Finite Group Databasebeta

← C3×#61844 #10680451

Subgroups — C3×#61844 #10680451

Order 768.

1756 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3522C211yesmaximal, characteristic
2C2×C3×Q82C211yesmaximal, characteristic, complemented
3C3×#3385C216yesmaximal, complemented
4C3×Q8×Q16C212yesmaximal, complemented
5C3×#3375C216yesmaximal, complemented
6C3×#3362C212yesmaximal, complemented
7C3×#3238C213yesmaximal, complemented
8C3×#3226C213yesmaximal, complemented
9C3×#3112C213yesmaximal, complemented
10C3×#3108C213yesmaximal, complemented
11C3×#2984C211yesmaximal, characteristic, complemented
12#61844C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#559C2213yes—
14C3×Q82—21no—
15C3×Q82—23no—
16C3×Q82C2216yes—
17C3×Q82C2212yes—
18C3×Q82C2211yescharacteristic
19C3×#554C2213yes—
20C3×#550C2211yescharacteristic
21C3×#548—21no—
22C3×#547—23no—
23C3×#542—23no—
24C3×#532C2213yes—
25C2×C3×C4⋊Q8—23no—
26C2×C3×C4⋊Q8C2213yes—
27C3×#529C2213yes—
28C3×#520—21no—
29C2×C3×C4×Q8—21no—
30C2×C3×C4×Q8C2213yes—
31C2×C3×C4×Q8C2211yescharacteristic
32C3×#501C2216yes—
33C3×#500C2216yes—
34C3×#489C2216yes—
35C3×#486C2216yes—
36C3×#483C2213yes—
37C3×#480C2213yes—
38C3×#480C2216yes—
39C3×#479C2216yes—
40C3×#477C2216yes—
41C3×#477C2216yes—
42C3×#476C2216yes—
43C3×#474C22112yes—
44C3×#464C2216yes—
45C3×#460C2216yes—
46C3×#451C2211yescharacteristic
47C3×#448C2213yes—
48C3×#446C2216yes—
49C3×C8×Q8C2212yes—
50C3×#440C22112yes—
51C3×C4×Q16C2216yes—
52C3×C4×SD16C2216yes—
53C3×#432C2213yes—
54C3×#431C2213yes—
55#3522C2×C311yescharacteristic
56C2×Q82C2×C311yescharacteristic, complemented
57#3385C2×C316yescomplemented
58Q8×Q16C2×C312yescomplemented
59#3375C2×C316yescomplemented
60#3362C2×C312yescomplemented
61#3238C2×C313yescomplemented
62#3226C2×C313yescomplemented
63#3112C2×C313yescomplemented
64#3108C2×C313yescomplemented
65#2984C2×C311yescharacteristic, complemented
66C2×C3×#41—21no—
67C2×C3×#41—23no—
68C22×C3×Q8—23no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×Q8C2311yescharacteristic
71C2×C3×Q16C2312yes—
72C2×C3×SD16C2316yes—
73C3×C4⋊Q8—26no—
74C3×C4⋊Q8—26no—
75C3×C4⋊Q8—212no—
76C3×C4⋊Q8—23no—
77C3×C4⋊Q8—23no—
78C3×C4⋊Q8—23no—
79C3×C4⋊Q8C2316yes—
80C3×C4⋊Q8C2316yes—
81C3×C4⋊Q8C2313yes—
82C3×#127C2313yes—
83C3×#126—23no—
84C3×#125C2313yes—
85C3×#124—23no—
86C3×#124—23no—
87C3×#124C2313yes—
88C3×#123—23no—
89C3×#123—26no—
90C3×C22⋊Q8—23no—
91C3×C22⋊Q8—26no—
92C3×C22⋊Q8C2311yescharacteristic
93C3×#121—23no—
94C3×#121—23no—
95C3×#121C2313yes—
96C3×C4×Q8—26no—
97C3×C4×Q8—22no—
98C3×C4×Q8—21no—
99C3×C4×Q8—23no—
100C3×C4×Q8—23no—
101C3×C4×Q8D411yescharacteristic
102C3×C4×Q8D413yes—
103C3×C4×Q8C2316yes—
104C3×C4×Q8C2316yes—
105C3×C4×Q8C2313yes—
106C3×C4×Q8C2311yescharacteristic
107C3×C4×Q8C2313yes—
108C3×C4×D4—23no—
109C3×C4×D4—23no—
110C3×C4×D4—23no—
111C3×C4×D4C2316yes—
112C3×C4×D4C2313yes—
113C3×#117—23no—
114C2×C3×C4⋊C4—23no—
115C2×C3×C4⋊C4—23no—
116C2×C3×C4⋊C4—23no—
117C2×C3×C4⋊C4D413yes—
118C2×C3×C4⋊C4C2313yes—
119C2×C3×C42—23no—
120C2×C3×C42C2313yes—
121C3×#107C2316yes—
122C3×#107C23112yes—
123C3×#106C2316yes—
124C3×C4⋊C8C2316yes—
125C3×C4⋊C8C2316yes—
126C3×#103C2312yes—
127C3×#103C2316yes—
128C3×#103C2316yes—
129C3×#102C2316yes—
130C3×#102C23112yes—
131C3×C22⋊C8C2311yescharacteristic
132C3×C22⋊C8C2313yes—
133C3×#97C2316yes—
134C3×C4×C8C2316yes—
135#559C22×C313yes—
136Q82—21no—
137Q82—23no—
138Q82C22×C312yes—
139Q82C22×C316yes—
140Q82C22×C311yescharacteristic
141#554C22×C313yes—
142#550C22×C311yescharacteristic
143#548—21no—
144#547—23no—
145#542—23no—
146#532C22×C313yes—
147C2×C4⋊Q8—23no—
148C2×C4⋊Q8C22×C313yes—
149#529C22×C313yes—
150#520—21no—
151C2×C4×Q8—21no—
152C2×C4×Q8C22×C313yes—
153C2×C4×Q8C22×C311yescharacteristic
154#501C22×C316yes—
155#500C22×C316yes—
156#489C22×C316yes—
157#486C22×C316yes—
158#483C22×C313yes—
159#480C22×C313yes—
160#480C22×C316yes—
161#479C22×C316yes—
162#477C22×C316yes—
163#477C22×C316yes—
164#476C22×C316yes—
165#474C22×C3112yes—
166#464C22×C316yes—
167#460C22×C316yes—
168#451C22×C311yescharacteristic
169#448C22×C313yes—
170#446C22×C316yes—
171C8×Q8C22×C312yes—
172#440C22×C3112yes—
173C4×Q16C22×C316yes—
174C4×SD16C22×C316yes—
175#432C22×C313yes—
176#431C22×C313yes—
177C3×#41—44no—
178C3×#41—86no—
179C2×C3×Q8—24no—
180C2×C3×Q8—24no—
181C2×C3×Q8—21no—
182C2×C3×Q8—23no—
183C2×C3×Q8—26no—
184C2×C3×Q8—26no—
185C2×C3×Q8—86no—
186C2×C3×Q8C2×D413yes—
187C2×C3×Q8C2×D411yescharacteristic
188C2×C3×Q8C2411yescharacteristic
189C2×C3×Q8C2412yes—
190C2×C3×D4—23no—
191C2×C3×D4—23no—
192C2×C3×D4—23no—
193C2×C3×D4C2413yes—
194C22×C3×C4—21no—
195C22×C3×C4—23no—
196C22×C3×C4—23no—
197C22×C3×C4—26no—
198C22×C3×C4C2×D413yes—
199C22×C3×C4C2×D413yes—
200C22×C3×C4C2411yescharacteristic
201C3×Q16—24no—
202C3×SD16—212no—
203C2×C3×C8C2412yes—
204C2×C3×C8C2416yes—
205C3×C4⋊C4—23no—
206C3×C4⋊C4—23no—
207C3×C4⋊C4—26no—
208C3×C4⋊C4—26no—
209C3×C4⋊C4—26no—
210C3×C4⋊C4—26no—
211C3×C4⋊C4—23no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—23no—
214C3×C4⋊C4—23no—
215C3×C4⋊C4C2×D413yes—
216C3×C4⋊C4C2×D413yes—
217C3×C4⋊C4C2×D416yes—
218C3×C4⋊C4C2416yes—
219C3×C4⋊C4C2411yescharacteristic
220C3×C4⋊C4C2413yes—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—23no—
223C3×C22⋊C4—23no—
224C3×C22⋊C4—23no—
225C3×C22⋊C4—23no—
226C3×C22⋊C4—26no—
227C3×C42—26no—
228C3×C42—26no—
229C3×C42—23no—
230C3×C42C2×D413yes—
231C3×C42C2×D413yes—
232C3×C42C2413yes—
233C3×C42C2413yes—
234C2×#41—21no—
235C2×#41—23no—
236C22×Q8—23no—
237C22×Q8C3×D411yescharacteristic
238C22×Q8C23×C311yescharacteristic
239C2×Q16C23×C312yes—
240C2×SD16C23×C316yes—
241C4⋊Q8—26no—
242C4⋊Q8—26no—
243C4⋊Q8—212no—
244C4⋊Q8—23no—
245C4⋊Q8—23no—
246C4⋊Q8—23no—
247C4⋊Q8C23×C316yes—
248C4⋊Q8C23×C316yes—
249C4⋊Q8C23×C313yes—
250#127C23×C313yes—
251#126—23no—
252#125C23×C313yes—
253#124—23no—
254#124—23no—
255#124C23×C313yes—
256#123—23no—
257#123—26no—
258C22⋊Q8—23no—
259C22⋊Q8—26no—
260C22⋊Q8C23×C311yescharacteristic
261#121—23no—
262#121—23no—
263#121C23×C313yes—
264C4×Q8—26no—
265C4×Q8—22no—
266C4×Q8—21no—
267C4×Q8—23no—
268C4×Q8—23no—
269C4×Q8C3×D411yescharacteristic
270C4×Q8C3×D413yes—
271C4×Q8C23×C316yes—
272C4×Q8C23×C316yes—
273C4×Q8C23×C313yes—
274C4×Q8C23×C311yescharacteristic
275C4×Q8C23×C313yes—
276C4×D4—23no—
277C4×D4—23no—
278C4×D4—23no—
279C4×D4C23×C313yes—
280C4×D4C23×C316yes—
281#117—23no—
282C2×C4⋊C4—23no—
283C2×C4⋊C4—23no—
284C2×C4⋊C4—23no—
285C2×C4⋊C4C3×D413yes—
286C2×C4⋊C4C23×C313yes—
287C2×C42—23no—
288C2×C42C23×C313yes—
289#107C23×C316yes—
290#107C23×C3112yes—
291#106C23×C316yes—
292C4⋊C8C23×C316yes—
293C4⋊C8C23×C316yes—
294#103C23×C312yes—
295#103C23×C316yes—
296#103C23×C316yes—
297#102C23×C316yes—
298#102C23×C3112yes—
299C22⋊C8C23×C311yescharacteristic
300C22⋊C8C23×C313yes—
301#97C23×C316yes—
302C4×C8C23×C316yes—
303C23×C3—23no—
304C23×C3C22×D411yescharacteristic
305C3×Q8—24no—
306C3×Q8—26no—
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—86no—
310C3×Q8—86no—
311C3×Q8—83no—
312C3×Q8#13714yes—
313C3×Q82−1+414yes—
314C3×D4—26no—
315C3×D4—46no—
316C3×D4—46no—
317C3×D4—83no—
318C2×C3×C4—24no—
319C2×C3×C4—26no—
320C2×C3×C4—22no—
321C2×C3×C4—26no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—26no—
325C2×C3×C4—23no—
326C2×C3×C4—21no—
327C2×C3×C4—46no—
328C2×C3×C4—46no—
329C2×C3×C4—86no—
330C2×C3×C4C22×D413yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D413yes—
333C2×C3×C4C22×D413yes—
334C2×C3×C4C22×D413yes—
335C2×C3×C4C2511yescharacteristic
336C3×C8—26no—
337C3×C8—22no—
338#41—44no—
339#41—86no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—21no—
343C2×Q8—23no—
344C2×Q8—26no—
345C2×Q8—26no—
346C2×Q8—86no—
347C2×Q8C2×C3×D413yes—
348C2×Q8C2×C3×D411yescharacteristic
349C2×Q8C24×C311yescharacteristic
350C2×Q8C24×C312yes—
351C2×D4—23no—
352C2×D4—23no—
353C2×D4—23no—
354C2×D4C24×C313yes—
355C22×C4—21no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—26no—
359C22×C4C2×C3×D413yes—
360C22×C4C2×C3×D413yes—
361C22×C4C24×C311yescharacteristic
362Q16—24no—
363SD16—212no—
364C2×C8C24×C312yes—
365C2×C8C24×C316yes—
366C4⋊C4—23no—
367C4⋊C4—23no—
368C4⋊C4—26no—
369C4⋊C4—26no—
370C4⋊C4—26no—
371C4⋊C4—26no—
372C4⋊C4—23no—
373C4⋊C4—21no—
374C4⋊C4—23no—
375C4⋊C4—23no—
376C4⋊C4C2×C3×D413yes—
377C4⋊C4C2×C3×D413yes—
378C4⋊C4C2×C3×D416yes—
379C4⋊C4C24×C316yes—
380C4⋊C4C24×C311yescharacteristic
381C4⋊C4C24×C313yes—
382C22⋊C4—21no—
383C22⋊C4—23no—
384C22⋊C4—23no—
385C22⋊C4—23no—
386C22⋊C4—23no—
387C22⋊C4—26no—
388C42—26no—
389C42—26no—
390C42—23no—
391C42C2×C3×D413yes—
392C42C2×C3×D413yes—
393C42C24×C313yes—
394C42C24×C313yes—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—43no—
398C22×C3—43no—
399C22×C3—43no—
400C22×C3#57612yes—
401C22×C3C23×D411yescharacteristic
402C3×C4—24no—
403C3×C4—21no—
404C3×C4—23no—
405C3×C4—23no—
406C3×C4—23no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—46no—
410C3×C4—43no—
411C3×C4C2×#13716yes—
412C3×C4C2×2−1+412yes—
413C23—23no—
414C23C22×C3×D411yescharacteristic
415Q8—24no—
416Q8—26no—
417Q8—22no—
418Q8—22no—
419Q8—86no—
420Q8—86no—
421Q8—83no—
422Q8C3×#13714yes—
423Q8C3×2−1+414yes—
424D4—26no—
425D4—46no—
426D4—46no—
427D4—83no—
428C2×C4—24no—
429C2×C4—26no—
430C2×C4—22no—
431C2×C4—26no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—26no—
435C2×C4—23no—
436C2×C4—21no—
437C2×C4—46no—
438C2×C4—46no—
439C2×C4—86no—
440C2×C4C22×C3×D413yes—
441C2×C4C22×C3×D411yescharacteristic
442C2×C4C22×C3×D413yes—
443C2×C4C22×C3×D413yes—
444C2×C4C22×C3×D413yes—
445C2×C4C25×C311yescharacteristic
446C8—26no—
447C8—22no—
448C2×C3—22no—
449C2×C3—83no—
450C2×C3#348611yescharacteristic
451C2×C3C22×#13711yescharacteristic
452C2×C3C2×#57611yescharacteristic
453C22—21no—
454C22—21no—
455C22—43no—
456C22—43no—
457C22—43no—
458C22C3×#57612yes—
459C22C23×C3×D411yescharacteristic
460C4—24no—
461C4—21no—
462C4—23no—
463C4—23no—
464C4—23no—
465C4—41no—
466C4—41no—
467C4—46no—
468C4—43no—
469C4C2×C3×#13716yes—
470C4C2×C3×2−1+412yes—
471C3#6184411yescharacteristic, direct product factor, Sylow
472C2—22no—
473C2—83no—
474C2C3×#348611yescharacteristic
475C2C22×C3×#13711yescharacteristic
476C2C2×C3×#57611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3522C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q82C2semidirect
3. C3×#3385C2semidirect
4. C3×Q8×Q16C2semidirect
5. C3×#3375C2semidirect
6. C3×#3362C2semidirect
7. C3×#3238C2semidirect
8. C3×#3226C2semidirect
9. C3×#3112C2semidirect
10. C3×#3108C2semidirect
11. C3×#2984C2semidirect
12. #61844C3direct (N × H)
56. C2×Q82C2×C3semidirect
57. #3385C2×C3semidirect
58. Q8×Q16C2×C3semidirect
59. #3375C2×C3semidirect
60. #3362C2×C3semidirect
61. #3238C2×C3semidirect
62. #3226C2×C3semidirect
63. #3112C2×C3semidirect
64. #3108C2×C3semidirect
65. #2984C2×C3semidirect
471. C3#61844direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35222
C2×C3×Q822
C3×#33852
C3×Q8×Q162
C3×#33752
C3×#33622
C3×#32382
C3×#32262
C3×#31122
C3×#31082
C3×#29842
#618443