Finite Group Databasebeta

← C3×#29681 #10648288

Subgroups — C3×#29681 #10648288

Order 768.

1038 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×#3461C211yesmaximal, characteristic
2C3×#3039C212yesmaximal, complemented
3C3×#2546C211yesmaximal, characteristic, complemented
4C3×#2028C212yesmaximal, complemented
5C3×#2007C212yesmaximal
6C3×#2005C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1801C211yesmaximal, characteristic, complemented
9#29681C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×#137C2214yes—
11C3×#544C2211yescharacteristic
12C3×#542—22no—
13C3×#541—22no—
14C3×#539—22no—
15C3×#537C2211yescharacteristic
16C3×#536C2212yes—
17C3×#533—22no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C3×#529—22no—
21C3×#529—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C3×#462—22no—
25C3×#462—22no—
26C3×#461—22no—
27C3×#461—22no—
28C3×#423C2212yes—
29C2×C3×#104C2214yes—
30C3×#419C2214yes—
31C2×C3×#101C2212yes—
32C3×#398—22no—
33C3×#391C2212yes—
34C3×#389—22no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3461C2×C311yescharacteristic
39#3039C2×C312yescomplemented
40#2546C2×C311yescharacteristic, complemented
41#2028C2×C312yescomplemented
42#2007C2×C312yes—
43#2005C2×C314yescomplemented
44#1806C2×C312yescomplemented
45#1801C2×C311yescharacteristic, complemented
46C2×C3×#41C2312yes—
47C22×C3×Q8C2312yes—
48C22×C3×Q8C2311yescharacteristic
49C3×#137—44no—
50C3×#137—44no—
51C2×C3×Q16—24no—
52C2×C3×SD16—24no—
53C2×C3×M16C2314yes—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—44no—
59C3×#126—24no—
60C3×#126—24no—
61C3×#125—22no—
62C3×#125—24no—
63C3×#125—22no—
64C3×#124—22no—
65C3×#124—22no—
66C3×#123—24no—
67C3×#123—24no—
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×#121—22no—
74C3×#121—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8D412yes—
79C3×C4×Q8D412yes—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4D412yes—
84C3×C4×D4D412yes—
85C3×#117—24no—
86C3×#117—22no—
87C3×#117C2312yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D412yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—22no—
95C2×C3×C42D412yes—
96C2×C3×C42C2312yes—
97C2×C3×C42C2311yescharacteristic
98C3×C4⋊C8—22no—
99C3×C4⋊C8—22no—
100C3×#104—44no—
101C3×#103—24no—
102C3×#103—24no—
103C3×#101—42no—
104C3×#95—21no—
105C3×#95—24no—
106C2×#137C22×C314yes—
107#544C22×C311yescharacteristic
108#542—22no—
109#541—22no—
110#539—22no—
111#537C22×C311yescharacteristic
112#536C22×C312yes—
113#533—22no—
114C2×C4⋊Q8C22×C311yescharacteristic
115C2×C4⋊Q8C22×C312yes—
116#529—22no—
117#529—22no—
118C4×#41C22×C312yes—
119C2×C4×Q8C22×C311yescharacteristic
120#462—22no—
121#462—22no—
122#461—22no—
123#461—22no—
124#423C22×C312yes—
125C2×#104C22×C314yes—
126#419C22×C314yes—
127C2×#101C22×C312yes—
128#398—22no—
129#391C22×C312yes—
130#389—22no—
131C4×C4⋊C4C22×C311yescharacteristic
132#343C22×C312yes—
133#339C22×C314yes—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—21no—
143C2×C3×Q8—24no—
144C2×C3×Q8—24no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8C2×D412yes—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C3×Q16—48no—
162C3×SD16—48no—
163C3×M16—24no—
164C3×M16—24no—
165C2×C3×C8—24no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—48no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4C2×D412yes—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—24no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196C2×#41C23×C312yes—
197C22×Q8C23×C312yes—
198C22×Q8C23×C311yescharacteristic
199#137—44no—
200#137—44no—
201C2×Q16—24no—
202C2×SD16—24no—
203C2×M16C23×C314yes—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—44no—
209#126—24no—
210#126—24no—
211#125—22no—
212#125—24no—
213#125—22no—
214#124—22no—
215#124—22no—
216#123—24no—
217#123—24no—
218C22⋊Q8—24no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223#121—22no—
224#121—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8C3×D412yes—
229C4×Q8C3×D412yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4C3×D412yes—
234C4×D4C3×D412yes—
235#117—24no—
236#117—22no—
237#117C23×C312yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C3×D412yes—
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C42—22no—
245C2×C42C3×D412yes—
246C2×C42C23×C312yes—
247C2×C42C23×C311yescharacteristic
248C4⋊C8—22no—
249C4⋊C8—22no—
250#104—44no—
251#103—24no—
252#103—24no—
253#101—42no—
254#95—21no—
255#95—24no—
256C23×C3—22no—
257C23×C3C22×D411yescharacteristic
258C3×Q8—22no—
259C3×Q8—22no—
260C3×Q8—42no—
261C3×Q8—42no—
262C3×Q8—44no—
263C3×Q8—44no—
264C3×Q8—82no—
265C3×D4—22no—
266C3×D4—22no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—82no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—44no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—82no—
290C2×C3×C4#12011yescharacteristic
291C2×C3×C4#12012yes—
292C2×C3×C4#12011yescharacteristic
293C2×C3×C4C22×D412yes—
294C2×C3×C4H(5,𝔽2)11yescharacteristic
295C2×C3×C4H(5,𝔽2)12yes—
296C2×C3×C42−1+411yescharacteristic
297C3×C8—44no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—21no—
305C2×Q8—22no—
306C2×Q8—21no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8C2×C3×D412yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4C2×C3×D412yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C24×C311yescharacteristic
325Q16—48no—
326SD16—48no—
327M16—24no—
328M16—24no—
329C2×C8—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—48no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4C2×C3×D412yes—
345C4⋊C4C2×C3×D412yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42C2×C3×D412yes—
359C42C2×C3×D412yes—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3#53711yescharacteristic
368C3×C4—21no—
369C3×C4—22no—
370C3×C4—44no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4—42no—
376C3×C4—82no—
377C3×C4#45612yes—
378C3×C4#53512yes—
379C3×C4#53712yes—
380C23—22no—
381C23C22×C3×D411yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—42no—
385Q8—42no—
386Q8—44no—
387Q8—44no—
388Q8—82no—
389D4—22no—
390D4—22no—
391D4—42no—
392D4—42no—
393D4—82no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4C3×#12011yescharacteristic
415C2×C4C3×#12012yes—
416C2×C4C3×#12011yescharacteristic
417C2×C4C22×C3×D412yes—
418C2×C4C3×H(5,𝔽2)11yescharacteristic
419C2×C4C3×H(5,𝔽2)12yes—
420C2×C4C3×2−1+411yescharacteristic
421C8—44no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—82no—
425C2×C3#261511yescharacteristic
426C2×C3C2×#45611yescharacteristic
427C2×C3#302211yescharacteristic
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22C2×C3×#12011yescharacteristic
434C22C3×#53511yescharacteristic
435C22C3×#53711yescharacteristic
436C4—21no—
437C4—22no—
438C4—44no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—82no—
445C4C3×#45612yes—
446C4C3×#53512yes—
447C4C3×#53712yes—
448C3#2968111yescharacteristic, direct product factor, Sylow
449C2—21no—
450C2—21no—
451C2—82no—
452C2C3×#261511yescharacteristic
453C2C2×C3×#45611yescharacteristic
454C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#537C22
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×#3039C2semidirect
3. C3×#2546C2semidirect
4. C3×#2028C2semidirect
6. C3×#2005C2semidirect
7. C3×#1806C2semidirect
8. C3×#1801C2semidirect
9. #29681C3direct (N × H)
39. #3039C2×C3semidirect
40. #2546C2×C3semidirect
41. #2028C2×C3semidirect
43. #2005C2×C3semidirect
44. #1806C2×C3semidirect
45. #1801C2×C3semidirect
448. C3#29681direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×#30392
C3×#25462
C3×#20282
C3×#20072
C3×#20052
C3×#18062
C3×#18012
#296813