Finite Group Databasebeta

← C3×#29804 #10648411

Subgroups — C3×#29804 #10648411

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#463C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C3×#2658C211yesmaximal, characteristic, complemented
4C3×#2319C211yesmaximal, characteristic, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#2013C212yesmaximal, complemented
8C3×#2012C212yesmaximal
9C3×#1983C211yesmaximal, characteristic, complemented
10C3×#1981C211yesmaximal, characteristic, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#29804C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×SD16C2211yescharacteristic
15C22×C3×D8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C3×#463—24no—
22C3×#461—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2212yes—
32C3×#399—21no—
33C3×#399—21no—
34C3×#399—22no—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#388—22no—
38C3×#381—21no—
39C3×#380—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#463C2×C311yescharacteristic
45C2×#461C2×C311yescharacteristic, complemented
46#2658C2×C311yescharacteristic, complemented
47#2319C2×C311yescharacteristic, complemented
48#2043C2×C312yescomplemented
49#2042C2×C312yescomplemented
50#2013C2×C312yescomplemented
51#2012C2×C312yes—
52#1983C2×C311yescharacteristic, complemented
53#1981C2×C311yescharacteristic, complemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×C3×SD16—24no—
61C2×C3×SD16—22no—
62C2×C3×SD16—42no—
63C2×C3×SD16—42no—
64C2×C3×SD16—42no—
65C2×C3×D8—22no—
66C2×C3×D8—42no—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—42no—
73C3×C4⋊Q8—42no—
74C3×#124—22no—
75C3×#124—22no—
76C3×#124—42no—
77C3×#124—42no—
78C3×C4×D4—22no—
79C3×C4×D4D414yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C42—21no—
95C2×C3×C42C2312yes—
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—22no—
98C3×C4⋊C8—22no—
99C3×#103—24no—
100C3×#103—22no—
101C3×#103—42no—
102C3×#103—42no—
103C3×#103—42no—
104C3×#102—22no—
105C3×#102—42no—
106C3×#95—21no—
107C3×#95—22no—
108C3×#95D411yescharacteristic
109C3×#95D411yescharacteristic
110C3×#95D412yes—
111C22×SD16C22×C312yes—
112C22×SD16C22×C311yescharacteristic
113C22×D8C22×C311yescharacteristic
114C2×C4⋊Q8C22×C311yescharacteristic
115C2×C4⋊Q8C22×C312yes—
116C2×#124C22×C311yescharacteristic
117C2×#124C22×C312yes—
118C2×C4×D4C22×C311yescharacteristic
119#463—24no—
120#461—24no—
121C2×C4⋊C8C22×C311yescharacteristic
122C2×C4⋊C8C22×C311yescharacteristic
123C2×#103C22×C312yes—
124C2×#103C22×C311yescharacteristic
125C2×#103C22×C312yes—
126C2×#103C22×C312yes—
127C2×#103C22×C312yes—
128C2×#102C22×C311yescharacteristic
129C2×#102C22×C312yes—
130#399—21no—
131#399—21no—
132#399—22no—
133#390C22×C312yes—
134#389C22×C312yes—
135#388—22no—
136#381—21no—
137#380—21no—
138C4×C4⋊C4C22×C311yescharacteristic
139#376C22×C311yescharacteristic
140#336C22×C312yes—
141#336C22×C312yes—
142C24×C3—21no—
143C24×C3—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4C2×D412yes—
162C22×C3×C4—21no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—82no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C3×SD16—44no—
174C3×SD16—48no—
175C3×D8—44no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4—44no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—82no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—82no—
198C3×C22⋊C4—82no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—81no—
202C3×C42C2×D412yes—
203C3×C42#4112yes—
204C22×Q8C23×C311yescharacteristic
205C22×Q8C23×C312yes—
206C22×D4C23×C311yescharacteristic
207C22×D4C23×C311yescharacteristic
208C23×C4—21no—
209C2×SD16—24no—
210C2×SD16—22no—
211C2×SD16—42no—
212C2×SD16—42no—
213C2×SD16—42no—
214C2×D8—22no—
215C2×D8—42no—
216C22×C8C23×C311yescharacteristic
217C22×C8C23×C311yescharacteristic
218C22×C8C23×C312yes—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223#124—22no—
224#124—22no—
225#124—42no—
226#124—42no—
227C4×D4—22no—
228C4×D4C3×D414yes—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—21no—
244C2×C42C23×C312yes—
245C2×C42C23×C311yescharacteristic
246C4⋊C8—22no—
247C4⋊C8—22no—
248#103—24no—
249#103—22no—
250#103—42no—
251#103—42no—
252#103—42no—
253#102—22no—
254#102—42no—
255#95—21no—
256#95—22no—
257#95C3×D411yescharacteristic
258#95C3×D411yescharacteristic
259#95C3×D412yes—
260C23×C3—22no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—42no—
267C23×C3—41no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—41no—
272C3×Q8—42no—
273C3×Q8—41no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—42no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—41no—
281C3×D4—41no—
282C3×D4—42no—
283C3×D4—42no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—84no—
304C2×C3×C4#12112yes—
305C2×C3×C4#12112yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)12yes—
309C2×C3×C4H(5,𝔽2)12yes—
310C3×C8—41no—
311C3×C8—41no—
312C3×C8—42no—
313C24—21no—
314C24—21no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4C2×C3×D412yes—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—82no—
339C22×C4C2×C3×D412yes—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C24×C311yescharacteristic
344SD16—44no—
345SD16—48no—
346D8—44no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—82no—
362C4⋊C4C2×C3×D412yes—
363C22⋊C4—24no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—22no—
371C42—22no—
372C42—81no—
373C42C2×C3×D412yes—
374C42C3×#4112yes—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—81no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3C2×#12111yescharacteristic
386C22×C3#53512yes—
387C22×C3#57711yescharacteristic
388C22×C3#57712yes—
389C22×C3#57811yescharacteristic
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—81no—
396C3×C4—82no—
397C3×C4—82no—
398C3×C4#53511yescharacteristic
399C3×C4#53511yescharacteristic
400C3×C4#54012yes—
401C23—22no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23C22×C3×D411yescharacteristic
412Q8—41no—
413Q8—42no—
414Q8—41no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419D4—22no—
420D4—22no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—42no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—84no—
445C2×C4C3×#12112yes—
446C2×C4C3×#12112yes—
447C2×C4C22×C3×D411yescharacteristic
448C2×C4C2×C3×#4111yescharacteristic
449C2×C4C3×H(5,𝔽2)12yes—
450C2×C4C3×H(5,𝔽2)12yes—
451C8—41no—
452C8—41no—
453C8—42no—
454C2×C3—82no—
455C2×C3—161no—
456C2×C3#261911yescharacteristic
457C2×C3#304211yescharacteristic
458C2×C3#304611yescharacteristic
459C2×C3#322112yes—
460C2×C3#322512yes—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22C2×C3×#12111yescharacteristic
472C22C3×#53512yes—
473C22C3×#57711yescharacteristic
474C22C3×#57712yes—
475C22C3×#57811yescharacteristic
476C4—22no—
477C4—22no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—82no—
483C4—82no—
484C4C3×#53511yescharacteristic
485C4C3×#53511yescharacteristic
486C4C3×#54012yes—
487C3#2980411yescharacteristic, direct product factor, Sylow
488C2—82no—
489C2—161no—
490C2C3×#261911yescharacteristic
491C2C3×#304211yescharacteristic
492C2C3×#304611yescharacteristic
493C2C3×#322112yes—
494C2C3×#322512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
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
2. C2×C3×#461C2semidirect
3. C3×#2658C2semidirect
4. C3×#2319C2semidirect
5. C3×#2043C2semidirect
6. C3×#2042C2semidirect
7. C3×#2013C2semidirect
9. C3×#1983C2semidirect
10. C3×#1981C2semidirect
11. C3×#1832C2semidirect
12. #29804C3direct (N × H)
45. C2×#461C2×C3semidirect
46. #2658C2×C3semidirect
47. #2319C2×C3semidirect
48. #2043C2×C3semidirect
49. #2042C2×C3semidirect
50. #2013C2×C3semidirect
52. #1983C2×C3semidirect
53. #1981C2×C3semidirect
54. #1832C2×C3semidirect
487. C3#29804direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4632
C2×C3×#4612
C3×#26582
C3×#23192
C3×#20432
C3×#20422
C3×#20132
C3×#20122
C3×#19832
C3×#19812
C3×#18322
#298043