Finite Group Databasebeta

← C3×#25646 #10644253

Subgroups — C3×#25646 #10644253

Order 768.

1034 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2042C214yesmaximal
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1865C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic
10#25646C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#556—24no—
12C3×#556—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#126—22no—
16C2×C3×#124C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C4×Q8C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#117—21no—
22C3×#487—24no—
23C3×#476—24no—
24C2×C3×#106C2212yes—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2214yes—
27C2×C3×#103C2214yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#395—22no—
32C3×#391—21no—
33C3×#389C2214yes—
34C3×#384—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#556C2×C311yescharacteristic
39C2×#487C2×C311yescharacteristic, complemented
40C2×#476C2×C311yescharacteristic
41#2567C2×C311yescharacteristic, complemented
42#2042C2×C314yes—
43#1984C2×C311yescharacteristic, complemented
44#1981C2×C311yescharacteristic, complemented
45#1865C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8D411yescharacteristic
49C22×C3×Q8C2312yes—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8D414yes—
58C3×#126—216no—
59C3×#124—24no—
60C3×#124—24no—
61C3×C22⋊Q8—28no—
62C3×C4×Q8—24no—
63C3×C4×Q8—24no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×D4—22no—
67C3×C4×D4D414yes—
68C3×#117—28no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D412yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—22no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#106—24no—
87C3×C4⋊C8—22no—
88C3×#103—28no—
89C3×#103—28no—
90C3×#102—24no—
91C3×#102—24no—
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—24no—
95#556—24no—
96#556—24no—
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×#126—22no—
100C2×#124C22×C312yes—
101C2×C22⋊Q8—21no—
102C2×C4×Q8C22×C312yes—
103C2×C4×Q8C22×C311yescharacteristic
104C2×C4×D4C22×C311yescharacteristic
105C2×#117—21no—
106#487—24no—
107#476—24no—
108C2×#106C22×C312yes—
109C2×C4⋊C8C22×C311yescharacteristic
110C2×#103C22×C314yes—
111C2×#103C22×C314yes—
112C2×#102C22×C312yes—
113C2×#102C22×C312yes—
114C2×C4×C8C22×C311yescharacteristic
115#395—22no—
116#391—21no—
117#389C22×C314yes—
118#384—21no—
119C4×C4⋊C4C22×C311yescharacteristic
120#340C22×C312yes—
121#336C22×C314yes—
122C24×C3—21no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—24no—
128C2×C3×Q8—24no—
129C2×C3×Q8—42no—
130C2×C3×Q8—82no—
131C2×C3×Q8C2×D414yes—
132C2×C3×D4—21no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4C2×D412yes—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2411yescharacteristic
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—24no—
152C2×C3×C8—22no—
153C3×C4⋊C4—28no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4—84no—
164C3×C4⋊C4C2×D412yes—
165C3×C4⋊C4#4114yes—
166C3×C22⋊C4—28no—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—24no—
169C3×C42—28no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—42no—
173C3×C42C2×D412yes—
174C3×C42C2×D412yes—
175C22×Q8—21no—
176C22×Q8C3×D411yescharacteristic
177C22×Q8C23×C312yes—
178C22×D4C23×C311yescharacteristic
179C23×C4—21no—
180C22×C8C23×C312yes—
181C22×C8C23×C312yes—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8C3×D414yes—
186#126—216no—
187#124—24no—
188#124—24no—
189C22⋊Q8—28no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×D4—22no—
195C4×D4C3×D414yes—
196#117—28no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4C3×D411yescharacteristic
203C2×C4⋊C4C3×D412yes—
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C312yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C23×C312yes—
213C2×C42C23×C311yescharacteristic
214#106—24no—
215C4⋊C8—22no—
216#103—28no—
217#103—28no—
218#102—24no—
219#102—24no—
220C4×C8—22no—
221#95—21no—
222#95—24no—
223C23×C3—22no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3C22×D411yescharacteristic
229C3×Q8—28no—
230C3×Q8—42no—
231C3×Q8—42no—
232C3×Q8—44no—
233C3×Q8—84no—
234C3×D4—22no—
235C3×D4—22no—
236C3×D4—42no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—48no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4#12012yes—
255C2×C3×C4#12012yes—
256C2×C3×C4#13512yes—
257C2×C3×C4#13712yes—
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4112yes—
261C2×C3×C42−1+412yes—
262C3×C8—42no—
263C3×C8—42no—
264C24—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—42no—
272C2×Q8—82no—
273C2×Q8C2×C3×D414yes—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4C2×C3×D412yes—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C2×C3×D412yes—
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×C8—22no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—84no—
306C4⋊C4C2×C3×D412yes—
307C4⋊C4C3×#4114yes—
308C22⋊C4—28no—
309C22⋊C4—28no—
310C22⋊C4—24no—
311C42—28no—
312C42—22no—
313C42—22no—
314C42—42no—
315C42C2×C3×D412yes—
316C42C2×C3×D412yes—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3C2×#12011yescharacteristic
323C22×C3#54812yes—
324C22×C3C2×#13511yescharacteristic
325C22×C3C2×#13711yescharacteristic
326C22×C3#57712yes—
327C3×C4—22no—
328C3×C4—44no—
329C3×C4—44no—
330C3×C4—42no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—82no—
334C3×C4#45214yes—
335C3×C4#53211yescharacteristic
336C3×C4#53611yescharacteristic
337C3×C4#54712yes—
338C23—22no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—42no—
343C23C22×C3×D411yescharacteristic
344Q8—28no—
345Q8—44no—
346Q8—42no—
347Q8—42no—
348Q8—84no—
349D4—22no—
350D4—22no—
351D4—42no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4C3×#12012yes—
370C2×C4C3×#12012yes—
371C2×C4C3×#13512yes—
372C2×C4C3×#13712yes—
373C2×C4C22×C3×D411yescharacteristic
374C2×C4C22×C3×D411yescharacteristic
375C2×C4C2×C3×#4112yes—
376C2×C4C3×2−1+412yes—
377C8—42no—
378C8—42no—
379C2×C3—82no—
380C2×C3#243211yescharacteristic
381C2×C3C2×#45211yescharacteristic
382C2×C3#301011yescharacteristic
383C2×C3#336012yes—
384C2×C3#336912yes—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—82no—
389C22—82no—
390C22C2×C3×#12011yescharacteristic
391C22C3×#54812yes—
392C22C2×C3×#13511yescharacteristic
393C22C2×C3×#13711yescharacteristic
394C22C3×#57712yes—
395C4—22no—
396C4—44no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—44no—
401C4—82no—
402C4C3×#45214yes—
403C4C3×#53211yescharacteristic
404C4C3×#53611yescharacteristic
405C4C3×#54712yes—
406C3#2564611yescharacteristic, direct product factor, Sylow
407C2—82no—
408C2C3×#243211yescharacteristic
409C2C2×C3×#45211yescharacteristic
410C2C3×#301011yescharacteristic
411C2C3×#336012yes—
412C2C3×#336912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#487C2semidirect
4. C3×#2567C2semidirect
6. C3×#1984C2semidirect
7. C3×#1981C2semidirect
8. C3×#1865C2semidirect
10. #25646C3direct (N × H)
39. C2×#487C2×C3semidirect
41. #2567C2×C3semidirect
43. #1984C2×C3semidirect
44. #1981C2×C3semidirect
45. #1865C2×C3semidirect
406. C3#25646direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4872
C2×C3×#4762
C3×#25672
C3×#20422
C3×#19842
C3×#19812
C3×#18652
C3×C4×#1022
#256463