Finite Group Databasebeta

← C3×#32993 #10651600

Subgroups — C3×#32993 #10651600

Order 768.

1444 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic
2C3×#3095C212yesmaximal, complemented
3C3×#2435C211yesmaximal, characteristic
4C3×#1885C214yesmaximal, complemented
5C3×#1856C214yesmaximal, complemented
6C3×#1826C212yesmaximal, complemented
7C3×#1783C211yesmaximal, characteristic
8#32993C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#124C2212yes—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C3×#518—22no—
13C3×#518—24no—
14C3×#518C2214yes—
15C2×C3×C4×D4—22no—
16C2×C3×C4×D4—22no—
17C2×C3×#117C2212yes—
18C22×C3×C22⋊C4—22no—
19C3×#485—24no—
20C3×#484—22no—
21C3×#483—22no—
22C3×#428C2214yescomplemented
23C3×#419C2214yescomplemented
24C2×C3×#100C2214yes—
25C3×#407C2212yes—
26C3×#394—22no—
27C3×#390C2212yes—
28C3×#379—22no—
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#339C2218yescomplemented
31C2×#518C2×C311yescharacteristic
32#3095C2×C312yescomplemented
33#2435C2×C311yescharacteristic
34#1885C2×C314yescomplemented
35#1856C2×C314yescomplemented
36#1826C2×C312yescomplemented
37#1783C2×C311yescharacteristic
38C25×C3D412yes—
39C22×C3×Q8C2311yescharacteristic
40C22×C3×D4—28no—
41C22×C3×D4—22no—
42C22×C3×D4—22no—
43C22×C3×D4D412yes—
44C22×C3×D4C2312yes—
45C23×C3×C4—22no—
46C23×C3×C4—22no—
47C23×C3×C4C2311yescharacteristic
48C2×C3×M16C2314yes—
49C3×#124—28no—
50C3×C22⋊Q8—22no—
51C3×C22⋊Q8—22no—
52C3×C4×D4—28no—
53C3×C4×D4—28no—
54C3×C4×D4—28no—
55C3×C4×D4—28no—
56C3×#117—22no—
57C3×#117—22no—
58C3×#117D414yes—
59C3×#117C2314yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—28no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—28no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C42—22no—
73C2×C3×C42C2312yes—
74C3×#107—24no—
75C3×#106—24no—
76C3×#103—24no—
77C3×#103—24no—
78C3×#100—28no—
79C3×C22⋊C8—22no—
80C3×C22⋊C8—22no—
81C3×#95—21no—
82C23×D4C22×C311yescharacteristic
83C2×#124C22×C312yes—
84C2×C22⋊Q8C22×C311yescharacteristic
85#518—22no—
86#518—24no—
87#518C22×C314yes—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×#117C22×C312yes—
91C22×C22⋊C4—22no—
92#485—24no—
93#484—22no—
94#483—22no—
95#428C22×C314yescomplemented
96#419C22×C314yescomplemented
97C2×#100C22×C314yes—
98#407C22×C312yes—
99#394—22no—
100#390C22×C312yes—
101#379—22no—
102C4×C22⋊C4C22×C311yescharacteristic
103#339C22×C318yescomplemented
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—28no—
107C24×C3—44no—
108C24×C3—82no—
109C24×C3C2×D411yescharacteristic
110C24×C3C2×D414yes—
111C2×C3×Q8—22no—
112C2×C3×Q8—24no—
113C2×C3×D4—216no—
114C2×C3×D4—216no—
115C2×C3×D4—28no—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4#4118yes—
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—28no—
124C22×C3×C4—28no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—84no—
129C22×C3×C4—84no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C3×M16—28no—
135C2×C3×C8—24no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4C2×D414yes—
141C3×C4⋊C4C2×D414yes—
142C3×C22⋊C4—216no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—48no—
149C3×C22⋊C4—84no—
150C3×C42—24no—
151C3×C42—24no—
152C3×C42—24no—
153C25C3×D412yes—
154C22×Q8C23×C311yescharacteristic
155C22×D4—28no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4C3×D412yes—
159C22×D4C23×C312yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C23×C311yescharacteristic
163C2×M16C23×C314yes—
164#124—28no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—28no—
171#117—22no—
172#117—22no—
173#117C3×D414yes—
174#117C23×C314yes—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C3×D412yes—
177C2×C4⋊C4C3×D412yes—
178C2×C4⋊C4C23×C311yescharacteristic
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—24no—
187C2×C42—22no—
188C2×C42C23×C312yes—
189#107—24no—
190#106—24no—
191#103—24no—
192#103—24no—
193#100—28no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—21no—
197C23×C3—28no—
198C23×C3—28no—
199C23×C3—24no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—24no—
203C23×C3—41no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3—44no—
208C23×C3—82no—
209C23×C3—82no—
210C23×C3—816no—
211C23×C3#12011yescharacteristic
212C23×C3#12011yescharacteristic
213C23×C3#12318yes—
214C23×C3C22×D411yescharacteristic
215C3×Q8—44no—
216C3×D4—216no—
217C3×D4—216no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—88no—
233C2×C3×C4—88no—
234C2×C3×C4#12012yes—
235C2×C3×C4C22×D412yes—
236C2×C3×C4C2×#4114yes—
237C3×C8—44no—
238C24—22no—
239C24—22no—
240C24—28no—
241C24—44no—
242C24—82no—
243C24C2×C3×D411yescharacteristic
244C24C2×C3×D414yes—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×D4—216no—
248C2×D4—216no—
249C2×D4—28no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4C3×#4118yes—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4C2×C3×D411yescharacteristic
265C22×C4C2×C3×D412yes—
266C22×C4C2×C3×D412yes—
267C22×C4C24×C311yescharacteristic
268M16—28no—
269C2×C8—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4C2×C3×D414yes—
275C4⋊C4C2×C3×D414yes—
276C22⋊C4—216no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—48no—
283C22⋊C4—84no—
284C42—24no—
285C42—24no—
286C42—24no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—28no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—44no—
296C22×C3—48no—
297C22×C3—416no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—84no—
301C22×C3—88no—
302C22×C3#45512yes—
303C22×C3C2×#12011yescharacteristic
304C22×C3C2×#12312yes—
305C3×C4—22no—
306C3×C4—44no—
307C3×C4—44no—
308C3×C4—82no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4#52514yes—
312C23—28no—
313C23—28no—
314C23—22no—
315C23—24no—
316C23—22no—
317C23—24no—
318C23—41no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—82no—
324C23—82no—
325C23—816no—
326C23C3×#12011yescharacteristic
327C23C3×#12011yescharacteristic
328C23C3×#12318yes—
329C23C22×C3×D411yescharacteristic
330Q8—44no—
331D4—216no—
332D4—216no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4—88no—
349C2×C4C3×#12012yes—
350C2×C4C22×C3×D412yes—
351C2×C4C2×C3×#4114yes—
352C8—44no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—41no—
356C2×C3—48no—
357C2×C3#284911yescharacteristic
358C2×C3C2×#45511yescharacteristic
359C2×C3#301911yescharacteristic
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—28no—
368C22—44no—
369C22—48no—
370C22—416no—
371C22—44no—
372C22—44no—
373C22—84no—
374C22—88no—
375C22C3×#45512yes—
376C22C2×C3×#12011yescharacteristic
377C22C2×C3×#12312yes—
378C4—22no—
379C4—44no—
380C4—44no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4C3×#52514yes—
385C3#3299311yescharacteristic, direct product factor, Sylow
386C2—22no—
387C2—22no—
388C2—41no—
389C2—48no—
390C2C3×#284911yescharacteristic
391C2C2×C3×#45511yescharacteristic
392C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3095C2semidirect
4. C3×#1885C2semidirect
5. C3×#1856C2semidirect
6. C3×#1826C2semidirect
8. #32993C3direct (N × H)
22. C3×#428C22semidirect
23. C3×#419C22semidirect
30. C3×#339C22semidirect
32. #3095C2×C3semidirect
34. #1885C2×C3semidirect
35. #1856C2×C3semidirect
36. #1826C2×C3semidirect
95. #428C22×C3semidirect
96. #419C22×C3semidirect
103. #339C22×C3semidirect
385. C3#32993direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C3×#30952
C3×#24352
C3×#18852
C3×#18562
C3×#18262
C3×#17832
#329933