Finite Group Databasebeta

← C3×#29340 #10647947

Subgroups — C3×#29340 #10647947

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2619C211yesmaximal, characteristic
2C3×#2316C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2041C212yesmaximal, complemented
5C3×#1936C212yesmaximal, complemented
6C3×#1934C212yesmaximal, complemented
7C3×#1875C212yesmaximal, complemented
8C3×#1874C212yesmaximal, complemented
9C3×#1750C211yesmaximal, characteristic, complemented
10#29340C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121C2212yes—
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×C3×#102C2212yes—
17C2×C3×#102C2212yes—
18C2×C3×#102C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×C22⋊C8C2212yes—
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#394—22no—
24C3×#392—22no—
25C3×#390C2212yes—
26C3×#389—22no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×#382C2211yescharacteristic
30C3×#380—22no—
31C3×#340C2212yes—
32C3×#340C2212yes—
33C3×#340C2212yes—
34C3×#340C2212yes—
35C3×#336C2212yes—
36#2619C2×C311yescharacteristic
37#2316C2×C311yescharacteristic, complemented
38#2101C2×C312yescomplemented
39#2041C2×C312yescomplemented
40#1936C2×C312yescomplemented
41#1934C2×C312yescomplemented
42#1875C2×C312yescomplemented
43#1874C2×C312yescomplemented
44#1750C2×C311yescharacteristic, complemented
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4C2312yes—
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C22×C3×C8C2312yes—
51C22×C3×C8C2312yes—
52C3×#127—41no—
53C3×#127—41no—
54C3×#121—24no—
55C3×#121—24no—
56C3×#117—41no—
57C3×#117—41no—
58C3×#117—41no—
59C3×#117—41no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4D412yes—
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C3×#102—24no—
77C3×#102—24no—
78C3×#102—42no—
79C3×#102—42no—
80C3×C22⋊C8—24no—
81C3×#97—41no—
82C3×C4×C8—41no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C2×#127C22×C311yescharacteristic
87C2×#121C22×C312yes—
88C2×#117C22×C311yescharacteristic
89C2×#117C22×C311yescharacteristic
90C22×C4⋊C4C22×C311yescharacteristic
91C2×#102C22×C312yes—
92C2×#102C22×C312yes—
93C2×#102C22×C312yes—
94C2×#102C22×C312yes—
95C2×C22⋊C8C22×C312yes—
96C2×#97C22×C311yescharacteristic
97C2×C4×C8C22×C311yescharacteristic
98#394—22no—
99#392—22no—
100#390C22×C312yes—
101#389—22no—
102#389C22×C312yes—
103#389C22×C312yes—
104#382C22×C311yescharacteristic
105#380—22no—
106#340C22×C312yes—
107#340C22×C312yes—
108#340C22×C312yes—
109#340C22×C312yes—
110#336C22×C312yes—
111C24×C3—22no—
112C24×C3C2×D411yescharacteristic
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—44no—
120C2×C3×D4—44no—
121C2×C3×D4—82no—
122C2×C3×D4—82no—
123C2×C3×D4—82no—
124C2×C3×D4—82no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—82no—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4114yes—
137C22×C3×C4C2411yescharacteristic
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—42no—
151C3×C4⋊C4—82no—
152C3×C4⋊C4—82no—
153C3×C4⋊C4#4114yes—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—44no—
157C3×C22⋊C4—82no—
158C3×C22⋊C4—82no—
159C3×C42—21no—
160C3×C42—21no—
161C3×C42—21no—
162C3×C42—21no—
163C3×C42—42no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4C23×C312yes—
167C23×C4—21no—
168C23×C4C23×C311yescharacteristic
169C22×C8C23×C312yes—
170C22×C8C23×C312yes—
171#127—41no—
172#127—41no—
173#121—24no—
174#121—24no—
175#117—41no—
176#117—41no—
177#117—41no—
178#117—41no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C3×D412yes—
184C2×C4⋊C4C23×C312yes—
185C2×C4⋊C4C23×C312yes—
186C2×C4⋊C4C23×C312yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C3×D412yes—
192C2×C42—21no—
193C2×C42C23×C311yescharacteristic
194C2×C42C23×C311yescharacteristic
195#102—24no—
196#102—24no—
197#102—42no—
198#102—42no—
199C22⋊C8—24no—
200#97—41no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—22no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3#12412yes—
217C23×C3C22×D411yescharacteristic
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—88no—
223C3×D4—88no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—84no—
244C2×C3×C4#12412yes—
245C2×C3×C4C2×#4112yes—
246C2×C3×C4C2×#4112yes—
247C2×C3×C4H(5,𝔽2)11yescharacteristic
248C2×C3×C4H(5,𝔽2)11yescharacteristic
249C3×C8—42no—
250C3×C8—42no—
251C24—22no—
252C24C2×C3×D411yescharacteristic
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—82no—
273C22×C4C2×C3×D412yes—
274C22×C4C2×C3×D412yes—
275C22×C4C2×C3×D411yescharacteristic
276C22×C4C3×#4114yes—
277C22×C4C24×C311yescharacteristic
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4C3×#4114yes—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—42no—
304C22×C3—22no—
305C22×C3—24no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—42no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3#52511yescharacteristic
317C22×C3C2×#12411yescharacteristic
318C22×C3#53411yescharacteristic
319C22×C3#57612yes—
320C22×C3#57712yes—
321C3×C4—22no—
322C3×C4—44no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—82no—
326C3×C4—82no—
327C3×C4#55312yes—
328C3×C4#55312yes—
329C23—22no—
330C23—22no—
331C23—21no—
332C23—21no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23C3×#12412yes—
341C23C22×C3×D411yescharacteristic
342D4—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—88no—
347D4—88no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—84no—
368C2×C4C3×#12412yes—
369C2×C4C2×C3×#4112yes—
370C2×C4C2×C3×#4112yes—
371C2×C4C3×H(5,𝔽2)11yescharacteristic
372C2×C4C3×H(5,𝔽2)11yescharacteristic
373C8—42no—
374C8—42no—
375C2×C3—42no—
376C2×C3—162no—
377C2×C3#259111yescharacteristic
378C2×C3#312211yescharacteristic
379C2×C3#312611yescharacteristic
380C2×C3#313612yes—
381C2×C3#319812yes—
382C22—24no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22C3×#52511yescharacteristic
395C22C2×C3×#12411yescharacteristic
396C22C3×#53411yescharacteristic
397C22C3×#57612yes—
398C22C3×#57712yes—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—82no—
404C4—82no—
405C4C3×#55312yes—
406C4C3×#55312yes—
407C3#2934011yescharacteristic, direct product factor, Sylow
408C2—42no—
409C2—162no—
410C2C3×#259111yescharacteristic
411C2C3×#312211yescharacteristic
412C2C3×#312611yescharacteristic
413C2C3×#313612yes—
414C2C3×#319812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2619C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2316C2semidirect
3. C3×#2101C2semidirect
4. C3×#2041C2semidirect
5. C3×#1936C2semidirect
6. C3×#1934C2semidirect
7. C3×#1875C2semidirect
8. C3×#1874C2semidirect
9. C3×#1750C2semidirect
10. #29340C3direct (N × H)
37. #2316C2×C3semidirect
38. #2101C2×C3semidirect
39. #2041C2×C3semidirect
40. #1936C2×C3semidirect
41. #1934C2×C3semidirect
42. #1875C2×C3semidirect
43. #1874C2×C3semidirect
44. #1750C2×C3semidirect
407. C3#29340direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26192
C3×#23162
C3×#21012
C3×#20412
C3×#19362
C3×#19342
C3×#18752
C3×#18742
C3×#17502
#293403