Finite Group Databasebeta

← C3×#25573 #10644180

Subgroups — C3×#25573 #10644180

Order 768.

1170 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C3×#2542C211yesmaximal, characteristic
5C3×#2001C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic
8C3×#1872C214yesmaximal, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25573C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2212yes—
12C3×#554—24no—
13C3×#554—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C22⋊Q8C2214yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C22×C3×C4⋊C4C2212yes—
24C3×#487—24no—
25C3×#462—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2214yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×C22⋊C8C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#399—22no—
33C3×#390—21no—
34C3×#384—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2214yes—
37C3×#340C2212yes—
38C2×#554C2×C311yescharacteristic
39C2×#487C2×C311yescharacteristic, complemented
40C2×#462C2×C311yescharacteristic, complemented
41#2542C2×C311yescharacteristic
42#2001C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic
45#1872C2×C314yescomplemented
46C4×#103C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8C2312yes—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—22no—
52C23×C3×C4C2312yes—
53C2×C3×Q16—24no—
54C2×C3×Q16—44no—
55C22×C3×C8C2312yes—
56C22×C3×C8C2312yes—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8D414yes—
61C3×#125—28no—
62C3×C22⋊Q8—216no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8—28no—
66C3×C22⋊Q8—44no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C3×C4×D4—28no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4D412yes—
88C2×C3×C42—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C3×C4⋊C8—22no—
92C3×#103—24no—
93C3×#103—28no—
94C3×#103—24no—
95C3×C22⋊C8—44no—
96C3×C4×C8—22no—
97C3×#95—21no—
98C22×Q16C22×C312yes—
99#554—24no—
100#554—24no—
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×#125—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C22×C312yes—
106C2×C22⋊Q8C22×C314yes—
107C2×C4×Q8C22×C311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4C22×C311yescharacteristic
110C22×C4⋊C4C22×C312yes—
111#487—24no—
112#462—24no—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×#103C22×C314yes—
115C2×#103C22×C312yes—
116C2×#103C22×C312yes—
117C2×C22⋊C8C22×C314yes—
118C2×C4×C8C22×C311yescharacteristic
119#399—22no—
120#390—21no—
121#384—21no—
122C4×C4⋊C4C22×C311yescharacteristic
123#340C22×C312yes—
124#340C22×C314yes—
125C24×C3C2×D412yes—
126C2×C3×Q8—24no—
127C2×C3×Q8—21no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—24no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—82no—
135C2×C3×Q8C2×D412yes—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4Q1614yescomplemented
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—28no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—84no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C3×Q16—48no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—24no—
160C3×C4⋊C4—28no—
161C3×C4⋊C4—28no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—48no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4C2×D412yes—
172C3×C4⋊C4C2×D414yes—
173C3×C4⋊C4#4114yes—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—48no—
177C3×C42—24no—
178C3×C42—42no—
179C3×C42C2×D412yes—
180C3×C42C2×D412yes—
181C22×Q8—21no—
182C22×Q8C23×C312yes—
183C22×Q8C23×C311yescharacteristic
184C22×D4C3×D411yescharacteristic
185C23×C4—22no—
186C23×C4C23×C312yes—
187C2×Q16—24no—
188C2×Q16—44no—
189C22×C8C23×C312yes—
190C22×C8C23×C312yes—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8C3×D414yes—
195#125—28no—
196C22⋊Q8—216no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8—28no—
200C22⋊Q8—44no—
201C4×Q8—22no—
202C4×Q8C3×D414yes—
203C4×D4—28no—
204C4×D4—22no—
205C4×D4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C3×D411yescharacteristic
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C312yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4C3×D412yes—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C23×C311yescharacteristic
225C4⋊C8—22no—
226#103—24no—
227#103—28no—
228#103—24no—
229C22⋊C8—44no—
230C4×C8—22no—
231#95—21no—
232C23×C3—22no—
233C23×C3—24no—
234C23×C3—22no—
235C23×C3—24no—
236C23×C3C2×Q1614yes—
237C23×C3C22×D411yescharacteristic
238C3×Q8—22no—
239C3×Q8—22no—
240C3×Q8—42no—
241C3×Q8—42no—
242C3×Q8—42no—
243C3×Q8—44no—
244C3×Q8—84no—
245C3×D4—28no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—44no—
261C2×C3×C4—42no—
262C2×C3×C4—44no—
263C2×C3×C4—88no—
264C2×C3×C4#12012yes—
265C2×C3×C4#12012yes—
266C2×C3×C4C2×Q1612yes—
267C2×C3×C4#13712yes—
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C2×#4112yes—
271C2×C3×C4H(5,𝔽2)12yes—
272C3×C8—42no—
273C3×C8—42no—
274C24C2×C3×D412yes—
275C2×Q8—24no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—82no—
284C2×Q8C2×C3×D412yes—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4C3×Q1614yescomplemented
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—28no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—84no—
299C22×C4C2×C3×D412yes—
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C2×C3×D411yescharacteristic
302C22×C4C24×C311yescharacteristic
303Q16—48no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4C2×C3×D412yes—
321C4⋊C4C2×C3×D414yes—
322C4⋊C4C3×#4114yes—
323C22⋊C4—28no—
324C22⋊C4—24no—
325C22⋊C4—48no—
326C42—24no—
327C42—42no—
328C42C2×C3×D412yes—
329C42C2×C3×D412yes—
330C22×C3—24no—
331C22×C3—28no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3C2×#12011yescharacteristic
336C22×C3#54612yes—
337C22×C3C22×Q1611yescharacteristic
338C22×C3C2×#13711yescharacteristic
339C22×C3#57712yes—
340C3×C4—22no—
341C3×C4—44no—
342C3×C4—42no—
343C3×C4—42no—
344C3×C4—42no—
345C3×C4—44no—
346C3×C4—82no—
347C3×C4#45114yes—
348C3×C4#53211yescharacteristic
349C3×C4#53511yescharacteristic
350C3×C4#55012yes—
351C23—22no—
352C23—24no—
353C23—22no—
354C23—24no—
355C23C2×C3×Q1614yes—
356C23C22×C3×D411yescharacteristic
357Q8—22no—
358Q8—22no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—44no—
363Q8—84no—
364D4—28no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—88no—
383C2×C4C3×#12012yes—
384C2×C4C3×#12012yes—
385C2×C4C2×C3×Q1612yes—
386C2×C4C3×#13712yes—
387C2×C4C22×C3×D411yescharacteristic
388C2×C4C22×C3×D411yescharacteristic
389C2×C4C2×C3×#4112yes—
390C2×C4C3×H(5,𝔽2)12yes—
391C8—42no—
392C8—42no—
393C2×C3—44no—
394C2×C3#243111yescharacteristic
395C2×C3C2×#45111yescharacteristic
396C2×C3#300911yescharacteristic
397C2×C3#329912yes—
398C2×C3#333112yes—
399C22—24no—
400C22—28no—
401C22—42no—
402C22—42no—
403C22—44no—
404C22C2×C3×#12011yescharacteristic
405C22C3×#54612yes—
406C22C22×C3×Q1611yescharacteristic
407C22C2×C3×#13711yescharacteristic
408C22C3×#57712yes—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—44no—
414C4—44no—
415C4—82no—
416C4C3×#45114yes—
417C4C3×#53211yescharacteristic
418C4C3×#53511yescharacteristic
419C4C3×#55012yes—
420C3#2557311yescharacteristic, direct product factor, Sylow
421C2—44no—
422C2C3×#243111yescharacteristic
423C2C2×C3×#45111yescharacteristic
424C2C3×#300911yescharacteristic
425C2C3×#329912yes—
426C2C3×#333112yes—
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×#487C2semidirect
3. C2×C3×#462C2semidirect
5. C3×#2001C2semidirect
6. C3×#1987C2semidirect
8. C3×#1872C2semidirect
9. C3×C4×#103C2semidirect
10. #25573C3direct (N × H)
39. C2×#487C2×C3semidirect
40. C2×#462C2×C3semidirect
42. #2001C2×C3semidirect
43. #1987C2×C3semidirect
45. #1872C2×C3semidirect
46. C4×#103C2×C3semidirect
138. C2×C3×D4Q16semidirect
287. C2×D4C3×Q16semidirect
420. C3#25573direct (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×#4872
C2×C3×#4622
C3×#25422
C3×#20012
C3×#19872
C3×#19542
C3×#18722
C3×C4×#1032
#255733