Finite Group Databasebeta

← C3×#61838 #10680445

Subgroups — C3×#61838 #10680445

Order 768.

1700 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3523C211yesmaximal, characteristic
2C2×C3×Q82C211yesmaximal, characteristic, complemented
3C3×#3387C216yesmaximal, complemented
4C3×Q8×SD16C212yesmaximal, complemented
5C3×#3369C216yesmaximal, complemented
6C3×#3363C212yesmaximal, complemented
7C3×#3238C213yesmaximal
8C3×#3229C213yesmaximal, complemented
9C3×#3113C213yesmaximal, complemented
10C3×#3108C213yesmaximal
11C3×#2984C211yesmaximal, characteristic, complemented
12#61838C311yesmaximal, characteristic, direct product factor, Sylow
13C3×Q82—21no—
14C3×Q82—23no—
15C3×Q82C2216yes—
16C3×Q82C2212yes—
17C3×#557C2211yescharacteristic
18C3×#557C2213yes—
19C3×#556C2213yes—
20C3×#552—23no—
21C3×D4×Q8—21no—
22C3×D4×Q8C2211yescharacteristic
23C3×#541—23no—
24C3×#533C2213yes—
25C2×C3×C4⋊Q8—23no—
26C2×C3×C4⋊Q8C2213yes—
27C3×#529C2213yes—
28C3×#519—21no—
29C2×C3×C4×Q8—21no—
30C2×C3×C4×Q8C2213yes—
31C2×C3×C4×Q8C2211yescharacteristic
32C3×#501C2216yes—
33C3×#498C2216yes—
34C3×#489C2216yes—
35C3×#487C2216yes—
36C3×#483C2213yes—
37C3×#483C2216yes—
38C3×#480C2213yes—
39C3×#479C2216yes—
40C3×#477C2216yes—
41C3×#476C2216yes—
42C3×#475C22112yes—
43C3×#475C2216yes—
44C3×#464C2216yes—
45C3×#462C2216yes—
46C3×#451C2213yes—
47C3×#448C2211yescharacteristic
48C3×#446C2216yes—
49C3×C8×Q8C2212yes—
50C3×#441C22112yes—
51C3×C4×Q16C2216yes—
52C3×C4×SD16C2216yes—
53C3×#432C2213yes—
54C3×#431C2213yes—
55#3523C2×C311yescharacteristic
56C2×Q82C2×C311yescharacteristic, complemented
57#3387C2×C316yescomplemented
58Q8×SD16C2×C312yescomplemented
59#3369C2×C316yescomplemented
60#3363C2×C312yescomplemented
61#3238C2×C313yes—
62#3229C2×C313yescomplemented
63#3113C2×C313yescomplemented
64#3108C2×C313yes—
65#2984C2×C311yescharacteristic, complemented
66C22×C3×Q8—21no—
67C22×C3×Q8—23no—
68C22×C3×Q8D411yescharacteristic
69C22×C3×Q8C2311yescharacteristic
70C2×C3×Q16C2316yes—
71C2×C3×SD16C2312yes—
72C3×C4⋊Q8—23no—
73C3×C4⋊Q8—26no—
74C3×C4⋊Q8—26no—
75C3×C4⋊Q8—212no—
76C3×C4⋊Q8—23no—
77C3×C4⋊Q8—23no—
78C3×C4⋊Q8C2313yes—
79C3×C4⋊Q8C2316yes—
80C3×C4⋊Q8C2316yes—
81C3×C4⋊Q8C2313yes—
82C3×#126—26no—
83C3×#126—23no—
84C3×#125—26no—
85C3×#125—23no—
86C3×#125—23no—
87C3×#125C2313yes—
88C3×#124C2313yes—
89C3×#123—23no—
90C3×C22⋊Q8—23no—
91C3×C22⋊Q8—23no—
92C3×C22⋊Q8—23no—
93C3×C22⋊Q8C2313yes—
94C3×#121C2311yescharacteristic
95C3×C4×Q8—26no—
96C3×C4×Q8—22no—
97C3×C4×Q8—21no—
98C3×C4×Q8—23no—
99C3×C4×Q8—23no—
100C3×C4×Q8—23no—
101C3×C4×Q8—23no—
102C3×C4×Q8D411yescharacteristic
103C3×C4×Q8D413yes—
104C3×C4×Q8C2316yes—
105C3×C4×Q8C2316yes—
106C3×C4×Q8C2313yes—
107C3×C4×Q8C2316yes—
108C3×C4×Q8C2311yescharacteristic
109C3×C4×Q8C2313yes—
110C3×C4×D4—23no—
111C3×C4×D4C2313yes—
112C3×#117—23no—
113C3×#117—23no—
114C3×#117—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×#106C2316yes—
123C3×#106C23112yes—
124C3×C4⋊C8C2316yes—
125C3×C4⋊C8C2316yes—
126C3×#103C2316yes—
127C3×#103C2312yes—
128C3×#103C2316yes—
129C3×#103C23112yes—
130C3×#102C2316yes—
131C3×C22⋊C8C2311yescharacteristic
132C3×C22⋊C8C2313yes—
133C3×#97C2316yes—
134C3×C4×C8C2316yes—
135Q82—21no—
136Q82—23no—
137Q82C22×C312yes—
138Q82C22×C316yes—
139#557C22×C311yescharacteristic
140#557C22×C313yes—
141#556C22×C313yes—
142#552—23no—
143D4×Q8—21no—
144D4×Q8C22×C311yescharacteristic
145#541—23no—
146#533C22×C313yes—
147C2×C4⋊Q8—23no—
148C2×C4⋊Q8C22×C313yes—
149#529C22×C313yes—
150#519—21no—
151C2×C4×Q8—21no—
152C2×C4×Q8C22×C313yes—
153C2×C4×Q8C22×C311yescharacteristic
154#501C22×C316yes—
155#498C22×C316yes—
156#489C22×C316yes—
157#487C22×C316yes—
158#483C22×C313yes—
159#483C22×C316yes—
160#480C22×C313yes—
161#479C22×C316yes—
162#477C22×C316yes—
163#476C22×C316yes—
164#475C22×C316yes—
165#475C22×C3112yes—
166#464C22×C316yes—
167#462C22×C316yes—
168#451C22×C313yes—
169#448C22×C311yescharacteristic
170#446C22×C316yes—
171C8×Q8C22×C312yes—
172#441C22×C3112yes—
173C4×Q16C22×C316yes—
174C4×SD16C22×C316yes—
175#432C22×C313yes—
176#431C22×C313yes—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8—21no—
180C2×C3×Q8—23no—
181C2×C3×Q8—26no—
182C2×C3×Q8—26no—
183C2×C3×Q8—44no—
184C2×C3×Q8—86no—
185C2×C3×Q8C2×D413yes—
186C2×C3×Q8C2×D411yescharacteristic
187C2×C3×Q8C2413yes—
188C2×C3×Q8C2412yes—
189C2×C3×D4—21no—
190C2×C3×D4C2411yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4—23no—
193C22×C3×C4—23no—
194C22×C3×C4C2×D413yes—
195C22×C3×C4C2×D413yes—
196C22×C3×C4C2411yescharacteristic
197C3×Q16—212no—
198C3×SD16—24no—
199C2×C3×C8C2412yes—
200C2×C3×C8C2416yes—
201C3×C4⋊C4—26no—
202C3×C4⋊C4—26no—
203C3×C4⋊C4—26no—
204C3×C4⋊C4—26no—
205C3×C4⋊C4—23no—
206C3×C4⋊C4—23no—
207C3×C4⋊C4—23no—
208C3×C4⋊C4—23no—
209C3×C4⋊C4—23no—
210C3×C4⋊C4—23no—
211C3×C4⋊C4—23no—
212C3×C4⋊C4—26no—
213C3×C4⋊C4—23no—
214C3×C4⋊C4C2×D413yes—
215C3×C4⋊C4C2×D413yes—
216C3×C4⋊C4C2×D416yes—
217C3×C4⋊C4C2416yes—
218C3×C4⋊C4C2413yes—
219C3×C4⋊C4C2411yescharacteristic
220C3×C22⋊C4—23no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—23no—
223C3×C22⋊C4—23no—
224C3×C22⋊C4—23no—
225C3×C42—26no—
226C3×C42—26no—
227C3×C42—26no—
228C3×C42—23no—
229C3×C42C2×D413yes—
230C3×C42C2×D413yes—
231C3×C42C2413yes—
232C3×C42C2413yes—
233C22×Q8—21no—
234C22×Q8—23no—
235C22×Q8C3×D411yescharacteristic
236C22×Q8C23×C311yescharacteristic
237C2×Q16C23×C316yes—
238C2×SD16C23×C312yes—
239C4⋊Q8—23no—
240C4⋊Q8—26no—
241C4⋊Q8—26no—
242C4⋊Q8—212no—
243C4⋊Q8—23no—
244C4⋊Q8—23no—
245C4⋊Q8C23×C313yes—
246C4⋊Q8C23×C316yes—
247C4⋊Q8C23×C316yes—
248C4⋊Q8C23×C313yes—
249#126—26no—
250#126—23no—
251#125—26no—
252#125—23no—
253#125—23no—
254#125C23×C313yes—
255#124C23×C313yes—
256#123—23no—
257C22⋊Q8—23no—
258C22⋊Q8—23no—
259C22⋊Q8—23no—
260C22⋊Q8C23×C313yes—
261#121C23×C311yescharacteristic
262C4×Q8—26no—
263C4×Q8—22no—
264C4×Q8—21no—
265C4×Q8—23no—
266C4×Q8—23no—
267C4×Q8—23no—
268C4×Q8—23no—
269C4×Q8C3×D411yescharacteristic
270C4×Q8C3×D413yes—
271C4×Q8C23×C316yes—
272C4×Q8C23×C316yes—
273C4×Q8C23×C313yes—
274C4×Q8C23×C316yes—
275C4×Q8C23×C311yescharacteristic
276C4×Q8C23×C313yes—
277C4×D4—23no—
278C4×D4C23×C313yes—
279#117—23no—
280#117—23no—
281#117—23no—
282C2×C4⋊C4—23no—
283C2×C4⋊C4—23no—
284C2×C4⋊C4C3×D413yes—
285C2×C4⋊C4C23×C313yes—
286C2×C42—23no—
287C2×C42C23×C313yes—
288#107C23×C316yes—
289#106C23×C316yes—
290#106C23×C3112yes—
291C4⋊C8C23×C316yes—
292C4⋊C8C23×C316yes—
293#103C23×C316yes—
294#103C23×C312yes—
295#103C23×C316yes—
296#103C23×C3112yes—
297#102C23×C316yes—
298C22⋊C8C23×C311yescharacteristic
299C22⋊C8C23×C313yes—
300#97C23×C316yes—
301C4×C8C23×C316yes—
302C23×C3—21no—
303C23×C3C22×D411yescharacteristic
304C3×Q8—24no—
305C3×Q8—26no—
306C3×Q8—22no—
307C3×Q8—26no—
308C3×Q8—46no—
309C3×Q8—86no—
310C3×Q8—86no—
311C3×Q8#13714yes—
312C3×Q82−1+414yes—
313C3×D4—22no—
314C3×D4—42no—
315C2×C3×C4—24no—
316C2×C3×C4—26no—
317C2×C3×C4—22no—
318C2×C3×C4—26no—
319C2×C3×C4—22no—
320C2×C3×C4—23no—
321C2×C3×C4—21no—
322C2×C3×C4—23no—
323C2×C3×C4—26no—
324C2×C3×C4—46no—
325C2×C3×C4—46no—
326C2×C3×C4C22×D413yes—
327C2×C3×C4C22×D413yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D413yes—
330C2×C3×C4C22×D413yes—
331C2×C3×C4C2511yescharacteristic
332C3×C8—22no—
333C3×C8—26no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—21no—
337C2×Q8—23no—
338C2×Q8—26no—
339C2×Q8—26no—
340C2×Q8—44no—
341C2×Q8—86no—
342C2×Q8C2×C3×D413yes—
343C2×Q8C2×C3×D411yescharacteristic
344C2×Q8C24×C313yes—
345C2×Q8C24×C312yes—
346C2×D4—21no—
347C2×D4C24×C311yescharacteristic
348C22×C4—21no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4C2×C3×D413yes—
352C22×C4C2×C3×D413yes—
353C22×C4C24×C311yescharacteristic
354Q16—212no—
355SD16—24no—
356C2×C8C24×C312yes—
357C2×C8C24×C316yes—
358C4⋊C4—26no—
359C4⋊C4—26no—
360C4⋊C4—26no—
361C4⋊C4—26no—
362C4⋊C4—23no—
363C4⋊C4—23no—
364C4⋊C4—23no—
365C4⋊C4—23no—
366C4⋊C4—23no—
367C4⋊C4—23no—
368C4⋊C4—23no—
369C4⋊C4—26no—
370C4⋊C4—23no—
371C4⋊C4C2×C3×D413yes—
372C4⋊C4C2×C3×D413yes—
373C4⋊C4C2×C3×D416yes—
374C4⋊C4C24×C316yes—
375C4⋊C4C24×C313yes—
376C4⋊C4C24×C311yescharacteristic
377C22⋊C4—23no—
378C22⋊C4—21no—
379C22⋊C4—23no—
380C22⋊C4—23no—
381C22⋊C4—23no—
382C42—26no—
383C42—26no—
384C42—26no—
385C42—23no—
386C42C2×C3×D413yes—
387C42C2×C3×D413yes—
388C42C24×C313yes—
389C42C24×C313yes—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3#57712yes—
396C22×C3C23×D411yescharacteristic
397C3×C4—24no—
398C3×C4—21no—
399C3×C4—23no—
400C3×C4—23no—
401C3×C4—23no—
402C3×C4—43no—
403C3×C4—43no—
404C3×C4—46no—
405C3×C4—41no—
406C3×C4C2×#13716yes—
407C3×C4C2×2−1+412yes—
408C23—21no—
409C23C22×C3×D411yescharacteristic
410Q8—24no—
411Q8—26no—
412Q8—22no—
413Q8—26no—
414Q8—46no—
415Q8—86no—
416Q8—86no—
417Q8C3×#13714yes—
418Q8C3×2−1+414yes—
419D4—22no—
420D4—42no—
421C2×C4—24no—
422C2×C4—26no—
423C2×C4—22no—
424C2×C4—26no—
425C2×C4—22no—
426C2×C4—23no—
427C2×C4—21no—
428C2×C4—23no—
429C2×C4—26no—
430C2×C4—46no—
431C2×C4—46no—
432C2×C4C22×C3×D413yes—
433C2×C4C22×C3×D413yes—
434C2×C4C22×C3×D411yescharacteristic
435C2×C4C22×C3×D413yes—
436C2×C4C22×C3×D413yes—
437C2×C4C25×C311yescharacteristic
438C8—22no—
439C8—26no—
440C2×C3—22no—
441C2×C3—81no—
442C2×C3#348611yescharacteristic
443C2×C3C22×#13711yescharacteristic
444C2×C3C2×#57711yescharacteristic
445C22—21no—
446C22—21no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22C3×#57712yes—
451C22C23×C3×D411yescharacteristic
452C4—24no—
453C4—21no—
454C4—23no—
455C4—23no—
456C4—23no—
457C4—43no—
458C4—43no—
459C4—46no—
460C4—41no—
461C4C2×C3×#13716yes—
462C4C2×C3×2−1+412yes—
463C3#6183811yescharacteristic, direct product factor, Sylow
464C2—22no—
465C2—81no—
466C2C3×#348611yescharacteristic
467C2C22×C3×#13711yescharacteristic
468C2C2×C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#121C23
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×#3387C2semidirect
4. C3×Q8×SD16C2semidirect
5. C3×#3369C2semidirect
6. C3×#3363C2semidirect
8. C3×#3229C2semidirect
9. C3×#3113C2semidirect
11. C3×#2984C2semidirect
12. #61838C3direct (N × H)
56. C2×Q82C2×C3semidirect
57. #3387C2×C3semidirect
58. Q8×SD16C2×C3semidirect
59. #3369C2×C3semidirect
60. #3363C2×C3semidirect
62. #3229C2×C3semidirect
63. #3113C2×C3semidirect
65. #2984C2×C3semidirect
463. C3#61838direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35232
C2×C3×Q822
C3×#33872
C3×Q8×SD162
C3×#33692
C3×#33632
C3×#32382
C3×#32292
C3×#31132
C3×#31082
C3×#29842
#618383