Finite Group Databasebeta

← C3×#32966 #10651573

Subgroups — C3×#32966 #10651573

Order 768.

1414 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic
2C2×C3×#478C212yesmaximal, complemented
3C3×#2629C211yesmaximal, characteristic, complemented
4C3×#1864C212yesmaximal, complemented
5C3×#1862C212yesmaximal, complemented
6C3×#1854C212yesmaximal, complemented
7C3×#1802C212yesmaximal, complemented
8C3×#1798C211yesmaximal, characteristic, complemented
9C3×#1786C212yesmaximal
10#32966C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C2×C3×#125C2211yescharacteristic, complemented
13C3×#518—28no—
14C2×C3×C4×D4—22no—
15C2×C3×C4×D4C2212yes—
16C2×C3×C4×D4C2212yes—
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C22⋊C4—22no—
20C3×#478—28no—
21C2×C3×#107C2212yescomplemented
22C2×C3×#106C2212yescomplemented
23C2×C3×C4⋊C8C2212yescomplemented
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#398—22no—
30C3×#396—22no—
31C3×#389C2212yescomplemented
32C3×#384C2211yescharacteristic, complemented
33C3×#380C2212yes—
34C3×#340C2212yescomplemented
35C3×#340C2212yescomplemented
36C3×#336C2211yescharacteristic, complemented
37C3×#336C2211yescharacteristic, complemented
38C2×#518C2×C311yescharacteristic
39C2×#478C2×C312yescomplemented
40#2629C2×C311yescharacteristic, complemented
41#1864C2×C312yescomplemented
42#1862C2×C312yescomplemented
43#1854C2×C312yescomplemented
44#1802C2×C312yescomplemented
45#1798C2×C311yescharacteristic, complemented
46#1786C2×C312yes—
47C25×C3—21no—
48C22×C3×D4—24no—
49C22×C3×D4—24no—
50C22×C3×D4—21no—
51C22×C3×D4—22no—
52C22×C3×D4Q814yes—
53C22×C3×D4C2312yes—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4C2311yescharacteristic
57C22×C3×C8C2312yes—
58C22×C3×C8C2312yes—
59C3×#125—24no—
60C3×C4×D4—216no—
61C3×C4×D4—24no—
62C3×C4×D4—24no—
63C3×C4×D4—24no—
64C3×C4×D4D418yes—
65C3×#117—24no—
66C3×#117—24no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—216no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D412yes—
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#107—24no—
84C3×#106—24no—
85C3×C4⋊C8—24no—
86C3×#102—24no—
87C3×#102—24no—
88C3×#102—24no—
89C3×#102—24no—
90C3×C22⋊C8—24no—
91C3×#95—22no—
92C3×#95—22no—
93C23×D4C22×C311yescharacteristic
94C2×#125C22×C311yescharacteristic, complemented
95#518—28no—
96C2×C4×D4—22no—
97C2×C4×D4C22×C312yes—
98C2×C4×D4C22×C312yes—
99C2×#117C22×C311yescharacteristic
100C2×#117C22×C311yescharacteristic
101C22×C22⋊C4—22no—
102#478—28no—
103C2×#107C22×C312yescomplemented
104C2×#106C22×C312yescomplemented
105C2×C4⋊C8C22×C312yescomplemented
106C2×#102C22×C312yes—
107C2×#102C22×C312yes—
108C2×#102C22×C312yes—
109C2×#102C22×C312yes—
110C2×C22⋊C8C22×C312yes—
111#398—22no—
112#396—22no—
113#389C22×C312yescomplemented
114#384C22×C311yescharacteristic, complemented
115#380C22×C312yes—
116#340C22×C312yescomplemented
117#340C22×C312yescomplemented
118#336C22×C311yescharacteristic, complemented
119#336C22×C311yescharacteristic, complemented
120C24×C3—24no—
121C24×C3—24no—
122C24×C3—22no—
123C24×C3—24no—
124C24×C3—82no—
125C24×C3—82no—
126C24×C3C2×D411yescharacteristic
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—216no—
130C2×C3×D4—216no—
131C2×C3×D4—24no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4C2×D414yes—
136C2×C3×D4C2×Q814yes—
137C2×C3×D4#4114yes—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—28no—
142C22×C3×C4—28no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—82no—
146C22×C3×C4—82no—
147C22×C3×C4—82no—
148C22×C3×C4—82no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×Q812yes—
153C22×C3×C4#4112yes—
154C22×C3×C4C2411yescharacteristic
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—44no—
168C3×C4⋊C4—82no—
169C3×C4⋊C4—82no—
170C3×C4⋊C4C2×D414yes—
171C3×C22⋊C4—24no—
172C3×C22⋊C4—28no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—82no—
176C3×C42—24no—
177C3×C42—22no—
178C3×C42C2×D414yes—
179C25—21no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4C3×Q814yes—
185C22×D4C23×C312yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4C23×C311yescharacteristic
189C22×C8C23×C312yes—
190C22×C8C23×C312yes—
191#125—24no—
192C4×D4—216no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4C3×D418yes—
197#117—24no—
198#117—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C3×D412yes—
203C2×C4⋊C4C23×C312yes—
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C22⋊C4—216no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C3×D412yes—
212C2×C42—21no—
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#107—24no—
216#106—24no—
217C4⋊C8—24no—
218#102—24no—
219#102—24no—
220#102—24no—
221#102—24no—
222C22⋊C8—24no—
223#95—22no—
224#95—22no—
225C23×C3—216no—
226C23×C3—21no—
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—21no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—88no—
238C23×C3—84no—
239C23×C3—84no—
240C23×C3—88no—
241C23×C3C22⋊Q812yes—
242C23×C3C22×D411yescharacteristic
243C3×D4—216no—
244C3×D4—24no—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4C22⋊Q818yes—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—88no—
267C2×C3×C4—88no—
268C2×C3×C4#12014yes—
269C2×C3×C4C22⋊Q812yes—
270C2×C3×C4C22×D412yes—
271C2×C3×C4C22×Q811yescharacteristic
272C2×C3×C4C2×#4112yes—
273C2×C3×C4C2×#4111yescharacteristic
274C3×C8—42no—
275C3×C8—42no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—82no—
281C24—82no—
282C24C2×C3×D411yescharacteristic
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—216no—
286C2×D4—216no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4C2×C3×D414yes—
292C2×D4C2×C3×Q814yes—
293C2×D4C3×#4114yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4C2×C3×D412yes—
306C22×C4C2×C3×D412yes—
307C22×C4C2×C3×D411yescharacteristic
308C22×C4C2×C3×Q812yes—
309C22×C4C3×#4112yes—
310C22×C4C24×C311yescharacteristic
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—44no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4C2×C3×D414yes—
327C22⋊C4—24no—
328C22⋊C4—28no—
329C22⋊C4—24no—
330C22⋊C4—28no—
331C22⋊C4—82no—
332C42—24no—
333C42—22no—
334C42C2×C3×D414yes—
335C22×C3—24no—
336C22×C3—22no—
337C22×C3—28no—
338C22×C3—44no—
339C22×C3—44no—
340C22×C3—41no—
341C22×C3—44no—
342C22×C3—41no—
343C22×C3—44no—
344C22×C3—44no—
345C22×C3—44no—
346C22×C3—42no—
347C22×C3—816no—
348C22×C3C2×#12011yescharacteristic
349C22×C3C2×C22⋊Q811yescharacteristic
350C22×C3#52511yescharacteristic
351C22×C3#57612yes—
352C22×C3#57712yes—
353C3×C4—22no—
354C3×C4—24no—
355C3×C4—42no—
356C3×C4—44no—
357C3×C4—82no—
358C3×C4—82no—
359C3×C4C2×C22⋊Q812yes—
360C3×C4#52512yes—
361C23—216no—
362C23—22no—
363C23—22no—
364C23—21no—
365C23—24no—
366C23—24no—
367C23—21no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—88no—
374C23—88no—
375C23—84no—
376C23—84no—
377C23C3×C22⋊Q812yes—
378C23C22×C3×D411yescharacteristic
379D4—216no—
380D4—24no—
381D4—24no—
382D4—24no—
383D4C3×C22⋊Q818yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—88no—
403C2×C4—88no—
404C2×C4C3×#12014yes—
405C2×C4C3×C22⋊Q812yes—
406C2×C4C22×C3×D412yes—
407C2×C4C22×C3×Q811yescharacteristic
408C2×C4C2×C3×#4112yes—
409C2×C4C2×C3×#4111yescharacteristic
410C8—42no—
411C8—42no—
412C2×C3—42no—
413C2×C3—48no—
414C2×C3#284811yescharacteristic
415C2×C3#301212yes—
416C2×C3#308112yes—
417C2×C3#312211yescharacteristic
418C2×C3#312311yescharacteristic
419C22—24no—
420C22—22no—
421C22—28no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—44no—
426C22—44no—
427C22—41no—
428C22—44no—
429C22—41no—
430C22—44no—
431C22—816no—
432C22C2×C3×#12011yescharacteristic
433C22C2×C3×C22⋊Q811yescharacteristic
434C22C3×#52511yescharacteristic
435C22C3×#57612yes—
436C22C3×#57712yes—
437C4—24no—
438C4—22no—
439C4—42no—
440C4—44no—
441C4—82no—
442C4—82no—
443C4C2×C3×C22⋊Q812yes—
444C4C3×#52512yes—
445C3#3296611yescharacteristic, direct product factor, Sylow
446C2—42no—
447C2—48no—
448C2C3×#284811yescharacteristic
449C2C3×#301212yes—
450C2C3×#308112yes—
451C2C3×#312211yescharacteristic
452C2C3×#312311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#478C2semidirect
3. C3×#2629C2semidirect
4. C3×#1864C2semidirect
5. C3×#1862C2semidirect
6. C3×#1854C2semidirect
7. C3×#1802C2semidirect
8. C3×#1798C2semidirect
10. #32966C3direct (N × H)
12. C2×C3×#125C22semidirect
21. C2×C3×#107C22semidirect
22. C2×C3×#106C22semidirect
23. C2×C3×C4⋊C8C22semidirect
31. C3×#389C22semidirect
32. C3×#384C22semidirect
34. C3×#340C22semidirect
35. C3×#340C22semidirect
36. C3×#336C22semidirect
37. C3×#336C22semidirect
39. C2×#478C2×C3semidirect
40. #2629C2×C3semidirect
41. #1864C2×C3semidirect
42. #1862C2×C3semidirect
43. #1854C2×C3semidirect
44. #1802C2×C3semidirect
45. #1798C2×C3semidirect
94. C2×#125C22×C3semidirect
103. C2×#107C22×C3semidirect
104. C2×#106C22×C3semidirect
105. C2×C4⋊C8C22×C3semidirect
113. #389C22×C3semidirect
114. #384C22×C3semidirect
116. #340C22×C3semidirect
117. #340C22×C3semidirect
118. #336C22×C3semidirect
119. #336C22×C3semidirect
445. C3#32966direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×C3×#4782
C3×#26292
C3×#18642
C3×#18622
C3×#18542
C3×#18022
C3×#17982
C3×#17862
#329663