Finite Group Databasebeta

← C3×#70008 #11792763

Subgroups — C3×#70008 #11792763

Order 1152.

763 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#951C212yesmaximal
2C22×C3×Dic12C211yesmaximal, characteristic, complemented
3C32×#408C211yesmaximal, characteristic
4C2×C3×#824C212yesmaximal, complemented
5C3×#4552C211yesmaximal, characteristic, complemented
6C3×#2020—31nomaximal
7#70008C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C22×C3×Dic6C2212yes—
10C2×C32×M16C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C3×#951—24no—
13C3×#951—24no—
14C2×C3×#216—22no—
15C2×C3×#214—22no—
16C2×C3×#208C2212yes—
17C2×C3×Dic12—24no—
18C2×C3×Dic12—22no—
19C32×C22⋊C8—22no—
20C3×#824—24no—
21C2×C3×#137—32no—
22C22×C3×Q16—31no—
23C2×#951C2×C312yes—
24C22×Dic12C2×C311yescharacteristic, complemented
25C2×C3×#101—32no—
26C3×#408—21no—
27C3×#408D311yescharacteristic
28C3×#408C2×C311yescharacteristic
29C2×#824C2×C312yescomplemented
30C3×#341—31no—
31#4552C2×C311yescharacteristic, complemented
32C32×#41—22no—
33C32×#41—22no—
34C2×C32×Q8D411yescharacteristic
35C2×C32×D4—21no—
36C2×C32×D4D411yescharacteristic
37C22×C32×C4—21no—
38C22×C32×C4C2311yescharacteristic
39C22×C3×Dic3—22no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6—22no—
43C2×C3×Dic6—22no—
44C2×C3×Dic6—22no—
45C2×C3×Dic6D414yes—
46C32×M16—22no—
47C2×C32×C8—22no—
48C2×C32×C8—21no—
49C2×C32×C8D412yes—
50C3×#216—28no—
51C3×#214—28no—
52C3×#208—22no—
53C3×#208D414yes—
54C2×C3×C3⋊C8—22no—
55C3×Dic12—28no—
56#2020—31noSylow
57C2×C3×#41—21no—
58C2×C3×#41C2×D311yescharacteristic
59C2×C3×#41C22×C311yescharacteristic
60C22×C3×Q8—32no—
61C22×Dic6C22×C312yes—
62C3×#137—64no—
63C3×#137—64no—
64C2×C3×Q16—64no—
65C2×C3×Q16—62no—
66C2×C3×Q16—62no—
67C2×C3×SD16—62no—
68C2×C3×M16—21no—
69C2×C3×M16—32no—
70C2×C3×M16C2×D311yescharacteristic
71C2×C3×M16C22×C311yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8C2×D311yescharacteristic
74C22×C3×C8C22×C311yescharacteristic
75#951—24no—
76#951—24no—
77C2×#216—22no—
78C2×#214—22no—
79C2×#208C22×C312yes—
80C2×Dic12—24no—
81C2×Dic12—22no—
82C3×#101—64no—
83C3×C22⋊C8—24no—
84C3×C22⋊C8—22no—
85C3×C22⋊C8—22no—
86#824—24no—
87C23×C32—21no—
88C23×C32#4111yescharacteristic
89C32×Q8—22no—
90C32×D4—22no—
91C32×D4—42no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D412yes—
97C2×C32×C4C2×D412yes—
98C2×C32×C4C2×D411yescharacteristic
99C2×C3×Dic3—24no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3—42no—
102C3×Dic6—24no—
103C3×Dic6—24no—
104C3×Dic6—42no—
105C3×Dic6—42no—
106C32×C8—22no—
107C32×C8—41no—
108C3×C3⋊C8—24no—
109C2×#137—32no—
110C22×Q16—31no—
111C2×#101—32no—
112#408C3×D311yescharacteristic
113#341—31no—
114C3×#41—22no—
115C3×#41—22no—
116C3×#41—22no—
117C3×#41—22no—
118C3×#41—42no—
119C3×#41—42no—
120C2×C3×Q8—21no—
121C2×C3×Q8—34no—
122C2×C3×Q8—62no—
123C2×C3×Q8—62no—
124C2×C3×Q8—62no—
125C2×C3×Q8—62no—
126C2×C3×Q8—62no—
127C2×C3×Q8D1211yescharacteristic
128C2×C3×Q8C3×D411yescharacteristic
129C2×C3×D4—22no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4D1211yescharacteristic
134C2×C3×D4C3×D411yescharacteristic
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—62no—
140C22×C3×C4C22×D311yescharacteristic
141C22×C3×C4C23×C311yescharacteristic
142C22×Dic3—22no—
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6C3×D414yes—
149C3×Q16—68no—
150C3×Q16—68no—
151C3×SD16—68no—
152C3×M16—22no—
153C3×M16—22no—
154C3×M16—34no—
155C3×M16—42no—
156C3×M16—62no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—22no—
163C2×C3×C8—42no—
164C2×C3×C8—62no—
165C2×C3×C8#6712yes—
166C2×C3×C8C3×D412yes—
167#216—28no—
168#214—28no—
169#208—22no—
170#208C3×D414yes—
171C2×C3⋊C8—22no—
172Dic12—28no—
173C22×C32—22no—
174C22×C32—41no—
175C22×C32—42no—
176C22×C32#12112yes—
177C22×C32#12111yescharacteristic
178C32×C4—41no—
179C32×C4#12012yes—
180C32×C4#12011yescharacteristic
181C32×C4#12711yescharacteristic
182C3×Dic3—44no—
183C2×#41C2×C3×D311yescharacteristic
184C22×Q8—32no—
185#137—64no—
186#137—64no—
187C2×Q16—64no—
188C2×Q16—62no—
189C2×Q16—62no—
190C2×SD16—62no—
191C2×M16—32no—
192C2×M16C2×C3×D311yescharacteristic
193C22×C8C2×C3×D311yescharacteristic
194#101—64no—
195C22⋊C8—22no—
196C23×C3—22no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3#23511yescharacteristic
201C23×C3C3×#4111yescharacteristic
202C3×Q8—22no—
203C3×Q8—22no—
204C3×Q8—42no—
205C3×Q8—64no—
206C3×Q8—64no—
207C3×Q8—122no—
208C3×Q8—122no—
209C3×D4—22no—
210C3×D4—22no—
211C3×D4—42no—
212C3×D4—44no—
213C3×D4—42no—
214C3×D4—42no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—64no—
229C2×C3×C4—122no—
230C2×C3×C4—122no—
231C2×C3×C4C2×D1211yescharacteristic
232C2×C3×C4D3×D412yes—
233C2×C3×C4D3×D412yes—
234C2×C3×C4C2×#6711yescharacteristic
235C2×C3×C4C2×C3×D411yescharacteristic
236C2×C3×C4C2×C3×D412yes—
237C2×C3×C4C2×C3×D412yes—
238C2×C3×C4C2×C3×D411yescharacteristic
239C2×Dic3—24no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242Dic6—24no—
243Dic6—24no—
244Dic6—42no—
245Dic6—42no—
246C3×C8—22no—
247C3×C8—22no—
248C3×C8—42no—
249C3×C8—42no—
250C3×C8—41no—
251C3×C8—41no—
252C3×C8—64no—
253C3⋊C8—24no—
254C2×C32—22no—
255C2×C32—81no—
256C2×C32#39211yescharacteristic
257C2×C32#47312yes—
258#41—22no—
259#41—22no—
260C2×Q8—34no—
261C2×Q8—62no—
262C2×Q8—62no—
263C2×Q8—62no—
264C2×Q8—62no—
265C2×Q8—62no—
266C2×Q8C3×D1211yescharacteristic
267C2×D4—21no—
268C2×D4C3×D1211yescharacteristic
269C22×C4—21no—
270C22×C4—62no—
271C22×C4C22×C3×D311yescharacteristic
272Q16—68no—
273Q16—68no—
274SD16—68no—
275M16—22no—
276M16—34no—
277M16—62no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—62no—
281C2×C8C3×#6712yes—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—44no—
289C22×C3—42no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—42no—
293C22×C3#88412yes—
294C22×C3#93011yescharacteristic
295C22×C3C3×#12112yes—
296C22×C3C3×#12111yescharacteristic
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—42no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—124no—
304C3×C4#88212yes—
305C3×C4#93711yescharacteristic
306C3×C4#94011yescharacteristic
307C3×C4C3×#12012yes—
308C3×C4C3×#12011yescharacteristic
309C3×C4C3×#12711yescharacteristic
310Dic3—44no—
311C32#202011yescharacteristic, complemented, Sylow
312C23—21no—
313C23C3×#23511yescharacteristic
314Q8—22no—
315Q8—64no—
316Q8—64no—
317Q8—122no—
318Q8—122no—
319D4—22no—
320D4—42no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—64no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4C2×C3×D1211yescharacteristic
328C2×C4C3×D3×D412yes—
329C2×C4C3×D3×D412yes—
330C2×C4C2×C3×#6711yescharacteristic
331C8—22no—
332C8—41no—
333C8—64no—
334C2×C3—22no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—42no—
339C2×C3—82no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3#496011yescharacteristic
343C2×C3#514312yes—
344C2×C3C3×#39211yescharacteristic
345C2×C3C3×#47312yes—
346C22—22no—
347C22—41no—
348C22—42no—
349C22C3×#88412yes—
350C22C3×#93011yescharacteristic
351C4—41no—
352C4—124no—
353C4C3×#88212yes—
354C4C3×#93711yescharacteristic
355C4C3×#94011yescharacteristic
356C3—21no—
357C3#7000811yescharacteristic, direct product factor
358C3C3×#202011yescharacteristic, complemented
359C2—22no—
360C2—81no—
361C2C3×#496011yescharacteristic
362C2C3×#514312yes—
subgroupG/H
centerC22×C3#930
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#408C2
socleC22×C32#121
prime radicalC2×C32×#41C22
hypercenterC3×#408D3
hypocenterC3C3×#2020
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic12C2semidirect
4. C2×C3×#824C2semidirect
5. C3×#4552C2semidirect
7. #70008C3direct (N × H)
24. C22×Dic12C2×C3semidirect
29. C2×#824C2×C3semidirect
31. #4552C2×C3semidirect
311. C32#2020semidirect
357. C3#70008direct (N × H)
358. C3C3×#2020semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9512
C22×C3×Dic122
C32×#4082
C2×C3×#8242
C3×#45522
C3×#20203
#700083