Finite Group Databasebeta

← C2×C3×#1610 #10636722

Subgroups — C2×C3×#1610 #10636722

Order 768.

1308 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic
2C2×C3×#480C212yesmaximal, complemented
3C2×C3×#398C211yesmaximal, characteristic, complemented
4C3×#1610C218yesmaximal, direct product factor
5C2×C3×#327C212yesmaximal, complemented
6C2×C3×#324C211yesmaximal, characteristic, complemented
7C2×#1610C311yesmaximal, characteristic, direct product factor, Sylow
8C22×C3×#41—21no—
9C3×#537—24no—
10C3×#537—22no—
11C3×#537C2214yes—
12C2×C3×C22⋊Q8—22no—
13C2×C3×#121—22no—
14C2×C3×#121—21no—
15C2×C3×#121C2212yes—
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#480—24no—
20C3×#480C2218yescomplemented
21C2×C3×#107—22no—
22C2×C3×#102—22no—
23C2×C3×C22⋊C8C2212yes—
24C3×#398—22no—
25C3×#398C2214yescomplemented
26C2×C3×#95—21no—
27C2×C3×#95C2211yescharacteristic
28C3×#327C2218yescomplemented
29C3×#324C2214yescomplemented
30C2×#537C2×C311yescharacteristic
31C2×#480C2×C312yescomplemented
32C2×#398C2×C311yescharacteristic, complemented
33#1610C2×C318yesdirect product factor
34C2×#327C2×C312yescomplemented
35C2×#324C2×C311yescharacteristic, complemented
36C2×C3×#41—24no—
37C2×C3×#41—24no—
38C2×C3×#41—44no—
39C2×C3×#41—44no—
40C22×C3×Q8—21no—
41C22×C3×D4—21no—
42C22×C3×D4—22no—
43C22×C3×D4—21no—
44C22×C3×D4D412yes—
45C23×C3×C4—22no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×C3×C8—22no—
52C3×C22⋊Q8—28no—
53C3×C22⋊Q8—28no—
54C3×#121—24no—
55C3×#121—28no—
56C3×#121—28no—
57C3×#121—24no—
58C3×#121—22no—
59C3×#121—22no—
60C3×#121D414yes—
61C3×#121C2314yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—44no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C3×#107—28no—
77C3×#102—28no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8C2314yes—
80C3×#95—24no—
81C3×#95—21no—
82C3×#95C2312yes—
83C22×#41—21no—
84#537—24no—
85#537—22no—
86#537C22×C314yes—
87C2×C22⋊Q8—22no—
88C2×#121—22no—
89C2×#121—21no—
90C2×#121C22×C312yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C4⋊C4C22×C311yescharacteristic
94#480—24no—
95#480C22×C318yescomplemented
96C2×#107—22no—
97C2×#102—22no—
98C2×C22⋊C8C22×C312yes—
99#398—22no—
100#398C22×C314yescomplemented
101C2×#95—21no—
102C2×#95C22×C311yescharacteristic
103#327C22×C318yescomplemented
104#324C22×C314yescomplemented
105C24×C3—22no—
106C24×C3C2×D411yescharacteristic
107C3×#41—48no—
108C3×#41—416no—
109C3×#41—48no—
110C2×C3×Q8—24no—
111C2×C3×Q8—44no—
112C2×C3×D4—24no—
113C2×C3×D4—24no—
114C2×C3×D4—24no—
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C2×C3×D4—44no—
118C2×C3×D4—44no—
119C2×C3×D4—44no—
120C2×C3×D4—44no—
121C2×C3×D4—82no—
122C2×C3×D4C2×D414yes—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—28no—
127C22×C3×C4—24no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—42no—
134C22×C3×C4—48no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4D814yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2411yescharacteristic
149C2×C3×C8—24no—
150C2×C3×C8—44no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—48no—
158C3×C4⋊C4C2×D414yes—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—24no—
162C2×#41—24no—
163C2×#41—24no—
164C2×#41—44no—
165C2×#41—44no—
166C22×Q8—21no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4C3×D412yes—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C3×D411yescharacteristic
175C23×C4C3×D411yescharacteristic
176C23×C4C23×C311yescharacteristic
177C22×C8—22no—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180#121—24no—
181#121—28no—
182#121—28no—
183#121—24no—
184#121—22no—
185#121—22no—
186#121C3×D414yes—
187#121C23×C314yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C3×D412yes—
197C2×C4⋊C4C3×D412yes—
198C2×C4⋊C4C23×C312yes—
199C2×C4⋊C4C23×C312yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202#107—28no—
203#102—28no—
204C22⋊C8—22no—
205C22⋊C8C23×C314yes—
206#95—24no—
207#95—21no—
208#95C23×C312yes—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—24no—
212C23×C3—44no—
213C23×C3—44no—
214C23×C3—44no—
215C23×C3#12011yescharacteristic
216C23×C3#12011yescharacteristic
217C23×C3C2×D812yes—
218C23×C3AGL(1,ℤ/8)12yes—
219C23×C3C22×D411yescharacteristic
220C3×Q8—48no—
221C3×D4—44no—
222C3×D4—48no—
223C3×D4—44no—
224C3×D4—44no—
225C3×D4—48no—
226C3×D4—82no—
227C3×D4—84no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—416no—
240C2×C3×C4—48no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#12012yes—
253C2×C3×C4C2×D814yes—
254C2×C3×C4C22×D412yes—
255C3×C8—44no—
256C24—22no—
257C24C2×C3×D411yescharacteristic
258#41—48no—
259#41—416no—
260#41—48no—
261C2×Q8—24no—
262C2×Q8—44no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—82no—
273C2×D4C2×C3×D414yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C3×D814yes—
294C22×C4C2×C3×D412yes—
295C22×C4C2×C3×D412yes—
296C22×C4C2×C3×D412yes—
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D412yes—
299C22×C4C24×C311yescharacteristic
300C2×C8—24no—
301C2×C8—44no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—48no—
309C4⋊C4C2×C3×D414yes—
310C22⋊C4—28no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—44no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—88no—
323C22×C3#44712yes—
324C22×C3#44712yes—
325C22×C3#45612yes—
326C22×C3#45812yes—
327C22×C3C2×#12011yescharacteristic
328C22×C3C22×D811yescharacteristic
329C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
330C3×C4—24no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—44no—
335C3×C4—42no—
336C3×C4—82no—
337C23—22no—
338C23—22no—
339C23—24no—
340C23—44no—
341C23—44no—
342C23—44no—
343C23C3×#12011yescharacteristic
344C23C3×#12011yescharacteristic
345C23C2×C3×D812yes—
346C23C3×AGL(1,ℤ/8)12yes—
347C23C22×C3×D411yescharacteristic
348Q8—48no—
349D4—44no—
350D4—48no—
351D4—44no—
352D4—44no—
353D4—48no—
354D4—82no—
355D4—84no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—416no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4C3×#12012yes—
381C2×C4C2×C3×D814yes—
382C2×C4C22×C3×D412yes—
383C8—44no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—22no—
387C2×C3—84no—
388C2×C3#161014yesdirect product factor
389C2×C3C2×#44711yescharacteristic
390C2×C3C2×#45611yescharacteristic
391C2×C3C2×#45811yescharacteristic
392C22—24no—
393C22—24no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—44no—
399C22—44no—
400C22—44no—
401C22—88no—
402C22C3×#44712yes—
403C22C3×#44712yes—
404C22C3×#45612yes—
405C22C3×#45812yes—
406C22C2×C3×#12011yescharacteristic
407C22C22×C3×D811yescharacteristic
408C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
409C4—24no—
410C4—22no—
411C4—22no—
412C4—44no—
413C4—44no—
414C4—42no—
415C4—82no—
416C3C2×#161011yescharacteristic, direct product factor, Sylow
417C2—21no—
418C2—21no—
419C2—22no—
420C2—84no—
421C2C3×#161014yesdirect product factor
422C2C2×C3×#44711yescharacteristic
423C2C2×C3×#45611yescharacteristic
424C2C2×C3×#45811yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#480C2semidirect
3. C2×C3×#398C2semidirect
4. C3×#1610C2direct (N × H)
5. C2×C3×#327C2semidirect
6. C2×C3×#324C2semidirect
7. C2×#1610C3direct (N × H)
20. C3×#480C22semidirect
25. C3×#398C22semidirect
28. C3×#327C22semidirect
29. C3×#324C22semidirect
31. C2×#480C2×C3semidirect
32. C2×#398C2×C3semidirect
33. #1610C2×C3direct (N × H)
34. C2×#327C2×C3semidirect
35. C2×#324C2×C3semidirect
95. #480C22×C3semidirect
100. #398C22×C3semidirect
103. #327C22×C3semidirect
104. #324C22×C3semidirect
388. C2×C3#1610direct (N × H)
416. C3C2×#1610direct (N × H)
421. C2C3×#1610direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C2×C3×#4802
C2×C3×#3982
C3×#16102
C2×C3×#3272
C2×C3×#3242
C2×#16103