Finite Group Databasebeta

← C3×#29240 #10647847

Subgroups — C3×#29240 #10647847

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic
3C3×#2088C212yesmaximal, complemented
4C3×#2066C212yesmaximal, complemented
5C3×#1919C212yesmaximal, complemented
6C3×#1872C211yesmaximal, characteristic
7C3×#1872C211yesmaximal, characteristic
8C3×#1865C212yesmaximal, complemented
9C3×#1853C211yesmaximal, characteristic
10C3×#1853C211yesmaximal, characteristic
11C3×#1797C211yesmaximal, characteristic
12#29240C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C4⋊C4—21no—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×C3×#106C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389—21no—
28C3×#389—21no—
29C3×#389C2212yes—
30C3×#389C2211yescharacteristic
31C3×#389C2212yes—
32C3×#387—21no—
33C3×#385—21no—
34C3×#380C2211yescharacteristic
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C2×C3×#95—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C4×C22⋊Q8C2×C311yescharacteristic
45C2×#389C2×C311yescharacteristic
46#2088C2×C312yescomplemented
47#2066C2×C312yescomplemented
48#1919C2×C312yescomplemented
49#1872C2×C311yescharacteristic
50#1872C2×C311yescharacteristic
51#1865C2×C312yescomplemented
52#1853C2×C311yescharacteristic
53#1853C2×C311yescharacteristic
54#1797C2×C311yescharacteristic
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8Q814yes—
65C3×C4×Q8—42no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—42no—
75C2×C3×C4⋊C4—44no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42—42no—
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C3×#106—24no—
91C3×#106—42no—
92C3×C4⋊C8—42no—
93C3×#103—24no—
94C3×#103—42no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—21no—
103C3×#95—21no—
104C2×C22⋊Q8C22×C311yescharacteristic
105C2×C4×Q8C22×C311yescharacteristic
106C22×C4⋊C4—21no—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C42C22×C311yescharacteristic
110C2×#106C22×C312yes—
111C2×#106C22×C312yes—
112C2×C4⋊C8C22×C312yes—
113C2×#103C22×C312yes—
114C2×#103C22×C312yes—
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C311yescharacteristic
117#391C22×C311yescharacteristic
118#389—21no—
119#389—21no—
120#389C22×C312yes—
121#389C22×C311yescharacteristic
122#389C22×C312yes—
123#387—21no—
124#385—21no—
125#380C22×C311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4C22×C311yescharacteristic
128C4×C22⋊C4—21no—
129C2×#95—21no—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8C2×Q812yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4SD1614yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×Q812yes—
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—44no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4C2×Q812yes—
188C3×C4⋊C4#4114yes—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—42no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—41no—
194C3×C42—41no—
195C3×C42—42no—
196C3×C42—42no—
197C22×Q8C23×C311yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200C23×C4C3×D411yescharacteristic
201C23×C4C3×D411yescharacteristic
202C23×C4C23×C311yescharacteristic
203C22×C8C23×C312yes—
204C22×C8C23×C312yes—
205C22⋊Q8—22no—
206C22⋊Q8C3×Q814yes—
207C4×Q8—42no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C4⋊C4C23×C312yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42C3×D411yescharacteristic
229C2×C42C3×D411yescharacteristic
230C2×C42C23×C311yescharacteristic
231C2×C42C23×C311yescharacteristic
232#106—24no—
233#106—42no—
234C4⋊C8—42no—
235#103—24no—
236#103—42no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3C22⋊Q812yes—
251C23×C3C2×SD1612yes—
252C23×C3#13712yes—
253C23×C3C22×D411yescharacteristic
254C3×Q8—24no—
255C3×Q8—42no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—44no—
274C2×C3×C4—48no—
275C2×C3×C4—44no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—84no—
292C2×C3×C4C22⋊Q812yes—
293C2×C3×C4C2×SD1612yes—
294C2×C3×C4C2×SD1612yes—
295C2×C3×C4C22×Q811yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×C3×C4H(5,𝔽2)12yes—
299C3×C8—42no—
300C3×C8—42no—
301C24C2×C3×D411yescharacteristic
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8C2×C3×Q812yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4C3×SD1614yes—
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×Q812yes—
332C22×C4C3×#4112yes—
333C22×C4C24×C311yescharacteristic
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4C2×C3×Q812yes—
354C4⋊C4C3×#4114yes—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C42—22no—
358C42—22no—
359C42—41no—
360C42—41no—
361C42—42no—
362C42—42no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3#47514yes—
370C22×C3C2×C22⋊Q811yescharacteristic
371C22×C3#52511yescharacteristic
372C22×C3#53411yescharacteristic
373C22×C3C22×SD1611yescharacteristic
374C22×C3C2×#13711yescharacteristic
375C22×C3#57511yescharacteristic
376C22×C3#57511yescharacteristic
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—44no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—81no—
386C3×C4—82no—
387C3×C4—81no—
388C3×C4#55312yes—
389C3×C4#55412yes—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23C3×C22⋊Q812yes—
395C23C2×C3×SD1612yes—
396C23C3×#13712yes—
397C23C22×C3×D411yescharacteristic
398Q8—24no—
399Q8—42no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—84no—
436C2×C4C3×C22⋊Q812yes—
437C2×C4C2×C3×SD1612yes—
438C2×C4C2×C3×SD1612yes—
439C2×C4C22×C3×Q811yescharacteristic
440C2×C4C2×C3×#4111yescharacteristic
441C2×C4C2×C3×#4112yes—
442C2×C4C3×H(5,𝔽2)12yes—
443C8—42no—
444C8—42no—
445C2×C3—24no—
446C2×C3#257911yescharacteristic
447C2×C3C2×#47511yescharacteristic
448C2×C3#308011yescharacteristic
449C2×C3#310111yescharacteristic
450C2×C3#311711yescharacteristic
451C2×C3#318711yescharacteristic
452C2×C3#319611yescharacteristic
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22C3×#47514yes—
460C22C2×C3×C22⋊Q811yescharacteristic
461C22C3×#52511yescharacteristic
462C22C3×#53411yescharacteristic
463C22C22×C3×SD1611yescharacteristic
464C22C2×C3×#13711yescharacteristic
465C22C3×#57511yescharacteristic
466C22C3×#57511yescharacteristic
467C4—22no—
468C4—22no—
469C4—22no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—42no—
474C4—44no—
475C4—82no—
476C4—81no—
477C4—81no—
478C4C3×#55312yes—
479C4C3×#55412yes—
480C3#2924011yescharacteristic, direct product factor, Sylow
481C2—24no—
482C2C3×#257911yescharacteristic
483C2C2×C3×#47511yescharacteristic
484C2C3×#308011yescharacteristic
485C2C3×#310111yescharacteristic
486C2C3×#311711yescharacteristic
487C2C3×#318711yescharacteristic
488C2C3×#319611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2088C2semidirect
4. C3×#2066C2semidirect
5. C3×#1919C2semidirect
8. C3×#1865C2semidirect
12. #29240C3direct (N × H)
46. #2088C2×C3semidirect
47. #2066C2×C3semidirect
48. #1919C2×C3semidirect
51. #1865C2×C3semidirect
480. C3#29240direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×C3×#3892
C3×#20882
C3×#20662
C3×#19192
C3×#18722
C3×#18652
C3×#18532
C3×#17972
#292403