Finite Group Databasebeta

← C3×#61665 #10680272

Subgroups — C3×#61665 #10680272

Order 768.

2008 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3550C211yesmaximal, characteristic, complemented
2C3×#3499C211yesmaximal, characteristic
3C3×#3328C214yesmaximal, complemented
4C3×#3302C214yesmaximal, complemented
5C3×D4×Q16C214yesmaximal, complemented
6C3×#3276C214yesmaximal, complemented
7C3×#3209C212yesmaximal, complemented
8C3×#3202C214yesmaximal, complemented
9C3×#3107C212yesmaximal, complemented
10C3×#3011C214yesmaximal, complemented
11C3×#2979C211yesmaximal, characteristic, complemented
12#61665C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2212yes—
14C2×C3×#137C2214yes—
15C22×C3×Q16C2214yes—
16C3×#563—22no—
17C3×#559—21no—
18C3×Q82—21no—
19C3×#556C2214yes—
20C3×D4×Q8C2214yes—
21C3×D4×Q8C2214yes—
22C3×#547—24no—
23C3×#547C2214yes—
24C3×#542—24no—
25C3×#536—22no—
26C3×#536C2212yes—
27C3×#536C2214yes—
28C3×#533C2212yes—
29C3×#532—21no—
30C3×#532C2211yescharacteristic
31C2×C3×C4⋊Q8—21no—
32C2×C3×C4⋊Q8C2211yescharacteristic
33C3×C4×#41C2211yescharacteristic
34C3×#494C2214yes—
35C3×#487C2214yes—
36C3×#485C2214yes—
37C3×#485C2218yes—
38C3×#475C2212yes—
39C3×#474C2212yes—
40C3×#470C2218yes—
41C3×#467C2218yes—
42C3×#462C2214yes—
43C3×#462C2214yes—
44C3×#462C2212yes—
45C3×#461C2212yes—
46C3×#452C2214yes—
47C3×#452C2218yes—
48C3×#451C2218yes—
49C3×#451C2218yes—
50C3×#451C2218yes—
51C3×C4×Q16C2214yes—
52C3×C4×SD16C2214yes—
53C3×#436C2214yes—
54C3×C8×D4C2214yes—
55C3×#431C2212yes—
56C3×#419C2214yes—
57C3×#416C2214yes—
58C3×#408C2214yes—
59#3550C2×C311yescharacteristic, complemented
60#3499C2×C311yescharacteristic
61#3328C2×C314yescomplemented
62#3302C2×C314yescomplemented
63D4×Q16C2×C314yescomplemented
64#3276C2×C314yescomplemented
65#3209C2×C312yescomplemented
66#3202C2×C314yescomplemented
67#3107C2×C312yescomplemented
68#3011C2×C314yescomplemented
69#2979C2×C311yescharacteristic, complemented
70C3×2−1+4—44no—
71C3×2−1+4—44no—
72C2×C3×#41—22no—
73C2×C3×#41—24no—
74C2×C3×#41C2311yescharacteristic
75C2×C3×#41C2314yes—
76C2×C3×#41C2312yes—
77C22×C3×Q8—22no—
78C22×C3×Q8C2314yes—
79C22×C3×Q8C2314yes—
80C22×C3×Q8C2312yes—
81C3×#137—48no—
82C2×C3×Q16—48no—
83C2×C3×Q16C2314yes—
84C2×C3×Q16C2318yes—
85C2×C3×Q16C2318yes—
86C2×C3×SD16C2314yes—
87C2×C3×M16C2314yes—
88C22×C3×C8C2314yes—
89C3×C4⋊Q8—24no—
90C3×C4⋊Q8—22no—
91C3×C4⋊Q8—24no—
92C3×C4⋊Q8—24no—
93C3×C4⋊Q8—21no—
94C3×C4⋊Q8—21no—
95C3×C4⋊Q8D411yescharacteristic
96C3×C4⋊Q8C2312yes—
97C3×C4⋊Q8C2314yes—
98C3×C4⋊Q8C2312yes—
99C3×#127D411yescharacteristic
100C3×#126—28no—
101C3×#125—22no—
102C3×#124—24no—
103C3×#124—22no—
104C3×#124—24no—
105C3×#124D412yes—
106C3×#123—24no—
107C3×#123—24no—
108C3×#123—24no—
109C3×#123—28no—
110C3×C22⋊Q8—22no—
111C3×C22⋊Q8—28no—
112C3×C22⋊Q8—28no—
113C3×C22⋊Q8C2314yes—
114C3×C22⋊Q8C2318yes—
115C3×C22⋊Q8C2318yes—
116C3×#121—24no—
117C3×#121D414yes—
118C3×C4×Q8—24no—
119C3×C4×Q8—22no—
120C3×C4×Q8C2314yes—
121C3×C4×Q8C2312yes—
122C3×C4×D4—24no—
123C3×C4×D4—24no—
124C3×C4×D4—22no—
125C3×C4×D4C2312yes—
126C3×C4×D4C2314yes—
127C3×#117—22no—
128C3×#117—21no—
129C3×#117C2314yes—
130C3×#117C2311yescharacteristic
131C2×C3×C4⋊C4—22no—
132C2×C3×C4⋊C4—24no—
133C2×C3×C42—21no—
134C2×C3×C42C2311yescharacteristic
135C3×#107C2314yes—
136C3×#106C2314yes—
137C3×C4⋊C8C2312yes—
138C3×C4⋊C8C2312yes—
139C3×#103C2318yes—
140C3×#103C2314yes—
141C3×#103C2318yes—
142C3×#103C2314yes—
143C3×#103C2314yes—
144C3×#102C2314yes—
145C3×C22⋊C8C2314yes—
146C3×C22⋊C8C2318yes—
147C3×C4×C8C2314yes—
148C2×2−1+4C22×C312yes—
149C2×#137C22×C314yes—
150C22×Q16C22×C314yes—
151#563—22no—
152#559—21no—
153Q82—21no—
154#556C22×C314yes—
155D4×Q8C22×C314yes—
156D4×Q8C22×C314yes—
157#547—24no—
158#547C22×C314yes—
159#542—24no—
160#536—22no—
161#536C22×C312yes—
162#536C22×C314yes—
163#533C22×C312yes—
164#532—21no—
165#532C22×C311yescharacteristic
166C2×C4⋊Q8—21no—
167C2×C4⋊Q8C22×C311yescharacteristic
168C4×#41C22×C311yescharacteristic
169#494C22×C314yes—
170#487C22×C314yes—
171#485C22×C318yes—
172#485C22×C314yes—
173#475C22×C312yes—
174#474C22×C312yes—
175#470C22×C318yes—
176#467C22×C318yes—
177#462C22×C314yes—
178#462C22×C312yes—
179#462C22×C314yes—
180#461C22×C312yes—
181#452C22×C314yes—
182#452C22×C318yes—
183#451C22×C318yes—
184#451C22×C318yes—
185#451C22×C318yes—
186C4×Q16C22×C314yes—
187C4×SD16C22×C314yes—
188#436C22×C314yes—
189C8×D4C22×C314yes—
190#431C22×C312yes—
191#419C22×C314yes—
192#416C22×C314yes—
193#408C22×C314yes—
194C3×#41—22no—
195C3×#41—22no—
196C3×#41—28no—
197C3×#41—42no—
198C3×#41—42no—
199C3×#41—44no—
200C3×#41—48no—
201C3×#41—416no—
202C2×C3×Q8—22no—
203C2×C3×Q8—24no—
204C2×C3×Q8—22no—
205C2×C3×Q8—24no—
206C2×C3×Q8—28no—
207C2×C3×Q8—44no—
208C2×C3×Q8—42no—
209C2×C3×Q8—42no—
210C2×C3×Q8—48no—
211C2×C3×Q8—44no—
212C2×C3×Q8—44no—
213C2×C3×Q8C2×D412yes—
214C2×C3×Q8C2414yes—
215C2×C3×Q8C2414yes—
216C2×C3×Q8C2411yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4—24no—
219C2×C3×D4C2×D414yes—
220C2×C3×D4C2×D414yes—
221C2×C3×D4C2×D412yes—
222C2×C3×D4C2411yescharacteristic
223C22×C3×C4—21no—
224C22×C3×C4—22no—
225C22×C3×C4—24no—
226C22×C3×C4—24no—
227C22×C3×C4—22no—
228C22×C3×C4—24no—
229C22×C3×C4C2×D412yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2414yes—
232C3×Q16—28no—
233C3×Q16—48no—
234C3×Q16—48no—
235C3×SD16—28no—
236C3×M16—44no—
237C2×C3×C8—44no—
238C2×C3×C8C2414yes—
239C2×C3×C8C2414yes—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—28no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4C2×D414yes—
251C3×C4⋊C4C2414yes—
252C3×C4⋊C4C2412yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—21no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—28no—
257C3×C22⋊C4—28no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4C2×D418yes—
260C3×C42—21no—
261C3×C42—22no—
262C3×C42—24no—
263C3×C42—24no—
264C3×C42C2×D411yescharacteristic
265C3×C42C2×D411yescharacteristic
266C3×C42C2412yes—
2672−1+4—44no—
2682−1+4—44no—
269C2×#41—22no—
270C2×#41—24no—
271C2×#41C23×C311yescharacteristic
272C2×#41C23×C314yes—
273C2×#41C23×C312yes—
274C22×Q8—22no—
275C22×Q8C23×C314yes—
276C22×Q8C23×C312yes—
277C22×Q8C23×C314yes—
278#137—48no—
279C2×Q16—48no—
280C2×Q16C23×C314yes—
281C2×Q16C23×C318yes—
282C2×Q16C23×C318yes—
283C2×SD16C23×C314yes—
284C2×M16C23×C314yes—
285C22×C8C23×C314yes—
286C4⋊Q8—22no—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8C3×D411yescharacteristic
293C4⋊Q8C23×C314yes—
294C4⋊Q8C23×C312yes—
295C4⋊Q8C23×C312yes—
296#127C3×D411yescharacteristic
297#126—28no—
298#125—22no—
299#124—22no—
300#124—24no—
301#124—24no—
302#124C3×D412yes—
303#123—24no—
304#123—24no—
305#123—24no—
306#123—28no—
307C22⋊Q8—22no—
308C22⋊Q8—28no—
309C22⋊Q8—28no—
310C22⋊Q8C23×C314yes—
311C22⋊Q8C23×C318yes—
312C22⋊Q8C23×C318yes—
313#121—24no—
314#121C3×D414yes—
315C4×Q8—24no—
316C4×Q8—22no—
317C4×Q8C23×C312yes—
318C4×Q8C23×C314yes—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4C23×C312yes—
323C4×D4C23×C314yes—
324#117—22no—
325#117—21no—
326#117C23×C311yescharacteristic
327#117C23×C314yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C42—21no—
331C2×C42C23×C311yescharacteristic
332#107C23×C314yes—
333#106C23×C314yes—
334C4⋊C8C23×C312yes—
335C4⋊C8C23×C312yes—
336#103C23×C318yes—
337#103C23×C314yes—
338#103C23×C318yes—
339#103C23×C314yes—
340#103C23×C314yes—
341#102C23×C314yes—
342C22⋊C8C23×C314yes—
343C22⋊C8C23×C318yes—
344C4×C8C23×C314yes—
345C23×C3—21no—
346C23×C3C22×D414yes—
347C23×C3C22×D411yescharacteristic
348C3×Q8—21no—
349C3×Q8—24no—
350C3×Q8—24no—
351C3×Q8—24no—
352C3×Q8—24no—
353C3×Q8—42no—
354C3×Q8—48no—
355C3×Q8—44no—
356C3×Q8—44no—
357C3×Q8—84no—
358C3×Q8H(5,𝔽2)12yes—
359C3×D4—24no—
360C3×D4—28no—
361C3×D4—21no—
362C3×D4—42no—
363C3×D4—44no—
364C3×D4—48no—
365C3×D4H(5,𝔽2)12yes—
366C2×C3×C4—24no—
367C2×C3×C4—21no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—48no—
378C2×C3×C4—48no—
379C2×C3×C4—44no—
380C2×C3×C4—44no—
381C2×C3×C4—84no—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D414yes—
384C2×C3×C4C22×D412yes—
385C2×C3×C4C22×D412yes—
386C2×C3×C4C2511yescharacteristic
387C3×C8—24no—
388C3×C8—24no—
389#41—22no—
390#41—22no—
391#41—28no—
392#41—42no—
393#41—42no—
394#41—44no—
395#41—48no—
396#41—416no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—28no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8C2×C3×D412yes—
409C2×Q8C24×C314yes—
410C2×Q8C24×C314yes—
411C2×Q8C24×C311yescharacteristic
412C2×D4—21no—
413C2×D4—24no—
414C2×D4C2×C3×D414yes—
415C2×D4C2×C3×D414yes—
416C2×D4C2×C3×D412yes—
417C2×D4C24×C311yescharacteristic
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4C2×C3×D412yes—
425C22×C4C24×C311yescharacteristic
426C22×C4C24×C314yes—
427Q16—28no—
428Q16—48no—
429Q16—48no—
430SD16—28no—
431M16—44no—
432C2×C8—44no—
433C2×C8C24×C314yes—
434C2×C8C24×C314yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4C2×C3×D414yes—
446C4⋊C4C24×C314yes—
447C4⋊C4C24×C312yes—
448C22⋊C4—24no—
449C22⋊C4—21no—
450C22⋊C4—24no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—22no—
454C22⋊C4C2×C3×D418yes—
455C42—21no—
456C42—22no—
457C42—24no—
458C42—24no—
459C42C2×C3×D411yescharacteristic
460C42C2×C3×D411yescharacteristic
461C42C24×C312yes—
462C22×C3—21no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—22no—
466C22×C3—28no—
467C22×C3—42no—
468C22×C3C23×D411yescharacteristic
469C3×C4—21no—
470C3×C4—22no—
471C3×C4—24no—
472C3×C4—22no—
473C3×C4—44no—
474C3×C4—44no—
475C3×C4—42no—
476C3×C4—44no—
477C3×C4#57814yes—
478C3×C4C2×H(5,𝔽2)11yescharacteristic
479C3×C4C2×H(5,𝔽2)11yescharacteristic
480C23—21no—
481C23C22×C3×D414yes—
482C23C22×C3×D411yescharacteristic
483Q8—21no—
484Q8—24no—
485Q8—24no—
486Q8—24no—
487Q8—24no—
488Q8—42no—
489Q8—48no—
490Q8—44no—
491Q8—44no—
492Q8—84no—
493Q8C3×H(5,𝔽2)12yes—
494D4—24no—
495D4—28no—
496D4—21no—
497D4—42no—
498D4—44no—
499D4—48no—
500D4C3×H(5,𝔽2)12yes—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—84no—
517C2×C4C22×C3×D411yescharacteristic
518C2×C4C22×C3×D414yes—
519C2×C4C22×C3×D412yes—
520C2×C4C22×C3×D412yes—
521C2×C4C25×C311yescharacteristic
522C8—24no—
523C8—24no—
524C2×C3—41no—
525C2×C3—44no—
526C2×C3—42no—
527C2×C3#348411yescharacteristic
528C2×C3C2×#57812yes—
529C22—21no—
530C22—22no—
531C22—24no—
532C22—22no—
533C22—28no—
534C22—42no—
535C22C23×C3×D411yescharacteristic
536C4—21no—
537C4—22no—
538C4—24no—
539C4—22no—
540C4—44no—
541C4—44no—
542C4—42no—
543C4—44no—
544C4C3×#57814yes—
545C4C2×C3×H(5,𝔽2)11yescharacteristic
546C4C2×C3×H(5,𝔽2)11yescharacteristic
547C3#6166511yescharacteristic, direct product factor, Sylow
548C2—41no—
549C2—44no—
550C2—42no—
551C2C3×#348411yescharacteristic
552C2C2×C3×#57812yes—
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3499C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#532 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3550C2semidirect
3. C3×#3328C2semidirect
4. C3×#3302C2semidirect
5. C3×D4×Q16C2semidirect
6. C3×#3276C2semidirect
7. C3×#3209C2semidirect
8. C3×#3202C2semidirect
9. C3×#3107C2semidirect
10. C3×#3011C2semidirect
11. C3×#2979C2semidirect
12. #61665C3direct (N × H)
59. #3550C2×C3semidirect
61. #3328C2×C3semidirect
62. #3302C2×C3semidirect
63. D4×Q16C2×C3semidirect
64. #3276C2×C3semidirect
65. #3209C2×C3semidirect
66. #3202C2×C3semidirect
67. #3107C2×C3semidirect
68. #3011C2×C3semidirect
69. #2979C2×C3semidirect
547. C3#61665direct (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×#34992
C3×#33282
C3×#33022
C3×D4×Q162
C3×#32762
C3×#32092
C3×#32022
C3×#31072
C3×#30112
C3×#29792
#616653