C32D811yescharacteristic
86C22×C3×D3—41no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4D811yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C3×#67—42no—
92C3×#67—42no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C3×D12—22no—
97C3×D12—41no—
98C3×C4×D3—42no—
99C3×C4×D3—42no—
100C3×Dic6—22no—
101C3×Dic6—41no—
102C3×Dic6—41no—
103C3×Dic6—41no—
104C32×C8—21no—
105C32×C8C2×D412yes—
106#575—31no—
107#510—32no—
108#470—31no—
109#360—32no—
110#349C3×D311yescharacteristic, complemented
111C3×#41—62no—
112C3×#41—62no—
113C3×#41—121no—
114C3×#41—121no—
115C2×C3×Q8—61no—
116C2×C3×Q8—61no—
117C2×C3×D4—61no—
118C2×C3×D4—121no—
119C22×C3×C4—21no—
120C22×C3×C4—121no—
121C22×C3×C4D1211yescharacteristic
122C22×C3×C4C3×D411yescharacteristic
123C2×#67—41no—
124#235—22no—
125#235—22no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×C4×D3—41no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C3×Q16—32no—
133C3×Q16—61no—
134C3×Q16—62no—
135C3×SD16—62no—
136C3×SD16—62no—
137C3×D8—32no—
138C3×D8—61no—
139C3×M16—22no—
140C3×M16—22no—
141C3×M16—42no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8D1211yescharacteristic
145C2×C3×C8C3×D411yescharacteristic
146C2×C3×C8C22×D311yescharacteristic
147C2×C3×C8C23×C311yescharacteristic
148C3×C4⋊C4—61no—
149C3×C4⋊C4—121no—
150C3×C22⋊C4—121no—
151#217—41no—
152C4⋊Dic3—21no—
153Dic3⋊C4—41no—
154Dic12—21no—
155Dic12—22no—
156Dic12C3×D412yes—
157D24—21no—
158D24C3×D412yes—
159#204—22no—
160#204—22no—
161C3×C16—22no—
162C3×C16—22no—
163C3×C16—42no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32C2×D811yescharacteristic
167C2×C3×D3—42no—
168C2×C3×D3—82no—
169C32×C4—21no—
170C32×C4#12011yescharacteristic
171C32×C4AGL(1,ℤ/8)11yescharacteristic
172C3×Dic3—42no—
173C3×Dic3—81no—
174C3×Dic3—81no—
175C2×#41—61no—
176#137—62no—
177AGL(1,ℤ/8)—62no—
178#135—32no—
179#135—61no—
180C2×Q16—31no—
181C2×M16C2×C3×D311yescharacteristic
182C22⋊Q8—61no—
183Q32—62no—
184SD32—62no—
185M32C2×C3×D312yescomplemented
186#106—31no—
187#103—61no—
188C23×C3—21no—
189C23×C3—121no—
190C23×C3D2411yescharacteristic
191C23×C3C3×D811yescharacteristic
192C22×D3—41no—
193C3×Q8—62no—
194C3×Q8—121no—
195C3×Q8—121no—
196C3×Q8—121no—
197C3×D4—62no—
198C3×D4—121no—
199C3×D4—122no—
200C3×D4—122no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—122no—
210C2×C3×C4—121no—
211C2×C3×C4—122no—
212C2×C3×C4—121no—
213C2×C3×C4—121no—
214C2×C3×C4D2411yescharacteristic
215C2×C3×C4C3×D811yescharacteristic
216C2×C3×C4C2×D1211yescharacteristic
217C2×C3×C4C2×C3×D411yescharacteristic
218#67—42no—
219#67—42no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223D12—22no—
224D12—41no—
225C4×D3—42no—
226C4×D3—42no—
227Dic6—22no—
228Dic6—41no—
229Dic6—41no—
230Dic6—41no—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8—22no—
234C3×C8—41no—
235C3×C8D3×D412yes—
236C3×C8C2×C3×D412yes—
237C2×C32—21no—
238C2×C32—41no—
239C2×C32#44711yescharacteristic
240C3×D3—82no—
241#41—62no—
242#41—62no—
243#41—121no—
244#41—121no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×D4—61no—
248C2×D4—121no—
249C22×C4—121no—
250C22×C4C3×D1211yescharacteristic
251Q16—32no—
252Q16—61no—
253Q16—62no—
254SD16—62no—
255SD16—62no—
256D8—32no—
257D8—61no—
258M16—22no—
259C2×C8C3×D1211yescharacteristic
260C2×C8C22×C3×D311yescharacteristic
261C4⋊C4—61no—
262C4⋊C4—121no—
263C22⋊C4—121no—
264C16—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—81no—
272C22×C3—122no—
273C22×C3—242no—
274C22×C3C2×D2411yescharacteristic
275C22×C3C2×C3×D811yescharacteristic
276C2×D3—42no—
277C2×D3—82no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—122no—
284C3×C4—241no—
285C3×C4—241no—
286C3×C4#88211yescharacteristic
287C3×C4#90811yescharacteristic
288C3×C4C3×#12011yescharacteristic
289C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
290Dic3—42no—
291Dic3—81no—
292Dic3—81no—
293C32#219111yescharacteristic, complemented, Sylow
294C23—121no—
295C23C3×D2411yescharacteristic
296Q8—62no—
297Q8—121no—
298Q8—121no—
299Q8—121no—
300D4—62no—
301D4—121no—
302D4—122no—
303D4—122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—122no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4C3×D2411yescharacteristic
312C2×C4C2×C3×D1211yescharacteristic
313C8—21no—
314C8C3×D3×D412yes—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—81no—
322C2×C3—242no—
323C2×C3#473211yescharacteristic
324C2×C3C3×#44711yescharacteristic
325D3—82no—
326C22—21no—
327C22—41no—
328C22—122no—
329C22—242no—
330C22C2×C3×D2411yescharacteristic
331C4—21no—
332C4—122no—
333C4—241no—
334C4—241no—
335C4C3×#88211yescharacteristic
336C4C3×#90811yescharacteristic
337C3—21no—
338C3#6912111yescharacteristic, direct product factor
339C3C3×#219111yescharacteristic, complemented
340C2—21no—
341C2—41no—
342C2—242no—
343C2C3×#473211yescharacteristic
subgroupG/H
centerC2×C3#4732
commutator, cosocleC2×C3×C8C23×C3
FrattiniC2×C8C22×C3×D3
FittingC32×#349C2
socleC2×C32#447
prime radicalC3×#5748C2
hypercenterC3×#349D3
hypocenterC3C3×#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ C3×#349

Complements

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

NH ≅ G/Ntype
2. C3×#5137C2semidirect
3. C3×#4909C2semidirect
4. C32×#349C2semidirect
5. C3×#4518C2semidirect
7. #69121C3direct (N × H)
13. C2×C3×Dic12C22semidirect
16. C32×M32C22semidirect
17. C3×#817C22semidirect
25. #5137C2×C3semidirect
26. #4909C2×C3semidirect
29. C3×#349D3semidirect
30. C3×#349C2×C3semidirect
31. #4518C2×C3semidirect
71. C2×Dic12C22×C3semidirect
77. C3×M32C2×D3semidirect
78. C3×M32C22×C3semidirect
81. #817C22×C3semidirect
110. #349C3×D3semidirect
185. M32C2×C3×D3semidirect
293. C32#2191semidirect
338. C3#69121direct (N × H)
339. C3C3×#2191semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
C3×#51372
C3×#49092
C32×#3492
C3×#45182
C3×#21913
#691213