Finite Group Databasebeta

← C3×#24645 #10643252

Subgroups — C3×#24645 #10643252

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#462C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic
4C3×#2356C211yesmaximal, characteristic, complemented
5C3×#2056C212yesmaximal, complemented
6C3×#2015C212yesmaximal
7C3×#1983C212yesmaximal
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1928C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic
12#24645C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#462—24no—
20C3×#440—24no—
21C2×C3×#107C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×C3×#97C2211yescharacteristic
31C3×#397—21no—
32C3×#397—21no—
33C3×#391C2212yes—
34C3×#391C2212yes—
35C3×#387—22no—
36C3×#383—21no—
37C3×#382C2211yescharacteristic
38C3×#380—22no—
39C3×C4×C4⋊C4C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376—21no—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#462C2×C311yescharacteristic, complemented
45C2×#440C2×C311yescharacteristic
46#2547C2×C311yescharacteristic
47#2356C2×C311yescharacteristic, complemented
48#2056C2×C312yescomplemented
49#2015C2×C312yes—
50#1983C2×C312yes—
51#1960C2×C311yescharacteristic, complemented
52#1940C2×C311yescharacteristic, complemented
53#1928C2×C312yescomplemented
54#1804C2×C311yescharacteristic
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×Q16—24no—
60C2×C3×Q16—42no—
61C2×C3×Q16—42no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×#124—24no—
71C3×#124—42no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—21no—
87C2×C3×C42—22no—
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C3×#107—22no—
91C3×C4⋊C8—22no—
92C3×C4⋊C8—42no—
93C3×#103—22no—
94C3×#103—24no—
95C3×#103—24no—
96C3×#103—42no—
97C3×#102—22no—
98C3×#102—42no—
99C3×#97—22no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—22no—
107C3×#95D411yescharacteristic
108C3×#95D411yescharacteristic
109C22×Q16C22×C312yes—
110C22×SD16C22×C311yescharacteristic
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×#124C22×C312yes—
113C2×C4×Q8C22×C311yescharacteristic
114C2×C4×D4C22×C311yescharacteristic
115#462—24no—
116#440—24no—
117C2×#107C22×C311yescharacteristic
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C312yes—
120C2×#103C22×C311yescharacteristic
121C2×#103C22×C312yes—
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#102C22×C311yescharacteristic
125C2×#102C22×C312yes—
126C2×#97C22×C311yescharacteristic
127#397—21no—
128#397—21no—
129#391C22×C312yes—
130#391C22×C312yes—
131#387—22no—
132#383—21no—
133#382C22×C311yescharacteristic
134#380—22no—
135C4×C4⋊C4C22×C312yes—
136C4×C4⋊C4C22×C311yescharacteristic
137#376—21no—
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D412yes—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4#4112yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—44no—
165C22×C3×C4—42no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C3×Q16—48no—
172C3×SD16—28no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8#4112yes—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—44no—
194C3×C22⋊C4—42no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—44no—
199C3×C42C2×D412yes—
200C3×C42#4112yes—
201C22×Q8C23×C312yes—
202C22×Q8C23×C311yescharacteristic
203C22×D4C23×C311yescharacteristic
204C23×C4—21no—
205C2×Q16—24no—
206C2×Q16—42no—
207C2×Q16—42no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×SD16—22no—
211C22×C8C23×C312yes—
212C22×C8C23×C311yescharacteristic
213C22×C8C23×C311yescharacteristic
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216#124—24no—
217#124—42no—
218C4×Q8—22no—
219C4×Q8C3×D414yes—
220C4×D4—22no—
221C4×D4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C3×D412yes—
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C312yes—
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42C23×C312yes—
235C2×C42C23×C311yescharacteristic
236#107—22no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239#103—22no—
240#103—24no—
241#103—24no—
242#103—42no—
243#102—22no—
244#102—42no—
245#97—22no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95C3×D411yescharacteristic
254#95C3×D411yescharacteristic
255C23×C3—22no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3C22×D411yescharacteristic
261C3×Q8—22no—
262C3×Q8—22no—
263C3×Q8—41no—
264C3×Q8—41no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×Q8—42no—
268C3×Q8—42no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—41no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—84no—
295C2×C3×C4#12112yes—
296C2×C3×C4#12112yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C3×C8—22no—
304C3×C8—41no—
305C3×C8—42no—
306C24—21no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8C2×C3×D412yes—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4C3×#4112yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4C2×C3×D412yes—
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C24×C311yescharacteristic
337Q16—48no—
338SD16—28no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8C3×#4112yes—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4C2×C3×D412yes—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C42—24no—
362C42—22no—
363C42—22no—
364C42—44no—
365C42C2×C3×D412yes—
366C42C3×#4112yes—
367C22×C3—42no—
368C22×C3—44no—
369C22×C3—82no—
370C22×C3—81no—
371C22×C3—81no—
372C22×C3C2×#12111yescharacteristic
373C22×C3#54611yescharacteristic
374C22×C3#55011yescharacteristic
375C22×C3#57512yes—
376C22×C3#57812yes—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—82no—
384C3×C4—82no—
385C3×C4—81no—
386C3×C4#52511yescharacteristic
387C3×C4#53511yescharacteristic
388C3×C4#55311yescharacteristic
389C3×C4#55511yescharacteristic
390C23—22no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23C22×C3×D411yescharacteristic
396Q8—22no—
397Q8—22no—
398Q8—41no—
399Q8—41no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404D4—22no—
405D4—22no—
406D4—41no—
407D4—41no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4C3×#12112yes—
431C2×C4C3×#12112yes—
432C2×C4C22×C3×D411yescharacteristic
433C2×C4C2×C3×#4111yescharacteristic
434C2×C4C2×C3×#4111yescharacteristic
435C2×C4C2×C3×#4111yescharacteristic
436C2×C4C3×H(5,𝔽2)11yescharacteristic
437C2×C4C3×H(5,𝔽2)11yescharacteristic
438C8—22no—
439C8—42no—
440C8—41no—
441C2×C3—82no—
442C2×C3#241411yescharacteristic
443C2×C3#304011yescharacteristic
444C2×C3#304411yescharacteristic
445C2×C3#331512yes—
446C2×C3#340212yes—
447C22—42no—
448C22—44no—
449C22—82no—
450C22—81no—
451C22—81no—
452C22C2×C3×#12111yescharacteristic
453C22C3×#54611yescharacteristic
454C22C3×#55011yescharacteristic
455C22C3×#57512yes—
456C22C3×#57812yes—
457C4—22no—
458C4—22no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—82no—
465C4—81no—
466C4C3×#52511yescharacteristic
467C4C3×#53511yescharacteristic
468C4C3×#55311yescharacteristic
469C4C3×#55511yescharacteristic
470C3#2464511yescharacteristic, direct product factor, Sylow
471C2—82no—
472C2C3×#241411yescharacteristic
473C2C3×#304011yescharacteristic
474C2C3×#304411yescharacteristic
475C2C3×#331512yes—
476C2C3×#340212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#462C2semidirect
4. C3×#2356C2semidirect
5. C3×#2056C2semidirect
8. C3×#1960C2semidirect
9. C3×#1940C2semidirect
10. C3×#1928C2semidirect
12. #24645C3direct (N × H)
44. C2×#462C2×C3semidirect
47. #2356C2×C3semidirect
48. #2056C2×C3semidirect
51. #1960C2×C3semidirect
52. #1940C2×C3semidirect
53. #1928C2×C3semidirect
470. C3#24645direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4622
C2×C3×#4402
C3×#25472
C3×#23562
C3×#20562
C3×#20152
C3×#19832
C3×#19602
C3×#19402
C3×#19282
C3×#18042
#246453