Finite Group Databasebeta

← C3×#25596 #10644203

Subgroups — C3×#25596 #10644203

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3510C211yesmaximal, characteristic
2C3×#3139C211yesmaximal, characteristic, complemented
3C3×#3039C211yesmaximal, characteristic, complemented
4C3×#2658C211yesmaximal, characteristic, complemented
5C3×#2005C212yesmaximal, complemented
6C3×#2004C212yesmaximal
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1967C211yesmaximal, characteristic, complemented
9C3×#1886C212yesmaximal, complemented
10C3×#1885C212yesmaximal, complemented
11C3×#1759C211yesmaximal, characteristic, complemented
12#25596C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#559—22no—
15C3×#556—22no—
16C3×#551—24no—
17C3×#542—22no—
18C3×#536C2211yescharacteristic
19C3×#535C2211yescharacteristic
20C3×#533C2211yescharacteristic
21C2×C3×C4⋊Q8C2212yes—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#529—22no—
24C2×C3×#124C2212yes—
25C2×C3×#124C2211yescharacteristic
26C3×#519C2211yescharacteristic
27C3×#518C2211yescharacteristic
28C3×C4×#41C2211yescharacteristic
29C3×#491—22no—
30C3×#487—22no—
31C3×#462—22no—
32C3×#461—22no—
33C3×#423C2211yescharacteristic
34C2×C3×#104C2212yes—
35C3×#419C2212yes—
36C3×#419C2212yes—
37C3×#419C2212yes—
38C2×C3×#101C2212yes—
39C2×C3×#100C2212yes—
40C3×C4×M16C2211yescharacteristic
41C3×#399—24no—
42C3×#376C2211yescharacteristic
43C3×#339C2212yes—
44C3×#339C2212yes—
45C3×#339C2212yes—
46#3510C2×C311yescharacteristic
47#3139C2×C311yescharacteristic, complemented
48#3039C2×C311yescharacteristic, complemented
49#2658C2×C311yescharacteristic, complemented
50#2005C2×C312yescomplemented
51#2004C2×C312yes—
52#1995C2×C311yescharacteristic, complemented
53#1967C2×C311yescharacteristic, complemented
54#1886C2×C312yescomplemented
55#1885C2×C312yescomplemented
56#1759C2×C311yescharacteristic, complemented
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C3×#137—42no—
62C3×#137—42no—
63C2×C3×Q16—22no—
64C2×C3×SD16—22no—
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—42no—
73C3×C4⋊Q8D412yes—
74C3×#127—21no—
75C3×#126—24no—
76C3×#126—24no—
77C3×#125D412yes—
78C3×#124—24no—
79C3×#124—21no—
80C3×#124—21no—
81C3×#124—22no—
82C3×#124—21no—
83C3×#124—21no—
84C3×#124—24no—
85C3×#124—42no—
86C3×#123—24no—
87C3×#123—24no—
88C3×#123—22no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×#121—22no—
93C3×#120—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8D412yes—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4D412yes—
103C3×#117—24no—
104C3×#117—22no—
105C3×#117—21no—
106C3×#117C2311yescharacteristic
107C3×#117C2311yescharacteristic
108C3×#117C2311yescharacteristic
109C2×C3×C4⋊C4—24no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C22⋊C4D414yes—
114C2×C3×C42—21no—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C3×C4⋊C8—22no—
118C3×#104—24no—
119C3×#103—24no—
120C3×#103—24no—
121C3×#103—24no—
122C3×#101—42no—
123C3×#100—42no—
124C3×#97—21no—
125C3×C4×C8—21no—
126C3×#95—22no—
127C2×#137C22×C312yes—
128#559—22no—
129#556—22no—
130#551—24no—
131#542—22no—
132#536C22×C311yescharacteristic
133#535C22×C311yescharacteristic
134#533C22×C311yescharacteristic
135C2×C4⋊Q8C22×C311yescharacteristic
136C2×C4⋊Q8C22×C312yes—
137#529—22no—
138C2×#124C22×C312yes—
139C2×#124C22×C311yescharacteristic
140#519C22×C311yescharacteristic
141#518C22×C311yescharacteristic
142C4×#41C22×C311yescharacteristic
143#491—22no—
144#487—22no—
145#462—22no—
146#461—22no—
147#423C22×C311yescharacteristic
148C2×#104C22×C312yes—
149#419C22×C312yes—
150#419C22×C312yes—
151#419C22×C312yes—
152C2×#101C22×C312yes—
153C2×#100C22×C312yes—
154C4×M16C22×C311yescharacteristic
155#399—24no—
156#376C22×C311yescharacteristic
157#339C22×C312yes—
158#339C22×C312yes—
159#339C22×C312yes—
160C24×C3C2×D412yes—
161C3×#41—22no—
162C3×#41—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8C2×D411yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4C2×D411yescharacteristic
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C3×Q16—44no—
191C3×SD16—44no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—48no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—48no—
220C3×C42—24no—
221C3×C42—22no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—22no—
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C3×C42C2×D412yes—
229C3×C42#4114yes—
230C2×#41C23×C311yescharacteristic
231C22×Q8C23×C312yes—
232C22×Q8C23×C311yescharacteristic
233C22×D4C23×C311yescharacteristic
234#137—42no—
235#137—42no—
236C2×Q16—22no—
237C2×SD16—22no—
238C2×M16C23×C312yes—
239C2×M16C23×C312yes—
240C4⋊Q8—21no—
241C4⋊Q8—22no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—24no—
245C4⋊Q8—42no—
246C4⋊Q8C3×D412yes—
247#127—21no—
248#126—24no—
249#126—24no—
250#125C3×D412yes—
251#124—24no—
252#124—21no—
253#124—21no—
254#124—21no—
255#124—21no—
256#124—22no—
257#124—24no—
258#124—42no—
259#123—24no—
260#123—24no—
261#123—22no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265#121—22no—
266#120—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8C3×D412yes—
271C4×D4—24no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4C3×D412yes—
276#117—24no—
277#117—22no—
278#117—21no—
279#117C23×C311yescharacteristic
280#117C23×C311yescharacteristic
281#117C23×C311yescharacteristic
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C3×D414yes—
287C2×C42—21no—
288C2×C42C23×C312yes—
289C2×C42C23×C311yescharacteristic
290C4⋊C8—22no—
291#104—24no—
292#103—24no—
293#103—24no—
294#103—24no—
295#101—42no—
296#100—42no—
297#97—21no—
298C4×C8—21no—
299#95—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—24no—
303C23×C3—44no—
304C23×C3C22×D411yescharacteristic
305C3×Q8—21no—
306C3×Q8—21no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—82no—
315C3×D4—21no—
316C3×D4—21no—
317C3×D4—42no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—42no—
321C3×D4—84no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—82no—
336C2×C3×C4—84no—
337C2×C3×C4#12012yes—
338C2×C3×C4#12012yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4112yes—
342C2×C3×C4H(5,𝔽2)12yes—
343C3×C8—42no—
344C3×C8—42no—
345C24C2×C3×D412yes—
346#41—22no—
347#41—22no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8C2×C3×D411yescharacteristic
360C2×D4—21no—
361C2×D4—24no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4C2×C3×D411yescharacteristic
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D412yes—
374C22×C4C24×C311yescharacteristic
375Q16—44no—
376SD16—44no—
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381C2×C8—22no—
382C2×C8—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4C2×C3×D412yes—
393C4⋊C4C2×C3×D412yes—
394C4⋊C4C2×C3×D412yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—21no—
399C22⋊C4—21no—
400C22⋊C4—21no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—48no—
405C42—24no—
406C42—22no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—22no—
411C42C2×C3×D411yescharacteristic
412C42C2×C3×D411yescharacteristic
413C42C2×C3×D412yes—
414C42C3×#4114yes—
415C22×C3—22no—
416C22×C3—41no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—44no—
420C22×C3—84no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54612yes—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—44no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—82no—
432C3×C4#53211yescharacteristic
433C3×C4#53511yescharacteristic
434C3×C4#55012yes—
435C23—22no—
436C23—21no—
437C23—24no—
438C23—44no—
439C23C22×C3×D411yescharacteristic
440Q8—21no—
441Q8—21no—
442Q8—41no—
443Q8—41no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—82no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—41no—
454D4—41no—
455D4—42no—
456D4—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—84no—
472C2×C4C3×#12012yes—
473C2×C4C3×#12012yes—
474C2×C4C22×C3×D411yescharacteristic
475C2×C4C22×C3×D411yescharacteristic
476C2×C4C2×C3×#4112yes—
477C2×C4C3×H(5,𝔽2)12yes—
478C8—42no—
479C8—42no—
480C2×C3—22no—
481C2×C3—81no—
482C2×C3—82no—
483C2×C3#243111yescharacteristic
484C2×C3#302212yes—
485C22—22no—
486C22—41no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—84no—
491C22C2×C3×#12011yescharacteristic
492C22C3×#54612yes—
493C4—22no—
494C4—22no—
495C4—44no—
496C4—42no—
497C4—41no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4C3×#53211yescharacteristic
503C4C3×#53511yescharacteristic
504C4C3×#55012yes—
505C3#2559611yescharacteristic, direct product factor, Sylow
506C2—22no—
507C2—82no—
508C2—81no—
509C2C3×#243111yescharacteristic
510C2C3×#302212yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3139C2semidirect
3. C3×#3039C2semidirect
4. C3×#2658C2semidirect
5. C3×#2005C2semidirect
7. C3×#1995C2semidirect
8. C3×#1967C2semidirect
9. C3×#1886C2semidirect
10. C3×#1885C2semidirect
11. C3×#1759C2semidirect
12. #25596C3direct (N × H)
47. #3139C2×C3semidirect
48. #3039C2×C3semidirect
49. #2658C2×C3semidirect
50. #2005C2×C3semidirect
52. #1995C2×C3semidirect
53. #1967C2×C3semidirect
54. #1886C2×C3semidirect
55. #1885C2×C3semidirect
56. #1759C2×C3semidirect
505. C3#25596direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35102
C3×#31392
C3×#30392
C3×#26582
C3×#20052
C3×#20042
C3×#19952
C3×#19672
C3×#18862
C3×#18852
C3×#17592
#255963