Finite Group Databasebeta

← C3×#25818 #10644425

Subgroups — C3×#25818 #10644425

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#501C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic
4C3×#2058C212yesmaximal, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#1992C212yesmaximal, complemented
7C3×#1854C214yesmaximal
8C3×#1749C211yesmaximal, characteristic
9#25818C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#533—28no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125C2212yes—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×#117C2212yes—
21C22×C3×C42C2211yescharacteristic
22C3×#501—28no—
23C2×C3×#107C2214yes—
24C2×C3×#106C2214yes—
25C2×C3×C22⋊C8C2212yes—
26C2×C3×#97C2212yes—
27C3×#391—21no—
28C3×#389—21no—
29C3×#389C2212yes—
30C3×#384C2211yescharacteristic
31C3×#380—21no—
32C3×#380C2212yes—
33C3×#340C2214yes—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#533C2×C311yescharacteristic
37C2×#501C2×C312yescomplemented
38#2857C2×C311yescharacteristic
39#2058C2×C312yescomplemented
40#2057C2×C312yescomplemented
41#1992C2×C312yescomplemented
42#1854C2×C314yes—
43#1749C2×C311yescharacteristic
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×C3×C8C2314yes—
51C3×C4⋊Q8—24no—
52C3×C4⋊Q8—24no—
53C3×C4⋊Q8—24no—
54C3×#125—24no—
55C3×#125D418yes—
56C3×C22⋊Q8—28no—
57C3×C22⋊Q8—28no—
58C3×C22⋊Q8—44no—
59C3×C4×Q8—28no—
60C3×C4×Q8—28no—
61C3×#117—24no—
62C3×#117—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—24no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#107—28no—
84C3×#106—28no—
85C3×C22⋊C8Q818yescomplemented
86C3×#97—24no—
87C3×#95—22no—
88C3×#95—22no—
89#533—28no—
90C2×C4⋊Q8C22×C311yescharacteristic
91C2×C4⋊Q8C22×C311yescharacteristic
92C2×C4⋊Q8C22×C311yescharacteristic
93C2×#125C22×C312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×#117C22×C312yes—
100C22×C42C22×C311yescharacteristic
101#501—28no—
102C2×#107C22×C314yes—
103C2×#106C22×C314yes—
104C2×C22⋊C8C22×C312yes—
105C2×#97C22×C312yes—
106#391—21no—
107#389—21no—
108#389C22×C312yes—
109#384C22×C311yescharacteristic
110#380—21no—
111#380C22×C312yes—
112#340C22×C314yes—
113#340C22×C314yes—
114#336C22×C312yes—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—24no—
117C2×C3×Q8—24no—
118C2×C3×Q8—42no—
119C2×C3×Q8—42no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×Q814yes—
143C22×C3×C4C2411yescharacteristic
144C2×C3×C8—24no—
145C2×C3×C8—24no—
146C2×C3×C8C2×Q818yes—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—28no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—44no—
159C3×C4⋊C4C2×D414yes—
160C3×C4⋊C4C2×D414yes—
161C3×C22⋊C4—28no—
162C3×C22⋊C4—44no—
163C3×C42—28no—
164C3×C42—22no—
165C3×C42—22no—
166C3×C42—22no—
167C3×C42C2×D414yes—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C23×C4C3×D411yescharacteristic
172C23×C4C3×D411yescharacteristic
173C23×C4C23×C311yescharacteristic
174C22×C8C23×C314yes—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178#125—24no—
179#125C3×D418yes—
180C22⋊Q8—28no—
181C22⋊Q8—28no—
182C22⋊Q8—44no—
183C4×Q8—28no—
184C4×Q8—28no—
185#117—24no—
186#117—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C23×C312yes—
196C2×C4⋊C4C23×C312yes—
197C2×C4⋊C4C23×C312yes—
198C2×C4⋊C4C23×C312yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42C3×D411yescharacteristic
204C2×C42C3×D411yescharacteristic
205C2×C42C23×C311yescharacteristic
206C2×C42C23×C311yescharacteristic
207#107—28no—
208#106—28no—
209C22⋊C8C3×Q818yescomplemented
210#97—24no—
211#95—22no—
212#95—22no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3C4⋊Q812yes—
220C23×C3C22×D411yescharacteristic
221C3×Q8—48no—
222C3×Q8—48no—
223C3×Q8—84no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4#12012yes—
250C2×C3×C4#12012yes—
251C2×C3×C4C4⋊Q812yes—
252C2×C3×C4AGL(1,ℤ/8)12yes—
253C2×C3×C4#13712yes—
254C2×C3×C4C22×D412yes—
255C2×C3×C4C22×Q812yes—
256C2×C3×C42−1+412yes—
257C3×C8—28no—
258C24C2×C3×D411yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D411yescharacteristic
282C22×C4C2×C3×D411yescharacteristic
283C22×C4C2×C3×D411yescharacteristic
284C22×C4C2×C3×D411yescharacteristic
285C22×C4C2×C3×Q814yes—
286C22×C4C24×C311yescharacteristic
287C2×C8—24no—
288C2×C8—24no—
289C2×C8C2×C3×Q818yes—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4C2×C3×D414yes—
303C4⋊C4C2×C3×D414yes—
304C22⋊C4—28no—
305C22⋊C4—44no—
306C42—28no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42C2×C3×D414yes—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3C2×#12011yescharacteristic
319C22×C3C2×C4⋊Q811yescharacteristic
320C22×C3#53611yescharacteristic
321C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
322C22×C3C2×#13711yescharacteristic
323C22×C3#57511yescharacteristic
324C22×C3#57511yescharacteristic
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—44no—
331C3×C4—44no—
332C3×C4—82no—
333C3×C4—82no—
334C3×C4D4×Q814yes—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23C3×C4⋊Q812yes—
342C23C22×C3×D411yescharacteristic
343Q8—48no—
344Q8—48no—
345Q8—84no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4C3×#12012yes—
372C2×C4C3×#12012yes—
373C2×C4C3×C4⋊Q812yes—
374C2×C4C3×AGL(1,ℤ/8)12yes—
375C2×C4C3×#13712yes—
376C2×C4C22×C3×D412yes—
377C2×C4C22×C3×Q812yes—
378C2×C4C3×2−1+412yes—
379C8—28no—
380C2×C3—42no—
381C2×C3#243411yescharacteristic
382C2×C3#300611yescharacteristic
383C2×C3#300711yescharacteristic
384C2×C3#316211yescharacteristic
385C2×C3#316511yescharacteristic
386C2×C3#323511yescharacteristic
387C2×C3#323711yescharacteristic
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22C2×C3×#12011yescharacteristic
396C22C2×C3×C4⋊Q811yescharacteristic
397C22C3×#53611yescharacteristic
398C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
399C22C2×C3×#13711yescharacteristic
400C22C3×#57511yescharacteristic
401C22C3×#57511yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—44no—
408C4—44no—
409C4—82no—
410C4—82no—
411C4C3×D4×Q814yes—
412C3#2581811yescharacteristic, direct product factor, Sylow
413C2—42no—
414C2C3×#243411yescharacteristic
415C2C3×#300611yescharacteristic
416C2C3×#300711yescharacteristic
417C2C3×#316211yescharacteristic
418C2C3×#316511yescharacteristic
419C2C3×#323511yescharacteristic
420C2C3×#323711yescharacteristic
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×#501C2semidirect
4. C3×#2058C2semidirect
5. C3×#2057C2semidirect
6. C3×#1992C2semidirect
9. #25818C3direct (N × H)
37. C2×#501C2×C3semidirect
39. #2058C2×C3semidirect
40. #2057C2×C3semidirect
41. #1992C2×C3semidirect
85. C3×C22⋊C8Q8semidirect
209. C22⋊C8C3×Q8semidirect
412. C3#25818direct (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×#5012
C3×#28572
C3×#20582
C3×#20572
C3×#19922
C3×#18542
C3×#17492
#258183