Finite Group Databasebeta

← C3×#32922 #10651529

Subgroups — C3×#32922 #10651529

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#3039C212yesmaximal, complemented
3C2×C3×#455C212yesmaximal
4C3×#2547C211yesmaximal, characteristic, complemented
5C3×#1885C212yesmaximal, complemented
6C3×#1882C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9#32922C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C2×C3×#137C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic, complemented
13C2×C3×#124C2212yes—
14C3×#520—22no—
15C3×#520C2212yes—
16C3×#518—22no—
17C3×#518—22no—
18C3×#518C2211yescharacteristic
19C3×C4×#41—22no—
20C3×C4×#41C2212yes—
21C2×C3×C4×D4—22no—
22C2×C3×C4×D4—22no—
23C2×C3×C4×D4C2211yescharacteristic
24C3×#462—22no—
25C3×#462—22no—
26C3×#461—22no—
27C3×#461—22no—
28C3×#455—28no—
29C3×#423C2212yescomplemented
30C2×C3×#104C2214yes—
31C3×#419C2214yescomplemented
32C2×C3×#100C2212yes—
33C3×#397—22no—
34C3×#391C2212yescomplemented
35C3×#380—22no—
36C3×C4×C4⋊C4C2211yescharacteristic, complemented
37C3×#343C2212yescomplemented
38C3×#339C2214yescomplemented
39C4×H(5,𝔽2)C2×C311yescharacteristic
40#3039C2×C312yescomplemented
41C2×#455C2×C312yes—
42#2547C2×C311yescharacteristic, complemented
43#1885C2×C312yescomplemented
44#1882C2×C314yescomplemented
45#1806C2×C312yescomplemented
46#1800C2×C311yescharacteristic, complemented
47C3×H(5,𝔽2)—24no—
48C3×H(5,𝔽2)—22no—
49C3×H(5,𝔽2)D414yes—
50C2×C3×#41—22no—
51C2×C3×#41C2312yes—
52C22×C3×Q8C2312yes—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—22no—
57C23×C3×C4D412yes—
58C3×#137—28no—
59C3×#137—28no—
60C2×C3×Q16—24no—
61C2×C3×SD16—24no—
62C2×C3×M16C2314yes—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×#124—24no—
66C3×#124—24no—
67C3×C4×Q8—22no—
68C3×C4×Q8D412yes—
69C3×C4×Q8D412yes—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4—24no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4D412yes—
83C3×C4×D4D412yes—
84C3×#117—24no—
85C3×#117—22no—
86C3×#117—22no—
87C3×#117C2312yes—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—24no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4D412yes—
99C2×C3×C42—22no—
100C2×C3×C42—22no—
101C2×C3×C42C2312yes—
102C2×C3×C42C2311yescharacteristic
103C3×C4⋊C8—22no—
104C3×C4⋊C8—22no—
105C3×#104—28no—
106C3×#103—24no—
107C3×#103—24no—
108C3×#100—24no—
109C3×#95—21no—
110C3×#95—24no—
111C2×H(5,𝔽2)C22×C311yescharacteristic
112C2×#137C22×C314yes—
113C2×C4⋊Q8C22×C311yescharacteristic, complemented
114C2×#124C22×C312yes—
115#520—22no—
116#520C22×C312yes—
117#518—22no—
118#518—22no—
119#518C22×C311yescharacteristic
120C4×#41—22no—
121C4×#41C22×C312yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C22×C311yescharacteristic
125#462—22no—
126#462—22no—
127#461—22no—
128#461—22no—
129#455—28no—
130#423C22×C312yescomplemented
131C2×#104C22×C314yes—
132#419C22×C314yescomplemented
133C2×#100C22×C312yes—
134#397—22no—
135#391C22×C312yescomplemented
136#380—22no—
137C4×C4⋊C4C22×C311yescharacteristic, complemented
138#343C22×C312yescomplemented
139#339C22×C314yescomplemented
140C24×C3—22no—
141C24×C3C2×D412yes—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—22no—
145C3×#41—22no—
146C3×#41C2×D414yes—
147C3×#41#4114yes—
148C2×C3×Q8—24no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—24no—
152C2×C3×Q8C2×D412yes—
153C2×C3×D4—24no—
154C2×C3×D4—28no—
155C2×C3×D4—28no—
156C2×C3×D4—28no—
157C2×C3×D4—22no—
158C2×C3×D4—28no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4C2×D412yes—
162C2×C3×D4C2×D412yes—
163C2×C3×D4#4112yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—44no—
176C22×C3×C4—84no—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C3×Q16—48no—
182C3×SD16—48no—
183C3×M16—24no—
184C3×M16—24no—
185C2×C3×C8—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—48no—
203C3×C42—24no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211H(5,𝔽2)—24no—
212H(5,𝔽2)—22no—
213H(5,𝔽2)C3×D414yes—
214C2×#41—22no—
215C2×#41C23×C312yes—
216C22×Q8C23×C312yes—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4C23×C311yescharacteristic
220C23×C4—22no—
221C23×C4C3×D412yes—
222#137—28no—
223#137—28no—
224C2×Q16—24no—
225C2×SD16—24no—
226C2×M16C23×C314yes—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229#124—24no—
230#124—24no—
231C4×Q8—22no—
232C4×Q8C3×D412yes—
233C4×Q8C3×D412yes—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4C3×D412yes—
247C4×D4C3×D412yes—
248#117—24no—
249#117—22no—
250#117—22no—
251#117C23×C312yes—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4C3×D412yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42C23×C312yes—
266C2×C42C23×C311yescharacteristic
267C4⋊C8—22no—
268C4⋊C8—22no—
269#104—28no—
270#103—24no—
271#103—24no—
272#100—24no—
273#95—21no—
274#95—24no—
275C23×C3—28no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—82no—
282C23×C3—82no—
283C23×C3#12114yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—22no—
286C3×Q8—44no—
287C3×Q8—44no—
288C3×Q8#12114yes—
289C3×D4—216no—
290C3×D4—24no—
291C3×D4—24no—
292C3×D4—24no—
293C3×D4—22no—
294C3×D4—24no—
295C3×D4#12114yes—
296C2×C3×C4—28no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—82no—
315C2×C3×C4—82no—
316C2×C3×C4—82no—
317C2×C3×C4—84no—
318C2×C3×C4—82no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12011yescharacteristic
321C2×C3×C4#12011yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D412yes—
324C2×C3×C4C2×#4112yes—
325C2×C3×C4C2×#4111yescharacteristic
326C3×C8—44no—
327C24—22no—
328C24C2×C3×D412yes—
329#41—28no—
330#41—28no—
331#41—22no—
332#41—22no—
333#41C2×C3×D414yes—
334#41C3×#4114yes—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8C2×C3×D412yes—
340C2×D4—24no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—22no—
345C2×D4—28no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4C2×C3×D412yes—
349C2×D4C2×C3×D412yes—
350C2×D4C3×#4112yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—44no—
363C22×C4—84no—
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C24×C311yescharacteristic
368Q16—48no—
369SD16—48no—
370M16—24no—
371M16—24no—
372C2×C8—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4C2×C3×D412yes—
382C4⋊C4C2×C3×D412yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—48no—
390C42—24no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42C2×C3×D412yes—
397C42C2×C3×D412yes—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—44no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—82no—
408C22×C3—88no—
409C22×C3C2×#12011yescharacteristic
410C22×C3C2×#12111yescharacteristic
411C22×C3#52511yescharacteristic
412C3×C4—24no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—82no—
419C3×C4—82no—
420C3×C4#45512yescomplemented
421C3×C4C2×#12112yes—
422C3×C4#52512yes—
423C23—28no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—82no—
430C23—82no—
431C23C3×#12114yes—
432C23C22×C3×D411yescharacteristic
433Q8—22no—
434Q8—44no—
435Q8—44no—
436Q8C3×#12114yes—
437D4—216no—
438D4—24no—
439D4—24no—
440D4—24no—
441D4—22no—
442D4—24no—
443D4C3×#12114yes—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4C3×#12012yes—
468C2×C4C3×#12011yescharacteristic
469C2×C4C3×#12011yescharacteristic
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C22×C3×D412yes—
472C2×C4C2×C3×#4112yes—
473C2×C4C2×C3×#4111yescharacteristic
474C8—44no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—44no—
478C2×C3—44no—
479C2×C3#284711yescharacteristic
480C2×C3C2×#45511yescharacteristic
481C2×C3#302211yescharacteristic
482C22—21no—
483C22—21no—
484C22—24no—
485C22—24no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—88no—
493C22C2×C3×#12011yescharacteristic
494C22C2×C3×#12111yescharacteristic
495C22C3×#52511yescharacteristic
496C4—21no—
497C4—22no—
498C4—24no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C4C3×#45512yescomplemented
505C4C2×C3×#12112yes—
506C4C3×#52512yes—
507C3#3292211yescharacteristic, direct product factor, Sylow
508C2—21no—
509C2—21no—
510C2—44no—
511C2—44no—
512C2C3×#284711yescharacteristic
513C2C2×C3×#45511yescharacteristic
514C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×H(5,𝔽2)C22
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
4. C3×#2547C2semidirect
5. C3×#1885C2semidirect
6. C3×#1882C2semidirect
7. C3×#1806C2semidirect
8. C3×#1800C2semidirect
9. #32922C3direct (N × H)
12. C2×C3×C4⋊Q8C22semidirect
29. C3×#423C22semidirect
31. C3×#419C22semidirect
34. C3×#391C22semidirect
36. C3×C4×C4⋊C4C22semidirect
37. C3×#343C22semidirect
38. C3×#339C22semidirect
40. #3039C2×C3semidirect
42. #2547C2×C3semidirect
43. #1885C2×C3semidirect
44. #1882C2×C3semidirect
45. #1806C2×C3semidirect
46. #1800C2×C3semidirect
113. C2×C4⋊Q8C22×C3semidirect
130. #423C22×C3semidirect
132. #419C22×C3semidirect
135. #391C22×C3semidirect
137. C4×C4⋊C4C22×C3semidirect
138. #343C22×C3semidirect
139. #339C22×C3semidirect
420. C3×C4#455semidirect
504. C4C3×#455semidirect
507. C3#32922direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C3×#30392
C2×C3×#4552
C3×#25472
C3×#18852
C3×#18822
C3×#18062
C3×#18002
#329223