Finite Group Databasebeta

← C3×#32669 #10651276

Subgroups — C3×#32669 #10651276

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2568C211yesmaximal, characteristic, complemented
2C3×#2548C211yesmaximal, characteristic
3C3×#2055C212yesmaximal, complemented
4C3×#2014C212yesmaximal
5C3×#2013C212yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1968C212yesmaximal, complemented
8C3×#1927C212yesmaximal, complemented
9C3×#1926C212yesmaximal, complemented
10#32669C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2212yes—
12C22×C3×SD16C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#124—21no—
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×C4×C8C2211yescharacteristic
24C3×#400—21no—
25C3×#400—21no—
26C3×#394—21no—
27C3×#394—21no—
28C3×#391—21no—
29C3×#391C2212yes—
30C3×#391C2212yes—
31C3×#390—21no—
32C3×#390C2212yes—
33C3×#390C2212yes—
34C3×#382—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×#340C2212yes—
38C3×#336C2212yes—
39C3×#336C2212yes—
40#2568C2×C311yescharacteristic, complemented
41#2548C2×C311yescharacteristic
42#2055C2×C312yescomplemented
43#2014C2×C312yes—
44#2013C2×C312yescomplemented
45#1984C2×C311yescharacteristic, complemented
46#1968C2×C312yescomplemented
47#1927C2×C312yescomplemented
48#1926C2×C312yescomplemented
49C22×C3×Q8D411yescharacteristic
50C22×C3×Q8C2312yes—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4C2312yes—
54C2×C3×SD16—24no—
55C2×C3×SD16—42no—
56C2×C3×SD16—42no—
57C2×C3×SD16—42no—
58C2×C3×SD16—42no—
59C2×C3×SD16—42no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—41no—
63C3×C4⋊Q8—41no—
64C3×#127—41no—
65C3×#127—41no—
66C3×#124—44no—
67C3×C4×Q8—41no—
68C3×C4×Q8—41no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—21no—
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C3×#106—42no—
86C3×C4⋊C8—41no—
87C3×#103—24no—
88C3×#103—42no—
89C3×#102—24no—
90C3×#102—42no—
91C3×C4×C8—41no—
92C3×#95—21no—
93C3×#95—21no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C22×SD16C22×C312yes—
99C22×SD16C22×C312yes—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×#127C22×C311yescharacteristic
102C2×#124—21no—
103C2×C4×Q8C22×C311yescharacteristic
104C2×#106C22×C312yes—
105C2×C4⋊C8C22×C311yescharacteristic
106C2×#103C22×C312yes—
107C2×#103C22×C312yes—
108C2×#102C22×C312yes—
109C2×#102C22×C312yes—
110C2×C4×C8C22×C311yescharacteristic
111#400—21no—
112#400—21no—
113#394—21no—
114#394—21no—
115#391—21no—
116#391C22×C312yes—
117#391C22×C312yes—
118#390—21no—
119#390C22×C312yes—
120#390C22×C312yes—
121#382—21no—
122C4×C4⋊C4C22×C311yescharacteristic
123C4×C4⋊C4C22×C311yescharacteristic
124#340C22×C312yes—
125#336C22×C312yes—
126#336C22×C312yes—
127C24×C3—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—41no—
135C2×C3×Q8—41no—
136C2×C3×Q8SD1614yescomplemented
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4—82no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C3×SD16—48no—
159C3×SD16—48no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8#4114yes—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—82no—
178C3×C22⋊C4—82no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—42no—
186C3×C42—42no—
187C22×Q8C3×D411yescharacteristic
188C22×Q8C23×C312yes—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C23×C312yes—
192C2×SD16—24no—
193C2×SD16—42no—
194C2×SD16—42no—
195C2×SD16—42no—
196C2×SD16—42no—
197C2×SD16—42no—
198C22×C8C23×C312yes—
199C22×C8C23×C312yes—
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202#127—41no—
203#127—41no—
204#124—44no—
205C4×Q8—41no—
206C4×Q8—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C3×D411yescharacteristic
210C2×C4⋊C4C3×D411yescharacteristic
211C2×C4⋊C4C3×D411yescharacteristic
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C23×C311yescharacteristic
221C2×C42C23×C311yescharacteristic
222C2×C42C23×C311yescharacteristic
223#106—42no—
224C4⋊C8—41no—
225#103—24no—
226#103—42no—
227#102—24no—
228#102—42no—
229C4×C8—41no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3C22×D411yescharacteristic
244C3×Q8—44no—
245C3×Q8—42no—
246C3×Q8—42no—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—42no—
253C3×D4—84no—
254C3×D4—84no—
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—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4C2×SD1612yes—
277C2×C3×C4C2×SD1612yes—
278C2×C3×C4C2×SD1612yes—
279C2×C3×C4C2×#4112yes—
280C2×C3×C4H(5,𝔽2)11yescharacteristic
281C2×C3×C4H(5,𝔽2)12yes—
282C2×C3×C4H(5,𝔽2)11yescharacteristic
283C3×C8—24no—
284C3×C8—42no—
285C24—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8C3×SD1614yescomplemented
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—82no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C24×C311yescharacteristic
316SD16—48no—
317SD16—48no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8C3×#4114yes—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—42no—
344C42—42no—
345C22×C3—82no—
346C22×C3—82no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3#53411yescharacteristic
353C22×C3#55012yes—
354C22×C3C22×SD1611yescharacteristic
355C22×C3#57511yescharacteristic
356C22×C3#57712yes—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—82no—
364C3×C4—82no—
365C3×C4—82no—
366C3×C4#53812yes—
367C3×C4#55511yescharacteristic
368C3×C4H(3,𝔽4)11yescharacteristic
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23C22×C3×D411yescharacteristic
377Q8—44no—
378Q8—42no—
379Q8—42no—
380Q8—42no—
381Q8—42no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—84no—
387D4—84no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4C2×C3×SD1612yes—
410C2×C4C2×C3×SD1612yes—
411C2×C4C2×C3×SD1612yes—
412C2×C4C2×C3×#4112yes—
413C2×C4C3×H(5,𝔽2)12yes—
414C2×C4C3×H(5,𝔽2)11yescharacteristic
415C2×C4C3×H(5,𝔽2)11yescharacteristic
416C8—24no—
417C8—42no—
418C2×C3—162no—
419C2×C3#273311yescharacteristic
420C2×C3#318711yescharacteristic
421C2×C3#320011yescharacteristic
422C2×C3#339212yes—
423C2×C3#340112yes—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22C3×#53411yescharacteristic
432C22C3×#55012yes—
433C22C22×C3×SD1611yescharacteristic
434C22C3×#57511yescharacteristic
435C22C3×#57712yes—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4C3×#53812yes—
446C4C3×#55511yescharacteristic
447C4C3×H(3,𝔽4)11yescharacteristic
448C3#3266911yescharacteristic, direct product factor, Sylow
449C2—162no—
450C2C3×#273311yescharacteristic
451C2C3×#318711yescharacteristic
452C2C3×#320011yescharacteristic
453C2C3×#339212yes—
454C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2568C2semidirect
3. C3×#2055C2semidirect
5. C3×#2013C2semidirect
6. C3×#1984C2semidirect
7. C3×#1968C2semidirect
8. C3×#1927C2semidirect
9. C3×#1926C2semidirect
10. #32669C3direct (N × H)
40. #2568C2×C3semidirect
42. #2055C2×C3semidirect
44. #2013C2×C3semidirect
45. #1984C2×C3semidirect
46. #1968C2×C3semidirect
47. #1927C2×C3semidirect
48. #1926C2×C3semidirect
136. C2×C3×Q8SD16semidirect
294. C2×Q8C3×SD16semidirect
448. C3#32669direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25682
C3×#25482
C3×#20552
C3×#20142
C3×#20132
C3×#19842
C3×#19682
C3×#19272
C3×#19262
#326693