Finite Group Databasebeta

← C3×#25706 #10644313

Subgroups — C3×#25706 #10644313

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#500C211yesmaximal, characteristic, complemented
3C2×C3×#477C211yesmaximal, characteristic, complemented
4C3×#2563C211yesmaximal, characteristic
5C3×#2036C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic
8C3×#1852C214yesmaximal, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25706C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125—21no—
16C2×C3×C22⋊Q8—22no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C4⋊C4C2212yes—
22C3×#500—24no—
23C3×#477—24no—
24C2×C3×#107C2214yes—
25C2×C3×#107C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2214yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#393—22no—
32C3×#390—21no—
33C3×#382—21no—
34C3×#380C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#340C2214yes—
38C2×#554C2×C311yescharacteristic
39C2×#500C2×C311yescharacteristic, complemented
40C2×#477C2×C311yescharacteristic, complemented
41#2563C2×C311yescharacteristic
42#2036C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic
45#1852C2×C314yescomplemented
46C4×#103C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—22no—
52C23×C3×C4C2312yes—
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8D414yes—
59C3×#125—28no—
60C3×C22⋊Q8—216no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—24no—
63C3×C4×Q8—22no—
64C3×C4×Q8D414yes—
65C3×C4×D4—28no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4D412yes—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#107—28no—
88C3×C4⋊C8—22no—
89C3×#103—24no—
90C3×#103—24no—
91C3×C22⋊C8—28no—
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—24no—
95#554—24no—
96#554—24no—
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C22×C312yes—
102C2×C4×Q8C22×C311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4C22×C311yescharacteristic
105C22×C4⋊C4C22×C312yes—
106#500—24no—
107#477—24no—
108C2×#107C22×C312yes—
109C2×#107C22×C314yes—
110C2×C4⋊C8C22×C311yescharacteristic
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×C22⋊C8C22×C314yes—
114C2×C4×C8C22×C311yescharacteristic
115#393—22no—
116#390—21no—
117#382—21no—
118#380C22×C314yes—
119C4×C4⋊C4C22×C311yescharacteristic
120#340C22×C312yes—
121#340C22×C314yes—
122C24×C3C2×D412yes—
123C2×C3×Q8—24no—
124C2×C3×Q8—21no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—42no—
128C2×C3×Q8—82no—
129C2×C3×Q8—82no—
130C2×C3×Q8C2×D412yes—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4Q1614yes—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—28no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—84no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—24no—
153C2×C3×C8#4114yes—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4C2×D412yes—
167C3×C4⋊C4C2×D414yes—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—24no—
170C3×C22⋊C4—84no—
171C3×C42—24no—
172C3×C42—42no—
173C3×C42C2×D412yes—
174C3×C42C2×D412yes—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8C23×C311yescharacteristic
178C22×D4C3×D411yescharacteristic
179C23×C4—22no—
180C23×C4C23×C312yes—
181C22×C8C23×C312yes—
182C22×C8C23×C312yes—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8C3×D414yes—
187#125—28no—
188C22⋊Q8—216no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C4×Q8—22no—
192C4×Q8C3×D414yes—
193C4×D4—28no—
194C4×D4—22no—
195C4×D4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C3×D411yescharacteristic
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C312yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4C3×D412yes—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C23×C311yescharacteristic
214#107—24no—
215#107—28no—
216C4⋊C8—22no—
217#103—24no—
218#103—24no—
219C22⋊C8—28no—
220C4×C8—22no—
221#95—21no—
222#95—24no—
223C23×C3—22no—
224C23×C3—24no—
225C23×C3—22no—
226C23×C3—24no—
227C23×C3C2×Q1614yes—
228C23×C3C22×D411yescharacteristic
229C3×Q8—22no—
230C3×Q8—22no—
231C3×Q8—42no—
232C3×Q8—84no—
233C3×Q8—84no—
234C3×D4—28no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—44no—
252C2×C3×C4—88no—
253C2×C3×C4#12012yes—
254C2×C3×C4#12012yes—
255C2×C3×C4C2×Q1612yes—
256C2×C3×C4#13712yes—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C2×#4112yes—
260C2×C3×C42−1+412yes—
261C3×C8—24no—
262C3×C8—42no—
263C24C2×C3×D412yes—
264C2×Q8—24no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8C2×C3×D412yes—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4C3×Q1614yes—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—84no—
286C22×C4C2×C3×D412yes—
287C22×C4C2×C3×D411yescharacteristic
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C24×C311yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—24no—
294C2×C8C3×#4114yes—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4C2×C3×D412yes—
308C4⋊C4C2×C3×D414yes—
309C22⋊C4—28no—
310C22⋊C4—24no—
311C22⋊C4—84no—
312C42—24no—
313C42—42no—
314C42C2×C3×D412yes—
315C42C2×C3×D412yes—
316C22×C3—28no—
317C22×C3—24no—
318C22×C3—44no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3C2×#12011yescharacteristic
322C22×C3#54712yes—
323C22×C3C22×Q1611yescharacteristic
324C22×C3C2×#13711yescharacteristic
325C22×C3#57612yes—
326C3×C4—22no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—42no—
330C3×C4—44no—
331C3×C4—82no—
332C3×C4—82no—
333C3×C4#45114yes—
334C3×C4#53211yescharacteristic
335C3×C4#53611yescharacteristic
336C3×C4#54812yes—
337C23—22no—
338C23—24no—
339C23—22no—
340C23—24no—
341C23C2×C3×Q1614yes—
342C23C22×C3×D411yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—42no—
346Q8—84no—
347Q8—84no—
348D4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—88no—
367C2×C4C3×#12012yes—
368C2×C4C3×#12012yes—
369C2×C4C2×C3×Q1612yes—
370C2×C4C3×#13712yes—
371C2×C4C22×C3×D411yescharacteristic
372C2×C4C22×C3×D411yescharacteristic
373C2×C4C2×C3×#4112yes—
374C2×C4C3×2−1+412yes—
375C8—24no—
376C8—42no—
377C2×C3—44no—
378C2×C3#243211yescharacteristic
379C2×C3C2×#45111yescharacteristic
380C2×C3#300811yescharacteristic
381C2×C3#333812yes—
382C2×C3#335712yes—
383C22—24no—
384C22—28no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22C2×C3×#12011yescharacteristic
389C22C3×#54712yes—
390C22C22×C3×Q1611yescharacteristic
391C22C2×C3×#13711yescharacteristic
392C22C3×#57612yes—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—44no—
398C4—82no—
399C4—82no—
400C4C3×#45114yes—
401C4C3×#53211yescharacteristic
402C4C3×#53611yescharacteristic
403C4C3×#54812yes—
404C3#2570611yescharacteristic, direct product factor, Sylow
405C2—44no—
406C2C3×#243211yescharacteristic
407C2C2×C3×#45111yescharacteristic
408C2C3×#300811yescharacteristic
409C2C3×#333812yes—
410C2C3×#335712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#500C2semidirect
3. C2×C3×#477C2semidirect
5. C3×#2036C2semidirect
6. C3×#1987C2semidirect
8. C3×#1852C2semidirect
9. C3×C4×#103C2semidirect
10. #25706C3direct (N × H)
39. C2×#500C2×C3semidirect
40. C2×#477C2×C3semidirect
42. #2036C2×C3semidirect
43. #1987C2×C3semidirect
45. #1852C2×C3semidirect
46. C4×#103C2×C3semidirect
404. C3#25706direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#5002
C2×C3×#4772
C3×#25632
C3×#20362
C3×#19872
C3×#19542
C3×#18522
C3×C4×#1032
#257063