Finite Group Databasebeta

← C3×#25620 #10644227

Subgroups — C3×#25620 #10644227

Order 768.

1136 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3512C211yesmaximal, characteristic, complemented
2C3×#3151C211yesmaximal, characteristic, complemented
3C3×#3038C211yesmaximal, characteristic, complemented
4C3×#2641C211yesmaximal, characteristic, complemented
5C3×#2006C214yesmaximal, complemented
6C3×#1998C211yesmaximal, characteristic, complemented
7C3×#1963C211yesmaximal, characteristic, complemented
8C3×#1883C214yesmaximal, complemented
9C3×#1759C211yesmaximal, characteristic, complemented
10#25620C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×AGL(1,ℤ/8)C2214yes—
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C3×#553—22no—
14C3×#553—24no—
15C3×#552—22no—
16C3×#541—21no—
17C3×#540—21no—
18C3×#537C2212yes—
19C3×#532C2211yescharacteristic
20C2×C3×#127C2211yescharacteristic
21C3×#529—22no—
22C2×C3×#125C2211yescharacteristic
23C3×#520C2212yes—
24C3×C4×#41C2211yescharacteristic
25C3×#496—22no—
26C3×#493—21no—
27C3×#492—21no—
28C3×#460—22no—
29C3×#459—22no—
30C3×#423C2211yescharacteristic
31C2×C3×#104C2212yes—
32C2×C3×#104C2214yes—
33C3×#418C2212yes—
34C3×C4×M16C2211yescharacteristic
35C3×#394—24no—
36C3×#390C2214yes—
37C3×#376C2211yescharacteristic
38C3×#343C2214yes—
39C3×#339C2212yes—
40#3512C2×C311yescharacteristic, complemented
41#3151C2×C311yescharacteristic, complemented
42#3038C2×C311yescharacteristic, complemented
43#2641C2×C311yescharacteristic, complemented
44#2006C2×C314yescomplemented
45#1998C2×C311yescharacteristic, complemented
46#1963C2×C311yescharacteristic, complemented
47#1883C2×C314yescomplemented
48#1759C2×C311yescharacteristic, complemented
49C2×C3×#41C2311yescharacteristic
50C2×C3×#41C2312yes—
51C22×C3×D4—22no—
52C22×C3×D4C2312yes—
53C3×AGL(1,ℤ/8)—28no—
54C3×AGL(1,ℤ/8)—44no—
55C3×AGL(1,ℤ/8)—44no—
56C2×C3×SD16—22no—
57C2×C3×SD16—24no—
58C2×C3×D8—22no—
59C2×C3×D8—24no—
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C3×C4⋊Q8D411yescharacteristic
63C3×#127—22no—
64C3×#127—21no—
65C3×#127—21no—
66C3×#127D411yescharacteristic
67C3×#126—24no—
68C3×#126—24no—
69C3×#125—24no—
70C3×#125—21no—
71C3×#125—21no—
72C3×#125—22no—
73C3×#124D412yes—
74C3×#123—24no—
75C3×#123—24no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—24no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—24no—
83C3×C4×Q8—22no—
84C3×C4×Q8D412yes—
85C3×C4×D4—24no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—24no—
89C3×C4×D4—22no—
90C3×C4×D4D412yes—
91C3×#117—24no—
92C3×#117—22no—
93C3×#117—21no—
94C3×#117C2311yescharacteristic
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4D414yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—28no—
99C2×C3×C42—21no—
100C2×C3×C42C2312yes—
101C2×C3×C42C2311yescharacteristic
102C3×C4⋊C8—22no—
103C3×#104—24no—
104C3×#104—44no—
105C3×#102—24no—
106C3×#97—21no—
107C3×C4×C8—21no—
108C3×#95—22no—
109C2×AGL(1,ℤ/8)C22×C314yes—
110C2×AGL(1,ℤ/8)C22×C312yes—
111#553—22no—
112#553—24no—
113#552—22no—
114#541—21no—
115#540—21no—
116#537C22×C312yes—
117#532C22×C311yescharacteristic
118C2×#127C22×C311yescharacteristic
119#529—22no—
120C2×#125C22×C311yescharacteristic
121#520C22×C312yes—
122C4×#41C22×C311yescharacteristic
123#496—22no—
124#493—21no—
125#492—21no—
126#460—22no—
127#459—22no—
128#423C22×C311yescharacteristic
129C2×#104C22×C312yes—
130C2×#104C22×C314yes—
131#418C22×C312yes—
132C4×M16C22×C311yescharacteristic
133#394—24no—
134#390C22×C314yes—
135#376C22×C311yescharacteristic
136#343C22×C314yes—
137#339C22×C312yes—
138C24×C3—22no—
139C3×#41—22no—
140C3×#41—24no—
141C3×#41—24no—
142C3×#41—22no—
143C2×C3×Q8C2×D412yes—
144C2×C3×Q8C2×D411yescharacteristic
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—42no—
154C2×C3×D4—42no—
155C2×C3×D4—84no—
156C2×C3×D4C2×D412yes—
157C2×C3×D4C2×D411yescharacteristic
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C3×SD16—48no—
167C3×SD16—44no—
168C3×D8—48no—
169C3×D8—44no—
170C3×M16—22no—
171C3×M16—24no—
172C3×M16#4114yes—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—88no—
192C3×C42—24no—
193C3×C42—22no—
194C3×C42—21no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D412yes—
200C2×#41C23×C311yescharacteristic
201C2×#41C23×C312yes—
202C22×D4—22no—
203C22×D4C23×C312yes—
204AGL(1,ℤ/8)—28no—
205AGL(1,ℤ/8)—44no—
206AGL(1,ℤ/8)—44no—
207C2×SD16—22no—
208C2×SD16—24no—
209C2×D8—22no—
210C2×D8—24no—
211C2×M16C23×C312yes—
212C2×M16C23×C312yes—
213C4⋊Q8C3×D411yescharacteristic
214#127—22no—
215#127—21no—
216#127—21no—
217#127C3×D411yescharacteristic
218#126—24no—
219#126—24no—
220#125—24no—
221#125—21no—
222#125—21no—
223#125—22no—
224#124C3×D412yes—
225#123—24no—
226#123—24no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229#121—24no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—24no—
234C4×Q8—22no—
235C4×Q8C3×D412yes—
236C4×D4—24no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—24no—
240C4×D4—22no—
241C4×D4C3×D412yes—
242#117—24no—
243#117—22no—
244#117—21no—
245#117C23×C311yescharacteristic
246C2×C4⋊C4—22no—
247C2×C4⋊C4C3×D414yes—
248C2×C4⋊C4C23×C312yes—
249C2×C22⋊C4—28no—
250C2×C42—21no—
251C2×C42C23×C312yes—
252C2×C42C23×C311yescharacteristic
253C4⋊C8—22no—
254#104—24no—
255#104—44no—
256#102—24no—
257#97—21no—
258C4×C8—21no—
259#95—22no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—48no—
263C23×C3—42no—
264C23×C3—44no—
265C23×C3C22×D411yescharacteristic
266C3×Q8—21no—
267C3×Q8—21no—
268C3×Q8—24no—
269C3×D4—21no—
270C3×D4—21no—
271C3×D4—24no—
272C3×D4—42no—
273C3×D4—44no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—44no—
277C3×D4—82no—
278C3×D4—84no—
279C3×D4—84no—
280C3×D4—84no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—24no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—44no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—82no—
293C2×C3×C4—84no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12012yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4112yes—
299C2×C3×C4H(5,𝔽2)12yes—
300C3×C8—24no—
301C3×C8—42no—
302C24—22no—
303#41—22no—
304#41—24no—
305#41—24no—
306#41—22no—
307C2×Q8C2×C3×D412yes—
308C2×Q8C2×C3×D411yescharacteristic
309C2×D4—21no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—84no—
320C2×D4C2×C3×D412yes—
321C2×D4C2×C3×D411yescharacteristic
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4C2×C3×D414yes—
328C22×C4C2×C3×D412yes—
329C22×C4C24×C311yescharacteristic
330SD16—48no—
331SD16—44no—
332D8—48no—
333D8—44no—
334M16—22no—
335M16—24no—
336M16C3×#4114yes—
337C2×C8—22no—
338C2×C8—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4C2×C3×D412yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—21no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—88no—
356C42—24no—
357C42—22no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42C2×C3×D411yescharacteristic
362C42C2×C3×D411yescharacteristic
363C42C2×C3×D412yes—
364C22×C3—22no—
365C22×C3—42no—
366C22×C3—41no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—88no—
371C22×C3—82no—
372C22×C3—84no—
373C22×C3C2×#12011yescharacteristic
374C22×C3#55012yes—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—42no—
378C3×C4—41no—
379C3×C4—44no—
380C3×C4—42no—
381C3×C4—82no—
382C3×C4#53211yescharacteristic
383C3×C4#53511yescharacteristic
384C3×C4#54612yes—
385C23—22no—
386C23—21no—
387C23—48no—
388C23—42no—
389C23—44no—
390C23C22×C3×D411yescharacteristic
391Q8—21no—
392Q8—21no—
393Q8—24no—
394D4—21no—
395D4—21no—
396D4—24no—
397D4—42no—
398D4—44no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—82no—
403D4—84no—
404D4—84no—
405D4—84no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—84no—
419C2×C4C3×#12012yes—
420C2×C4C3×#12012yes—
421C2×C4C22×C3×D411yescharacteristic
422C2×C4C22×C3×D411yescharacteristic
423C2×C4C2×C3×#4112yes—
424C2×C4C3×H(5,𝔽2)12yes—
425C8—24no—
426C8—42no—
427C2×C3—22no—
428C2×C3—82no—
429C2×C3—81no—
430C2×C3—162no—
431C2×C3#243111yescharacteristic
432C2×C3#302112yes—
433C22—22no—
434C22—42no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—88no—
440C22—82no—
441C22—84no—
442C22C2×C3×#12011yescharacteristic
443C22C3×#55012yes—
444C4—22no—
445C4—22no—
446C4—42no—
447C4—41no—
448C4—44no—
449C4—42no—
450C4—82no—
451C4C3×#53211yescharacteristic
452C4C3×#53511yescharacteristic
453C4C3×#54612yes—
454C3#2562011yescharacteristic, direct product factor, Sylow
455C2—22no—
456C2—82no—
457C2—81no—
458C2—162no—
459C2C3×#243111yescharacteristic
460C2C3×#302112yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C3×#3512C2semidirect
2. C3×#3151C2semidirect
3. C3×#3038C2semidirect
4. C3×#2641C2semidirect
5. C3×#2006C2semidirect
6. C3×#1998C2semidirect
7. C3×#1963C2semidirect
8. C3×#1883C2semidirect
9. C3×#1759C2semidirect
10. #25620C3direct (N × H)
40. #3512C2×C3semidirect
41. #3151C2×C3semidirect
42. #3038C2×C3semidirect
43. #2641C2×C3semidirect
44. #2006C2×C3semidirect
45. #1998C2×C3semidirect
46. #1963C2×C3semidirect
47. #1883C2×C3semidirect
48. #1759C2×C3semidirect
454. C3#25620direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35122
C3×#31512
C3×#30382
C3×#26412
C3×#20062
C3×#19982
C3×#19632
C3×#18832
C3×#17592
#256203