Finite Group Databasebeta

← C2×C3×#2191 #10652136

Subgroups — C2×C3×#2191 #10652136

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#575C211yesmaximal, characteristic
2C2×C3×#510C212yesmaximal, complemented
3C2×C3×#470C211yesmaximal, characteristic, complemented
4C3×#2191C218yesmaximal, direct product factor
5C2×C3×#360C212yesmaximal, complemented
6C2×C3×#349C211yesmaximal, characteristic, complemented
7C2×#2191C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×#41—21no—
9C3×#575—24no—
10C3×#575—22no—
11C3×#575C2214yes—
12C2×C3×#137—22no—
13C2×C3×AGL(1,ℤ/8)—22no—
14C2×C3×#135—21no—
15C2×C3×#135C2212yes—
16C22×C3×Q16C2211yescharacteristic, complemented
17C22×C3×M16C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C3×#510—24no—
20C3×#510C2218yescomplemented
21C2×C3×Q32—22no—
22C2×C3×SD32—22no—
23C2×C3×M32C2212yescomplemented
24C3×#470—22no—
25C3×#470C2214yescomplemented
26C2×C3×#106C2211yescharacteristic, complemented
27C2×C3×#103—21no—
28C3×#360C2218yescomplemented
29C3×#349C2214yescomplemented
30C2×#575C2×C311yescharacteristic
31C2×#510C2×C312yescomplemented
32C2×#470C2×C311yescharacteristic, complemented
33#2191C2×C318yesdirect product factor
34C2×#360C2×C312yescomplemented
35C2×#349C2×C311yescharacteristic, complemented
36C2×C3×#41—22no—
37C2×C3×#41—24no—
38C2×C3×#41—24no—
39C2×C3×#41—22no—
40C2×C3×#41—41no—
41C2×C3×#41—42no—
42C2×C3×#41—41no—
43C2×C3×#41—42no—
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×D4—21no—
47C22×C3×D4—41no—
48C23×C3×C4—41no—
49C23×C3×C4D411yescharacteristic
50C3×#137—28no—
51C3×#137—28no—
52C3×AGL(1,ℤ/8)—28no—
53C3×AGL(1,ℤ/8)—28no—
54C3×#135—24no—
55C3×#135—24no—
56C3×#135—22no—
57C3×#135—22no—
58C3×#135D414yes—
59C3×#135C2314yes—
60C2×C3×Q16—21no—
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×Q16—21no—
64C2×C3×Q16D412yes—
65C2×C3×Q16C2312yescomplemented
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C2×C3×D8—21no—
69C2×C3×D8D412yes—
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16D412yes—
73C2×C3×M16C2312yes—
74C22×C3×C8D411yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—42no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—41no—
80C2×C3×C22⋊C4—41no—
81C3×Q32—28no—
82C3×SD32—28no—
83C3×M32—22no—
84C3×M32C2314yescomplemented
85C2×C3×C16—22no—
86C3×#106—21no—
87C3×#106C2312yescomplemented
88C3×#103—24no—
89C22×#41—21no—
90#575—24no—
91#575—22no—
92#575C22×C314yes—
93C2×#137—22no—
94C2×AGL(1,ℤ/8)—22no—
95C2×#135—21no—
96C2×#135C22×C312yes—
97C22×Q16C22×C311yescharacteristic, complemented
98C22×M16C22×C311yescharacteristic
99C2×C22⋊Q8—21no—
100#510—24no—
101#510C22×C318yescomplemented
102C2×Q32—22no—
103C2×SD32—22no—
104C2×M32C22×C312yescomplemented
105#470—22no—
106#470C22×C314yescomplemented
107C2×#106C22×C311yescharacteristic, complemented
108C2×#103—21no—
109#360C22×C318yescomplemented
110#349C22×C314yescomplemented
111C24×C3—41no—
112C24×C3D811yescharacteristic
113C3×#41—28no—
114C3×#41—24no—
115C3×#41—24no—
116C3×#41—44no—
117C3×#41—44no—
118C3×#41—44no—
119C3×#41—44no—
120C3×#41—44no—
121C3×#41—42no—
122C3×#41—42no—
123C2×C3×Q8—24no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—41no—
127C2×C3×Q8—42no—
128C2×C3×Q8—41no—
129C2×C3×Q8—41no—
130C2×C3×Q8—41no—
131C2×C3×Q8—41no—
132C2×C3×Q8—41no—
133C2×C3×D4—24no—
134C2×C3×D4—22no—
135C2×C3×D4—41no—
136C2×C3×D4—42no—
137C2×C3×D4—44no—
138C2×C3×D4—42no—
139C2×C3×D4—44no—
140C2×C3×D4—42no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—41no—
149C22×C3×C4—42no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4D811yescharacteristic
153C22×C3×C4D812yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C3×Q16—24no—
157C3×Q16—22no—
158C3×Q16—24no—
159C3×Q16—42no—
160C3×Q16C2×D414yes—
161C3×SD16—28no—
162C3×SD16—24no—
163C3×SD16—24no—
164C3×D8—24no—
165C3×D8—22no—
166C3×D8C2×D414yes—
167C3×M16—24no—
168C3×M16—24no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8C2×D412yes—
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D412yes—
174C2×C3×C8C2×D412yes—
175C2×C3×C8C2411yescharacteristic
176C3×C4⋊C4—22no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C16—24no—
181C2×#41—22no—
182C2×#41—24no—
183C2×#41—22no—
184C2×#41—24no—
185C2×#41—41no—
186C2×#41—42no—
187C2×#41—41no—
188C2×#41—42no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×D4—21no—
192C22×D4—41no—
193C23×C4—41no—
194C23×C4C3×D411yescharacteristic
195#137—28no—
196#137—28no—
197AGL(1,ℤ/8)—28no—
198AGL(1,ℤ/8)—28no—
199#135—24no—
200#135—24no—
201#135—22no—
202#135—22no—
203#135C3×D414yes—
204#135C23×C314yes—
205C2×Q16—21no—
206C2×Q16—22no—
207C2×Q16—22no—
208C2×Q16—21no—
209C2×Q16C3×D412yes—
210C2×Q16C23×C312yescomplemented
211C2×SD16—22no—
212C2×SD16—22no—
213C2×D8—21no—
214C2×D8C3×D412yes—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16C3×D412yes—
218C2×M16C23×C312yes—
219C22×C8C3×D411yescharacteristic
220C22×C8C23×C311yescharacteristic
221C22⋊Q8—24no—
222C22⋊Q8—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C22⋊C4—41no—
226Q32—28no—
227SD32—28no—
228M32—22no—
229M32C23×C314yescomplemented
230C2×C16—22no—
231#106—21no—
232#106C23×C312yescomplemented
233#103—24no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—84no—
240C23×C3C2×D811yescharacteristic
241C23×C3C2×D812yes—
242C3×Q8—24no—
243C3×Q8—44no—
244C3×Q8—42no—
245C3×Q8—42no—
246C3×Q8—42no—
247C3×Q8—81no—
248C3×Q8—81no—
249C3×D4—24no—
250C3×D4—44no—
251C3×D4—48no—
252C3×D4—44no—
253C3×D4—44no—
254C3×D4—42no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—48no—
262C2×C3×C4—44no—
263C2×C3×C4—42no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—82no—
269C2×C3×C4—81no—
270C2×C3×C4—81no—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4C2×D812yes—
274C2×C3×C4C2×D811yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
277C2×C3×C4C22×D411yescharacteristic
278C3×C8—22no—
279C3×C8#12012yes—
280C3×C8C22×D412yes—
281C24—41no—
282C24C3×D811yescharacteristic
283#41—28no—
284#41—24no—
285#41—24no—
286#41—44no—
287#41—44no—
288#41—44no—
289#41—44no—
290#41—44no—
291#41—42no—
292#41—42no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—42no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4C3×D811yescharacteristic
323C22×C4C3×D812yes—
324C22×C4C2×C3×D412yes—
325C22×C4C2×C3×D411yescharacteristic
326Q16—24no—
327Q16—22no—
328Q16—24no—
329Q16—42no—
330Q16C2×C3×D414yes—
331SD16—28no—
332SD16—24no—
333SD16—24no—
334D8—24no—
335D8—22no—
336D8C2×C3×D414yes—
337M16—24no—
338M16—24no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8C2×C3×D412yes—
342C2×C8C2×C3×D411yescharacteristic
343C2×C8C2×C3×D412yes—
344C2×C8C2×C3×D412yes—
345C2×C8C24×C311yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—44no—
349C22⋊C4—44no—
350C16—24no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—44no—
356C22×C3—81no—
357C22×C3—88no—
358C22×C3—84no—
359C22×C3#44711yescharacteristic
360C22×C3#44711yescharacteristic
361C22×C3C22×D811yescharacteristic
362C3×C4—22no—
363C3×C4—44no—
364C3×C4—82no—
365C3×C4—82no—
366C3×C4#44711yescharacteristic
367C3×C4#44711yescharacteristic
368C3×C4C2×#12011yescharacteristic
369C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
370C23—21no—
371C23—21no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—84no—
376C23C2×C3×D811yescharacteristic
377C23C2×C3×D812yes—
378Q8—24no—
379Q8—44no—
380Q8—42no—
381Q8—42no—
382Q8—42no—
383Q8—81no—
384Q8—81no—
385D4—24no—
386D4—44no—
387D4—48no—
388D4—44no—
389D4—44no—
390D4—42no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C3×#12011yescharacteristic
408C2×C4C3×#12011yescharacteristic
409C2×C4C2×C3×D812yes—
410C2×C4C2×C3×D811yescharacteristic
411C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
413C2×C4C22×C3×D411yescharacteristic
414C8—22no—
415C8C3×#12012yes—
416C8C22×C3×D412yes—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—42no—
420C2×C3—84no—
421C2×C3#219112yesdirect product factor
422C2×C3C2×#44711yescharacteristic
423C22—22no—
424C22—22no—
425C22—42no—
426C22—42no—
427C22—44no—
428C22—81no—
429C22—88no—
430C22—84no—
431C22C3×#44711yescharacteristic
432C22C3×#44711yescharacteristic
433C22C22×C3×D811yescharacteristic
434C4—22no—
435C4—44no—
436C4—82no—
437C4—82no—
438C4C3×#44711yescharacteristic
439C4C3×#44711yescharacteristic
440C4C2×C3×#12011yescharacteristic
441C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
442C3C2×#219111yescharacteristic, direct product factor, Sylow
443C2—21no—
444C2—21no—
445C2—42no—
446C2—84no—
447C2C3×#219112yesdirect product factor
448C2C2×C3×#44711yescharacteristic
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC2×C3×#575C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#510C2semidirect
3. C2×C3×#470C2semidirect
4. C3×#2191C2direct (N × H)
5. C2×C3×#360C2semidirect
6. C2×C3×#349C2semidirect
7. C2×#2191C3direct (N × H)
16. C22×C3×Q16C22semidirect
20. C3×#510C22semidirect
23. C2×C3×M32C22semidirect
25. C3×#470C22semidirect
26. C2×C3×#106C22semidirect
28. C3×#360C22semidirect
29. C3×#349C22semidirect
31. C2×#510C2×C3semidirect
32. C2×#470C2×C3semidirect
33. #2191C2×C3direct (N × H)
34. C2×#360C2×C3semidirect
35. C2×#349C2×C3semidirect
65. C2×C3×Q16C23semidirect
84. C3×M32C23semidirect
87. C3×#106C23semidirect
97. C22×Q16C22×C3semidirect
101. #510C22×C3semidirect
104. C2×M32C22×C3semidirect
106. #470C22×C3semidirect
107. C2×#106C22×C3semidirect
109. #360C22×C3semidirect
110. #349C22×C3semidirect
210. C2×Q16C23×C3semidirect
229. M32C23×C3semidirect
232. #106C23×C3semidirect
421. C2×C3#2191direct (N × H)
442. C3C2×#2191direct (N × H)
447. C2C3×#2191direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5752
C2×C3×#5102
C2×C3×#4702
C3×#21912
C2×C3×#3602
C2×C3×#3492
C2×#21913