Finite Group Databasebeta

← C3×#29500 #10648107

Subgroups — C3×#29500 #10648107

Order 768.

770 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
1C3×#2860C211yesmaximal, characteristic
2C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#2083C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2013C211yesmaximal, characteristic
7C3×#1973C212yesmaximal
8C3×#1935C212yesmaximal, complemented
9C3×#1880C212yesmaximal
10C3×#1863C212yesmaximal, complemented
11C3×#1745C211yesmaximal, characteristic, complemented
12#29500C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#126—21no—
16C2×C3×C4×Q8C2212yes—
17C2×C3×#107C2212yes—
18C2×C3×#103C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#97C2212yes—
23C3×#391C2212yes—
24C3×#390C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389—21no—
28C3×#389—21no—
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#389C2211yescharacteristic
32C3×#388—21no—
33C3×#388—21no—
34C3×#388—21no—
35C3×#384C2211yescharacteristic
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4—21no—
38C3×C43C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2860C2×C311yescharacteristic
45C4×C4⋊Q8C2×C311yescharacteristic, complemented
46#2083C2×C311yescharacteristic, complemented
47#2071C2×C311yescharacteristic
48#2043C2×C311yescharacteristic, complemented
49#2013C2×C311yescharacteristic
50#1973C2×C312yes—
51#1935C2×C312yescomplemented
52#1880C2×C312yes—
53#1863C2×C312yescomplemented
54#1745C2×C311yescharacteristic, complemented
55C22×C3×Q8C2312yes—
56C22×C3×D4C2311yescharacteristic
57C2×C3×SD16—24no—
58C2×C3×SD16—24no—
59C2×C3×SD16—42no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—24no—
63C3×#126—44no—
64C3×C4×Q8—42no—
65C3×C4×Q8—42no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—22no—
85C2×C3×C42D411yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—42no—
93C3×#103—24no—
94C3×#103—24no—
95C3×#103—42no—
96C3×#102—42no—
97C3×#97—24no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—21no—
103C22×SD16C22×C312yes—
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×#126—21no—
106C2×C4×Q8C22×C312yes—
107C2×#107C22×C312yes—
108C2×#103C22×C312yes—
109C2×#103C22×C312yes—
110C2×#103C22×C312yes—
111C2×#102C22×C312yes—
112C2×#97C22×C312yes—
113#391C22×C312yes—
114#390C22×C311yescharacteristic
115#390C22×C311yescharacteristic
116#390C22×C311yescharacteristic
117#389—21no—
118#389—21no—
119#389C22×C311yescharacteristic
120#389C22×C311yescharacteristic
121#389C22×C311yescharacteristic
122#388—21no—
123#388—21no—
124#388—21no—
125#384C22×C311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C43C22×C311yescharacteristic
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C311yescharacteristic
132#336C22×C311yescharacteristic
133#336C22×C312yes—
134C24×C3—21no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8#4114yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C3×SD16—48no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8#4114yes—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C22⋊C4—42no—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42—41no—
192C3×C42—44no—
193C22×Q8C23×C312yes—
194C22×D4C23×C311yescharacteristic
195C2×SD16—24no—
196C2×SD16—24no—
197C2×SD16—42no—
198C22×C8C23×C312yes—
199C22×C8C23×C312yes—
200C4⋊Q8—24no—
201#126—44no—
202C4×Q8—42no—
203C4×Q8—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—22no—
223C2×C42C3×D411yescharacteristic
224C2×C42C3×D411yescharacteristic
225C2×C42C3×D411yescharacteristic
226C2×C42C3×D411yescharacteristic
227C2×C42C23×C311yescharacteristic
228C2×C42C23×C311yescharacteristic
229C2×C42C23×C311yescharacteristic
230#107—42no—
231#103—24no—
232#103—24no—
233#103—42no—
234#102—42no—
235#97—24no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—21no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3C22×D411yescharacteristic
249C3×Q8—24no—
250C3×Q8—24no—
251C3×Q8—42no—
252C3×Q8—42no—
253C3×D4—42no—
254C3×D4—42no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4AGL(1,ℤ/8)12yes—
286C2×C3×C4#13712yes—
287C2×C3×C4#13712yes—
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4112yes—
290C2×C3×C4H(5,𝔽2)11yescharacteristic
291C2×C3×C4H(5,𝔽2)11yescharacteristic
292C3×C8—24no—
293C3×C8—42no—
294C24—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8C3×#4114yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C24×C311yescharacteristic
317SD16—48no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8C3×#4114yes—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—21no—
350C42—21no—
351C42—41no—
352C42—44no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3#52511yescharacteristic
361C22×C3#53211yescharacteristic
362C22×C3#53511yescharacteristic
363C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
364C22×C3C2×#13711yescharacteristic
365C22×C3C2×#13711yescharacteristic
366C22×C3#57511yescharacteristic
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—44no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—81no—
375C3×C4—81no—
376C3×C4—82no—
377C3×C4—81no—
378C3×C4#55112yes—
379C3×C4#55312yes—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C22×C3×D411yescharacteristic
388Q8—24no—
389Q8—24no—
390Q8—42no—
391Q8—42no—
392D4—42no—
393D4—42no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4C3×AGL(1,ℤ/8)12yes—
425C2×C4C3×#13712yes—
426C2×C4C3×#13712yes—
427C2×C4C2×C3×#4112yes—
428C2×C4C2×C3×#4112yes—
429C2×C4C3×H(5,𝔽2)11yescharacteristic
430C2×C4C3×H(5,𝔽2)11yescharacteristic
431C8—24no—
432C8—42no—
433C2×C3—161no—
434C2×C3#260311yescharacteristic
435C2×C3#311211yescharacteristic
436C2×C3#311711yescharacteristic
437C2×C3#317211yescharacteristic
438C2×C3#318211yescharacteristic
439C2×C3#321411yescharacteristic
440C2×C3#321811yescharacteristic
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22C3×#52511yescharacteristic
449C22C3×#53211yescharacteristic
450C22C3×#53511yescharacteristic
451C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
452C22C2×C3×#13711yescharacteristic
453C22C2×C3×#13711yescharacteristic
454C22C3×#57511yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—44no—
462C4—81no—
463C4—81no—
464C4—82no—
465C4—81no—
466C4C3×#55112yes—
467C4C3×#55312yes—
468C3#2950011yescharacteristic, direct product factor, Sylow
469C2—161no—
470C2C3×#260311yescharacteristic
471C2C3×#311211yescharacteristic
472C2C3×#311711yescharacteristic
473C2C3×#317211yescharacteristic
474C2C3×#318211yescharacteristic
475C2C3×#321411yescharacteristic
476C2C3×#321811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C4×C4⋊Q8C2semidirect
3. C3×#2083C2semidirect
5. C3×#2043C2semidirect
8. C3×#1935C2semidirect
10. C3×#1863C2semidirect
11. C3×#1745C2semidirect
12. #29500C3direct (N × H)
45. C4×C4⋊Q8C2×C3semidirect
46. #2083C2×C3semidirect
48. #2043C2×C3semidirect
51. #1935C2×C3semidirect
53. #1863C2×C3semidirect
54. #1745C2×C3semidirect
468. C3#29500direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28602
C3×C4×C4⋊Q82
C3×#20832
C3×#20712
C3×#20432
C3×#20132
C3×#19732
C3×#19352
C3×#18802
C3×#18632
C3×#17452
#295003