Finite Group Databasebeta

← C3×#10053 #10628660

Subgroups — C3×#10053 #10628660

Order 768.

634 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic
3C3×#2568C211yesmaximal, characteristic, complemented
4C3×#1982C211yesmaximal, characteristic
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10053C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C3×#486—22no—
14C3×#486—22no—
15C3×#460—22no—
16C3×#460—22no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#102—22no—
20C2×C3×#102—21no—
21C2×C3×#102—22no—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#400—22no—
24C3×#391—21no—
25C3×#391—22no—
26C3×#382—21no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×#332—22no—
29C3×Q8⋊C8—22no—
30C2×#486C2×C311yescharacteristic
31C2×#460C2×C311yescharacteristic
32#2568C2×C311yescharacteristic, complemented
33#1982C2×C311yescharacteristic
34C2×#332C2×C311yescharacteristic, complemented
35C2×Q8⋊C8C2×C311yescharacteristic, complemented
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×C3×SD16—24no—
43C2×C3×SD16—42no—
44C2×C3×SD16—42no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×#127—21no—
53C3×#127—21no—
54C3×#127—21no—
55C3×#127D412yes—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C4⋊C4D412yes—
65C2×C3×C42—21no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#102—24no—
73C3×#102—28no—
74C3×#102—44no—
75C3×C4×C8—21no—
76C3×C4×C8D412yes—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—21no—
80C22×SD16—21no—
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×#127C22×C311yescharacteristic
83C2×C4×Q8C22×C311yescharacteristic
84#486—22no—
85#486—22no—
86#460—22no—
87#460—22no—
88C2×C4⋊C8C22×C311yescharacteristic
89C2×C4⋊C8C22×C311yescharacteristic
90C2×#102—22no—
91C2×#102—21no—
92C2×#102—22no—
93C2×C4×C8C22×C311yescharacteristic
94#400—22no—
95#391—21no—
96#391—22no—
97#382—21no—
98C4×C4⋊C4C22×C311yescharacteristic
99#332—22no—
100Q8⋊C8—22no—
101C24×C3—41no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8SD1612yes—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—44no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C3×SD16—48no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—42no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8—42no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C42—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158C22×Q8—21no—
159C22×Q8C3×D411yescharacteristic
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C2×SD16—24no—
164C2×SD16—42no—
165C2×SD16—42no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#127C3×D412yes—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C3×D411yescharacteristic
184C2×C4⋊C4C3×D412yes—
185C2×C4⋊C4C3×D412yes—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42C23×C311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8C3×D412yes—
193#102—24no—
194#102—28no—
195#102—44no—
196C4×C8—21no—
197C4×C8C3×D412yes—
198#95—24no—
199#95—24no—
200#95—21no—
201C23×C3—42no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3#12011yescharacteristic
207C3×Q8—22no—
208C3×Q8—22no—
209C3×Q8—41no—
210C3×Q8—41no—
211C3×Q8—82no—
212C3×Q8—81no—
213C3×Q8—81no—
214C3×D4—44no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—44no—
218C3×D4—82no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—44no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×C3×C4C2×SD1611yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4#13512yes—
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)12yes—
245C2×C3×C4#13711yescharacteristic
246C3×C8—42no—
247C3×C8—42no—
248C3×C8—81no—
249C24—41no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8C3×SD1612yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—44no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4C2×C3×D411yescharacteristic
278C22×C4C2×C3×D411yescharacteristic
279C22×C4C2×C3×D411yescharacteristic
280SD16—48no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42C2×C3×D411yescharacteristic
303C42C2×C3×D411yescharacteristic
304C42C2×C3×D411yescharacteristic
305C42C3×#4111yescharacteristic
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—162no—
310C22×C3—162no—
311C22×C3#39211yescharacteristic
312C22×C3#44811yescharacteristic
313C22×C3#44912yes—
314C22×C3#44911yescharacteristic
315C22×C3#45412yes—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—82no—
320C3×C4—81no—
321C3×C4—82no—
322C3×C4#45011yescharacteristic
323C3×C4#45211yescharacteristic
324C3×C4#46312yes—
325C3×C4#46511yescharacteristic
326C3×C4#46811yescharacteristic
327C3×C4#49511yescharacteristic
328C3×C4#49611yescharacteristic
329C23—42no—
330C23—81no—
331C23—81no—
332C23—82no—
333C23—82no—
334C23C3×#12011yescharacteristic
335Q8—22no—
336Q8—22no—
337Q8—41no—
338Q8—41no—
339Q8—82no—
340Q8—81no—
341Q8—81no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—44no—
346D4—82no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4C3×#12011yescharacteristic
363C2×C4C3×#12011yescharacteristic
364C2×C4C3×#12111yescharacteristic
365C2×C4C3×#12111yescharacteristic
366C2×C4C3×#12111yescharacteristic
367C2×C4C3×#12711yescharacteristic
368C2×C4C2×C3×SD1611yescharacteristic
369C2×C4C3×#13511yescharacteristic
370C2×C4C3×#13512yes—
371C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C3×AGL(1,ℤ/8)12yes—
373C2×C4C3×#13711yescharacteristic
374C8—42no—
375C8—42no—
376C8—81no—
377C2×C3—162no—
378C2×C3#162612yes—
379C2×C3#162812yes—
380C2×C3#200011yescharacteristic
381C2×C3#200311yescharacteristic
382C2×C3#201211yescharacteristic
383C22—82no—
384C22—82no—
385C22—82no—
386C22—162no—
387C22—162no—
388C22C3×#39211yescharacteristic
389C22C3×#44811yescharacteristic
390C22C3×#44911yescharacteristic
391C22C3×#44912yes—
392C22C3×#45412yes—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—81no—
397C4—82no—
398C4—82no—
399C4C3×#45011yescharacteristic
400C4C3×#45211yescharacteristic
401C4C3×#46312yes—
402C4C3×#46511yescharacteristic
403C4C3×#46811yescharacteristic
404C4C3×#49511yescharacteristic
405C4C3×#49611yescharacteristic
406C3#1005311yescharacteristic, direct product factor, Sylow
407C2—162no—
408C2C3×#162612yes—
409C2C3×#162812yes—
410C2C3×#200011yescharacteristic
411C2C3×#200311yescharacteristic
412C2C3×#201211yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2568C2semidirect
5. C2×C3×#332C2semidirect
6. C2×C3×Q8⋊C8C2semidirect
7. C3×#1279C2semidirect
8. #10053C3direct (N × H)
32. #2568C2×C3semidirect
34. C2×#332C2×C3semidirect
35. C2×Q8⋊C8C2×C3semidirect
36. #1279C2×C3semidirect
406. C3#10053direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×#4602
C3×#25682
C3×#19822
C2×C3×#3322
C2×C3×Q8⋊C82
C3×#12792
#100533