Finite Group Databasebeta

← C3×#29663 #10648270

Subgroups — C3×#29663 #10648270

Order 768.

1412 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#461C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#2024C212yesmaximal, complemented
5C3×#2014C212yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1802C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29663C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41—21no—
11C22×C3×SD16C2214yes—
12C3×#532—28no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C3×#461—28no—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2214yes—
26C2×C3×#102C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#391C2212yes—
29C3×#389—21no—
30C3×#389—21no—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380—21no—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#532C2×C311yescharacteristic
37C2×#461C2×C312yescomplemented
38#2857C2×C311yescharacteristic, complemented
39#2024C2×C312yescomplemented
40#2014C2×C312yescomplemented
41#2000C2×C314yescomplemented
42#1802C2×C312yescomplemented
43#1797C2×C311yescharacteristic, complemented
44C2×C3×#41—28no—
45C2×C3×#41—44no—
46C2×C3×#41—44no—
47C22×C3×Q8—21no—
48C22×C3×Q8C2312yes—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4—22no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C2×C3×SD16—28no—
58C2×C3×SD16—44no—
59C2×C3×SD16—44no—
60C22×C3×C8C2314yes—
61C3×C4⋊Q8—24no—
62C3×C4⋊Q8—24no—
63C3×#127—24no—
64C3×#124—28no—
65C3×C22⋊Q8—44no—
66C3×#121—28no—
67C3×#121—24no—
68C3×#121—24no—
69C3×C4×D4—28no—
70C3×C4×D4—24no—
71C3×C4×D4D418yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—24no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×C4⋊C8—24no—
89C3×#103—28no—
90C3×#102—44no—
91C3×C22⋊C8—42no—
92C3×#95—22no—
93C3×#95—22no—
94C22×#41—21no—
95C22×SD16C22×C314yes—
96#532—28no—
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×#127C22×C311yescharacteristic
100C2×#124—21no—
101C2×C22⋊Q8C22×C312yes—
102C2×#121—21no—
103C2×#121C22×C312yes—
104C2×C4×D4—21no—
105C2×C4×D4C22×C312yes—
106C22×C42C22×C311yescharacteristic
107#461—28no—
108C2×C4⋊C8C22×C312yes—
109C2×#103C22×C314yes—
110C2×#102C22×C314yes—
111C2×C22⋊C8C22×C312yes—
112#391C22×C312yes—
113#389—21no—
114#389—21no—
115#384C22×C311yescharacteristic
116#380—21no—
117#380—21no—
118#340C22×C314yes—
119#336C22×C312yes—
120C24×C3—22no—
121C24×C3C2×D411yescharacteristic
122C3×#41—432no—
123C2×C3×Q8—24no—
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×D4—24no—
131C2×C3×D4—28no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—44no—
141C2×C3×D4—44no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4C2×D414yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—28no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—41no—
158C22×C3×C4—44no—
159C22×C3×C4—44no—
160C22×C3×C4SD1614yescomplemented
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C3×SD16—416no—
168C2×C3×C8—24no—
169C2×C3×C8—24no—
170C2×C3×C8—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4C2×D414yes—
180C3×C22⋊C4—28no—
181C3×C22⋊C4—28no—
182C3×C22⋊C4—44no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42C2×D414yes—
187C2×#41—28no—
188C2×#41—44no—
189C2×#41—44no—
190C22×Q8—21no—
191C22×Q8C23×C312yes—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C23×C312yes—
196C23×C4—22no—
197C23×C4C3×D411yescharacteristic
198C23×C4C3×D411yescharacteristic
199C23×C4C23×C311yescharacteristic
200C2×SD16—28no—
201C2×SD16—44no—
202C2×SD16—44no—
203C22×C8C23×C314yes—
204C4⋊Q8—24no—
205C4⋊Q8—24no—
206#127—24no—
207#124—28no—
208C22⋊Q8—44no—
209#121—28no—
210#121—24no—
211#121—24no—
212C4×D4—28no—
213C4×D4—24no—
214C4×D4C3×D418yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C42—24no—
227C2×C42C3×D411yescharacteristic
228C2×C42C3×D411yescharacteristic
229C2×C42C23×C311yescharacteristic
230C2×C42C23×C311yescharacteristic
231C4⋊C8—24no—
232#103—28no—
233#102—44no—
234C22⋊C8—42no—
235#95—22no—
236#95—22no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—24no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3C2×SD1612yes—
251C23×C3C22×D411yescharacteristic
252C3×Q8—48no—
253C3×Q8—44no—
254C3×Q8—44no—
255C3×D4—24no—
256C3×D4—24no—
257C3×D4—48no—
258C3×D4—48no—
259C3×D4—416no—
260C3×D4—42no—
261C3×D4—42no—
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—28no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—44no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—416no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4#12012yes—
286C2×C3×C4#12012yes—
287C2×C3×C4C2×SD1612yes—
288C2×C3×C4C2×SD1612yes—
289C2×C3×C4AGL(1,ℤ/8)12yes—
290C2×C3×C4#13712yes—
291C2×C3×C4C22×D412yes—
292C2×C3×C4H(5,𝔽2)11yescharacteristic
293C2×C3×C4H(5,𝔽2)12yes—
294C2×C3×C42−1+411yescharacteristic
295C3×C8—44no—
296C24—22no—
297C24C2×C3×D411yescharacteristic
298#41—432no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×D4—24no—
307C2×D4—28no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4C2×C3×D414yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4C3×SD1614yescomplemented
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C2×C3×D411yescharacteristic
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C24×C311yescharacteristic
343SD16—416no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4C2×C3×D414yes—
356C22⋊C4—28no—
357C22⋊C4—28no—
358C22⋊C4—44no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42C2×C3×D414yes—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—44no—
371C22×C3—44no—
372C22×C3—44no—
373C22×C3—82no—
374C22×C3—82no—
375C22×C3—82no—
376C22×C3—82no—
377C22×C3C2×#12011yescharacteristic
378C22×C3#53511yescharacteristic
379C22×C3#53711yescharacteristic
380C22×C3C22×SD1611yescharacteristic
381C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
382C22×C3C2×#13711yescharacteristic
383C22×C3#57511yescharacteristic
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—41no—
388C3×C4—44no—
389C3×C4—82no—
390C3×C4—82no—
391C3×C4#45014yes—
392C3×C4#53512yes—
393C3×C4#53712yes—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23C2×C3×SD1612yes—
408C23C22×C3×D411yescharacteristic
409Q8—48no—
410Q8—44no—
411Q8—44no—
412D4—24no—
413D4—24no—
414D4—48no—
415D4—48no—
416D4—416no—
417D4—42no—
418D4—42no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—28no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—416no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4C3×#12012yes—
443C2×C4C3×#12012yes—
444C2×C4C2×C3×SD1612yes—
445C2×C4C2×C3×SD1612yes—
446C2×C4C3×AGL(1,ℤ/8)12yes—
447C2×C4C3×#13712yes—
448C2×C4C22×C3×D412yes—
449C2×C4C3×H(5,𝔽2)11yescharacteristic
450C2×C4C3×H(5,𝔽2)12yes—
451C2×C4C3×2−1+411yescharacteristic
452C8—44no—
453C2×C3—42no—
454C2×C3—84no—
455C2×C3#261511yescharacteristic
456C2×C3C2×#45011yescharacteristic
457C2×C3#300711yescharacteristic
458C2×C3#320611yescharacteristic
459C2×C3#321411yescharacteristic
460C2×C3#324611yescharacteristic
461C2×C3#325411yescharacteristic
462C22—22no—
463C22—22no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22C2×C3×#12011yescharacteristic
477C22C3×#53511yescharacteristic
478C22C3×#53711yescharacteristic
479C22C22×C3×SD1611yescharacteristic
480C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
481C22C2×C3×#13711yescharacteristic
482C22C3×#57511yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—44no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4C3×#45014yes—
491C4C3×#53512yes—
492C4C3×#53712yes—
493C3#2966311yescharacteristic, direct product factor, Sylow
494C2—42no—
495C2—84no—
496C2C3×#261511yescharacteristic
497C2C2×C3×#45011yescharacteristic
498C2C3×#300711yescharacteristic
499C2C3×#320611yescharacteristic
500C2C3×#321411yescharacteristic
501C2C3×#324611yescharacteristic
502C2C3×#325411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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
2. C2×C3×#461C2semidirect
3. C3×#2857C2semidirect
4. C3×#2024C2semidirect
5. C3×#2014C2semidirect
6. C3×#2000C2semidirect
7. C3×#1802C2semidirect
8. C3×#1797C2semidirect
9. #29663C3direct (N × H)
37. C2×#461C2×C3semidirect
38. #2857C2×C3semidirect
39. #2024C2×C3semidirect
40. #2014C2×C3semidirect
41. #2000C2×C3semidirect
42. #1802C2×C3semidirect
43. #1797C2×C3semidirect
160. C22×C3×C4SD16semidirect
336. C22×C4C3×SD16semidirect
493. C3#29663direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4612
C3×#28572
C3×#20242
C3×#20142
C3×#20002
C3×#18022
C3×#17972
#296633