Finite Group Databasebeta

← C3×#33571 #10652178

Subgroups — C3×#33571 #10652178

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3550C211yesmaximal, characteristic, complemented
2C3×#3022C212yesmaximal, complemented
3C3×#3018C211yesmaximal, characteristic
4C3×#2205C214yesmaximal, complemented
5C3×#2203C214yesmaximal, complemented
6C3×#2135C212yesmaximal, complemented
7C3×#2132C211yesmaximal, characteristic, complemented
8#33571C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#585C2211yescharacteristic
10C3×#575—21no—
11C2×C3×#137—22no—
12C3×#563—21no—
13C3×#563C2212yescomplemented
14C3×#559—21no—
15C3×Q82C2212yescomplemented
16C3×#556—22no—
17C3×D4×Q8—22no—
18C3×#542—24no—
19C3×#536—21no—
20C3×#536C2212yes—
21C3×#533—21no—
22C3×#532C2211yescharacteristic
23C3×#456C2212yes—
24C3×#456C2218yes—
25C3×#455—21no—
26C3×#455—24no—
27C3×#454C2212yes—
28C3×#453—21no—
29C3×#421—22no—
30C2×C3×#104—21no—
31C3×#413C2212yes—
32C3×#413C2211yescharacteristic
33C3×#356C2212yescomplemented
34C3×#356C2218yescomplemented
35C3×#355C2212yescomplemented
36#3550C2×C311yescharacteristic, complemented
37#3022C2×C312yescomplemented
38#3018C2×C311yescharacteristic
39#2205C2×C314yescomplemented
40#2203C2×C314yescomplemented
41#2135C2×C312yescomplemented
42#2132C2×C311yescharacteristic, complemented
43C3×2−1+4—41no—
44C3×2−1+4C2312yes—
45C3×H(5,𝔽2)—21no—
46C3×H(5,𝔽2)—22no—
47C3×H(5,𝔽2)—41no—
48C2×C3×#41—22no—
49C2×C3×#41—21no—
50C2×C3×#41—42no—
51C2×C3×#41D411yescharacteristic
52C2×C3×#41C2311yescharacteristic
53C22×C3×Q8—21no—
54C22×C3×Q8D412yes—
55C3×#137—28no—
56C3×#137—22no—
57C3×#137—41no—
58C3×#137—44no—
59C3×AGL(1,ℤ/8)—22no—
60C3×AGL(1,ℤ/8)—41no—
61C3×#135—42no—
62C2×C3×Q16—42no—
63C2×C3×SD16—42no—
64C2×C3×M16—22no—
65C2×C3×M16—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—24no—
70C3×C4⋊Q8C2311yescharacteristic
71C3×C4⋊Q8C2314yes—
72C3×#127D411yescharacteristic
73C3×#126—24no—
74C3×#126—24no—
75C3×#125—22no—
76C3×#124—24no—
77C3×#124—22no—
78C3×#124D411yescharacteristic
79C3×#124D414yes—
80C3×#124C2311yescharacteristic
81C3×#123—22no—
82C3×#123—22no—
83C3×#123—22no—
84C3×#123—22no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—24no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×#121—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8—24no—
95C3×C4×D4—24no—
96C3×C4×D4—22no—
97C3×#117—22no—
98C3×#117—21no—
99C3×#117D412yes—
100C2×C3×C42D411yescharacteristic
101C3×#104—22no—
102C3×#104—22no—
103C3×#104—28no—
104C3×#101C2312yes—
105C3×#101C2314yes—
106C3×#100—21no—
107C3×#100—22no—
108#585C22×C311yescharacteristic
109#575—21no—
110C2×#137—22no—
111#563—21no—
112#563C22×C312yescomplemented
113#559—21no—
114Q82C22×C312yescomplemented
115#556—22no—
116D4×Q8—22no—
117#542—24no—
118#536—21no—
119#536C22×C312yes—
120#533—21no—
121#532C22×C311yescharacteristic
122#456C22×C318yes—
123#456C22×C312yes—
124#455—21no—
125#455—24no—
126#454C22×C312yes—
127#453—21no—
128#421—22no—
129C2×#104—21no—
130#413C22×C312yes—
131#413C22×C311yescharacteristic
132#356C22×C312yescomplemented
133#356C22×C318yescomplemented
134#355C22×C312yescomplemented
135C3×#41—22no—
136C3×#41—24no—
137C3×#41—22no—
138C3×#41—24no—
139C3×#41—41no—
140C3×#41—41no—
141C3×#41—42no—
142C3×#41—41no—
143C3×#41—41no—
144C3×#41—41no—
145C3×#41—42no—
146C3×#41—42no—
147C3×#41—44no—
148C3×#41—41no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—42no—
155C2×C3×Q8—42no—
156C2×C3×Q8—42no—
157C2×C3×Q8—81no—
158C2×C3×Q8—81no—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8C2×D411yescharacteristic
161C2×C3×Q8C2411yescharacteristic
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—42no—
167C2×C3×D4—44no—
168C2×C3×D4—42no—
169C2×C3×D4—41no—
170C2×C3×D4C2×D411yescharacteristic
171C2×C3×D4C2×D412yes—
172C2×C3×D4C2×D411yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—41no—
178C22×C3×C4—42no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C3×Q16—48no—
182C3×Q16—42no—
183C3×SD16—48no—
184C3×SD16—42no—
185C3×SD16—42no—
186C3×D8—42no—
187C3×M16—22no—
188C3×M16—24no—
189C3×M16—41no—
190C3×M16—42no—
191C2×C3×C8—42no—
192C2×C3×C8—41no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C42—24no—
210C3×C42—22no—
211C3×C42—21no—
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D414yes—
2152−1+4—41no—
2162−1+4C23×C312yes—
217H(5,𝔽2)—21no—
218H(5,𝔽2)—22no—
219H(5,𝔽2)—41no—
220C2×#41—22no—
221C2×#41—21no—
222C2×#41—42no—
223C2×#41C3×D411yescharacteristic
224C2×#41C23×C311yescharacteristic
225C22×Q8—21no—
226C22×Q8C3×D412yes—
227#137—28no—
228#137—22no—
229#137—41no—
230#137—44no—
231AGL(1,ℤ/8)—22no—
232AGL(1,ℤ/8)—41no—
233#135—42no—
234C2×Q16—42no—
235C2×SD16—42no—
236C2×M16—22no—
237C2×M16—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8C23×C311yescharacteristic
243C4⋊Q8C23×C314yes—
244#127C3×D411yescharacteristic
245#126—24no—
246#126—24no—
247#125—22no—
248#124—22no—
249#124—24no—
250#124C3×D411yescharacteristic
251#124C3×D414yes—
252#124C23×C311yescharacteristic
253#123—22no—
254#123—22no—
255#123—22no—
256#123—22no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263#121—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—24no—
267C4×D4—24no—
268C4×D4—22no—
269#117—22no—
270#117—21no—
271#117C3×D412yes—
272C2×C42C3×D411yescharacteristic
273#104—22no—
274#104—22no—
275#104—28no—
276#101C23×C314yes—
277#101C23×C312yes—
278#100—21no—
279#100—22no—
280C23×C3—21no—
281C23×C3—42no—
282C23×C3—41no—
283C23×C3#12012yes—
284C23×C3#12011yescharacteristic
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8—41no—
288C3×Q8—44no—
289C3×Q8—41no—
290C3×Q8—44no—
291C3×Q8—41no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×Q8—82no—
295C3×D4—22no—
296C3×D4—24no—
297C3×D4—41no—
298C3×D4—42no—
299C3×D4—41no—
300C3×D4—44no—
301C3×D4—44no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—41no—
305C3×D4—81no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—24no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—24no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—82no—
324C2×C3×C4—82no—
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4C22×D412yes—
327C2×C3×C4C22×D411yescharacteristic
328C3×C8—42no—
329C3×C8—44no—
330#41—22no—
331#41—24no—
332#41—22no—
333#41—24no—
334#41—41no—
335#41—41no—
336#41—42no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—42no—
341#41—42no—
342#41—44no—
343#41—41no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8C2×C3×D414yes—
355C2×Q8C2×C3×D411yescharacteristic
356C2×Q8C24×C311yescharacteristic
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4C2×C3×D411yescharacteristic
366C2×D4C2×C3×D412yes—
367C2×D4C2×C3×D411yescharacteristic
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D412yes—
376Q16—48no—
377Q16—42no—
378SD16—48no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—22no—
383M16—24no—
384M16—41no—
385M16—42no—
386C2×C8—42no—
387C2×C8—41no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C42—24no—
405C42—22no—
406C42—21no—
407C42C2×C3×D411yescharacteristic
408C42C2×C3×D411yescharacteristic
409C42C2×C3×D414yes—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—41no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—41no—
417C22×C3—162no—
418C22×C3—161no—
419C22×C3C2×#12011yescharacteristic
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—41no—
424C3×C4—44no—
425C3×C4—41no—
426C3×C4—44no—
427C3×C4—42no—
428C3×C4B(4,𝔽2)11yescharacteristic
429C3×C4B(4,𝔽2)11yescharacteristic
430C23—21no—
431C23—42no—
432C23—41no—
433C23C3×#12012yes—
434C23C3×#12011yescharacteristic
435Q8—22no—
436Q8—24no—
437Q8—41no—
438Q8—44no—
439Q8—41no—
440Q8—44no—
441Q8—41no—
442Q8—42no—
443Q8—42no—
444Q8—82no—
445D4—22no—
446D4—24no—
447D4—41no—
448D4—42no—
449D4—41no—
450D4—44no—
451D4—44no—
452D4—42no—
453D4—42no—
454D4—41no—
455D4—81no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4C3×#12011yescharacteristic
476C2×C4C22×C3×D412yes—
477C2×C4C22×C3×D411yescharacteristic
478C8—42no—
479C8—44no—
480C2×C3—21no—
481C2×C3—42no—
482C2×C3—41no—
483C2×C3—81no—
484C2×C3—82no—
485C2×C3C2×B(4,𝔽2)11yescharacteristic
486C22—22no—
487C22—21no—
488C22—41no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—41no—
493C22—162no—
494C22—161no—
495C22C2×C3×#12011yescharacteristic
496C4—22no—
497C4—21no—
498C4—22no—
499C4—41no—
500C4—44no—
501C4—41no—
502C4—44no—
503C4—42no—
504C4C3×B(4,𝔽2)11yescharacteristic
505C4C3×B(4,𝔽2)11yescharacteristic
506C3#3357111yescharacteristic, direct product factor, Sylow
507C2—21no—
508C2—42no—
509C2—41no—
510C2—81no—
511C2—82no—
512C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C3
prime radicalC3×#3018C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3550C2semidirect
2. C3×#3022C2semidirect
4. C3×#2205C2semidirect
5. C3×#2203C2semidirect
6. C3×#2135C2semidirect
7. C3×#2132C2semidirect
8. #33571C3direct (N × H)
13. C3×#563C22semidirect
15. C3×Q82C22semidirect
33. C3×#356C22semidirect
34. C3×#356C22semidirect
35. C3×#355C22semidirect
36. #3550C2×C3semidirect
37. #3022C2×C3semidirect
39. #2205C2×C3semidirect
40. #2203C2×C3semidirect
41. #2135C2×C3semidirect
42. #2132C2×C3semidirect
112. #563C22×C3semidirect
114. Q82C22×C3semidirect
132. #356C22×C3semidirect
133. #356C22×C3semidirect
134. #355C22×C3semidirect
506. C3#33571direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35502
C3×#30222
C3×#30182
C3×#22052
C3×#22032
C3×#21352
C3×#21322
#335713