Finite Group Databasebeta

← C3×#32995 #10651602

Subgroups — C3×#32995 #10651602

Order 768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic
2C3×#3095C212yesmaximal, complemented
3C3×#2432C211yesmaximal, characteristic
4C3×#1886C212yesmaximal, complemented
5C3×#1885C212yesmaximal, complemented
6C3×#1857C212yesmaximal, complemented
7C3×#1856C212yesmaximal, complemented
8C3×#1826C212yesmaximal, complemented
9C3×#1785C211yesmaximal, characteristic
10#32995C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C3×#520—22no—
16C3×#520—22no—
17C3×#519C2212yes—
18C3×#518—22no—
19C3×#518C2212yes—
20C3×C4×#41—24no—
21C2×C3×#117—22no—
22C2×C3×#117C2212yes—
23C3×#485—24no—
24C3×#484—22no—
25C3×#483—22no—
26C3×#428C2214yes—
27C3×#419C2214yes—
28C2×C3×#101C2212yes—
29C2×C3×#100C2212yes—
30C3×#407C2212yes—
31C3×#394—22no—
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#385—22no—
35C3×C4×C22⋊C4C2211yescharacteristic
36C3×#339C2214yes—
37C3×#339C2214yes—
38#3428C2×C311yescharacteristic
39#3095C2×C312yescomplemented
40#2432C2×C311yescharacteristic
41#1886C2×C312yescomplemented
42#1885C2×C312yescomplemented
43#1857C2×C312yescomplemented
44#1856C2×C312yescomplemented
45#1826C2×C312yescomplemented
46#1785C2×C311yescharacteristic
47C2×C3×#41—24no—
48C2×C3×#41—24no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C22×C3×Q8C2311yescharacteristic
52C22×C3×Q8C2311yescharacteristic
53C22×C3×D4D412yes—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4D412yes—
56C23×C3×C4C2311yescharacteristic
57C2×C3×M16C2314yes—
58C3×C4⋊Q8—24no—
59C3×#124—24no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—22no—
62C3×C4×Q8—24no—
63C3×C4×Q8—24no—
64C3×C4×D4—28no—
65C3×C4×D4—28no—
66C3×C4×D4—24no—
67C3×C4×D4—24no—
68C3×#117—24no—
69C3×#117—24no—
70C3×#117—22no—
71C3×#117—24no—
72C3×#117—22no—
73C3×#117D414yes—
74C3×#117C2314yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—24no—
89C2×C3×C42—22no—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—24no—
93C3×#106—24no—
94C3×#103—24no—
95C3×#103—24no—
96C3×#101—24no—
97C3×#100—24no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×#95—21no—
101C3×#95—22no—
102C22×#41C22×C311yescharacteristic
103C2×C4⋊Q8C22×C311yescharacteristic
104C2×#124C22×C311yescharacteristic
105C2×C22⋊Q8C22×C311yescharacteristic
106#520—22no—
107#520—22no—
108#519C22×C312yes—
109#518—22no—
110#518C22×C312yes—
111C4×#41—24no—
112C2×#117—22no—
113C2×#117C22×C312yes—
114#485—24no—
115#484—22no—
116#483—22no—
117#428C22×C314yes—
118#419C22×C314yes—
119C2×#101C22×C312yes—
120C2×#100C22×C312yes—
121#407C22×C312yes—
122#394—22no—
123#391C22×C311yescharacteristic
124#390C22×C311yescharacteristic
125#385—22no—
126C4×C22⋊C4C22×C311yescharacteristic
127#339C22×C314yes—
128#339C22×C314yes—
129C24×C3C2×D411yescharacteristic
130C24×C3C2×D412yes—
131C3×#41—216no—
132C3×#41—216no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8#4114yes—
139C2×C3×D4—24no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4#4114yes—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2411yescharacteristic
164C3×M16—24no—
165C3×M16—24no—
166C2×C3×C8—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4C2×D414yes—
176C3×C4⋊C4C2×D414yes—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—44no—
184C3×C42—28no—
185C3×C42—24no—
186C3×C42—24no—
187C3×C42—22no—
188C3×C42—22no—
189C2×#41—24no—
190C2×#41—24no—
191C2×#41—22no—
192C2×#41—22no—
193C22×Q8C23×C311yescharacteristic
194C22×Q8C23×C311yescharacteristic
195C22×D4C3×D412yes—
196C22×D4C23×C311yescharacteristic
197C23×C4C3×D412yes—
198C23×C4C23×C311yescharacteristic
199C2×M16C23×C314yes—
200C4⋊Q8—24no—
201#124—24no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—24no—
209C4×D4—24no—
210#117—24no—
211#117—24no—
212#117—22no—
213#117—24no—
214#117—22no—
215#117C3×D414yes—
216#117C23×C314yes—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C3×D412yes—
222C2×C4⋊C4C3×D412yes—
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42C23×C311yescharacteristic
233C2×C42C23×C311yescharacteristic
234#107—24no—
235#106—24no—
236#103—24no—
237#103—24no—
238#101—24no—
239#100—24no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242#95—21no—
243#95—22no—
244C23×C3—22no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3#12012yes—
253C23×C3#12314yes—
254C23×C3C22×D411yescharacteristic
255C3×Q8—28no—
256C3×Q8—44no—
257C3×D4—216no—
258C3×D4—28no—
259C2×C3×C4—28no—
260C2×C3×C4—28no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—82no—
281C2×C3×C4—82no—
282C2×C3×C4—84no—
283C2×C3×C4—84no—
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4#12011yescharacteristic
286C2×C3×C4#12314yes—
287C2×C3×C4C22×D412yes—
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4112yes—
290C3×C8—44no—
291C24C2×C3×D411yescharacteristic
292C24C2×C3×D412yes—
293#41—216no—
294#41—216no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8C3×#4114yes—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4C3×#4114yes—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D412yes—
324C22×C4C2×C3×D412yes—
325C22×C4C24×C311yescharacteristic
326M16—24no—
327M16—24no—
328C2×C8—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4C2×C3×D414yes—
338C4⋊C4C2×C3×D414yes—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C42—28no—
347C42—24no—
348C42—24no—
349C42—22no—
350C42—22no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—22no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—44no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3—81no—
361C22×C3—82no—
362C22×C3—81no—
363C22×C3C2×#12011yescharacteristic
364C22×C3C2×#12311yescharacteristic
365C22×C3C2×#12311yescharacteristic
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—24no—
369C3×C4—42no—
370C3×C4—44no—
371C3×C4—44no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4—81no—
375C3×C4#52514yes—
376C23—22no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23C3×#12012yes—
385C23C3×#12314yes—
386C23C22×C3×D411yescharacteristic
387Q8—28no—
388Q8—44no—
389D4—216no—
390D4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4C3×#12011yescharacteristic
417C2×C4C3×#12011yescharacteristic
418C2×C4C3×#12314yes—
419C2×C4C22×C3×D412yes—
420C2×C4C2×C3×#4112yes—
421C2×C4C2×C3×#4112yes—
422C8—44no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—42no—
426C2×C3—44no—
427C2×C3#284911yescharacteristic
428C2×C3#301811yescharacteristic
429C2×C3#302211yescharacteristic
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—24no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—82no—
441C22—81no—
442C22C2×C3×#12011yescharacteristic
443C22C2×C3×#12311yescharacteristic
444C22C2×C3×#12311yescharacteristic
445C4—21no—
446C4—21no—
447C4—24no—
448C4—42no—
449C4—44no—
450C4—44no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4C3×#52514yes—
455C3#3299511yescharacteristic, direct product factor, Sylow
456C2—21no—
457C2—21no—
458C2—42no—
459C2—44no—
460C2C3×#284911yescharacteristic
461C2C3×#301811yescharacteristic
462C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3095C2semidirect
4. C3×#1886C2semidirect
5. C3×#1885C2semidirect
6. C3×#1857C2semidirect
7. C3×#1856C2semidirect
8. C3×#1826C2semidirect
10. #32995C3direct (N × H)
39. #3095C2×C3semidirect
41. #1886C2×C3semidirect
42. #1885C2×C3semidirect
43. #1857C2×C3semidirect
44. #1856C2×C3semidirect
45. #1826C2×C3semidirect
455. C3#32995direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34282
C3×#30952
C3×#24322
C3×#18862
C3×#18852
C3×#18572
C3×#18562
C3×#18262
C3×#17852
#329953