Finite Group Databasebeta

← C3×#29768 #10648375

Subgroups — C3×#29768 #10648375

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#481C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic, complemented
3C3×#2620C211yesmaximal, characteristic
4C3×#2316C211yesmaximal, characteristic, complemented
5C3×#2084C212yesmaximal, complemented
6C3×#2066C212yesmaximal, complemented
7C3×#2065C212yesmaximal, complemented
8C3×#2042C212yesmaximal, complemented
9C3×#1875C211yesmaximal, characteristic
10C3×#1873C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29768C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#483—24no—
20C3×#481—24no—
21C2×C3×#106C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#396—22no—
31C3×#394—22no—
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#389—22no—
35C3×#389C2212yes—
36C3×#389C2212yes—
37C3×#384C2211yescharacteristic
38C3×#380—22no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42C2×#481C2×C311yescharacteristic
43C2×#483C2×C311yescharacteristic, complemented
44#2620C2×C311yescharacteristic
45#2316C2×C311yescharacteristic, complemented
46#2084C2×C312yescomplemented
47#2066C2×C312yescomplemented
48#2065C2×C312yescomplemented
49#2042C2×C312yescomplemented
50#1875C2×C311yescharacteristic
51#1873C2×C311yescharacteristic, complemented
52#1798C2×C311yescharacteristic, complemented
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4—21no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×#124—42no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×#121—22no—
64C3×#121—22no—
65C3×#117—42no—
66C3×#117—42no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#106—24no—
89C3×#106—24no—
90C3×C4⋊C8—42no—
91C3×#103—24no—
92C3×#103—42no—
93C3×#102—24no—
94C3×#102—42no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C2×#124C22×C311yescharacteristic
101C2×C22⋊Q8C22×C311yescharacteristic
102C2×#121C22×C311yescharacteristic
103C2×#117C22×C311yescharacteristic
104C2×#117C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106#483—24no—
107#481—24no—
108C2×#106C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C311yescharacteristic
117#396—22no—
118#394—22no—
119#391C22×C311yescharacteristic
120#390C22×C311yescharacteristic
121#389—22no—
122#389C22×C312yes—
123#389C22×C312yes—
124#384C22×C311yescharacteristic
125#380—22no—
126#340C22×C312yes—
127#340C22×C312yes—
128#336C22×C312yes—
129C24×C3—21no—
130C24×C3C2×D411yescharacteristic
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—42no—
140C2×C3×D4—82no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—82no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4#4112yes—
155C22×C3×C4C2411yescharacteristic
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—82no—
173C3×C4⋊C4—82no—
174C3×C4⋊C4—82no—
175C3×C4⋊C4C2×D414yes—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—44no—
178C3×C22⋊C4—42no—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—82no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—42no—
185C22×Q8C23×C311yescharacteristic
186C22×D4—21no—
187C22×D4C23×C311yescharacteristic
188C23×C4—21no—
189C23×C4C23×C311yescharacteristic
190C22×C8C23×C312yes—
191C22×C8C23×C312yes—
192#124—42no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195#121—22no—
196#121—22no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C3×D414yes—
206C2×C4⋊C4C3×D412yes—
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C312yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4C3×D412yes—
217C2×C42—21no—
218C2×C42C23×C311yescharacteristic
219C2×C42C23×C311yescharacteristic
220#106—24no—
221#106—24no—
222C4⋊C8—42no—
223#103—24no—
224#103—42no—
225#102—24no—
226#102—42no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3#12112yes—
242C23×C3C22×D411yescharacteristic
243C3×Q8—42no—
244C3×Q8—42no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—84no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—84no—
272C2×C3×C4#12112yes—
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4H(5,𝔽2)12yes—
276C2×C3×C42−1+412yes—
277C3×C8—42no—
278C3×C8—42no—
279C24—21no—
280C24C2×C3×D411yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—42no—
290C2×D4—82no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—82no—
300C22×C4C2×C3×D412yes—
301C22×C4C2×C3×D412yes—
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C2×C3×D412yes—
304C22×C4C3×#4112yes—
305C22×C4C24×C311yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4C2×C3×D414yes—
326C22⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—82no—
332C42—22no—
333C42—22no—
334C42—42no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—81no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3C2×#12111yescharacteristic
346C22×C3#53411yescharacteristic
347C22×C3#53611yescharacteristic
348C22×C3#57712yes—
349C22×C3#57712yes—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—44no—
354C3×C4—81no—
355C3×C4—82no—
356C3×C4—81no—
357C3×C4—81no—
358C3×C4#53712yes—
359C3×C4#53912yes—
360C23—22no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23C3×#12112yes—
370C23C22×C3×D411yescharacteristic
371Q8—42no—
372Q8—42no—
373D4—42no—
374D4—42no—
375D4—84no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—84no—
400C2×C4C3×#12112yes—
401C2×C4C22×C3×D411yescharacteristic
402C2×C4C2×C3×#4111yescharacteristic
403C2×C4C3×H(5,𝔽2)12yes—
404C2×C4C3×2−1+412yes—
405C8—42no—
406C8—42no—
407C2×C3—42no—
408C2×C3—161no—
409C2×C3#261811yescharacteristic
410C2×C3#304211yescharacteristic
411C2×C3#304311yescharacteristic
412C2×C3#320412yes—
413C2×C3#324412yes—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—81no—
420C22—82no—
421C22—82no—
422C22—81no—
423C22—81no—
424C22C2×C3×#12111yescharacteristic
425C22C3×#53411yescharacteristic
426C22C3×#53611yescharacteristic
427C22C3×#57712yes—
428C22C3×#57712yes—
429C4—22no—
430C4—42no—
431C4—42no—
432C4—44no—
433C4—81no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4C3×#53712yes—
438C4C3×#53912yes—
439C3#2976811yescharacteristic, direct product factor, Sylow
440C2—42no—
441C2—161no—
442C2C3×#261811yescharacteristic
443C2C3×#304211yescharacteristic
444C2C3×#304311yescharacteristic
445C2C3×#320412yes—
446C2C3×#324412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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. C2×C3×#483C2semidirect
4. C3×#2316C2semidirect
5. C3×#2084C2semidirect
6. C3×#2066C2semidirect
7. C3×#2065C2semidirect
8. C3×#2042C2semidirect
10. C3×#1873C2semidirect
11. C3×#1798C2semidirect
12. #29768C3direct (N × H)
43. C2×#483C2×C3semidirect
45. #2316C2×C3semidirect
46. #2084C2×C3semidirect
47. #2066C2×C3semidirect
48. #2065C2×C3semidirect
49. #2042C2×C3semidirect
51. #1873C2×C3semidirect
52. #1798C2×C3semidirect
439. C3#29768direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4812
C2×C3×#4832
C3×#26202
C3×#23162
C3×#20842
C3×#20662
C3×#20652
C3×#20422
C3×#18752
C3×#18732
C3×#17982
#297683