Finite Group Databasebeta

← C3×#29829 #10648436

Subgroups — C3×#29829 #10648436

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 394 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×#2293C211yesmaximal, characteristic
5C3×#2068C214yesmaximal, complemented
6C3×#2055C214yesmaximal
7C3×#1988C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1842C211yesmaximal, characteristic, complemented
10#29829C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×#117—21no—
16C3×#474—24no—
17C3×#461—24no—
18C2×C3×#107C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2214yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2214yes—
24C2×C3×#102C2212yes—
25C3×#391—21no—
26C3×#391—21no—
27C3×#391C2214yes—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#388—22no—
31C3×#382—22no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C3×C43C2211yescharacteristic
35C3×#340C2214yes—
36C2×#474C2×C311yescharacteristic
37C2×#461C2×C311yescharacteristic
38#2862C2×C311yescharacteristic, complemented
39#2293C2×C311yescharacteristic
40#2068C2×C314yescomplemented
41#2055C2×C314yes—
42#1988C2×C311yescharacteristic, complemented
43#1981C2×C311yescharacteristic, complemented
44#1842C2×C311yescharacteristic, complemented
45C22×C3×Q8—21no—
46C22×C3×Q8C2312yes—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—21no—
49C2×C3×SD16—24no—
50C2×C3×SD16—44no—
51C22×C3×C8C2312yes—
52C22×C3×C8C2312yes—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4×D4—22no—
58C3×C4×D4D414yes—
59C3×#117—44no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—21no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D412yes—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C3×#107—24no—
77C3×C4⋊C8—22no—
78C3×C4⋊C8—22no—
79C3×C4⋊C8—44no—
80C3×#103—24no—
81C3×#103—44no—
82C3×#102—24no—
83C3×#95—22no—
84C3×#95—21no—
85C3×#95—24no—
86C3×#95—24no—
87C22×SD16C22×C312yes—
88C2×C4⋊Q8C22×C311yescharacteristic
89C2×C4⋊Q8C22×C311yescharacteristic
90C2×C4×D4C22×C311yescharacteristic
91C2×#117—21no—
92#474—24no—
93#461—24no—
94C2×#107C22×C312yes—
95C2×C4⋊C8C22×C311yescharacteristic
96C2×C4⋊C8C22×C311yescharacteristic
97C2×C4⋊C8C22×C314yes—
98C2×#103C22×C312yes—
99C2×#103C22×C314yes—
100C2×#102C22×C312yes—
101#391—21no—
102#391—21no—
103#391C22×C314yes—
104#390C22×C312yes—
105#389C22×C312yes—
106#388—22no—
107#382—22no—
108C4×C4⋊C4C22×C311yescharacteristic
109C4×C22⋊C4—21no—
110C43C22×C311yescharacteristic
111#340C22×C314yes—
112C24×C3—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—42no—
118C2×C3×Q8—42no—
119C2×C3×Q8—42no—
120C2×C3×D4—21no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4C2×D412yes—
124C22×C3×C4—21no—
125C22×C3×C4—24no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2411yescharacteristic
136C3×SD16—48no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—24no—
140C2×C3×C8—24no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—44no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—42no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4C2×D412yes—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—44no—
153C3×C22⋊C4—44no—
154C3×C42—22no—
155C3×C42—22no—
156C3×C42—24no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—42no—
160C3×C42C2×D412yes—
161C3×C42#4112yes—
162C22×Q8—21no—
163C22×Q8C23×C312yes—
164C22×D4C23×C311yescharacteristic
165C23×C4—21no—
166C2×SD16—24no—
167C2×SD16—44no—
168C22×C8C23×C312yes—
169C22×C8C23×C312yes—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4×D4—22no—
175C4×D4C3×D414yes—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C23×C312yes—
182C2×C4⋊C4C23×C312yes—
183C2×C4⋊C4C23×C311yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C42—21no—
188C2×C42C3×D411yescharacteristic
189C2×C42C3×D411yescharacteristic
190C2×C42C3×D412yes—
191C2×C42C23×C312yes—
192C2×C42C23×C311yescharacteristic
193#107—24no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—44no—
197#103—24no—
198#103—44no—
199#102—24no—
200#95—22no—
201#95—21no—
202#95—24no—
203#95—24no—
204C23×C3—22no—
205C23×C3—42no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3C22×D411yescharacteristic
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—44no—
213C3×Q8—84no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—42no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—48no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—44no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4#12112yes—
236C2×C3×C4#12112yes—
237C2×C3×C4#13512yes—
238C2×C3×C4AGL(1,ℤ/8)12yes—
239C2×C3×C4#13714yes—
240C2×C3×C4C22×D411yescharacteristic
241C2×C3×C4C2×#4111yescharacteristic
242C2×C3×C4H(5,𝔽2)12yes—
243C2×C3×C42−1+412yes—
244C3×C8—42no—
245C3×C8—42no—
246C24—21no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C2×C3×D412yes—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C3×D412yes—
266C22×C4C2×C3×D411yescharacteristic
267C22×C4C2×C3×D412yes—
268C22×C4C2×C3×D411yescharacteristic
269C22×C4C24×C311yescharacteristic
270SD16—48no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—24no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4C2×C3×D412yes—
285C22⋊C4—24no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—42no—
294C42C2×C3×D412yes—
295C42C3×#4112yes—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3C2×#12111yescharacteristic
302C22×C3#53712yes—
303C22×C3C2×#13511yescharacteristic
304C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C3C2×#13712yes—
306C3×C4—24no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—82no—
311C3×C4—82no—
312C3×C4—82no—
313C3×C4#46314yes—
314C3×C4#53511yescharacteristic
315C3×C4#53611yescharacteristic
316C3×C4#54212yes—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23C22×C3×D411yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—44no—
326Q8—84no—
327D4—22no—
328D4—22no—
329D4—42no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4C3×#12112yes—
349C2×C4C3×#12112yes—
350C2×C4C3×#13512yes—
351C2×C4C3×AGL(1,ℤ/8)12yes—
352C2×C4C3×#13714yes—
353C2×C4C22×C3×D411yescharacteristic
354C2×C4C2×C3×#4111yescharacteristic
355C2×C4C3×H(5,𝔽2)12yes—
356C2×C4C3×2−1+412yes—
357C8—42no—
358C8—42no—
359C2×C3—82no—
360C2×C3#262011yescharacteristic
361C2×C3C2×#46311yescharacteristic
362C2×C3#303911yescharacteristic
363C2×C3#324812yes—
364C2×C3#325412yes—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—82no—
370C22C2×C3×#12111yescharacteristic
371C22C3×#53712yes—
372C22C2×C3×#13511yescharacteristic
373C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
374C22C2×C3×#13712yes—
375C4—24no—
376C4—22no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—82no—
381C4—82no—
382C4C3×#46314yes—
383C4C3×#53511yescharacteristic
384C4C3×#53611yescharacteristic
385C4C3×#54212yes—
386C3#2982911yescharacteristic, direct product factor, Sylow
387C2—82no—
388C2C3×#262011yescharacteristic
389C2C2×C3×#46311yescharacteristic
390C2C3×#303911yescharacteristic
391C2C3×#324812yes—
392C2C3×#325412yes—
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×#2068C2semidirect
7. C3×#1988C2semidirect
8. C3×#1981C2semidirect
9. C3×#1842C2semidirect
10. #29829C3direct (N × H)
38. #2862C2×C3semidirect
40. #2068C2×C3semidirect
42. #1988C2×C3semidirect
43. #1981C2×C3semidirect
44. #1842C2×C3semidirect
386. C3#29829direct (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×#22932
C3×#20682
C3×#20552
C3×#19882
C3×#19812
C3×#18422
#298293