Finite Group Databasebeta

← C3×#32972 #10651579

Subgroups — C3×#32972 #10651579

Order 768.

1210 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic
2C3×#3079C212yesmaximal, complemented
3C3×#2567C211yesmaximal, characteristic, complemented
4C3×#1859C214yesmaximal, complemented
5C3×#1857C212yesmaximal, complemented
6C3×#1806C212yesmaximal, complemented
7C3×#1801C211yesmaximal, characteristic, complemented
8C3×#1793C212yesmaximal
9#32972C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#520—22no—
13C3×#520—22no—
14C3×#520C2212yes—
15C3×#519—22no—
16C3×#519C2211yescharacteristic
17C3×C4×#41—22no—
18C3×C4×#41—22no—
19C3×C4×#41C2212yes—
20C2×C3×C4×Q8—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C3×#477—22no—
23C3×#476—22no—
24C3×#475—22no—
25C3×#474—22no—
26C3×#428C2214yes—
27C3×#423C2212yes—
28C2×C3×#104C2214yes—
29C3×#417C2214yes—
30C2×C3×#101C2212yes—
31C3×#395—22no—
32C3×#391C2212yes—
33C3×#389—22no—
34C3×#384C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38C4×2−1+4C2×C311yescharacteristic
39#3079C2×C312yescomplemented
40#2567C2×C311yescharacteristic, complemented
41#1859C2×C314yescomplemented
42#1857C2×C312yescomplemented
43#1806C2×C312yescomplemented
44#1801C2×C311yescharacteristic, complemented
45#1793C2×C312yes—
46C3×2−1+4—24no—
47C3×2−1+4—22no—
48C3×2−1+4Q814yescomplemented
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41C2312yes—
52C22×C3×Q8—22no—
53C22×C3×Q8—21no—
54C22×C3×Q8C2311yescharacteristic
55C2×C3×M16C2314yes—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4×Q8—22no—
59C3×C4×Q8—24no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8D412yes—
67C3×C4×Q8D412yes—
68C3×C4×D4—22no—
69C3×C4×D4—24no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4D412yes—
76C3×C4×D4D412yes—
77C3×#117—24no—
78C3×#117—24no—
79C3×#117—22no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117C2312yes—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4D412yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—24no—
92C2×C3×C42—22no—
93C2×C3×C42D412yes—
94C2×C3×C42C2312yes—
95C2×C3×C42C2311yescharacteristic
96C3×#107—24no—
97C3×#106—24no—
98C3×C4⋊C8—22no—
99C3×C4⋊C8—22no—
100C3×#104—28no—
101C3×#103—24no—
102C3×#102—24no—
103C3×#101—24no—
104C3×#95—21no—
105C3×#95—24no—
106C2×2−1+4C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108#520—22no—
109#520—22no—
110#520C22×C312yes—
111#519—22no—
112#519C22×C311yescharacteristic
113C4×#41—22no—
114C4×#41—22no—
115C4×#41C22×C312yes—
116C2×C4×Q8—22no—
117C2×C4×Q8C22×C311yescharacteristic
118#477—22no—
119#476—22no—
120#475—22no—
121#474—22no—
122#428C22×C314yes—
123#423C22×C312yes—
124C2×#104C22×C314yes—
125#417C22×C314yes—
126C2×#101C22×C312yes—
127#395—22no—
128#391C22×C312yes—
129#389—22no—
130#384C22×C312yes—
131C4×C4⋊C4C22×C311yescharacteristic
132#343C22×C312yes—
133#339C22×C314yes—
134C3×#41—28no—
135C3×#41—28no—
136C3×#41—28no—
137C3×#41—28no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41C2×Q814yes—
141C3×#41#4114yes—
142C2×C3×Q8—22no—
143C2×C3×Q8—28no—
144C2×C3×Q8—28no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—42no—
148C2×C3×Q8—82no—
149C2×C3×Q8C2×D412yes—
150C2×C3×Q8C2×Q812yes—
151C2×C3×Q8#4112yes—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4C2×D412yes—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C3×M16—24no—
168C3×M16—24no—
169C2×C3×C8—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—84no—
182C3×C4⋊C4C2×D412yes—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C42—24no—
189C3×C42—24no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—24no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
1982−1+4—24no—
1992−1+4—22no—
2002−1+4C3×Q814yescomplemented
201C2×#41—22no—
202C2×#41—22no—
203C2×#41C23×C312yes—
204C22×Q8—22no—
205C22×Q8—21no—
206C22×Q8C23×C311yescharacteristic
207C2×M16C23×C314yes—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4×Q8—22no—
211C4×Q8—24no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C3×D412yes—
219C4×Q8C3×D412yes—
220C4×D4—22no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—24no—
226C4×D4—22no—
227C4×D4C3×D412yes—
228C4×D4C3×D412yes—
229#117—24no—
230#117—24no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117C23×C312yes—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4C3×D412yes—
241C2×C4⋊C4C23×C312yes—
242C2×C4⋊C4C23×C311yescharacteristic
243C2×C42—24no—
244C2×C42—22no—
245C2×C42C3×D412yes—
246C2×C42C23×C312yes—
247C2×C42C23×C311yescharacteristic
248#107—24no—
249#106—24no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252#104—28no—
253#103—24no—
254#102—24no—
255#101—24no—
256#95—21no—
257#95—24no—
258C23×C3—22no—
259C23×C3C22×D411yescharacteristic
260C3×Q8—28no—
261C3×Q8—24no—
262C3×Q8—22no—
263C3×Q8—24no—
264C3×Q8—84no—
265C3×Q8C22⋊Q814yes—
266C3×D4—28no—
267C3×D4—24no—
268C3×D4—24no—
269C3×D4—22no—
270C3×D4C22⋊Q814yes—
271C2×C3×C4—216no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—44no—
286C2×C3×C4—42no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4—82no—
290C2×C3×C4—84no—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12012yes—
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4C22⋊Q814yes—
295C2×C3×C4C22×D412yes—
296C2×C3×C4C22×Q811yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×C3×C4C2×#4111yescharacteristic
299C3×C8—44no—
300#41—28no—
301#41—28no—
302#41—28no—
303#41—28no—
304#41—22no—
305#41—22no—
306#41C2×C3×Q814yes—
307#41C3×#4114yes—
308C2×Q8—22no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—42no—
314C2×Q8—82no—
315C2×Q8C2×C3×D412yes—
316C2×Q8C2×C3×Q812yes—
317C2×Q8C3×#4112yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4C2×C3×D412yes—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D412yes—
330C22×C4C2×C3×D412yes—
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C24×C311yescharacteristic
333M16—24no—
334M16—24no—
335C2×C8—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—84no—
348C4⋊C4C2×C3×D412yes—
349C4⋊C4C2×C3×D412yes—
350C22⋊C4—24no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C42—24no—
355C42—24no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42C2×C3×D412yes—
363C42C2×C3×D412yes—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—24no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3C2×#12011yescharacteristic
370C22×C3C2×C22⋊Q811yescharacteristic
371C22×C3#52511yescharacteristic
372C3×C4—24no—
373C3×C4—24no—
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—82no—
381C3×C4#45612yes—
382C3×C4C2×C22⋊Q812yes—
383C3×C4#52512yes—
384C23—22no—
385C23C22×C3×D411yescharacteristic
386Q8—28no—
387Q8—24no—
388Q8—22no—
389Q8—24no—
390Q8—84no—
391Q8C3×C22⋊Q814yes—
392D4—28no—
393D4—24no—
394D4—24no—
395D4—22no—
396D4C3×C22⋊Q814yes—
397C2×C4—216no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—84no—
417C2×C4C3×#12011yescharacteristic
418C2×C4C3×#12012yes—
419C2×C4C3×#12011yescharacteristic
420C2×C4C3×C22⋊Q814yes—
421C2×C4C22×C3×D412yes—
422C2×C4C22×C3×Q811yescharacteristic
423C2×C4C2×C3×#4112yes—
424C2×C4C2×C3×#4111yescharacteristic
425C8—44no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—44no—
429C2×C3#284811yescharacteristic
430C2×C3C2×#45611yescharacteristic
431C2×C3#302111yescharacteristic
432C22—21no—
433C22—21no—
434C22—24no—
435C22—42no—
436C22—42no—
437C22C2×C3×#12011yescharacteristic
438C22C2×C3×C22⋊Q811yescharacteristic
439C22C3×#52511yescharacteristic
440C4—21no—
441C4—22no—
442C4—24no—
443C4—24no—
444C4—44no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—82no—
449C4C3×#45612yes—
450C4C2×C3×C22⋊Q812yes—
451C4C3×#52512yes—
452C3#3297211yescharacteristic, direct product factor, Sylow
453C2—21no—
454C2—21no—
455C2—44no—
456C2C3×#284811yescharacteristic
457C2C2×C3×#45611yescharacteristic
458C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×2−1+4C22
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×#3079C2semidirect
3. C3×#2567C2semidirect
4. C3×#1859C2semidirect
5. C3×#1857C2semidirect
6. C3×#1806C2semidirect
7. C3×#1801C2semidirect
9. #32972C3direct (N × H)
39. #3079C2×C3semidirect
40. #2567C2×C3semidirect
41. #1859C2×C3semidirect
42. #1857C2×C3semidirect
43. #1806C2×C3semidirect
44. #1801C2×C3semidirect
48. C3×2−1+4Q8semidirect
200. 2−1+4C3×Q8semidirect
452. C3#32972direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C3×#30792
C3×#25672
C3×#18592
C3×#18572
C3×#18062
C3×#18012
C3×#17932
#329723