Finite Group Databasebeta

← C3×#25839 #10644446

Subgroups — C3×#25839 #10644446

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 410 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×#476C211yesmaximal, characteristic
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2059C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1862C214yesmaximal
9C3×C4×#102C211yesmaximal, characteristic
10#25839C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2212yes—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C4⋊C4—21no—
22C3×#500—24no—
23C3×#476—24no—
24C2×C3×#107C2214yes—
25C2×C3×#106C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#395—22no—
31C3×#391—21no—
32C3×#389C2214yes—
33C3×#384—21no—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C2×#554C2×C311yescharacteristic
39C2×#500C2×C311yescharacteristic, complemented
40C2×#476C2×C311yescharacteristic
41#2567C2×C311yescharacteristic, complemented
42#2059C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1984C2×C311yescharacteristic, complemented
45#1862C2×C314yes—
46C4×#102C2×C311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8D411yescharacteristic
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4—22no—
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8D414yes—
59C3×#125—24no—
60C3×#125—24no—
61C3×C22⋊Q8—28no—
62C3×C22⋊Q8—216no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C3×C4×D4—22no—
68C3×C4×D4D414yes—
69C2×C3×C4⋊C4—28no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#107—28no—
87C3×#106—24no—
88C3×C4⋊C8—22no—
89C3×#102—24no—
90C3×#102Q818yescomplemented
91C3×C4×C8—22no—
92C3×#95—21no—
93C3×#95—24no—
94#554—24no—
95#554—24no—
96C2×C4⋊Q8C22×C311yescharacteristic
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×#125C22×C312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×C4×Q8C22×C311yescharacteristic
102C2×C4×D4C22×C312yes—
103C2×C4×D4C22×C311yescharacteristic
104C22×C4⋊C4—21no—
105#500—24no—
106#476—24no—
107C2×#107C22×C314yes—
108C2×#106C22×C312yes—
109C2×C4⋊C8C22×C311yescharacteristic
110C2×#102C22×C312yes—
111C2×#102C22×C312yes—
112C2×C4×C8C22×C311yescharacteristic
113#395—22no—
114#391—21no—
115#389C22×C314yes—
116#384—21no—
117C4×C4⋊C4C22×C311yescharacteristic
118#340C22×C312yes—
119#340C22×C314yes—
120#336C22×C314yes—
121C24×C3—21no—
122C2×C3×Q8—22no—
123C2×C3×Q8—24no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—42no—
127C2×C3×Q8—82no—
128C2×C3×Q8—82no—
129C2×C3×D4—21no—
130C2×C3×D4—22no—
131C2×C3×D4C2×D412yes—
132C2×C3×D4C2×Q814yes—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—28no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—84no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—24no—
151C2×C3×C8C2×Q814yes—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—28no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
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⋊C4C2×D414yes—
166C3×C4⋊C4C2×Q814yes—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—24no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—42no—
172C3×C42C2×D412yes—
173C3×C42C2×D412yes—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8C3×D411yescharacteristic
177C22×D4C23×C311yescharacteristic
178C23×C4—21no—
179C23×C4—22no—
180C22×C8C23×C312yes—
181C22×C8C23×C312yes—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8C3×D414yes—
186#125—24no—
187#125—24no—
188C22⋊Q8—28no—
189C22⋊Q8—216no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—22no—
195C4×D4C3×D414yes—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
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×C4⋊C4C23×C312yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42C23×C312yes—
212C2×C42C23×C311yescharacteristic
213#107—28no—
214#106—24no—
215C4⋊C8—22no—
216#102—24no—
217#102C3×Q818yescomplemented
218C4×C8—22no—
219#95—21no—
220#95—24no—
221C23×C3—22no—
222C23×C3—24no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3C22×D411yescharacteristic
227C3×Q8—44no—
228C3×Q8—84no—
229C3×Q8—84no—
230C3×D4—22no—
231C3×D4—24no—
232C3×D4C4⋊Q814yes—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—44no—
247C2×C3×C4—48no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—88no—
251C2×C3×C4#12012yes—
252C2×C3×C4#12012yes—
253C2×C3×C4#13512yes—
254C2×C3×C4#13712yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C22×Q812yes—
258C2×C3×C42−1+412yes—
259C3×C8—24no—
260C3×C8—42no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—42no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×D4—21no—
270C2×D4—22no—
271C2×D4C2×C3×D412yes—
272C2×D4C2×C3×Q814yes—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4C2×C3×D412yes—
283C22×C4C2×C3×D412yes—
284C22×C4C2×C3×D411yescharacteristic
285C22×C4C2×C3×D411yescharacteristic
286C22×C4C24×C311yescharacteristic
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8C2×C3×Q814yes—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—84no—
304C4⋊C4C2×C3×D412yes—
305C4⋊C4C2×C3×D414yes—
306C4⋊C4C2×C3×Q814yes—
307C22⋊C4—28no—
308C22⋊C4—24no—
309C42—22no—
310C42—22no—
311C42—42no—
312C42C2×C3×D412yes—
313C42C2×C3×D412yes—
314C22×C3—24no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3C2×#12011yescharacteristic
320C22×C3D4×Q812yes—
321C22×C3C2×#13511yescharacteristic
322C22×C3C2×#13711yescharacteristic
323C22×C3#57612yes—
324C3×C4—22no—
325C3×C4—44no—
326C3×C4—42no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—44no—
330C3×C4—82no—
331C3×C4#45214yes—
332C3×C4C2×C4⋊Q811yescharacteristic
333C3×C4#53611yescharacteristic
334C3×C4D4×Q812yes—
335C23—22no—
336C23—24no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23C22×C3×D411yescharacteristic
341Q8—44no—
342Q8—84no—
343Q8—84no—
344D4—22no—
345D4—24no—
346D4C3×C4⋊Q814yes—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—88no—
365C2×C4C3×#12012yes—
366C2×C4C3×#12012yes—
367C2×C4C3×#13512yes—
368C2×C4C3×#13712yes—
369C2×C4C22×C3×D411yescharacteristic
370C2×C4C22×C3×D411yescharacteristic
371C2×C4C22×C3×Q812yes—
372C2×C4C3×2−1+412yes—
373C8—24no—
374C8—42no—
375C2×C3—44no—
376C2×C3#243411yescharacteristic
377C2×C3C2×#45211yescharacteristic
378C2×C3#300811yescharacteristic
379C2×C3#338012yes—
380C2×C3#338512yes—
381C22—24no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—44no—
386C22C2×C3×#12011yescharacteristic
387C22C3×D4×Q812yes—
388C22C2×C3×#13511yescharacteristic
389C22C2×C3×#13711yescharacteristic
390C22C3×#57612yes—
391C4—22no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—44no—
396C4—44no—
397C4—82no—
398C4C3×#45214yes—
399C4C2×C3×C4⋊Q811yescharacteristic
400C4C3×#53611yescharacteristic
401C4C3×D4×Q812yes—
402C3#2583911yescharacteristic, direct product factor, Sylow
403C2—44no—
404C2C3×#243411yescharacteristic
405C2C2×C3×#45211yescharacteristic
406C2C3×#300811yescharacteristic
407C2C3×#338012yes—
408C2C3×#338512yes—
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×#500C2semidirect
4. C3×#2567C2semidirect
5. C3×#2059C2semidirect
6. C3×#1987C2semidirect
7. C3×#1984C2semidirect
10. #25839C3direct (N × H)
39. C2×#500C2×C3semidirect
41. #2567C2×C3semidirect
42. #2059C2×C3semidirect
43. #1987C2×C3semidirect
44. #1984C2×C3semidirect
90. C3×#102Q8semidirect
217. #102C3×Q8semidirect
402. C3#25839direct (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×#4762
C3×#25672
C3×#20592
C3×#19872
C3×#19842
C3×#18622
C3×C4×#1022
#258393