Finite Group Databasebeta

← C3×#33530 #10652137

Subgroups — C3×#33530 #10652137

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#575C211yesmaximal, characteristic, complemented
2C3×#3414C212yesmaximal, complemented
3C3×#3054C211yesmaximal, characteristic, complemented
4C3×#2191C214yesmaximal, complemented
5C3×#2190C214yesmaximal, complemented
6C3×#2145C212yesmaximal, complemented
7C2×C3×#349C211yesmaximal, characteristic, complemented
8#33530C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41—21no—
10C3×#575—24no—
11C3×#575—22no—
12C3×#575C2214yes—
13C2×C3×#137—22no—
14C2×C3×AGL(1,ℤ/8)—22no—
15C2×C3×#135—21no—
16C2×C3×#135C2212yes—
17C2×C3×#135C2211yescharacteristic, complemented
18C22×C3×M16C2211yescharacteristic
19C3×#525—21no—
20C3×#510—22no—
21C3×#510C2214yescomplemented
22C3×#509—22no—
23C3×#509C2214yescomplemented
24C3×#508—24no—
25C2×C3×M32C2212yescomplemented
26C3×#470C2212yescomplemented
27C3×#469C2212yescomplemented
28C3×#468—22no—
29C3×#427C2211yescharacteristic, complemented
30C3×#416—21no—
31C3×#360C2218yescomplemented
32C3×#349C2214yescomplemented
33C2×#575C2×C311yescharacteristic, complemented
34#3414C2×C312yescomplemented
35#3054C2×C311yescharacteristic, complemented
36#2191C2×C314yescomplemented
37#2190C2×C314yescomplemented
38#2145C2×C312yescomplemented
39C2×#349C2×C311yescharacteristic, complemented
40C2×C3×#41—22no—
41C2×C3×#41—24no—
42C2×C3×#41—24no—
43C2×C3×#41—22no—
44C2×C3×#41—21no—
45C2×C3×#41—41no—
46C2×C3×#41—42no—
47C2×C3×#41—42no—
48C2×C3×#41—41no—
49C22×C3×Q8—21no—
50C22×C3×D4—21no—
51C22×C3×D4—41no—
52C23×C3×C4—41no—
53C23×C3×C4D411yescharacteristic
54C3×#137—28no—
55C3×#137—28no—
56C3×AGL(1,ℤ/8)—28no—
57C3×AGL(1,ℤ/8)—28no—
58C3×#135—24no—
59C3×#135—24no—
60C3×#135—22no—
61C3×#135—22no—
62C3×#135—22no—
63C3×#135—22no—
64C3×#135D414yescomplemented
65C3×#135C2314yes—
66C2×C3×Q16—21no—
67C2×C3×Q16D412yescomplemented
68C2×C3×Q16C2311yescharacteristic, complemented
69C2×C3×SD16—22no—
70C2×C3×SD16—22no—
71C2×C3×SD16—21no—
72C2×C3×D8—21no—
73C2×C3×D8D412yescomplemented
74C2×C3×D8C2311yescharacteristic, complemented
75C2×C3×M16—22no—
76C2×C3×M16—22no—
77C2×C3×M16D412yes—
78C2×C3×M16C2312yes—
79C22×C3×C8D411yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#123—41no—
82C3×C22⋊Q8—22no—
83C3×#121—22no—
84C3×#120—41no—
85C3×C4×D4—42no—
86C3×#117—21no—
87C3×Q32—24no—
88C3×SD32—24no—
89C3×SD32—24no—
90C3×D16—24no—
91C3×M32—22no—
92C3×M32C2314yescomplemented
93C2×C3×C16—22no—
94C3×#107—21no—
95C3×#106C2312yescomplemented
96C3×#103—22no—
97C3×#102—22no—
98C22×#41—21no—
99#575—24no—
100#575—22no—
101#575C22×C314yes—
102C2×#137—22no—
103C2×AGL(1,ℤ/8)—22no—
104C2×#135—21no—
105C2×#135C22×C312yes—
106C2×#135C22×C311yescharacteristic, complemented
107C22×M16C22×C311yescharacteristic
108#525—21no—
109#510—22no—
110#510C22×C314yescomplemented
111#509—22no—
112#509C22×C314yescomplemented
113#508—24no—
114C2×M32C22×C312yescomplemented
115#470C22×C312yescomplemented
116#469C22×C312yescomplemented
117#468—22no—
118#427C22×C311yescharacteristic, complemented
119#416—21no—
120#360C22×C318yescomplemented
121#349C22×C314yescomplemented
122C24×C3—41no—
123C24×C3D811yescharacteristic
124C3×#41—28no—
125C3×#41—24no—
126C3×#41—24no—
127C3×#41—44no—
128C3×#41—44no—
129C3×#41—44no—
130C3×#41—44no—
131C3×#41—44no—
132C3×#41—42no—
133C3×#41—42no—
134C3×#41—41no—
135C3×#41—41no—
136C3×#41—41no—
137C3×#41—41no—
138C2×C3×Q8—24no—
139C2×C3×Q8—22no—
140C2×C3×Q8—21no—
141C2×C3×Q8—41no—
142C2×C3×Q8—42no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—41no—
147C2×C3×D4—42no—
148C2×C3×D4—44no—
149C2×C3×D4—42no—
150C2×C3×D4—44no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—41no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—42no—
159C22×C3×C4—44no—
160C22×C3×C4—44no—
161C22×C3×C4—41no—
162C22×C3×C4—42no—
163C22×C3×C4—41no—
164C22×C3×C4D812yes—
165C22×C3×C4D811yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C3×Q16—24no—
169C3×Q16—22no—
170C3×Q16—22no—
171C3×Q16C2×D414yes—
172C3×SD16—28no—
173C3×SD16—24no—
174C3×SD16—24no—
175C3×SD16—42no—
176C3×D8—24no—
177C3×D8—22no—
178C3×D8—22no—
179C3×D8C2×D414yes—
180C3×M16—24no—
181C3×M16—24no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8C2×D412yes—
185C2×C3×C8C2×D411yescharacteristic
186C2×C3×C8C2×D412yes—
187C2×C3×C8C2×D412yes—
188C2×C3×C8C2411yescharacteristic
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C22⋊C4—41no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C42—41no—
195C3×C16—24no—
196C2×#41—22no—
197C2×#41—24no—
198C2×#41—22no—
199C2×#41—24no—
200C2×#41—21no—
201C2×#41—41no—
202C2×#41—42no—
203C2×#41—42no—
204C2×#41—41no—
205C22×Q8—21no—
206C22×D4—21no—
207C22×D4—41no—
208C23×C4—41no—
209C23×C4C3×D411yescharacteristic
210#137—28no—
211#137—28no—
212AGL(1,ℤ/8)—28no—
213AGL(1,ℤ/8)—28no—
214#135—24no—
215#135—24no—
216#135—22no—
217#135—22no—
218#135—22no—
219#135—22no—
220#135C3×D414yescomplemented
221#135C23×C314yes—
222C2×Q16—21no—
223C2×Q16C3×D412yescomplemented
224C2×Q16C23×C311yescharacteristic, complemented
225C2×SD16—22no—
226C2×SD16—22no—
227C2×SD16—21no—
228C2×D8—21no—
229C2×D8C3×D412yescomplemented
230C2×D8C23×C311yescharacteristic, complemented
231C2×M16—22no—
232C2×M16—22no—
233C2×M16C3×D412yes—
234C2×M16C23×C312yes—
235C22×C8C3×D411yescharacteristic
236C22×C8C23×C311yescharacteristic
237#123—41no—
238C22⋊Q8—22no—
239#121—22no—
240#120—41no—
241C4×D4—42no—
242#117—21no—
243Q32—24no—
244SD32—24no—
245SD32—24no—
246D16—24no—
247M32—22no—
248M32C23×C314yescomplemented
249C2×C16—22no—
250#107—21no—
251#106C23×C312yescomplemented
252#103—22no—
253#102—22no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—44no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—81no—
260C23×C3—84no—
261C23×C3C2×D812yes—
262C23×C3C2×D811yescharacteristic
263C3×Q8—24no—
264C3×Q8—44no—
265C3×Q8—42no—
266C3×Q8—41no—
267C3×Q8—41no—
268C3×D4—24no—
269C3×D4—44no—
270C3×D4—48no—
271C3×D4—44no—
272C3×D4—44no—
273C3×D4—41no—
274C3×D4—41no—
275C3×D4—42no—
276C3×D4—84no—
277C3×D4—81no—
278C3×D4—81no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—48no—
286C2×C3×C4—44no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—81no—
293C2×C3×C4—81no—
294C2×C3×C4#12011yescharacteristic
295C2×C3×C4#12011yescharacteristic
296C2×C3×C4C2×D812yes—
297C2×C3×C4C2×D811yescharacteristic
298C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
299C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C3×C8—22no—
302C3×C8#12012yes—
303C3×C8C22×D412yes—
304C24—41no—
305C24C3×D811yescharacteristic
306#41—28no—
307#41—24no—
308#41—24no—
309#41—44no—
310#41—44no—
311#41—44no—
312#41—44no—
313#41—44no—
314#41—42no—
315#41—42no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—42no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4C3×D812yes—
347C22×C4C3×D811yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D411yescharacteristic
350Q16—24no—
351Q16—22no—
352Q16—22no—
353Q16C2×C3×D414yes—
354SD16—28no—
355SD16—24no—
356SD16—24no—
357SD16—42no—
358D8—24no—
359D8—22no—
360D8—22no—
361D8C2×C3×D414yes—
362M16—24no—
363M16—24no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8C2×C3×D412yes—
367C2×C8C2×C3×D411yescharacteristic
368C2×C8C2×C3×D412yes—
369C2×C8C2×C3×D412yes—
370C2×C8C24×C311yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C42—41no—
377C16—24no—
378C22×C3—22no—
379C22×C3—41no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—44no—
383C22×C3—81no—
384C22×C3—84no—
385C22×C3—82no—
386C22×C3—88no—
387C22×C3—81no—
388C22×C3#44711yescharacteristic
389C22×C3#44711yescharacteristic
390C22×C3C22×D811yescharacteristic
391C3×C4—22no—
392C3×C4—44no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4#44711yescharacteristic
396C3×C4#44711yescharacteristic
397C3×C4C2×#12011yescharacteristic
398C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
399C23—21no—
400C23—21no—
401C23—44no—
402C23—42no—
403C23—41no—
404C23—81no—
405C23—84no—
406C23C2×C3×D812yes—
407C23C2×C3×D811yescharacteristic
408Q8—24no—
409Q8—44no—
410Q8—42no—
411Q8—41no—
412Q8—41no—
413D4—24no—
414D4—44no—
415D4—48no—
416D4—44no—
417D4—44no—
418D4—41no—
419D4—42no—
420D4—41no—
421D4—84no—
422D4—81no—
423D4—81no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4C3×#12011yescharacteristic
440C2×C4C3×#12011yescharacteristic
441C2×C4C2×C3×D812yes—
442C2×C4C2×C3×D811yescharacteristic
443C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
445C2×C4C22×C3×D411yescharacteristic
446C8—22no—
447C8C3×#12012yes—
448C8C22×C3×D412yes—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—42no—
453C2×C3—84no—
454C2×C3—161no—
455C2×C3C2×#44711yescharacteristic
456C22—22no—
457C22—41no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—81no—
462C22—84no—
463C22—82no—
464C22—88no—
465C22—81no—
466C22C3×#44711yescharacteristic
467C22C3×#44711yescharacteristic
468C22C22×C3×D811yescharacteristic
469C4—22no—
470C4—44no—
471C4—82no—
472C4—81no—
473C4C3×#44711yescharacteristic
474C4C3×#44711yescharacteristic
475C4C2×C3×#12011yescharacteristic
476C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
477C3#3353011yescharacteristic, direct product factor, Sylow
478C2—21no—
479C2—21no—
480C2—21no—
481C2—42no—
482C2—84no—
483C2—161no—
484C2C2×C3×#44711yescharacteristic
subgroupG/H
centerC3×C4#447
commutator, Frattini, cosocleC2×C8C24×C3
socleC2×C3C2×#447
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#575C2semidirect
2. C3×#3414C2semidirect
3. C3×#3054C2semidirect
4. C3×#2191C2semidirect
5. C3×#2190C2semidirect
6. C3×#2145C2semidirect
7. C2×C3×#349C2semidirect
8. #33530C3direct (N × H)
17. C2×C3×#135C22semidirect
21. C3×#510C22semidirect
23. C3×#509C22semidirect
25. C2×C3×M32C22semidirect
26. C3×#470C22semidirect
27. C3×#469C22semidirect
29. C3×#427C22semidirect
31. C3×#360C22semidirect
32. C3×#349C22semidirect
33. C2×#575C2×C3semidirect
34. #3414C2×C3semidirect
35. #3054C2×C3semidirect
36. #2191C2×C3semidirect
37. #2190C2×C3semidirect
38. #2145C2×C3semidirect
39. C2×#349C2×C3semidirect
64. C3×#135D4semidirect
67. C2×C3×Q16D4semidirect
68. C2×C3×Q16C23semidirect
73. C2×C3×D8D4semidirect
74. C2×C3×D8C23semidirect
92. C3×M32C23semidirect
95. C3×#106C23semidirect
106. C2×#135C22×C3semidirect
110. #510C22×C3semidirect
112. #509C22×C3semidirect
114. C2×M32C22×C3semidirect
115. #470C22×C3semidirect
116. #469C22×C3semidirect
118. #427C22×C3semidirect
120. #360C22×C3semidirect
121. #349C22×C3semidirect
220. #135C3×D4semidirect
223. C2×Q16C3×D4semidirect
224. C2×Q16C23×C3semidirect
229. C2×D8C3×D4semidirect
230. C2×D8C23×C3semidirect
248. M32C23×C3semidirect
251. #106C23×C3semidirect
477. C3#33530direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5752
C3×#34142
C3×#30542
C3×#21912
C3×#21902
C3×#21452
C2×C3×#3492
#335303