Finite Group Databasebeta

← C3×#60143 #10678750

Subgroups — C3×#60143 #10678750

Order 768.

2706 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#568C212yesmaximal, complemented
2C3×#3449C211yesmaximal, characteristic
3C3×#2933C2116yesmaximal, complemented
4C3×#2932C214yesmaximal, complemented
5C3×#2931C214yesmaximal, complemented
6C3×#2905C211yesmaximal, characteristic, complemented
7C2×C3×#408C212yesmaximal, complemented
8C2×C3×#405C211yesmaximal, characteristic, complemented
9#60143C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2212yes—
11C3×#568—216no—
12C2×C3×#131C2218yes—
13C2×C3×#131C2218yescomplemented
14C22×C3×M16C2212yes—
15C22×C3×M16C2211yescharacteristic
16C22×C3×M16C2218yescomplemented
17C23×C3×C8C2211yescharacteristic
18C3×#537C414yes—
19C3×#536C414yes—
20C3×#535C414yes—
21C3×#534C414yes—
22C3×#532C2214yes—
23C3×#525C2218yes—
24C2×C3×#117C2211yescharacteristic
25C3×#436C22116yescomplemented
26C3×#435C22132yescomplemented
27C3×C8×D4C22116yescomplemented
28C3×#424C2218yescomplemented
29C3×#423C2214yescomplemented
30C3×#408C2218yes—
31C3×#408C2214yescomplemented
32C3×#408C2214yescomplemented
33C2×C3×C22⋊C8C2218yescomplemented
34C3×#405C2218yescomplemented
35C2×C3×#97C2212yescomplemented
36C2×C3×C4×C8C2212yescomplemented
37C2×#568C2×C312yescomplemented
38#3449C2×C311yescharacteristic
39#2933C2×C3116yescomplemented
40#2932C2×C314yescomplemented
41#2931C2×C314yescomplemented
42#2905C2×C311yescharacteristic, complemented
43C2×#408C2×C312yescomplemented
44C2×#405C2×C311yescharacteristic, complemented
45C2×C3×#41—28no—
46C2×C3×#41—28no—
47C2×C3×#41C2×C414yes—
48C2×C3×#41C2×C414yes—
49C2×C3×#41C2314yes—
50C2×C3×#41C2314yes—
51C22×C3×Q8C2×C412yes—
52C22×C3×D4C2×C414yes—
53C22×C3×D4C2×C412yes—
54C23×C3×C4C2314yes—
55C23×C3×C4C2311yescharacteristic
56C3×#131—232no—
57C3×#131—232no—
58C2×C3×M16—28no—
59C2×C3×M16—232no—
60C2×C3×M16D418yescomplemented
61C2×C3×M16C2314yes—
62C2×C3×M16C2314yes—
63C2×C3×M16C23116yes—
64C2×C3×M16C2314yes—
65C22×C3×C8—24no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2314yes—
69C22×C3×C8C2314yes—
70C22×C3×C8C23116yes—
71C22×C3×C8C2312yes—
72C3×C4⋊Q8C2×C414yes—
73C3×#127C2×C414yes—
74C3×#124C2×C418yes—
75C3×#123C2×C4116yes—
76C3×C22⋊Q8C2×C4116yes—
77C3×#121C2×C4116yes—
78C3×#121C2×C4116yes—
79C3×#120C2×C4116yes—
80C3×C4×D4C23116yes—
81C3×#117C2×C414yes—
82C3×#117C2314yes—
83C2×C3×C4⋊C4C2×C412yes—
84C2×C3×C22⋊C4C2×C412yes—
85C2×C3×C42C2312yes—
86C3×C4⋊C8C23116yes—
87C3×C22⋊C8C23116yes—
88C3×C22⋊C8C23116yes—
89C3×#97C2318yes—
90C3×C4×C8C2318yes—
91C22×#41C22×C312yes—
92#568—216no—
93C2×#131C22×C318yes—
94C2×#131C22×C318yescomplemented
95C22×M16C22×C312yes—
96C22×M16C22×C311yescharacteristic
97C22×M16C22×C318yescomplemented
98C23×C8C22×C311yescharacteristic
99#537C3×C414yes—
100#536C3×C414yes—
101#535C3×C414yes—
102#534C3×C414yes—
103#532C22×C314yes—
104#525C22×C318yes—
105C2×#117C22×C311yescharacteristic
106#436C22×C3116yescomplemented
107#435C22×C3132yescomplemented
108C8×D4C22×C3116yescomplemented
109#424C22×C318yescomplemented
110#423C22×C314yescomplemented
111#408C22×C318yes—
112#408C22×C314yescomplemented
113#408C22×C314yescomplemented
114C2×C22⋊C8C22×C318yescomplemented
115#405C22×C318yescomplemented
116C2×#97C22×C312yescomplemented
117C2×C4×C8C22×C312yescomplemented
118C24×C3C22×C414yes—
119C24×C3C22×C411yescharacteristic
120C3×#41—216no—
121C3×#41—216no—
122C3×#41—28no—
123C3×#41—28no—
124C3×#41—28no—
125C3×#41—28no—
126C2×C3×Q8—28no—
127C2×C3×Q8C22×C414yes—
128C2×C3×Q8C22×C414yes—
129C2×C3×D4—216no—
130C2×C3×D4—28no—
131C2×C3×D4C22×C418yes—
132C2×C3×D4C22×C418yes—
133C2×C3×D4C22×C414yes—
134C2×C3×D4C22×C414yes—
135C2×C3×D4C22×C4116yes—
136C22×C3×C4—28no—
137C22×C3×C4—28no—
138C22×C3×C4C22×C418yes—
139C22×C3×C4C22×C414yes—
140C22×C3×C4C22×C414yes—
141C22×C3×C4C22×C411yescharacteristic
142C22×C3×C4C22×C412yes—
143C22×C3×C4C2×D414yes—
144C22×C3×C4#4114yes—
145C22×C3×C4C2418yes—
146C22×C3×C4C2411yescharacteristic
147C22×C3×C4C2412yes—
148C3×M16—28no—
149C3×M16—28no—
150C3×M16—232no—
151C3×M16C2×D4116yes—
152C2×C3×C8—28no—
153C2×C3×C8—28no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—232no—
157C2×C3×C8C2×D418yes—
158C2×C3×C8C2412yes—
159C2×C3×C8C2412yes—
160C2×C3×C8C2418yes—
161C2×C3×C8C2414yes—
162C3×C4⋊C4C22×C418yes—
163C3×C4⋊C4C22×C4116yes—
164C3×C22⋊C4C22×C418yes—
165C3×C22⋊C4C22×C4132yes—
166C3×C42C22×C414yes—
167C3×C42C2414yes—
168C2×#41—28no—
169C2×#41—28no—
170C2×#41C2×C3×C414yes—
171C2×#41C2×C3×C414yes—
172C2×#41C23×C314yes—
173C2×#41C23×C314yes—
174C22×Q8C2×C3×C412yes—
175C22×D4C2×C3×C414yes—
176C22×D4C2×C3×C412yes—
177C23×C4C23×C314yes—
178C23×C4C23×C311yescharacteristic
179#131—232no—
180#131—232no—
181C2×M16—28no—
182C2×M16—232no—
183C2×M16C3×D418yescomplemented
184C2×M16C23×C314yes—
185C2×M16C23×C314yes—
186C2×M16C23×C3116yes—
187C2×M16C23×C314yes—
188C22×C8—24no—
189C22×C8C23×C311yescharacteristic
190C22×C8C23×C311yescharacteristic
191C22×C8C23×C314yes—
192C22×C8C23×C314yes—
193C22×C8C23×C3116yes—
194C22×C8C23×C312yes—
195C4⋊Q8C2×C3×C414yes—
196#127C2×C3×C414yes—
197#124C2×C3×C418yes—
198#123C2×C3×C4116yes—
199C22⋊Q8C2×C3×C4116yes—
200#121C2×C3×C4116yes—
201#121C2×C3×C4116yes—
202#120C2×C3×C4116yes—
203C4×D4C23×C3116yes—
204#117C2×C3×C414yes—
205#117C23×C314yes—
206C2×C4⋊C4C2×C3×C412yes—
207C2×C22⋊C4C2×C3×C412yes—
208C2×C42C23×C312yes—
209C4⋊C8C23×C3116yes—
210C22⋊C8C23×C3116yes—
211C22⋊C8C23×C3116yes—
212#97C23×C318yes—
213C4×C8C23×C318yes—
214C23×C3—28no—
215C23×C3—42no—
216C23×C3—48no—
217C23×C3C4×D414yes—
218C23×C3C23×C418yes—
219C23×C3C23×C412yes—
220C23×C3C23×C411yescharacteristic
221C3×Q8—28no—
222C3×Q8—28no—
223C3×D4—216no—
224C3×D4—216no—
225C3×D4—28no—
226C3×D4—28no—
227C3×D4—416no—
228C2×C3×C4—28no—
229C2×C3×C4—216no—
230C2×C3×C4—28no—
231C2×C3×C4—28no—
232C2×C3×C4—44no—
233C2×C3×C4C4×D418yes—
234C2×C3×C4C4×D414yes—
235C2×C3×C4C23×C418yes—
236C2×C3×C4C23×C418yes—
237C2×C3×C4C23×C411yescharacteristic
238C2×C3×C4C23×C412yes—
239C2×C3×C4C22×D414yes—
240C2×C3×C4C22×D412yes—
241C2×C3×C4C2×#4114yes—
242C2×C3×C4C2×#4112yes—
243C2×C3×C4C2511yescharacteristic
244C3×C8—28no—
245C3×C8—22no—
246C3×C8—22no—
247C3×C8C22×D418yes—
248C24C22×C3×C414yes—
249C24C22×C3×C411yescharacteristic
250#41—216no—
251#41—216no—
252#41—28no—
253#41—28no—
254#41—28no—
255#41—28no—
256C2×Q8—28no—
257C2×Q8C22×C3×C414yes—
258C2×Q8C22×C3×C414yes—
259C2×D4—216no—
260C2×D4—28no—
261C2×D4C22×C3×C418yes—
262C2×D4C22×C3×C418yes—
263C2×D4C22×C3×C414yes—
264C2×D4C22×C3×C414yes—
265C2×D4C22×C3×C4116yes—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4C22×C3×C418yes—
269C22×C4C22×C3×C414yes—
270C22×C4C22×C3×C414yes—
271C22×C4C22×C3×C411yescharacteristic
272C22×C4C22×C3×C412yes—
273C22×C4C2×C3×D414yes—
274C22×C4C3×#4114yes—
275C22×C4C24×C318yes—
276C22×C4C24×C311yescharacteristic
277C22×C4C24×C312yes—
278M16—28no—
279M16—28no—
280M16—232no—
281M16C2×C3×D4116yes—
282C2×C8—28no—
283C2×C8—28no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—232no—
287C2×C8C2×C3×D418yes—
288C2×C8C24×C312yes—
289C2×C8C24×C312yes—
290C2×C8C24×C318yes—
291C2×C8C24×C314yes—
292C4⋊C4C22×C3×C418yes—
293C4⋊C4C22×C3×C4116yes—
294C22⋊C4C22×C3×C418yes—
295C22⋊C4C22×C3×C4132yes—
296C42C22×C3×C414yes—
297C42C24×C314yes—
298C22×C3—22no—
299C22×C3—24no—
300C22×C3—28no—
301C22×C3—216no—
302C22×C3—44no—
303C22×C3—88no—
304C22×C3C2×C4×D414yes—
305C22×C3C2×C4×D412yes—
306C22×C3C24×C411yescharacteristic
307C3×C4—28no—
308C3×C4—28no—
309C3×C4C2×C4×D414yes—
310C3×C4C2×C4×D412yes—
311C3×C4C23×D411yescharacteristic
312C3×C4C22×#4111yescharacteristic
313C23—28no—
314C23—42no—
315C23—48no—
316C23C3×C4×D414yes—
317C23C23×C3×C418yes—
318C23C23×C3×C412yes—
319C23C23×C3×C411yescharacteristic
320Q8—28no—
321Q8—28no—
322D4—216no—
323D4—216no—
324D4—28no—
325D4—28no—
326D4—416no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—216no—
331C2×C4—44no—
332C2×C4C3×C4×D418yes—
333C2×C4C3×C4×D414yes—
334C2×C4C23×C3×C418yes—
335C2×C4C23×C3×C418yes—
336C2×C4C23×C3×C411yescharacteristic
337C2×C4C23×C3×C412yes—
338C2×C4C22×C3×D414yes—
339C2×C4C22×C3×D412yes—
340C2×C4C2×C3×#4114yes—
341C2×C4C2×C3×#4112yes—
342C2×C4C25×C311yescharacteristic
343C8—28no—
344C8—22no—
345C8—22no—
346C8C22×C3×D418yes—
347C2×C3—24no—
348C2×C3—22no—
349C2×C3—48no—
350C2×C3C22×C4×D411yescharacteristic
351C2×C3#357312yes—
352C22—22no—
353C22—24no—
354C22—28no—
355C22—216no—
356C22—44no—
357C22—88no—
358C22C2×C3×C4×D414yes—
359C22C2×C3×C4×D412yes—
360C22C24×C3×C411yescharacteristic
361C4—28no—
362C4—28no—
363C4C2×C3×C4×D414yes—
364C4C2×C3×C4×D412yes—
365C4C23×C3×D411yescharacteristic
366C4C22×C3×#4111yescharacteristic
367C3#6014311yescharacteristic, direct product factor, Sylow
368C2—24no—
369C2—22no—
370C2—48no—
371C2C22×C3×C4×D411yescharacteristic
372C2C3×#357312yes—
subgroupG/H
centerC2×C3×C4C25
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC22×C3C24×C4
prime radicalC3×#3449C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#568C2semidirect
3. C3×#2933C2semidirect
4. C3×#2932C2semidirect
5. C3×#2931C2semidirect
6. C3×#2905C2semidirect
7. C2×C3×#408C2semidirect
8. C2×C3×#405C2semidirect
9. #60143C3direct (N × H)
13. C2×C3×#131C22semidirect
16. C22×C3×M16C22semidirect
25. C3×#436C22semidirect
26. C3×#435C22semidirect
27. C3×C8×D4C22semidirect
28. C3×#424C22semidirect
29. C3×#423C22semidirect
31. C3×#408C22semidirect
32. C3×#408C22semidirect
33. C2×C3×C22⋊C8C22semidirect
34. C3×#405C22semidirect
35. C2×C3×#97C22semidirect
36. C2×C3×C4×C8C22semidirect
37. C2×#568C2×C3semidirect
39. #2933C2×C3semidirect
40. #2932C2×C3semidirect
41. #2931C2×C3semidirect
42. #2905C2×C3semidirect
43. C2×#408C2×C3semidirect
44. C2×#405C2×C3semidirect
60. C2×C3×M16D4semidirect
94. C2×#131C22×C3semidirect
97. C22×M16C22×C3semidirect
106. #436C22×C3semidirect
107. #435C22×C3semidirect
108. C8×D4C22×C3semidirect
109. #424C22×C3semidirect
110. #423C22×C3semidirect
112. #408C22×C3semidirect
113. #408C22×C3semidirect
114. C2×C22⋊C8C22×C3semidirect
115. #405C22×C3semidirect
116. C2×#97C22×C3semidirect
117. C2×C4×C8C22×C3semidirect
183. C2×M16C3×D4semidirect
367. C3#60143direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5682
C3×#34492
C3×#29332
C3×#29322
C3×#29312
C3×#29052
C2×C3×#4082
C2×C3×#4052
#601433