Finite Group Databasebeta

← C3×#61825 #10680432

Subgroups — C3×#61825 #10680432

Order 768.

1990 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3497C211yesmaximal, characteristic, complemented
2C2×C3×#554C211yesmaximal, characteristic, complemented
3C3×#3384C218yesmaximal, complemented
4C3×#3366C218yesmaximal, complemented
5C3×#3234C214yesmaximal, complemented
6C3×#3227C212yesmaximal, complemented
7C3×#3109C214yesmaximal, complemented
8C3×#3100C212yesmaximal, complemented
9C3×#2983C211yesmaximal, characteristic, complemented
10#61825C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4—21no—
12C3×#554—24no—
13C3×#554C2218yes—
14C3×#550C2214yescomplemented
15C3×D4×Q8C2214yescomplemented
16C3×#548—24no—
17C3×#537—21no—
18C3×#537—22no—
19C3×#532C2212yes—
20C3×#532C2214yes—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C2×C3×#125C2212yes—
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8—22no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2212yes—
28C22×C3×C4⋊C4—21no—
29C3×#501C2218yescomplemented
30C3×#498C2214yescomplemented
31C3×#490C2218yescomplemented
32C3×#486C2214yescomplemented
33C3×#481C2218yes—
34C3×#480C2214yes—
35C3×#478C22116yescomplemented
36C3×#476C2218yes—
37C3×#476C2218yescomplemented
38C3×#474C2214yes—
39C3×#459C2214yes—
40C3×#459C2218yes—
41C3×#447C2214yes—
42C3×#446C2218yescomplemented
43C3×#442C22116yes—
44C3×C4×D8C2218yes—
45C3×#432C2214yes—
46C3×#431C2212yes—
47#3497C2×C311yescharacteristic, complemented
48C2×#554C2×C311yescharacteristic, complemented
49#3384C2×C318yescomplemented
50#3366C2×C318yescomplemented
51#3234C2×C314yescomplemented
52#3227C2×C312yescomplemented
53#3109C2×C314yescomplemented
54#3100C2×C312yescomplemented
55#2983C2×C311yescharacteristic, complemented
56C3×2−1+4—48no—
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C22×C3×Q8—24no—
61C22×C3×Q8—21no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—22no—
66C2×C3×D8C2318yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8C2312yes—
69C3×C4⋊Q8C2314yes—
70C3×C4⋊Q8C2314yes—
71C3×#127C2312yes—
72C3×#127C2314yes—
73C3×#125—24no—
74C3×#125C2318yes—
75C3×#124—24no—
76C3×#124—22no—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—216no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—28no—
81C3×#121—22no—
82C3×#121—24no—
83C3×#121—24no—
84C3×#121—28no—
85C3×#121C2314yes—
86C3×C4×Q8D414yes—
87C3×C4×Q8C2314yes—
88C3×C4×D4—24no—
89C3×C4×D4—22no—
90C3×C4×D4—24no—
91C3×C4×D4—22no—
92C3×C4×D4C2314yes—
93C3×C4×D4C2314yes—
94C3×C4×D4C2318yes—
95C3×C4×D4C2318yes—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—28no—
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4D412yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2312yes—
108C3×#107C2318yes—
109C3×#106C23116yes—
110C3×C4⋊C8C2314yes—
111C3×C4⋊C8C2318yes—
112C3×#102C2318yes—
113C3×#102C2318yes—
114C3×#102C23116yes—
115C3×C22⋊C8C2314yes—
116C3×#97C2318yes—
117C3×C4×C8C2314yes—
118C2×2−1+4—21no—
119#554—24no—
120#554C22×C318yes—
121#550C22×C314yescomplemented
122D4×Q8C22×C314yescomplemented
123#548—24no—
124#537—21no—
125#537—22no—
126#532C22×C312yes—
127#532C22×C314yes—
128C2×C4⋊Q8C22×C311yescharacteristic
129C2×#125C22×C312yes—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—22no—
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134C2×C4×D4C22×C312yes—
135C22×C4⋊C4—21no—
136#501C22×C318yescomplemented
137#498C22×C314yescomplemented
138#490C22×C318yescomplemented
139#486C22×C314yescomplemented
140#481C22×C318yes—
141#480C22×C314yes—
142#478C22×C3116yescomplemented
143#476C22×C318yes—
144#476C22×C318yescomplemented
145#474C22×C314yes—
146#459C22×C314yes—
147#459C22×C318yes—
148#447C22×C314yes—
149#446C22×C318yescomplemented
150#442C22×C3116yes—
151C4×D8C22×C318yes—
152#432C22×C314yes—
153#431C22×C312yes—
154C24×C3—21no—
155C3×#41—416no—
156C3×#41—416no—
157C3×#41—48no—
158C2×C3×Q8—24no—
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—24no—
162C2×C3×Q8—416no—
163C2×C3×Q8—82no—
164C2×C3×Q8C2×D414yes—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—24no—
172C2×C3×D4C2414yes—
173C2×C3×D4C2412yes—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—28no—
178C22×C3×C4—24no—
179C22×C3×C4—28no—
180C22×C3×C4—44no—
181C22×C3×C4—84no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C3×D8—216no—
188C2×C3×C8C2418yes—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4C2×D414yes—
197C3×C4⋊C4C2×D418yes—
198C3×C4⋊C4C2414yes—
199C3×C4⋊C4C2412yes—
200C3×C4⋊C4C2414yes—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—28no—
206C3×C42C2×D414yes—
207C3×C42C2×D412yes—
208C3×C42C2412yes—
209C3×C42C2414yes—
2102−1+4—48no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—22no—
214C22×Q8—24no—
215C22×Q8—21no—
216C22×Q8C3×D411yescharacteristic
217C22×D4C23×C311yescharacteristic
218C23×C4—21no—
219C23×C4—22no—
220C2×D8C23×C318yes—
221C4⋊Q8—22no—
222C4⋊Q8C23×C314yes—
223C4⋊Q8C23×C312yes—
224C4⋊Q8C23×C314yes—
225#127C23×C312yes—
226#127C23×C314yes—
227#125—24no—
228#125C23×C318yes—
229#124—24no—
230#124—22no—
231C22⋊Q8—28no—
232C22⋊Q8—216no—
233C22⋊Q8—24no—
234C22⋊Q8—28no—
235#121—22no—
236#121—24no—
237#121—24no—
238#121—28no—
239#121C23×C314yes—
240C4×Q8C3×D414yes—
241C4×Q8C23×C314yes—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4—24no—
245C4×D4—22no—
246C4×D4C23×C314yes—
247C4×D4C23×C318yes—
248C4×D4C23×C314yes—
249C4×D4C23×C318yes—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4C3×D411yescharacteristic
255C2×C4⋊C4C3×D412yes—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C312yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C42C23×C312yes—
261C2×C42C23×C311yescharacteristic
262#107C23×C318yes—
263#106C23×C3116yes—
264C4⋊C8C23×C314yes—
265C4⋊C8C23×C318yes—
266#102C23×C318yes—
267#102C23×C318yes—
268#102C23×C3116yes—
269C22⋊C8C23×C314yes—
270#97C23×C318yes—
271C4×C8C23×C314yes—
272C23×C3—24no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3C22×D411yescharacteristic
278C3×Q8—28no—
279C3×Q8—48no—
280C3×Q8—44no—
281C3×Q8—84no—
282C3×D4—24no—
283C3×D4—22no—
284C3×D4—28no—
285C3×D4—48no—
286C3×D4—48no—
287C3×D4—44no—
288C3×D4—84no—
289C3×D4—82no—
290C3×D42−1+414yescomplemented
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—416no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—48no—
302C2×C3×C4—48no—
303C2×C3×C4—88no—
304C2×C3×C4AGL(1,ℤ/8)14yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D412yes—
307C2×C3×C4C22×D414yes—
308C2×C3×C4C22×D414yes—
309C2×C3×C4C22×D412yes—
310C2×C3×C4C2511yescharacteristic
311C3×C8—28no—
312C24—21no—
313#41—416no—
314#41—416no—
315#41—48no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—416no—
321C2×Q8—82no—
322C2×Q8C2×C3×D414yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—21no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4C24×C314yes—
331C2×D4C24×C312yes—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C24×C311yescharacteristic
345D8—216no—
346C2×C8C24×C318yes—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4C2×C3×D414yes—
355C4⋊C4C2×C3×D418yes—
356C4⋊C4C24×C314yes—
357C4⋊C4C24×C312yes—
358C4⋊C4C24×C314yes—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—28no—
364C42C2×C3×D414yes—
365C42C2×C3×D412yes—
366C42C24×C312yes—
367C42C24×C314yes—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—24no—
371C22×C3—48no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3C2×AGL(1,ℤ/8)12yes—
378C22×C3C23×D411yescharacteristic
379C3×C4—24no—
380C3×C4—22no—
381C3×C4—21no—
382C3×C4—24no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—44no—
386C3×C4C2×AGL(1,ℤ/8)14yes—
387C3×C4C2×2−1+412yes—
388C23—24no—
389C23—22no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23C22×C3×D411yescharacteristic
394Q8—28no—
395Q8—48no—
396Q8—44no—
397Q8—84no—
398D4—24no—
399D4—22no—
400D4—28no—
401D4—48no—
402D4—48no—
403D4—44no—
404D4—84no—
405D4—82no—
406D4C3×2−1+414yescomplemented
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—416no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—88no—
420C2×C4C3×AGL(1,ℤ/8)14yes—
421C2×C4C22×C3×D411yescharacteristic
422C2×C4C22×C3×D412yes—
423C2×C4C22×C3×D414yes—
424C2×C4C22×C3×D414yes—
425C2×C4C22×C3×D412yes—
426C2×C4C25×C311yescharacteristic
427C8—28no—
428C2×C3—22no—
429C2×C3—44no—
430C2×C3—84no—
431C2×C3#348611yescharacteristic
432C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
433C2×C3#358511yescharacteristic
434C22—21no—
435C22—21no—
436C22—24no—
437C22—48no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22C2×C3×AGL(1,ℤ/8)12yes—
444C22C23×C3×D411yescharacteristic
445C4—21no—
446C4—24no—
447C4—24no—
448C4—22no—
449C4—44no—
450C4—42no—
451C4—44no—
452C4C2×C3×AGL(1,ℤ/8)14yes—
453C4C2×C3×2−1+412yes—
454C3#6182511yescharacteristic, direct product factor, Sylow
455C2—22no—
456C2—44no—
457C2—84no—
458C2C3×#348611yescharacteristic
459C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
460C2C3×#358511yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
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
1. C3×#3497C2semidirect
2. C2×C3×#554C2semidirect
3. C3×#3384C2semidirect
4. C3×#3366C2semidirect
5. C3×#3234C2semidirect
6. C3×#3227C2semidirect
7. C3×#3109C2semidirect
8. C3×#3100C2semidirect
9. C3×#2983C2semidirect
10. #61825C3direct (N × H)
14. C3×#550C22semidirect
15. C3×D4×Q8C22semidirect
29. C3×#501C22semidirect
30. C3×#498C22semidirect
31. C3×#490C22semidirect
32. C3×#486C22semidirect
35. C3×#478C22semidirect
37. C3×#476C22semidirect
42. C3×#446C22semidirect
47. #3497C2×C3semidirect
48. C2×#554C2×C3semidirect
49. #3384C2×C3semidirect
50. #3366C2×C3semidirect
51. #3234C2×C3semidirect
52. #3227C2×C3semidirect
53. #3109C2×C3semidirect
54. #3100C2×C3semidirect
55. #2983C2×C3semidirect
121. #550C22×C3semidirect
122. D4×Q8C22×C3semidirect
136. #501C22×C3semidirect
137. #498C22×C3semidirect
138. #490C22×C3semidirect
139. #486C22×C3semidirect
142. #478C22×C3semidirect
144. #476C22×C3semidirect
149. #446C22×C3semidirect
290. C3×D42−1+4semidirect
406. D4C3×2−1+4semidirect
454. C3#61825direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34972
C2×C3×#5542
C3×#33842
C3×#33662
C3×#32342
C3×#32272
C3×#31092
C3×#31002
C3×#29832
#618253