Finite Group Databasebeta

← C3×#29269 #10647876

Subgroups — C3×#29269 #10647876

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2330C211yesmaximal, characteristic, complemented
2C3×#2326C211yesmaximal, characteristic
3C3×#2096C211yesmaximal, characteristic, complemented
4C3×#2094C211yesmaximal, characteristic
5C3×#2026C211yesmaximal, characteristic, complemented
6C3×#2025C211yesmaximal, characteristic
7C3×#1983C212yesmaximal
8C3×#1948C212yesmaximal, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1864C212yesmaximal
11C3×#1832C211yesmaximal, characteristic, complemented
12#29269C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C3×#391C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#389—22no—
27C3×#389C2212yes—
28C3×#388—22no—
29C3×#387—22no—
30C3×#385—22no—
31C3×#384C2211yescharacteristic
32C3×#384C2211yescharacteristic
33C3×C4×C4⋊C4C2212yes—
34C3×#376C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#336C2211yescharacteristic
38C3×#336C2211yescharacteristic
39C3×#336C2212yes—
40#2330C2×C311yescharacteristic, complemented
41#2326C2×C311yescharacteristic
42#2096C2×C311yescharacteristic, complemented
43#2094C2×C311yescharacteristic
44#2026C2×C311yescharacteristic, complemented
45#2025C2×C311yescharacteristic
46#1983C2×C312yes—
47#1948C2×C312yescomplemented
48#1865C2×C312yescomplemented
49#1864C2×C312yes—
50#1832C2×C311yescharacteristic, complemented
51C22×C3×Q8C2311yescharacteristic
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4—21no—
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×#125—24no—
57C3×#124—22no—
58C3×#124Q814yes—
59C3×C4×Q8—42no—
60C3×C4×D4—42no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C42—21no—
74C2×C3×C42—22no—
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C3×#106—42no—
79C3×#106—42no—
80C3×C4⋊C8—24no—
81C3×#103—24no—
82C3×#103—42no—
83C3×#102—24no—
84C3×#102—42no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95D412yes—
89C3×#95D412yes—
90C2×#125C22×C311yescharacteristic
91C2×#124C22×C311yescharacteristic
92C2×C4×Q8C22×C311yescharacteristic
93C2×C4×D4C22×C311yescharacteristic
94C2×#106C22×C312yes—
95C2×#106C22×C312yes—
96C2×C4⋊C8C22×C312yes—
97C2×#103C22×C312yes—
98C2×#103C22×C312yes—
99C2×#102C22×C312yes—
100C2×#102C22×C312yes—
101#391C22×C311yescharacteristic
102#390C22×C311yescharacteristic
103#389—22no—
104#389C22×C312yes—
105#388—22no—
106#387—22no—
107#385—22no—
108#384C22×C311yescharacteristic
109#384C22×C311yescharacteristic
110C4×C4⋊C4C22×C312yes—
111#376C22×C311yescharacteristic
112#340C22×C312yes—
113#340C22×C312yes—
114#336C22×C311yescharacteristic
115#336C22×C311yescharacteristic
116#336C22×C312yes—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8C2×Q812yes—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—22no—
125C2×C3×D4C2×Q812yes—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4—82no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D412yes—
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—28no—
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—82no—
154C3×C4⋊C4#4114yes—
155C3×C22⋊C4—28no—
156C3×C22⋊C4—42no—
157C3×C22⋊C4—82no—
158C3×C42—22no—
159C3×C42—22no—
160C3×C42—42no—
161C3×C42—82no—
162C3×C42C2×Q812yes—
163C3×C42#4112yes—
164C22×Q8C23×C311yescharacteristic
165C22×D4C23×C311yescharacteristic
166C23×C4—21no—
167C22×C8C23×C312yes—
168C22×C8C23×C312yes—
169#125—24no—
170#124—22no—
171#124C3×Q814yes—
172C4×Q8—42no—
173C4×D4—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C23×C311yescharacteristic
180C2×C4⋊C4C23×C311yescharacteristic
181C2×C4⋊C4C23×C312yes—
182C2×C4⋊C4C23×C312yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42C23×C311yescharacteristic
189C2×C42C23×C311yescharacteristic
190C2×C42C23×C311yescharacteristic
191#106—42no—
192#106—42no—
193C4⋊C8—24no—
194#103—24no—
195#103—42no—
196#102—24no—
197#102—42no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95C3×D412yes—
202#95C3×D412yes—
203C23×C3—22no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3C22×D411yescharacteristic
209C3×Q8—24no—
210C3×Q8—42no—
211C3×D4—24no—
212C3×D4—42no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
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—42no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—84no—
236C2×C3×C4C22⋊Q814yes—
237C2×C3×C4C22×Q811yescharacteristic
238C2×C3×C4C2×#4112yes—
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4H(5,𝔽2)12yes—
241C3×C8—42no—
242C3×C8—42no—
243C24—21no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8C2×C3×Q812yes—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—22no—
251C2×D4C2×C3×Q812yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—82no—
259C22×C4C2×C3×D411yescharacteristic
260C22×C4C2×C3×D412yes—
261C22×C4C2×C3×D411yescharacteristic
262C22×C4C2×C3×D412yes—
263C22×C4C24×C311yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C4⋊C4—28no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4C3×#4114yes—
281C22⋊C4—28no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—22no—
285C42—22no—
286C42—42no—
287C42—82no—
288C42C2×C3×Q812yes—
289C42C3×#4112yes—
290C22×C3—44no—
291C22×C3—42no—
292C22×C3—82no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3#52511yescharacteristic
297C22×C3#53511yescharacteristic
298C22×C3#57712yes—
299C22×C3#57712yes—
300C3×C4—24no—
301C3×C4—42no—
302C3×C4—42no—
303C3×C4—44no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—82no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4#55412yes—
310C3×C4#55912yes—
311C23—22no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23C22×C3×D411yescharacteristic
317Q8—24no—
318Q8—42no—
319D4—24no—
320D4—42no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4C3×C22⋊Q814yes—
345C2×C4C22×C3×Q811yescharacteristic
346C2×C4C2×C3×#4111yescharacteristic
347C2×C4C2×C3×#4112yes—
348C2×C4C3×H(5,𝔽2)12yes—
349C8—42no—
350C8—42no—
351C2×C3—82no—
352C2×C3#258011yescharacteristic
353C2×C3#308412yes—
354C2×C3#312512yes—
355C2×C3#322111yescharacteristic
356C2×C3#322211yescharacteristic
357C22—44no—
358C22—42no—
359C22—81no—
360C22—81no—
361C22—82no—
362C22C2×C3×C22⋊Q811yescharacteristic
363C22C3×#52511yescharacteristic
364C22C3×#53511yescharacteristic
365C22C3×#57712yes—
366C22C3×#57712yes—
367C4—24no—
368C4—41no—
369C4—41no—
370C4—42no—
371C4—42no—
372C4—44no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4C3×#55412yes—
377C4C3×#55912yes—
378C3#2926911yescharacteristic, direct product factor, Sylow
379C2—82no—
380C2C3×#258011yescharacteristic
381C2C3×#308412yes—
382C2C3×#312512yes—
383C2C3×#322111yescharacteristic
384C2C3×#322211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2330C2semidirect
3. C3×#2096C2semidirect
5. C3×#2026C2semidirect
8. C3×#1948C2semidirect
9. C3×#1865C2semidirect
11. C3×#1832C2semidirect
12. #29269C3direct (N × H)
40. #2330C2×C3semidirect
42. #2096C2×C3semidirect
44. #2026C2×C3semidirect
47. #1948C2×C3semidirect
48. #1865C2×C3semidirect
50. #1832C2×C3semidirect
378. C3#29269direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23302
C3×#23262
C3×#20962
C3×#20942
C3×#20262
C3×#20252
C3×#19832
C3×#19482
C3×#18652
C3×#18642
C3×#18322
#292693