Finite Group Databasebeta

← C3×#70041 #11785481

Subgroups — C3×#70041 #11785481

Order 1152.

599 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#948C211yesmaximal, characteristic
2C32×#410C211yesmaximal, characteristic
3C3×#5240C211yesmaximal, characteristic
4C3×#4998C212yesmaximal, complemented
5C3×#4531C212yesmaximal, complemented
6C3×#1812—31nomaximal
7#70041C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C32×#117C2212yes—
10C3×#948—22no—
11C3×#948—22no—
12C3×#926C2212yes—
13C2×C3×#208C412yescomplemented
14C2×C3×#208C2211yescharacteristic
15C22×C3×C3⋊C8C412yescomplemented
16C22×C3×C3⋊C8C2211yescharacteristic
17C32×#99—22no—
18C3×#834—22no—
19C3×#808—22no—
20C3×#807—22no—
21C2×C3×#131—31no—
22C2×#948C2×C311yescharacteristic
23C3×#427—32no—
24C3×#410—21no—
25C3×#410—31no—
26C3×#410D311yescharacteristic
27C3×#410C2×C311yescharacteristic
28#5240C2×C311yescharacteristic
29#4998C2×C312yescomplemented
30C3×H(3,ℤ/4)—32no—
31#4531C2×C312yescomplemented
32C32×#41—22no—
33C32×#41—22no—
34C2×C32×Q8Q811yescharacteristic
35C2×C32×D4D412yes—
36C2×C32×D4Q811yescharacteristic
37C22×C32×C4C2×C412yes—
38C22×C32×C4C2311yescharacteristic
39C32×C4⋊C4—22no—
40C32×C22⋊C4—22no—
41C32×C42—22no—
42C3×#217—22no—
43C3×C4⋊Dic3—22no—
44C3×C4×Dic3—22no—
45C3×#208—22no—
46C3×#208—24no—
47C2×C3×C3⋊C8—22no—
48C2×C3×C3⋊C8—22no—
49C2×C3×C3⋊C8C2×C412yes—
50C2×C3×C3⋊C8C2×C411yescharacteristic
51C2×C3×C3⋊C8C2×C411yescharacteristic
52C2×C3×C3⋊C8D412yes—
53C2×C3×C3⋊C8D412yes—
54#1812—31noSylow
55C2×C3×#41—21no—
56C2×C3×#41C2×D311yescharacteristic
57C2×C3×#41C22×C311yescharacteristic
58C3×#131—62no—
59C3×#131—62no—
60C2×C3×M16—32no—
61C2×C3×M16—31no—
62C22×C3×C8—31no—
63C22×C3×C8—32no—
64C3×#117—22no—
65C3×#117—32no—
66C3×#117C2×D312yes—
67C3×#117C22×C312yes—
68#948—22no—
69#948—22no—
70#926C22×C312yes—
71C2×#208C3×C412yescomplemented
72C2×#208C22×C311yescharacteristic
73C22×C3⋊C8C3×C412yescomplemented
74C22×C3⋊C8C22×C311yescharacteristic
75C3×#107—62no—
76C3×#106—62no—
77C3×#99—24no—
78C3×#99—22no—
79C3×#99—22no—
80C3×#99—62no—
81#834—22no—
82#808—22no—
83#807—22no—
84C23×C32C4⋊C412yes—
85C23×C32C2×D411yescharacteristic
86C32×Q8—22no—
87C32×D4—24no—
88C32×D4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—42no—
92C2×C32×C4C4⋊C412yes—
93C2×C32×C4C22×C411yescharacteristic
94C2×C32×C4C2×D412yes—
95C2×C32×C4C2×Q811yescharacteristic
96C2×C32×C4#4112yes—
97C2×C3×Dic3—42no—
98C3×C3⋊C8—22no—
99C3×C3⋊C8C22⋊C411yescharacteristic
100C3×C3⋊C8C22⋊C411yescharacteristic
101C3×C3⋊C8C22⋊C412yes—
102C2×#131—31no—
103#427—32no—
104#410—31no—
105#410C3×D311yescharacteristic
106H(3,ℤ/4)—32no—
107C3×#41—22no—
108C3×#41—22no—
109C3×#41—22no—
110C3×#41—22no—
111C3×#41—42no—
112C3×#41—42no—
113C2×C3×Q8—21no—
114C2×C3×Q8Dic611yescharacteristic
115C2×C3×Q8C3×Q811yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—22no—
118C2×C3×D4Dic611yescharacteristic
119C2×C3×D4#6712yes—
120C2×C3×D4C3×D412yes—
121C2×C3×D4C3×Q811yescharacteristic
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4C4×D312yes—
125C22×C3×C4C2×C3×C412yes—
126C22×C3×C4C22×D311yescharacteristic
127C22×C3×C4C23×C311yescharacteristic
128C3×M16—64no—
129C3×M16—62no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—32no—
133C2×C3×C8—32no—
134C2×C3×C8—32no—
135C2×C3×C8—62no—
136C2×C3×C8—62no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4—62no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—62no—
145C3×C42—22no—
146C3×C42—22no—
147C3×C42—42no—
148C3×C42—62no—
149#217—22no—
150C4⋊Dic3—22no—
151C4×Dic3—22no—
152#208—22no—
153#208—24no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8C2×C3×C412yes—
157C2×C3⋊C8C2×C3×C411yescharacteristic
158C2×C3⋊C8C2×C3×C411yescharacteristic
159C2×C3⋊C8C3×D412yes—
160C2×C3⋊C8C3×D412yes—
161C22×C32—22no—
162C22×C32—41no—
163C22×C32—42no—
164C22×C32C2×C4⋊C411yescharacteristic
165C22×C32#12112yes—
166C32×C4—22no—
167C32×C4—82no—
168C32×C4C2×C22⋊C411yescharacteristic
169C32×C4#11711yescharacteristic
170C32×C4C22⋊Q812yes—
171C3×Dic3—82no—
172C2×#41C2×C3×D311yescharacteristic
173#131—62no—
174#131—62no—
175C2×M16—32no—
176C2×M16—31no—
177C22×C8—31no—
178C22×C8—32no—
179#117—32no—
180#117C2×C3×D312yes—
181#107—62no—
182#106—62no—
183#99—22no—
184#99—62no—
185C23×C3—22no—
186C23×C3—21no—
187C23×C3Dic3⋊C412yes—
188C23×C3C3×C4⋊C412yes—
189C23×C3C2×#6711yescharacteristic
190C23×C3C2×C3×D411yescharacteristic
191C3×Q8—22no—
192C3×Q8—22no—
193C3×Q8—42no—
194C3×D4—24no—
195C3×D4—24no—
196C3×D4—22no—
197C3×D4—22no—
198C3×D4—42no—
199C3×D4—44no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—44no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—122no—
215C2×C3×C4Dic3⋊C412yes—
216C2×C3×C4C3×C4⋊C412yes—
217C2×C3×C4C2×Dic611yescharacteristic
218C2×C3×C4C2×C4×D311yescharacteristic
219C2×C3×C4D3×D412yes—
220C2×C3×C4#23712yes—
221C2×C3×C4C22×C3×C411yescharacteristic
222C2×C3×C4C2×C3×D412yes—
223C2×C3×C4C2×C3×Q811yescharacteristic
224C2×C3×C4C3×#4112yes—
225C2×Dic3—42no—
226C3×C8—31no—
227C3×C8—31no—
228C3×C8—32no—
229C3×C8—62no—
230C3⋊C8—22no—
231C3⋊C8C3×C22⋊C411yescharacteristic
232C3⋊C8C3×C22⋊C411yescharacteristic
233C3⋊C8C3×C22⋊C412yes—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—42no—
237C2×C32#38011yescharacteristic
238#41—22no—
239#41—22no—
240C2×Q8C3×Dic611yescharacteristic
241C2×D4C3×Dic611yescharacteristic
242C2×D4C3×#6712yes—
243C22×C4C3×C4×D312yes—
244C22×C4C22×C3×D311yescharacteristic
245M16—64no—
246M16—62no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—32no—
250C2×C8—32no—
251C2×C8—32no—
252C2×C8—62no—
253C2×C8—62no—
254C4⋊C4—22no—
255C4⋊C4—62no—
256C22⋊C4—22no—
257C22⋊C4—62no—
258C42—22no—
259C42—62no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—42no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3—81no—
270C22×C3—82no—
271C22×C3C2×Dic3⋊C411yescharacteristic
272C22×C3#93912yes—
273C22×C3C2×C3×C4⋊C411yescharacteristic
274C22×C3C3×#12112yes—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—42no—
281C3×C4—82no—
282C3×C4—82no—
283C3×C4—84no—
284C3×C4—242no—
285C3×C4#87711yescharacteristic
286C3×C4#87812yes—
287C3×C4D3×C22⋊C411yescharacteristic
288C3×C4C2×C3×C22⋊C411yescharacteristic
289C3×C4C3×#11711yescharacteristic
290C3×C4C3×C22⋊Q812yes—
291Dic3—82no—
292C32#181211yescharacteristic, complemented, Sylow
293C23C3×Dic3⋊C412yes—
294C23C2×C3×#6711yescharacteristic
295Q8—22no—
296D4—24no—
297D4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—122no—
302C2×C4C3×Dic3⋊C412yes—
303C2×C4C2×C3×Dic611yescharacteristic
304C2×C4C2×C3×C4×D311yescharacteristic
305C2×C4C3×D3×D412yes—
306C2×C4C3×#23712yes—
307C8—31no—
308C8—31no—
309C8—32no—
310C8—62no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—42no—
317C2×C3—41no—
318C2×C3—42no—
319C2×C3—42no—
320C2×C3—82no—
321C2×C3#494211yescharacteristic
322C2×C3C3×#38011yescharacteristic
323C22—22no—
324C22—41no—
325C22—42no—
326C22C2×C3×Dic3⋊C411yescharacteristic
327C22C3×#93912yes—
328C4—22no—
329C4—82no—
330C4—242no—
331C4C3×#87711yescharacteristic
332C4C3×#87812yes—
333C4C3×D3×C22⋊C411yescharacteristic
334C3—21no—
335C3#7004111yescharacteristic, direct product factor
336C3C3×#181211yescharacteristic, complemented
337C2—22no—
338C2—21no—
339C2—42no—
340C2C3×#494211yescharacteristic
subgroupG/H
centerC3×C4#877
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#410C2
socleC2×C32#380
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#410D3
hypocenterC3C3×#1812
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
4. C3×#4998C2semidirect
5. C3×#4531C2semidirect
7. #70041C3direct (N × H)
13. C2×C3×#208C4semidirect
15. C22×C3×C3⋊C8C4semidirect
29. #4998C2×C3semidirect
31. #4531C2×C3semidirect
71. C2×#208C3×C4semidirect
73. C22×C3⋊C8C3×C4semidirect
292. C32#1812semidirect
335. C3#70041direct (N × H)
336. C3C3×#1812semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9482
C32×#4102
C3×#52402
C3×#49982
C3×#45312
C3×#18123
#700413