Finite Group Databasebeta

← C3×#19085 #10637692

Subgroups — C3×#19085 #10637692

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3553C211yesmaximal, characteristic
2C3×#3181C212yesmaximal, complemented
3C3×#3166C211yesmaximal, characteristic, complemented
4C3×#1664C212yesmaximal, complemented
5C3×#1663C212yesmaximal, complemented
6C3×#1661C212yesmaximal, complemented
7C3×#1660C212yesmaximal, complemented
8C3×#1524C212yesmaximal, complemented
9C2×C3×#330C211yesmaximal, characteristic, complemented
10#19085C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#564C2211yescharacteristic
12C3×#563C2212yes—
13C3×#562C2211yescharacteristic
14C3×#554—22no—
15C3×#553—22no—
16C3×#552—22no—
17C3×#541—22no—
18C3×#539—22no—
19C3×#539—22no—
20C3×#533C2211yescharacteristic
21C3×#529C2212yes—
22C2×C3×#125C2211yescharacteristic
23C3×#501C2212yes—
24C3×#501C2212yes—
25C3×#501C2212yes—
26C3×#499—22no—
27C3×#497C2212yes—
28C3×#491C2212yes—
29C3×#489C2212yes—
30C3×#467—22no—
31C3×#465—22no—
32C3×#441—22no—
33C3×#440—22no—
34C3×#428—22no—
35C3×C4×M16C2211yescharacteristic
36C2×C3×#97C2212yes—
37C3×#330C2211yescharacteristic
38C3×#330C2212yes—
39C3×#330C2211yescharacteristic
40C3×#330C2212yes—
41C3×#330C2212yes—
42C3×#329C2212yes—
43C3×#329C2212yes—
44#3553C2×C311yescharacteristic
45#3181C2×C312yescomplemented
46#3166C2×C311yescharacteristic, complemented
47#1664C2×C312yescomplemented
48#1663C2×C312yescomplemented
49#1661C2×C312yescomplemented
50#1660C2×C312yescomplemented
51#1524C2×C312yescomplemented
52C2×#330C2×C311yescharacteristic, complemented
53C2×C3×Q16—22no—
54C2×C3×SD16—22no—
55C2×C3×M16—21no—
56C22×C3×C8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8C2312yes—
59C3×#126—22no—
60C3×#126—22no—
61C3×#126—22no—
62C3×#126—22no—
63C3×#126—22no—
64C3×#125—22no—
65C3×#125—22no—
66C3×#125—22no—
67C3×#125—22no—
68C3×#125—22no—
69C3×#125—22no—
70C3×#125—21no—
71C3×#125D412yes—
72C3×#125C2311yescharacteristic
73C3×#125C2311yescharacteristic
74C3×#125C2312yes—
75C3×#124C2312yes—
76C3×#123—22no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8D412yes—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121D412yes—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8D412yes—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4D412yes—
100C3×#117—22no—
101C3×#117—22no—
102C3×#117—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4D412yes—
106C2×C3×C42C2311yescharacteristic
107C3×#107—22no—
108C3×#107—22no—
109C3×#107—22no—
110C3×#106—22no—
111C3×#106—22no—
112C3×#106—22no—
113C3×#103—22no—
114C3×#103—22no—
115C3×#103—22no—
116C3×#102—22no—
117C3×#97—22no—
118C3×#97C2312yes—
119C3×#97C2312yes—
120C3×#97C2312yes—
121C3×C4×C8—21no—
122#564C22×C311yescharacteristic
123#563C22×C312yes—
124#562C22×C311yescharacteristic
125#554—22no—
126#553—22no—
127#552—22no—
128#541—22no—
129#539—22no—
130#539—22no—
131#533C22×C311yescharacteristic
132#529C22×C312yes—
133C2×#125C22×C311yescharacteristic
134#501C22×C312yes—
135#501C22×C312yes—
136#501C22×C312yes—
137#499—22no—
138#497C22×C312yes—
139#491C22×C312yes—
140#489C22×C312yes—
141#467—22no—
142#465—22no—
143#441—22no—
144#440—22no—
145#428—22no—
146C4×M16C22×C311yescharacteristic
147C2×#97C22×C312yes—
148#330C22×C311yescharacteristic
149#330C22×C312yes—
150#330C22×C312yes—
151#330C22×C312yes—
152#330C22×C311yescharacteristic
153#329C22×C312yes—
154#329C22×C312yes—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D412yes—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4C2×D412yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C3×Q16—44no—
168C3×SD16—44no—
169C3×M16—24no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—44no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—41no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42—41no—
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D412yes—
212C3×C42C2411yescharacteristic
213C2×Q16—22no—
214C2×SD16—22no—
215C2×M16—21no—
216C22×C8—22no—
217C4⋊Q8—22no—
218C4⋊Q8C23×C312yes—
219#126—22no—
220#126—22no—
221#126—22no—
222#126—22no—
223#126—22no—
224#125—22no—
225#125—22no—
226#125—22no—
227#125—22no—
228#125—22no—
229#125—22no—
230#125—21no—
231#125C3×D412yes—
232#125C23×C311yescharacteristic
233#125C23×C311yescharacteristic
234#125C23×C312yes—
235#124C23×C312yes—
236#123—22no—
237#123—22no—
238#123—22no—
239#123—22no—
240#123—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8C3×D412yes—
249#121—22no—
250#121—22no—
251#121C3×D412yes—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8C3×D412yes—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4C3×D412yes—
260#117—22no—
261#117—22no—
262#117—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C3×D412yes—
266C2×C42C23×C311yescharacteristic
267#107—22no—
268#107—22no—
269#107—22no—
270#106—22no—
271#106—22no—
272#106—22no—
273#103—22no—
274#103—22no—
275#103—22no—
276#102—22no—
277#97—22no—
278#97C23×C312yes—
279#97C23×C312yes—
280#97C23×C312yes—
281C4×C8—21no—
282C23×C3—22no—
283C23×C3#12011yescharacteristic
284C3×Q8—42no—
285C3×Q8—42no—
286C3×Q8—42no—
287C3×Q8—82no—
288C3×D4—42no—
289C3×D4—44no—
290C3×D4—82no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—82no—
300C2×C3×C4—82no—
301C2×C3×C4—82no—
302C2×C3×C4—82no—
303C2×C3×C4#12012yes—
304C2×C3×C4#12011yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D412yes—
307C3×C8—24no—
308C3×C8—24no—
309C3×C8—42no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8C2×C3×D412yes—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4C2×C3×D412yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4C2×C3×D411yescharacteristic
321C22×C4C2×C3×D412yes—
322Q16—44no—
323SD16—44no—
324M16—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—44no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4C2×C3×D412yes—
348C4⋊C4C2×C3×D412yes—
349C4⋊C4C2×C3×D412yes—
350C4⋊C4C2×C3×D412yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—41no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—41no—
365C42C2×C3×D411yescharacteristic
366C42C2×C3×D412yes—
367C42C24×C311yescharacteristic
368C22×C3—22no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3#45612yes—
373C22×C3C2×#12011yescharacteristic
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—82no—
384C23—22no—
385C23C3×#12011yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—42no—
389Q8—82no—
390D4—42no—
391D4—44no—
392D4—82no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4C3×#12012yes—
406C2×C4C3×#12011yescharacteristic
407C2×C4C22×C3×D411yescharacteristic
408C2×C4C22×C3×D412yes—
409C8—24no—
410C8—24no—
411C8—42no—
412C2×C3—22no—
413C2×C3—82no—
414C2×C3C2×#45611yescharacteristic
415C2×C3#301812yes—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22C3×#45612yes—
421C22C2×C3×#12011yescharacteristic
422C4—21no—
423C4—21no—
424C4—22no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—82no—
432C3#1908511yescharacteristic, direct product factor, Sylow
433C2—22no—
434C2—82no—
435C2C2×C3×#45611yescharacteristic
436C2C3×#301812yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3553C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3181C2semidirect
3. C3×#3166C2semidirect
4. C3×#1664C2semidirect
5. C3×#1663C2semidirect
6. C3×#1661C2semidirect
7. C3×#1660C2semidirect
8. C3×#1524C2semidirect
9. C2×C3×#330C2semidirect
10. #19085C3direct (N × H)
45. #3181C2×C3semidirect
46. #3166C2×C3semidirect
47. #1664C2×C3semidirect
48. #1663C2×C3semidirect
49. #1661C2×C3semidirect
50. #1660C2×C3semidirect
51. #1524C2×C3semidirect
52. C2×#330C2×C3semidirect
432. C3#19085direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35532
C3×#31812
C3×#31662
C3×#16642
C3×#16632
C3×#16612
C3×#16602
C3×#15242
C2×C3×#3302
#190853