Finite Group Databasebeta

← C3×#25597 #10644204

Subgroups — C3×#25597 #10644204

Order 768.

1438 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic, complemented
5C3×#2000C214yesmaximal, complemented
6C3×#1990C211yesmaximal, characteristic, complemented
7C3×#1957C211yesmaximal, characteristic, complemented
8C3×#1877C214yesmaximal, complemented
9C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
10#25597C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C22×C3×SD16C2214yes—
13C3×#532—24no—
14C3×#532—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#492—24no—
26C3×#487—24no—
27C2×C3×#103C2214yes—
28C2×C3×#102C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C3×#391C2212yes—
33C3×#389—21no—
34C3×#389—21no—
35C3×#384C2211yescharacteristic
36C3×#380—21no—
37C3×#380—21no—
38C3×#340C2214yes—
39C3×#336C2212yes—
40C2×#532C2×C311yescharacteristic
41C2×#492C2×C311yescharacteristic, complemented
42C2×#487C2×C311yescharacteristic, complemented
43#2857C2×C311yescharacteristic, complemented
44#2000C2×C314yescomplemented
45#1990C2×C311yescharacteristic, complemented
46#1957C2×C311yescharacteristic, complemented
47#1877C2×C314yescomplemented
48C4×C22⋊C8C2×C311yescharacteristic, complemented
49C2×C3×#41—28no—
50C2×C3×#41—44no—
51C2×C3×#41—44no—
52C22×C3×Q8—21no—
53C22×C3×Q8C2312yes—
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C23×C3×C4—22no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×SD16—28no—
63C2×C3×SD16—28no—
64C2×C3×SD16—44no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×#127—22no—
72C3×#127D414yes—
73C3×#124—28no—
74C3×C22⋊Q8—44no—
75C3×#121—28no—
76C3×#121—24no—
77C3×#121—24no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C42—22no—
93C2×C3×C42—22no—
94C2×C3×C42D411yescharacteristic
95C2×C3×C42D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×#103—28no—
99C3×#102—44no—
100C3×C22⋊C8—24no—
101C3×C4×C8—22no—
102C3×C4×C8—22no—
103C3×#95—22no—
104C3×#95—22no—
105C22×#41—21no—
106C22×SD16C22×C314yes—
107#532—24no—
108#532—24no—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×C4⋊Q8C22×C311yescharacteristic
111C2×#127C22×C311yescharacteristic
112C2×#124—21no—
113C2×C22⋊Q8C22×C312yes—
114C2×#121—21no—
115C2×#121C22×C312yes—
116C2×C4×D4—21no—
117C2×C4×D4C22×C312yes—
118C22×C42C22×C311yescharacteristic
119#492—24no—
120#487—24no—
121C2×#103C22×C314yes—
122C2×#102C22×C314yes—
123C2×C22⋊C8C22×C312yes—
124C2×C4×C8C22×C311yescharacteristic
125C2×C4×C8C22×C311yescharacteristic
126#391C22×C312yes—
127#389—21no—
128#389—21no—
129#384C22×C311yescharacteristic
130#380—21no—
131#380—21no—
132#340C22×C314yes—
133#336C22×C312yes—
134C24×C3—22no—
135C24×C3C2×D411yescharacteristic
136C3×#41—432no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—24no—
141C2×C3×Q8—24no—
142C2×C3×Q8—42no—
143C2×C3×Q8—42no—
144C2×C3×D4—28no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4C2×D414yes—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—28no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—44no—
171C22×C3×C4—44no—
172C22×C3×C4SD1614yescomplemented
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C3×SD16—416no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8#4114yescomplemented
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—44no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—21no—
201C3×C42—21no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204C2×#41—28no—
205C2×#41—44no—
206C2×#41—44no—
207C22×Q8—21no—
208C22×Q8C23×C312yes—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C23×C312yes—
213C23×C4—22no—
214C23×C4C3×D411yescharacteristic
215C23×C4C3×D411yescharacteristic
216C23×C4C23×C311yescharacteristic
217C2×SD16—28no—
218C2×SD16—28no—
219C2×SD16—44no—
220C22×C8C23×C312yes—
221C22×C8C23×C312yes—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8C3×D414yes—
226#127—22no—
227#127C3×D414yes—
228#124—28no—
229C22⋊Q8—44no—
230#121—28no—
231#121—24no—
232#121—24no—
233C4×D4—28no—
234C4×D4—24no—
235C4×D4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C23×C312yes—
243C2×C4⋊C4C23×C312yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C3×D411yescharacteristic
250C2×C42C3×D411yescharacteristic
251C2×C42C23×C311yescharacteristic
252C2×C42C23×C311yescharacteristic
253#103—28no—
254#102—44no—
255C22⋊C8—24no—
256C4×C8—22no—
257C4×C8—22no—
258#95—22no—
259#95—22no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—24no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3C2×SD1612yes—
274C23×C3C22×D411yescharacteristic
275C3×Q8—48no—
276C3×Q8—44no—
277C3×Q8—44no—
278C3×D4—28no—
279C3×D4—416no—
280C3×D4—44no—
281C3×D4—48no—
282C3×D4—48no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—28no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—416no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4#12012yes—
306C2×C3×C4#12012yes—
307C2×C3×C4C2×SD1612yes—
308C2×C3×C4C2×SD1612yes—
309C2×C3×C4#13512yes—
310C2×C3×C4AGL(1,ℤ/8)12yes—
311C2×C3×C4#13712yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4H(5,𝔽2)12yes—
316C3×C8—24no—
317C3×C8—42no—
318C24—22no—
319C24C2×C3×D411yescharacteristic
320#41—432no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4C2×C3×D414yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4C3×SD1614yescomplemented
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C24×C311yescharacteristic
363SD16—416no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8C3×#4114yescomplemented
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4C2×C3×D414yes—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—44no—
382C42—22no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42C2×C3×D412yes—
387C42C2×C3×D412yes—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#53211yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3C22×SD1611yescharacteristic
406C22×C3C2×#13511yescharacteristic
407C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
408C22×C3C2×#13711yescharacteristic
409C3×C4—22no—
410C3×C4—41no—
411C3×C4—44no—
412C3×C4—41no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4#45014yes—
416C3×C4#45214yes—
417C3×C4#54612yes—
418C3×C4#55012yes—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—24no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23C2×C3×SD1612yes—
433C23C22×C3×D411yescharacteristic
434Q8—48no—
435Q8—44no—
436Q8—44no—
437D4—28no—
438D4—416no—
439D4—48no—
440D4—48no—
441D4—44no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—416no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4C3×#12012yes—
465C2×C4C3×#12012yes—
466C2×C4C2×C3×SD1612yes—
467C2×C4C2×C3×SD1612yes—
468C2×C4C3×#13512yes—
469C2×C4C3×AGL(1,ℤ/8)12yes—
470C2×C4C3×#13712yes—
471C2×C4C22×C3×D411yescharacteristic
472C2×C4C22×C3×D411yescharacteristic
473C2×C4C2×C3×#4112yes—
474C2×C4C3×H(5,𝔽2)12yes—
475C8—24no—
476C8—42no—
477C2×C3—42no—
478C2×C3—84no—
479C2×C3#243111yescharacteristic
480C2×C3C2×#45011yescharacteristic
481C2×C3C2×#45211yescharacteristic
482C2×C3#316711yescharacteristic
483C2×C3#318211yescharacteristic
484C2×C3#320611yescharacteristic
485C2×C3#320711yescharacteristic
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22C2×C3×#12011yescharacteristic
501C22C3×#53211yescharacteristic
502C22C3×#53511yescharacteristic
503C22C22×C3×SD1611yescharacteristic
504C22C2×C3×#13511yescharacteristic
505C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
506C22C2×C3×#13711yescharacteristic
507C4—22no—
508C4—41no—
509C4—44no—
510C4—41no—
511C4—82no—
512C4—82no—
513C4C3×#45014yes—
514C4C3×#45214yes—
515C4C3×#54612yes—
516C4C3×#55012yes—
517C3#2559711yescharacteristic, direct product factor, Sylow
518C2—42no—
519C2—84no—
520C2C3×#243111yescharacteristic
521C2C2×C3×#45011yescharacteristic
522C2C2×C3×#45211yescharacteristic
523C2C3×#316711yescharacteristic
524C2C3×#318211yescharacteristic
525C2C3×#320611yescharacteristic
526C2C3×#320711yescharacteristic
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×#492C2semidirect
3. C2×C3×#487C2semidirect
4. C3×#2857C2semidirect
5. C3×#2000C2semidirect
6. C3×#1990C2semidirect
7. C3×#1957C2semidirect
8. C3×#1877C2semidirect
9. C3×C4×C22⋊C8C2semidirect
10. #25597C3direct (N × H)
41. C2×#492C2×C3semidirect
42. C2×#487C2×C3semidirect
43. #2857C2×C3semidirect
44. #2000C2×C3semidirect
45. #1990C2×C3semidirect
46. #1957C2×C3semidirect
47. #1877C2×C3semidirect
48. C4×C22⋊C8C2×C3semidirect
172. C22×C3×C4SD16semidirect
185. C2×C3×C8#41semidirect
356. C22×C4C3×SD16semidirect
369. C2×C8C3×#41semidirect
517. C3#25597direct (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×#4922
C2×C3×#4872
C3×#28572
C3×#20002
C3×#19902
C3×#19572
C3×#18772
C3×C4×C22⋊C82
#255973