Finite Group Databasebeta

← C3×#33032 #10651639

Subgroups — C3×#33032 #10651639

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic
3C3×#2862C211yesmaximal, characteristic, complemented
4C3×C42×D4C211yesmaximal, characteristic
5C3×#1988C211yesmaximal, characteristic, complemented
6C3×#1981C211yesmaximal, characteristic, complemented
7C3×#1928C214yesmaximal, complemented
8C3×#1877C214yesmaximal
9C3×#1831C211yesmaximal, characteristic, complemented
10#33032C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C42—21no—
17C3×#474—24no—
18C3×#461—24no—
19C2×C3×#107C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2214yes—
23C2×C3×#103C2212yes—
24C2×C3×#102C2212yes—
25C3×#391—21no—
26C3×#391—21no—
27C3×#391C2214yes—
28C3×#382—22no—
29C3×C4×C4⋊C4C2212yes—
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4—22no—
32C3×C4×C22⋊C4—21no—
33C3×C43C2211yescharacteristic
34C3×#340C2214yes—
35C3×#336C2214yes—
36C2×#474C2×C311yescharacteristic
37C2×#461C2×C311yescharacteristic
38#2862C2×C311yescharacteristic, complemented
39C42×D4C2×C311yescharacteristic
40#1988C2×C311yescharacteristic, complemented
41#1981C2×C311yescharacteristic, complemented
42#1928C2×C314yescomplemented
43#1877C2×C314yes—
44#1831C2×C311yescharacteristic, complemented
45C22×C3×Q8—21no—
46C22×C3×Q8C2312yes—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—22no—
49C23×C3×C4—21no—
50C2×C3×SD16—24no—
51C2×C3×SD16—28no—
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4×D4—24no—
59C3×C4×D4—24no—
60C3×C4×D4—22no—
61C3×C4×D4D414yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—22no—
71C2×C3×C42—21no—
72C2×C3×C42—44no—
73C2×C3×C42D412yes—
74C2×C3×C42D411yescharacteristic
75C2×C3×C42D411yescharacteristic
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78C3×#107—24no—
79C3×C4⋊C8—22no—
80C3×C4⋊C8—22no—
81C3×#103—24no—
82C3×#103—28no—
83C3×#102—24no—
84C3×#95—22no—
85C3×#95—21no—
86C3×#95—24no—
87C3×#95—24no—
88C22×SD16C22×C312yes—
89C2×C4⋊Q8C22×C311yescharacteristic
90C2×C4⋊Q8C22×C311yescharacteristic
91C2×C4×D4C22×C312yes—
92C2×C4×D4C22×C311yescharacteristic
93C22×C42—21no—
94#474—24no—
95#461—24no—
96C2×#107C22×C312yes—
97C2×C4⋊C8C22×C311yescharacteristic
98C2×C4⋊C8C22×C311yescharacteristic
99C2×#103C22×C314yes—
100C2×#103C22×C312yes—
101C2×#102C22×C312yes—
102#391—21no—
103#391—21no—
104#391C22×C314yes—
105#382—22no—
106C4×C4⋊C4C22×C312yes—
107C4×C4⋊C4C22×C311yescharacteristic
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C43C22×C311yescharacteristic
111#340C22×C314yes—
112#336C22×C314yes—
113C24×C3—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—24no—
118C2×C3×Q8—42no—
119C2×C3×Q8—42no—
120C2×C3×Q8—42no—
121C2×C3×D4—21no—
122C2×C3×D4—22no—
123C2×C3×D4C2×D412yes—
124C2×C3×D4#4114yes—
125C22×C3×C4—28no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—44no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—44no—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C3×SD16—48no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—24no—
144C2×C3×C8—24no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—42no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4C2×D412yes—
153C3×C4⋊C4#4114yes—
154C3×C22⋊C4—28no—
155C3×C22⋊C4—24no—
156C3×C42—22no—
157C3×C42—22no—
158C3×C42—24no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—44no—
162C3×C42—42no—
163C3×C42C2×D412yes—
164C3×C42#4112yes—
165C22×Q8—21no—
166C22×Q8C23×C312yes—
167C22×D4C23×C311yescharacteristic
168C23×C4—22no—
169C23×C4—21no—
170C2×SD16—24no—
171C2×SD16—28no—
172C22×C8C23×C312yes—
173C22×C8C23×C312yes—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4C3×D414yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C23×C312yes—
186C2×C4⋊C4C23×C311yescharacteristic
187C2×C4⋊C4C23×C312yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—44no—
193C2×C42C3×D412yes—
194C2×C42C3×D411yescharacteristic
195C2×C42C3×D411yescharacteristic
196C2×C42C23×C312yes—
197C2×C42C23×C311yescharacteristic
198#107—24no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201#103—24no—
202#103—28no—
203#102—24no—
204#95—22no—
205#95—21no—
206#95—24no—
207#95—24no—
208C23×C3—22no—
209C23×C3—24no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3C22×D411yescharacteristic
214C3×Q8—42no—
215C3×Q8—44no—
216C3×Q8—42no—
217C3×Q8—84no—
218C3×D4—24no—
219C3×D4—22no—
220C3×D4#12414yes—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—24no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—416no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—48no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—44no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12112yes—
241C2×C3×C4#12112yes—
242C2×C3×C4#13512yes—
243C2×C3×C4AGL(1,ℤ/8)12yes—
244C2×C3×C4#13714yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C3×C8—42no—
250C3×C8—42no—
251C24—21no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×D4—21no—
260C2×D4—22no—
261C2×D4C2×C3×D412yes—
262C2×D4C3×#4114yes—
263C22×C4—28no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4C2×C3×D412yes—
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C2×C3×D412yes—
276C22×C4C2×C3×D411yescharacteristic
277C22×C4C24×C311yescharacteristic
278SD16—48no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—24no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4C2×C3×D412yes—
291C4⋊C4C3×#4114yes—
292C22⋊C4—28no—
293C22⋊C4—24no—
294C42—22no—
295C42—22no—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—44no—
300C42—42no—
301C42C2×C3×D412yes—
302C42C3×#4112yes—
303C22×C3—24no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3C2×#12111yescharacteristic
309C22×C3#52512yes—
310C22×C3C2×#13511yescharacteristic
311C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C3C2×#13712yes—
313C3×C4—24no—
314C3×C4—22no—
315C3×C4—42no—
316C3×C4—44no—
317C3×C4—42no—
318C3×C4—82no—
319C3×C4—82no—
320C3×C4#46314yes—
321C3×C4C2×#12411yescharacteristic
322C3×C4#52912yes—
323C3×C4#53211yescharacteristic
324C23—22no—
325C23—24no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23C22×C3×D411yescharacteristic
330Q8—42no—
331Q8—44no—
332Q8—42no—
333Q8—84no—
334D4—24no—
335D4—22no—
336D4C3×#12414yes—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—416no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4C3×#12112yes—
357C2×C4C3×#12112yes—
358C2×C4C3×#13512yes—
359C2×C4C3×AGL(1,ℤ/8)12yes—
360C2×C4C3×#13714yes—
361C2×C4C22×C3×D411yescharacteristic
362C2×C4C2×C3×#4112yes—
363C2×C4C2×C3×#4111yescharacteristic
364C2×C4C2×C3×#4112yes—
365C8—42no—
366C8—42no—
367C2×C3—44no—
368C2×C3#285011yescharacteristic
369C2×C3C2×#46311yescharacteristic
370C2×C3#303911yescharacteristic
371C2×C3#310712yes—
372C2×C3#311012yes—
373C22—24no—
374C22—42no—
375C22—42no—
376C22—44no—
377C22—44no—
378C22C2×C3×#12111yescharacteristic
379C22C3×#52512yes—
380C22C2×C3×#13511yescharacteristic
381C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
382C22C2×C3×#13712yes—
383C4—22no—
384C4—24no—
385C4—42no—
386C4—42no—
387C4—44no—
388C4—82no—
389C4—82no—
390C4C3×#46314yes—
391C4C2×C3×#12411yescharacteristic
392C4C3×#52912yes—
393C4C3×#53211yescharacteristic
394C3#3303211yescharacteristic, direct product factor, Sylow
395C2—44no—
396C2C3×#285011yescharacteristic
397C2C2×C3×#46311yescharacteristic
398C2C3×#303911yescharacteristic
399C2C3×#310712yes—
400C2C3×#311012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2862C2semidirect
5. C3×#1988C2semidirect
6. C3×#1981C2semidirect
7. C3×#1928C2semidirect
9. C3×#1831C2semidirect
10. #33032C3direct (N × H)
38. #2862C2×C3semidirect
40. #1988C2×C3semidirect
41. #1981C2×C3semidirect
42. #1928C2×C3semidirect
44. #1831C2×C3semidirect
394. C3#33032direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×#4612
C3×#28622
C3×C42×D42
C3×#19882
C3×#19812
C3×#19282
C3×#18772
C3×#18312
#330323