Finite Group Databasebeta

← C3×#61798 #10680405

Subgroups — C3×#61798 #10680405

Order 768.

1674 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3550C211yesmaximal, characteristic, complemented
2C3×#3512C211yesmaximal, characteristic
3C3×#3350C214yesmaximal, complemented
4C3×#3349C214yesmaximal, complemented
5C3×#3320C214yesmaximal, complemented
6C3×#3312C214yesmaximal, complemented
7C3×#3236C212yesmaximal, complemented
8C3×#3201C214yesmaximal, complemented
9C3×#3116C212yesmaximal, complemented
10C3×#3097C214yesmaximal, complemented
11C3×#2980C211yesmaximal, characteristic, complemented
12#61798C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#563—22no—
14C3×#559—21no—
15C3×Q82—21no—
16C3×#556C2214yes—
17C3×#553—24no—
18C3×#553C2214yes—
19C3×#552C2214yes—
20C3×D4×Q8C2214yes—
21C3×#542—24no—
22C3×#541—21no—
23C3×#540—21no—
24C3×#537C2212yes—
25C3×#536C2214yes—
26C3×#533C2212yes—
27C3×#532C2211yescharacteristic
28C3×#529—22no—
29C2×C3×#125C2211yescharacteristic
30C3×#520C2212yes—
31C3×C4×#41C2211yescharacteristic
32C3×#501C2214yes—
33C3×#491C2214yes—
34C3×#485C2214yes—
35C3×#485C2214yes—
36C3×#485C2214yes—
37C3×#485C2214yes—
38C3×#484C2214yes—
39C3×#484C2214yes—
40C3×#483C2214yes—
41C3×#483C2214yes—
42C3×#482C2214yes—
43C3×#479C2214yes—
44C3×#479C2214yes—
45C3×#477C2212yes—
46C3×#476C2212yes—
47C3×#470C2214yes—
48C3×#468C2214yes—
49C3×#467C2214yes—
50C3×#465C2214yes—
51C3×#462C2212yes—
52C3×#461C2212yes—
53C3×#452C2214yes—
54C3×#451C2214yes—
55C3×#448C2214yes—
56C3×#441C2214yes—
57C3×#440C2214yes—
58C3×#435C2214yes—
59C3×#435C2214yes—
60C3×#432C2212yes—
61C3×#428C2214yes—
62C3×#427C2214yes—
63C3×#419C2214yes—
64C2×C3×#103C2214yes—
65C3×#408C2214yes—
66#3550C2×C311yescharacteristic, complemented
67#3512C2×C311yescharacteristic
68#3350C2×C314yescomplemented
69#3349C2×C314yescomplemented
70#3320C2×C314yescomplemented
71#3312C2×C314yescomplemented
72#3236C2×C312yescomplemented
73#3201C2×C314yescomplemented
74#3116C2×C312yescomplemented
75#3097C2×C314yescomplemented
76#2980C2×C311yescharacteristic, complemented
77C2×C3×#41C2311yescharacteristic
78C2×C3×#41C2312yes—
79C22×C3×Q8C2314yes—
80C2×C3×Q16C2314yes—
81C2×C3×SD16C2314yes—
82C2×C3×M16C2314yes—
83C22×C3×C8C2314yes—
84C3×C4⋊Q8—24no—
85C3×C4⋊Q8—24no—
86C3×C4⋊Q8D411yescharacteristic
87C3×C4⋊Q8C2312yes—
88C3×C4⋊Q8C2312yes—
89C3×#127D411yescharacteristic
90C3×#126—24no—
91C3×#126—24no—
92C3×#126—24no—
93C3×#126—24no—
94C3×#125—24no—
95C3×#125—22no—
96C3×#125—21no—
97C3×#125—21no—
98C3×#125C2314yes—
99C3×#124—24no—
100C3×#124—24no—
101C3×#124D412yes—
102C3×#123—24no—
103C3×#123—24no—
104C3×#123—24no—
105C3×#123—24no—
106C3×C22⋊Q8—22no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—24no—
109C3×C22⋊Q8—24no—
110C3×C22⋊Q8C2314yes—
111C3×C22⋊Q8C2314yes—
112C3×C22⋊Q8C2314yes—
113C3×C22⋊Q8C2314yes—
114C3×#121—22no—
115C3×#121—24no—
116C3×#121—24no—
117C3×#121D414yes—
118C3×#121C2314yes—
119C3×C4×Q8—22no—
120C3×C4×Q8—24no—
121C3×C4×Q8—22no—
122C3×C4×Q8C2314yes—
123C3×C4×Q8C2312yes—
124C3×C4×D4—24no—
125C3×C4×D4—22no—
126C3×C4×D4—24no—
127C3×C4×D4—22no—
128C3×C4×D4—24no—
129C3×C4×D4C2312yes—
130C3×C4×D4C2314yes—
131C3×#117—22no—
132C3×#117—24no—
133C3×#117—21no—
134C3×#117C2314yes—
135C3×#117C2314yes—
136C3×#117C2311yescharacteristic
137C2×C3×C4⋊C4—22no—
138C2×C3×C4⋊C4C2314yes—
139C2×C3×C4⋊C4C2312yes—
140C2×C3×C42—21no—
141C2×C3×C42C2311yescharacteristic
142C3×#107C2314yes—
143C3×#107C2314yes—
144C3×#107C2314yes—
145C3×#106C2314yes—
146C3×#106C2314yes—
147C3×#106C2314yes—
148C3×C4⋊C8C2312yes—
149C3×C4⋊C8C2312yes—
150C3×#103C2314yes—
151C3×#103C2314yes—
152C3×#103C2314yes—
153C3×#103C2314yes—
154C3×#103C2314yes—
155C3×#103C2314yes—
156C3×#103C2314yes—
157C3×#102C2314yes—
158C3×C22⋊C8C2314yes—
159C3×C22⋊C8C2314yes—
160C3×C22⋊C8C2314yes—
161C3×#97C2314yes—
162#563—22no—
163#559—21no—
164Q82—21no—
165#556C22×C314yes—
166#553—24no—
167#553C22×C314yes—
168#552C22×C314yes—
169D4×Q8C22×C314yes—
170#542—24no—
171#541—21no—
172#540—21no—
173#537C22×C312yes—
174#536C22×C314yes—
175#533C22×C312yes—
176#532C22×C311yescharacteristic
177#529—22no—
178C2×#125C22×C311yescharacteristic
179#520C22×C312yes—
180C4×#41C22×C311yescharacteristic
181#501C22×C314yes—
182#491C22×C314yes—
183#485C22×C314yes—
184#485C22×C314yes—
185#485C22×C314yes—
186#485C22×C314yes—
187#484C22×C314yes—
188#484C22×C314yes—
189#483C22×C314yes—
190#483C22×C314yes—
191#482C22×C314yes—
192#479C22×C314yes—
193#479C22×C314yes—
194#477C22×C312yes—
195#476C22×C312yes—
196#470C22×C314yes—
197#468C22×C314yes—
198#467C22×C314yes—
199#465C22×C314yes—
200#462C22×C312yes—
201#461C22×C312yes—
202#452C22×C314yes—
203#451C22×C314yes—
204#448C22×C314yes—
205#441C22×C314yes—
206#440C22×C314yes—
207#435C22×C314yes—
208#435C22×C314yes—
209#432C22×C312yes—
210#428C22×C314yes—
211#427C22×C314yes—
212#419C22×C314yes—
213C2×#103C22×C314yes—
214#408C22×C314yes—
215C3×#41—42no—
216C3×#41—42no—
217C3×#41—42no—
218C2×C3×Q8—22no—
219C2×C3×Q8—24no—
220C2×C3×Q8—44no—
221C2×C3×Q8—44no—
222C2×C3×Q8C2×D412yes—
223C2×C3×Q8C2414yes—
224C2×C3×Q8C2411yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4—24no—
227C2×C3×D4C2×D414yes—
228C2×C3×D4C2×D414yes—
229C2×C3×D4C2×D412yes—
230C2×C3×D4C2411yescharacteristic
231C22×C3×C4—22no—
232C22×C3×C4—24no—
233C22×C3×C4—24no—
234C22×C3×C4—21no—
235C22×C3×C4—22no—
236C22×C3×C4C2×D412yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2414yes—
239C3×Q16—44no—
240C3×SD16—44no—
241C3×M16—44no—
242C2×C3×C8—44no—
243C2×C3×C8C2414yes—
244C2×C3×C8C2414yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4C2×D414yes—
259C3×C4⋊C4C2414yes—
260C3×C4⋊C4C2412yes—
261C3×C4⋊C4C2414yes—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—24no—
271C3×C22⋊C4C2×D414yes—
272C3×C22⋊C4C2×D414yes—
273C3×C42—22no—
274C3×C42—21no—
275C3×C42—24no—
276C3×C42—24no—
277C3×C42—24no—
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D411yescharacteristic
280C3×C42C2412yes—
281C2×#41C23×C311yescharacteristic
282C2×#41C23×C312yes—
283C22×Q8C23×C314yes—
284C2×Q16C23×C314yes—
285C2×SD16C23×C314yes—
286C2×M16C23×C314yes—
287C22×C8C23×C314yes—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8C3×D411yescharacteristic
291C4⋊Q8C23×C312yes—
292C4⋊Q8C23×C312yes—
293#127C3×D411yescharacteristic
294#126—24no—
295#126—24no—
296#126—24no—
297#126—24no—
298#125—24no—
299#125—21no—
300#125—21no—
301#125—22no—
302#125C23×C314yes—
303#124—24no—
304#124—24no—
305#124C3×D412yes—
306#123—24no—
307#123—24no—
308#123—24no—
309#123—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8C23×C314yes—
315C22⋊Q8C23×C314yes—
316C22⋊Q8C23×C314yes—
317C22⋊Q8C23×C314yes—
318#121—22no—
319#121—24no—
320#121—24no—
321#121C3×D414yes—
322#121C23×C314yes—
323C4×Q8—22no—
324C4×Q8—24no—
325C4×Q8—22no—
326C4×Q8C23×C312yes—
327C4×Q8C23×C314yes—
328C4×D4—22no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—22no—
332C4×D4—24no—
333C4×D4C23×C312yes—
334C4×D4C23×C314yes—
335#117—22no—
336#117—24no—
337#117—21no—
338#117C23×C311yescharacteristic
339#117C23×C314yes—
340#117C23×C314yes—
341C2×C4⋊C4—22no—
342C2×C4⋊C4C23×C312yes—
343C2×C4⋊C4C23×C314yes—
344C2×C42—21no—
345C2×C42C23×C311yescharacteristic
346#107C23×C314yes—
347#107C23×C314yes—
348#107C23×C314yes—
349#106C23×C314yes—
350#106C23×C314yes—
351#106C23×C314yes—
352C4⋊C8C23×C312yes—
353C4⋊C8C23×C312yes—
354#103C23×C314yes—
355#103C23×C314yes—
356#103C23×C314yes—
357#103C23×C314yes—
358#103C23×C314yes—
359#103C23×C314yes—
360#103C23×C314yes—
361#102C23×C314yes—
362C22⋊C8C23×C314yes—
363C22⋊C8C23×C314yes—
364C22⋊C8C23×C314yes—
365#97C23×C314yes—
366C23×C3—21no—
367C23×C3C22×D414yes—
368C23×C3C22×D411yescharacteristic
369C3×Q8—24no—
370C3×Q8—22no—
371C3×Q8—24no—
372C3×Q8—42no—
373C3×Q8—44no—
374C3×Q8—84no—
375C3×D4—22no—
376C3×D4—44no—
377C3×D4—42no—
378C3×D4—44no—
379C3×D4—42no—
380C3×D4—84no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—21no—
387C2×C3×C4—24no—
388C2×C3×C4—24no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—82no—
392C2×C3×C4—84no—
393C2×C3×C4—82no—
394C2×C3×C4—84no—
395C2×C3×C4C22×D412yes—
396C2×C3×C4C22×D414yes—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D412yes—
399C2×C3×C4C2511yescharacteristic
400C3×C8—24no—
401C3×C8—24no—
402#41—42no—
403#41—42no—
404#41—42no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8C2×C3×D412yes—
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—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4C2×C3×D412yes—
424C22×C4C24×C311yescharacteristic
425C22×C4C24×C314yes—
426Q16—44no—
427SD16—44no—
428M16—44no—
429C2×C8—44no—
430C2×C8C24×C314yes—
431C2×C8C24×C314yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4C2×C3×D414yes—
446C4⋊C4C24×C314yes—
447C4⋊C4C24×C312yes—
448C4⋊C4C24×C314yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—21no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4C2×C3×D414yes—
459C22⋊C4C2×C3×D414yes—
460C42—22no—
461C42—21no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42C2×C3×D411yescharacteristic
466C42C2×C3×D411yescharacteristic
467C42C24×C312yes—
468C22×C3—22no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—21no—
473C22×C3—42no—
474C22×C3—41no—
475C22×C3C23×D411yescharacteristic
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—21no—
479C3×C4—24no—
480C3×C4—42no—
481C3×C4—44no—
482C3×C4—44no—
483C3×C4—41no—
484C3×C4—44no—
485C3×C4#58512yes—
486C23—21no—
487C23C22×C3×D414yes—
488C23C22×C3×D411yescharacteristic
489Q8—24no—
490Q8—22no—
491Q8—24no—
492Q8—42no—
493Q8—44no—
494Q8—84no—
495D4—22no—
496D4—42no—
497D4—44no—
498D4—44no—
499D4—42no—
500D4—84no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4C22×C3×D412yes—
516C2×C4C22×C3×D414yes—
517C2×C4C22×C3×D411yescharacteristic
518C2×C4C22×C3×D412yes—
519C2×C4C25×C311yescharacteristic
520C8—24no—
521C8—24no—
522C2×C3—41no—
523C2×C3—44no—
524C2×C3—81no—
525C2×C3#348511yescharacteristic
526C2×C3#358612yes—
527C22—24no—
528C22—21no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—41no—
533C22—42no—
534C22C23×C3×D411yescharacteristic
535C4—21no—
536C4—24no—
537C4—22no—
538C4—22no—
539C4—41no—
540C4—44no—
541C4—42no—
542C4—44no—
543C4—44no—
544C4C3×#58512yes—
545C3#6179811yescharacteristic, direct product factor, Sylow
546C2—44no—
547C2—41no—
548C2—81no—
549C2C3×#348511yescharacteristic
550C2C3×#358612yes—
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3512C2
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×#3350C2semidirect
4. C3×#3349C2semidirect
5. C3×#3320C2semidirect
6. C3×#3312C2semidirect
7. C3×#3236C2semidirect
8. C3×#3201C2semidirect
9. C3×#3116C2semidirect
10. C3×#3097C2semidirect
11. C3×#2980C2semidirect
12. #61798C3direct (N × H)
66. #3550C2×C3semidirect
68. #3350C2×C3semidirect
69. #3349C2×C3semidirect
70. #3320C2×C3semidirect
71. #3312C2×C3semidirect
72. #3236C2×C3semidirect
73. #3201C2×C3semidirect
74. #3116C2×C3semidirect
75. #3097C2×C3semidirect
76. #2980C2×C3semidirect
545. C3#61798direct (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×#35122
C3×#33502
C3×#33492
C3×#33202
C3×#33122
C3×#32362
C3×#32012
C3×#31162
C3×#30972
C3×#29802
#617983