Finite Group Databasebeta

← C3×#29765 #10648372

Subgroups — C3×#29765 #10648372

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#482C212yesmaximal, complemented
2C3×#2619C211yesmaximal, characteristic
3C3×#2317C211yesmaximal, characteristic, complemented
4C3×#2085C212yesmaximal, complemented
5C3×#2067C212yesmaximal, complemented
6C3×#2065C212yesmaximal, complemented
7C3×#2041C212yesmaximal, complemented
8C3×#1874C212yesmaximal, complemented
9C3×#1798C211yesmaximal, characteristic, complemented
10#29765C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121C2212yes—
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C3×#482—28no—
17C2×C3×#107C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×C22⋊C8C2212yes—
26C3×#394—21no—
27C3×#394—21no—
28C3×#392—21no—
29C3×#392—21no—
30C3×#390C2212yes—
31C3×#389C2212yes—
32C3×#389C2212yes—
33C3×#385—21no—
34C3×#385—21no—
35C3×#384—21no—
36C3×#382—21no—
37C3×#382C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2212yes—
41C2×#482C2×C312yescomplemented
42#2619C2×C311yescharacteristic
43#2317C2×C311yescharacteristic, complemented
44#2085C2×C312yescomplemented
45#2067C2×C312yescomplemented
46#2065C2×C312yescomplemented
47#2041C2×C312yescomplemented
48#1874C2×C312yescomplemented
49#1798C2×C311yescharacteristic, complemented
50C22×C3×D4—22no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4C2312yes—
54C23×C3×C4—21no—
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C3×#127—41no—
59C3×#127—41no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#117—22no—
63C3×#117—41no—
64C3×#117—41no—
65C3×#117D414yescomplemented
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—41no—
70C2×C3×C4⋊C4—41no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C42—21no—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C3×#107—24no—
86C3×#106—24no—
87C3×C4⋊C8—41no—
88C3×C4⋊C8—41no—
89C3×#102—24no—
90C3×#102—24no—
91C3×#102—42no—
92C3×#102—42no—
93C3×C22⋊C8—24no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—21no—
97C3×#95—21no—
98C2×#127C22×C311yescharacteristic
99C2×#121C22×C312yes—
100C2×#117C22×C311yescharacteristic
101C2×#117C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103#482—28no—
104C2×#107C22×C312yes—
105C2×#106C22×C312yes—
106C2×C4⋊C8C22×C311yescharacteristic
107C2×C4⋊C8C22×C311yescharacteristic
108C2×#102C22×C312yes—
109C2×#102C22×C312yes—
110C2×#102C22×C312yes—
111C2×#102C22×C312yes—
112C2×C22⋊C8C22×C312yes—
113#394—21no—
114#394—21no—
115#392—21no—
116#392—21no—
117#390C22×C312yes—
118#389C22×C312yes—
119#389C22×C312yes—
120#385—21no—
121#385—21no—
122#384—21no—
123#382—21no—
124#382C22×C311yescharacteristic
125#340C22×C312yes—
126#340C22×C312yes—
127#336C22×C312yes—
128C24×C3—22no—
129C24×C3C2×D411yescharacteristic
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—44no—
139C2×C3×D4—82no—
140C2×C3×D4—82no—
141C2×C3×D4—82no—
142C2×C3×D4—82no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—82no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—82no—
175C3×C4⋊C4—82no—
176C3×C4⋊C4C2×D414yes—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—82no—
182C3×C22⋊C4—82no—
183C3×C42—24no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—21no—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C23×C312yes—
192C23×C4—21no—
193C23×C4C23×C311yescharacteristic
194C22×C8C23×C312yes—
195C22×C8C23×C312yes—
196#127—41no—
197#127—41no—
198#121—24no—
199#121—24no—
200#117—22no—
201#117—41no—
202#117—41no—
203#117C3×D414yescomplemented
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C3×D411yescharacteristic
210C2×C4⋊C4C3×D411yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C3×D411yescharacteristic
219C2×C22⋊C4C3×D411yescharacteristic
220C2×C42—21no—
221C2×C42C23×C311yescharacteristic
222C2×C42C23×C311yescharacteristic
223#107—24no—
224#106—24no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227#102—24no—
228#102—24no—
229#102—42no—
230#102—42no—
231C22⋊C8—24no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3#12112yes—
248C23×C3C22×D411yescharacteristic
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—42no—
253C3×D4—84no—
254C3×D4—84no—
255C3×D4—88no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—84no—
280C2×C3×C4#12112yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4H(5,𝔽2)11yescharacteristic
284C2×C3×C4H(5,𝔽2)11yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287C3×C8—42no—
288C3×C8—42no—
289C24—22no—
290C24C2×C3×D411yescharacteristic
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—44no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—82no—
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D412yes—
319C22×C4C3×#4112yes—
320C22×C4C24×C311yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4C2×C3×D414yes—
338C22⋊C4—24no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—44no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—24no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C22×C3—22no—
350C22×C3—24no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3C2×#12111yescharacteristic
362C22×C3#53411yescharacteristic
363C22×C3#53611yescharacteristic
364C22×C3#57612yes—
365C22×C3#57712yes—
366C3×C4—22no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—44no—
372C3×C4—82no—
373C3×C4—82no—
374C3×C4#53711yescharacteristic
375C3×C4#53711yescharacteristic
376C3×C4#53911yescharacteristic
377C3×C4#53911yescharacteristic
378C23—22no—
379C23—22no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23C3×#12112yes—
390C23C22×C3×D411yescharacteristic
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—84no—
396D4—84no—
397D4—88no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—84no—
422C2×C4C3×#12112yes—
423C2×C4C22×C3×D411yescharacteristic
424C2×C4C2×C3×#4111yescharacteristic
425C2×C4C3×H(5,𝔽2)11yescharacteristic
426C2×C4C3×H(5,𝔽2)11yescharacteristic
427C2×C4C3×2−1+411yescharacteristic
428C2×C4C3×2−1+411yescharacteristic
429C8—42no—
430C8—42no—
431C2×C3—42no—
432C2×C3—162no—
433C2×C3#261811yescharacteristic
434C2×C3#304512yes—
435C2×C3#319911yescharacteristic
436C2×C3#320011yescharacteristic
437C2×C3#324112yes—
438C22—22no—
439C22—24no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22C2×C3×#12111yescharacteristic
451C22C3×#53411yescharacteristic
452C22C3×#53611yescharacteristic
453C22C3×#57612yes—
454C22C3×#57712yes—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—44no—
461C4—82no—
462C4—82no—
463C4C3×#53711yescharacteristic
464C4C3×#53711yescharacteristic
465C4C3×#53911yescharacteristic
466C4C3×#53911yescharacteristic
467C3#2976511yescharacteristic, direct product factor, Sylow
468C2—42no—
469C2—162no—
470C2C3×#261811yescharacteristic
471C2C3×#304512yes—
472C2C3×#319911yescharacteristic
473C2C3×#320011yescharacteristic
474C2C3×#324112yes—
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
1. C2×C3×#482C2semidirect
3. C3×#2317C2semidirect
4. C3×#2085C2semidirect
5. C3×#2067C2semidirect
6. C3×#2065C2semidirect
7. C3×#2041C2semidirect
8. C3×#1874C2semidirect
9. C3×#1798C2semidirect
10. #29765C3direct (N × H)
41. C2×#482C2×C3semidirect
43. #2317C2×C3semidirect
44. #2085C2×C3semidirect
45. #2067C2×C3semidirect
46. #2065C2×C3semidirect
47. #2041C2×C3semidirect
48. #1874C2×C3semidirect
49. #1798C2×C3semidirect
65. C3×#117D4semidirect
203. #117C3×D4semidirect
467. C3#29765direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4822
C3×#26192
C3×#23172
C3×#20852
C3×#20672
C3×#20652
C3×#20412
C3×#18742
C3×#17982
#297653