Finite Group Databasebeta

← C3×#25600 #10644207

Subgroups — C3×#25600 #10644207

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#2857C211yesmaximal, characteristic
5C3×#2001C214yesmaximal
6C3×#1990C211yesmaximal, characteristic, complemented
7C3×#1956C211yesmaximal, characteristic, complemented
8C3×#1878C214yesmaximal, complemented
9C3×C4×C22⋊C8C211yesmaximal, characteristic
10#25600C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2214yes—
12C3×#533—24no—
13C3×#533—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×#117—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#494—24no—
26C3×#487—24no—
27C2×C3×#103C2214yes—
28C2×C3×#103C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C3×#391C2212yes—
33C3×#389—21no—
34C3×#389—21no—
35C3×#384C2211yescharacteristic
36C3×#380—21no—
37C3×#380—21no—
38C3×#340C2214yes—
39C3×#336C2212yes—
40C2×#533C2×C311yescharacteristic
41C2×#494C2×C311yescharacteristic, complemented
42C2×#487C2×C311yescharacteristic, complemented
43#2857C2×C311yescharacteristic
44#2001C2×C314yes—
45#1990C2×C311yescharacteristic, complemented
46#1956C2×C311yescharacteristic, complemented
47#1878C2×C314yescomplemented
48C4×C22⋊C8C2×C311yescharacteristic
49C22×C3×Q8—21no—
50C22×C3×Q8C2312yes—
51C22×C3×Q8C2312yes—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C2×C3×Q16—28no—
56C2×C3×Q16—28no—
57C2×C3×Q16—44no—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×C4⋊Q8D414yes—
66C3×#125—28no—
67C3×C22⋊Q8—28no—
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—44no—
71C3×C4×Q8—28no—
72C3×C4×Q8—24no—
73C3×C4×Q8—24no—
74C3×#117—28no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C42—22no—
90C2×C3×C42—22no—
91C2×C3×C42—22no—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C3×#103—28no—
97C3×#103—44no—
98C3×C22⋊C8—24no—
99C3×C4×C8—22no—
100C3×C4×C8—22no—
101C3×#95—22no—
102C3×#95—22no—
103C22×Q16C22×C314yes—
104#533—24no—
105#533—24no—
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×C4⋊Q8C22×C311yescharacteristic
109C2×#125—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C22×C312yes—
112C2×C22⋊Q8C22×C312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C22×C312yes—
115C2×#117—21no—
116C22×C42C22×C311yescharacteristic
117#494—24no—
118#487—24no—
119C2×#103C22×C314yes—
120C2×#103C22×C314yes—
121C2×C22⋊C8C22×C312yes—
122C2×C4×C8C22×C311yescharacteristic
123C2×C4×C8C22×C311yescharacteristic
124#391C22×C312yes—
125#389—21no—
126#389—21no—
127#384C22×C311yescharacteristic
128#380—21no—
129#380—21no—
130#340C22×C314yes—
131#336C22×C312yes—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C2×C3×Q8—24no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×Q8C2×D414yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4Q1614yescomplemented
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×Q16—416no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8#4114yes—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4C2×D414yes—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—44no—
186C3×C42—28no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42C2×D412yes—
192C3×C42C2×D412yes—
193C22×Q8—21no—
194C22×Q8C23×C312yes—
195C22×Q8C23×C312yes—
196C23×C4C3×D411yescharacteristic
197C23×C4C3×D411yescharacteristic
198C23×C4C23×C311yescharacteristic
199C2×Q16—28no—
200C2×Q16—28no—
201C2×Q16—44no—
202C22×C8C23×C312yes—
203C22×C8C23×C312yes—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8C3×D414yes—
209C4⋊Q8C3×D414yes—
210#125—28no—
211C22⋊Q8—28no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—44no—
215C4×Q8—28no—
216C4×Q8—24no—
217C4×Q8—24no—
218#117—28no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C312yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C3×D411yescharacteristic
237C2×C42C3×D411yescharacteristic
238C2×C42C23×C311yescharacteristic
239C2×C42C23×C311yescharacteristic
240#103—28no—
241#103—44no—
242C22⋊C8—24no—
243C4×C8—22no—
244C4×C8—22no—
245#95—22no—
246#95—22no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3C2×Q1612yes—
254C23×C3C22×D411yescharacteristic
255C3×Q8—28no—
256C3×Q8—44no—
257C3×Q8—48no—
258C3×Q8—44no—
259C3×Q8—44no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—28no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C2×Q1612yes—
287C2×C3×C4C2×Q1612yes—
288C2×C3×C4#13512yes—
289C2×C3×C4#13712yes—
290C2×C3×C4#13712yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4H(5,𝔽2)12yes—
295C3×C8—24no—
296C3×C8—42no—
297C24C2×C3×D411yescharacteristic
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8C2×C3×D414yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4C3×Q1614yescomplemented
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C24×C311yescharacteristic
329Q16—416no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8C3×#4114yes—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4C2×C3×D414yes—
349C22⋊C4—28no—
350C22⋊C4—44no—
351C42—28no—
352C42—22no—
353C42—22no—
354C42—21no—
355C42—21no—
356C42C2×C3×D412yes—
357C42C2×C3×D412yes—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#53211yescharacteristic
367C22×C3#53511yescharacteristic
368C22×C3C22×Q1611yescharacteristic
369C22×C3C2×#13511yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—44no—
375C3×C4—41no—
376C3×C4—44no—
377C3×C4—82no—
378C3×C4—82no—
379C3×C4#45114yes—
380C3×C4#45214yes—
381C3×C4#54612yes—
382C3×C4#55012yes—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C2×C3×Q1612yes—
390C23C22×C3×D411yescharacteristic
391Q8—28no—
392Q8—48no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4C3×#12012yes—
421C2×C4C3×#12012yes—
422C2×C4C2×C3×Q1612yes—
423C2×C4C2×C3×Q1612yes—
424C2×C4C3×#13512yes—
425C2×C4C3×#13712yes—
426C2×C4C3×#13712yes—
427C2×C4C22×C3×D411yescharacteristic
428C2×C4C22×C3×D411yescharacteristic
429C2×C4C2×C3×#4112yes—
430C2×C4C3×H(5,𝔽2)12yes—
431C8—24no—
432C8—42no—
433C2×C3—42no—
434C2×C3#243111yescharacteristic
435C2×C3C2×#45111yescharacteristic
436C2×C3C2×#45211yescharacteristic
437C2×C3#317011yescharacteristic
438C2×C3#318511yescharacteristic
439C2×C3#320911yescharacteristic
440C2×C3#321011yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22C2×C3×#12011yescharacteristic
449C22C3×#53211yescharacteristic
450C22C3×#53511yescharacteristic
451C22C22×C3×Q1611yescharacteristic
452C22C2×C3×#13511yescharacteristic
453C22C2×C3×#13711yescharacteristic
454C22C2×C3×#13711yescharacteristic
455C4—22no—
456C4—41no—
457C4—44no—
458C4—41no—
459C4—44no—
460C4—82no—
461C4—82no—
462C4C3×#45114yes—
463C4C3×#45214yes—
464C4C3×#54612yes—
465C4C3×#55012yes—
466C3#2560011yescharacteristic, direct product factor, Sylow
467C2—42no—
468C2C3×#243111yescharacteristic
469C2C2×C3×#45111yescharacteristic
470C2C2×C3×#45211yescharacteristic
471C2C3×#317011yescharacteristic
472C2C3×#318511yescharacteristic
473C2C3×#320911yescharacteristic
474C2C3×#321011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#494C2semidirect
3. C2×C3×#487C2semidirect
6. C3×#1990C2semidirect
7. C3×#1956C2semidirect
8. C3×#1878C2semidirect
10. #25600C3direct (N × H)
41. C2×#494C2×C3semidirect
42. C2×#487C2×C3semidirect
45. #1990C2×C3semidirect
46. #1956C2×C3semidirect
47. #1878C2×C3semidirect
157. C22×C3×C4Q16semidirect
322. C22×C4C3×Q16semidirect
466. C3#25600direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4942
C2×C3×#4872
C3×#28572
C3×#20012
C3×#19902
C3×#19562
C3×#18782
C3×C4×C22⋊C82
#256003