Finite Group Databasebeta

← C3×#25690 #10644297

Subgroups — C3×#25690 #10644297

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic
2C3×#3166C211yesmaximal, characteristic, complemented
3C3×#3080C211yesmaximal, characteristic, complemented
4C3×#2641C211yesmaximal, characteristic, complemented
5C3×#2037C214yesmaximal, complemented
6C3×#1995C211yesmaximal, characteristic, complemented
7C3×#1965C211yesmaximal, characteristic, complemented
8C3×#1856C214yesmaximal, complemented
9C3×#1759C211yesmaximal, characteristic, complemented
10#25690C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—22no—
12C3×#550—22no—
13C3×#546—24no—
14C3×#540—22no—
15C3×#535C2212yes—
16C3×#533C2211yescharacteristic
17C3×#532—22no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C3×#518C2212yes—
21C3×C4×#41C2211yescharacteristic
22C3×#501—22no—
23C3×#499—22no—
24C3×#479—22no—
25C3×#478—22no—
26C3×#428C2214yes—
27C3×#428C2212yes—
28C3×#423C2211yescharacteristic
29C2×C3×#104C2212yes—
30C3×#417C2212yes—
31C2×C3×#100C2214yes—
32C3×C4×M16C2211yescharacteristic
33C3×#394—24no—
34C3×#390C2214yes—
35C3×#376C2211yescharacteristic
36C3×#339C2214yes—
37C3×#339C2212yes—
38#3508C2×C311yescharacteristic
39#3166C2×C311yescharacteristic, complemented
40#3080C2×C311yescharacteristic, complemented
41#2641C2×C311yescharacteristic, complemented
42#2037C2×C314yescomplemented
43#1995C2×C311yescharacteristic, complemented
44#1965C2×C311yescharacteristic, complemented
45#1856C2×C314yescomplemented
46#1759C2×C311yescharacteristic, complemented
47C2×C3×#41—24no—
48C2×C3×#41C2311yescharacteristic
49C22×C3×D4D414yes—
50C22×C3×D4C2312yes—
51C2×C3×M16C2312yes—
52C2×C3×M16C2312yes—
53C3×C4⋊Q8D412yes—
54C3×#127—22no—
55C3×#127—22no—
56C3×#127—21no—
57C3×#127—21no—
58C3×#125—22no—
59C3×#125—21no—
60C3×#125—21no—
61C3×#125D412yes—
62C3×#124—24no—
63C3×#124—22no—
64C3×#123—28no—
65C3×C22⋊Q8—22no—
66C3×#121—24no—
67C3×#121—28no—
68C3×#121—24no—
69C3×#120—24no—
70C3×C4×Q8—22no—
71C3×C4×Q8D412yes—
72C3×C4×D4—24no—
73C3×C4×D4—22no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C3×C4×D4D412yes—
77C3×#117—21no—
78C3×#117C2311yescharacteristic
79C3×#117C2312yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—28no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#107—22no—
88C3×#107—24no—
89C3×#106—22no—
90C3×#106—24no—
91C3×C4⋊C8—22no—
92C3×#104—24no—
93C3×#103—22no—
94C3×#102—22no—
95C3×#100—28no—
96C3×#97—21no—
97C3×C4×C8—21no—
98C3×#95—22no—
99#559—22no—
100#550—22no—
101#546—24no—
102#540—22no—
103#535C22×C312yes—
104#533C22×C311yescharacteristic
105#532—22no—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108#518C22×C312yes—
109C4×#41C22×C311yescharacteristic
110#501—22no—
111#499—22no—
112#479—22no—
113#478—22no—
114#428C22×C312yes—
115#428C22×C314yes—
116#423C22×C311yescharacteristic
117C2×#104C22×C312yes—
118#417C22×C312yes—
119C2×#100C22×C314yes—
120C4×M16C22×C311yescharacteristic
121#394—24no—
122#390C22×C314yes—
123#376C22×C311yescharacteristic
124#339C22×C312yes—
125#339C22×C314yes—
126C24×C3C2×D414yes—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—88no—
130C2×C3×Q8—22no—
131C2×C3×Q8C2×D411yescharacteristic
132C2×C3×D4—21no—
133C2×C3×D4—28no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—84no—
141C2×C3×D4C2×D411yescharacteristic
142C22×C3×C4—28no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2411yescharacteristic
149C3×M16—22no—
150C3×M16—24no—
151C3×M16#4114yescomplemented
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4C2×D412yes—
162C3×C4⋊C4C2×D414yes—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—28no—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—88no—
169C3×C42—22no—
170C3×C42—21no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D411yescharacteristic
176C3×C42C2×D412yes—
177C2×#41—24no—
178C2×#41C23×C311yescharacteristic
179C22×D4C3×D414yes—
180C22×D4C23×C312yes—
181C2×M16C23×C312yes—
182C2×M16C23×C312yes—
183C4⋊Q8C3×D412yes—
184#127—22no—
185#127—21no—
186#127—21no—
187#127—22no—
188#125—22no—
189#125—21no—
190#125—21no—
191#125C3×D412yes—
192#124—24no—
193#124—22no—
194#123—28no—
195C22⋊Q8—22no—
196#121—24no—
197#121—28no—
198#121—24no—
199#120—24no—
200C4×Q8—22no—
201C4×Q8C3×D412yes—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4C3×D412yes—
207#117—21no—
208#117C23×C311yescharacteristic
209#117C23×C312yes—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C23×C312yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C42—21no—
215C2×C42C23×C312yes—
216C2×C42C23×C311yescharacteristic
217#107—22no—
218#107—24no—
219#106—22no—
220#106—24no—
221C4⋊C8—22no—
222#104—24no—
223#103—22no—
224#102—22no—
225#100—28no—
226#97—21no—
227C4×C8—21no—
228#95—22no—
229C23×C3—24no—
230C23×C3—21no—
231C23×C3—28no—
232C23×C3—48no—
233C23×C3C22×D411yescharacteristic
234C3×Q8—21no—
235C3×Q8—21no—
236C3×Q8—44no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—48no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—44no—
244C3×D4—84no—
245C3×D4—88no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—42no—
250C2×C3×C4—44no—
251C2×C3×C4—42no—
252C2×C3×C4—44no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—88no—
257C2×C3×C4#12012yes—
258C2×C3×C4#12012yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C2×#4112yes—
262C2×C3×C42−1+412yes—
263C3×C8—24no—
264C3×C8—42no—
265C24C2×C3×D414yes—
266#41—22no—
267#41—22no—
268#41—88no—
269C2×Q8—22no—
270C2×Q8C2×C3×D411yescharacteristic
271C2×D4—21no—
272C2×D4—28no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—84no—
280C2×D4C2×C3×D411yescharacteristic
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4C2×C3×D412yes—
287C22×C4C24×C311yescharacteristic
288M16—22no—
289M16—24no—
290M16C3×#4114yescomplemented
291C2×C8—22no—
292C2×C8—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4C2×C3×D412yes—
301C4⋊C4C2×C3×D414yes—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—21no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—88no—
308C42—22no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42C2×C3×D411yescharacteristic
314C42C2×C3×D411yescharacteristic
315C42C2×C3×D412yes—
316C22×C3—22no—
317C22×C3—41no—
318C22×C3—44no—
319C22×C3—42no—
320C22×C3—48no—
321C22×C3—88no—
322C22×C3C2×#12011yescharacteristic
323C22×C3#54712yes—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—44no—
329C3×C4—82no—
330C3×C4—82no—
331C3×C4#53211yescharacteristic
332C3×C4#53611yescharacteristic
333C3×C4#54812yes—
334C23—24no—
335C23—21no—
336C23—28no—
337C23—48no—
338C23C22×C3×D411yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—44no—
342D4—21no—
343D4—21no—
344D4—48no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—44no—
349D4—84no—
350D4—88no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—88no—
362C2×C4C3×#12012yes—
363C2×C4C3×#12012yes—
364C2×C4C22×C3×D411yescharacteristic
365C2×C4C22×C3×D411yescharacteristic
366C2×C4C2×C3×#4112yes—
367C2×C4C3×2−1+412yes—
368C8—24no—
369C8—42no—
370C2×C3—22no—
371C2×C3—81no—
372C2×C3—84no—
373C2×C3#243211yescharacteristic
374C2×C3#301912yes—
375C22—22no—
376C22—41no—
377C22—44no—
378C22—42no—
379C22—48no—
380C22—88no—
381C22C2×C3×#12011yescharacteristic
382C22C3×#54712yes—
383C4—22no—
384C4—22no—
385C4—41no—
386C4—42no—
387C4—44no—
388C4—82no—
389C4—82no—
390C4C3×#53211yescharacteristic
391C4C3×#53611yescharacteristic
392C4C3×#54812yes—
393C3#2569011yescharacteristic, direct product factor, Sylow
394C2—22no—
395C2—84no—
396C2—81no—
397C2C3×#243211yescharacteristic
398C2C3×#301912yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3508C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3166C2semidirect
3. C3×#3080C2semidirect
4. C3×#2641C2semidirect
5. C3×#2037C2semidirect
6. C3×#1995C2semidirect
7. C3×#1965C2semidirect
8. C3×#1856C2semidirect
9. C3×#1759C2semidirect
10. #25690C3direct (N × H)
39. #3166C2×C3semidirect
40. #3080C2×C3semidirect
41. #2641C2×C3semidirect
42. #2037C2×C3semidirect
43. #1995C2×C3semidirect
44. #1965C2×C3semidirect
45. #1856C2×C3semidirect
46. #1759C2×C3semidirect
151. C3×M16#41semidirect
290. M16C3×#41semidirect
393. C3#25690direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35082
C3×#31662
C3×#30802
C3×#26412
C3×#20372
C3×#19952
C3×#19652
C3×#18562
C3×#17592
#256903