Finite Group Databasebeta

← C3×#10206 #10628813

Subgroups — C3×#10206 #10628813

Order 768.

560 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3246C211yesmaximal, characteristic
2C3×#3229C211yesmaximal, characteristic, complemented
3C3×#3152C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5C3×#1986C211yesmaximal, characteristic, complemented
6C3×#1502C211yesmaximal, characteristic
7C3×#1283C211yesmaximal, characteristic, complemented
8#10206C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—22no—
10C3×#533C2211yescharacteristic
11C3×#532C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C3×#498—21no—
14C3×#497—22no—
15C3×#494—21no—
16C3×#492—21no—
17C3×#487—21no—
18C3×#483—22no—
19C3×#476—21no—
20C3×#475—22no—
21C3×#475—21no—
22C3×#465—22no—
23C3×#461—21no—
24C3×#460—21no—
25C3×#431C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C3×C4×M16C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#340—22no—
30C3×#339—22no—
31C3×#327—22no—
32#3246C2×C311yescharacteristic
33#3229C2×C311yescharacteristic, complemented
34#3152C2×C311yescharacteristic
35#1988C2×C311yescharacteristic, complemented
36#1986C2×C311yescharacteristic, complemented
37#1502C2×C311yescharacteristic
38#1283C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C3×#137—44no—
42C3×#137—44no—
43C2×C3×Q16—24no—
44C2×C3×SD16—24no—
45C2×C3×SD16—22no—
46C2×C3×M16D412yes—
47C22×C3×C8D412yes—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8D411yescharacteristic
52C3×C4⋊Q8D412yes—
53C3×C4⋊Q8D411yescharacteristic
54C3×C4⋊Q8D411yescharacteristic
55C3×#127D411yescharacteristic
56C3×#125—21no—
57C3×#124—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×C4×Q8—21no—
63C3×C4×Q8—21no—
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C3×#117—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72C3×#106—22no—
73C3×#106—24no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×#103—22no—
78C3×#103—24no—
79C3×#102—22no—
80C3×C22⋊C8D412yes—
81C3×#97—21no—
82C3×C4×C8—21no—
83C3×C4×C8—21no—
84C3×#95—21no—
85C2×#137—22no—
86#533C22×C311yescharacteristic
87#532C22×C311yescharacteristic
88C2×C4⋊Q8C22×C311yescharacteristic
89#498—21no—
90#497—22no—
91#494—21no—
92#492—21no—
93#487—21no—
94#483—22no—
95#476—21no—
96#475—22no—
97#475—21no—
98#465—22no—
99#461—21no—
100#460—21no—
101#431C22×C311yescharacteristic
102C2×C4⋊C8C22×C311yescharacteristic
103C4×M16C22×C311yescharacteristic
104C4×C4⋊C4C22×C311yescharacteristic
105#340—22no—
106#339—22no—
107#327—22no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—41no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C3×Q16—48no—
128C3×SD16—48no—
129C3×SD16—82no—
130C3×M16—22no—
131C3×M16SD1614yescomplemented
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—42no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—41no—
154C3×C42—42no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C2×#41—21no—
160C22×Q8—21no—
161#137—44no—
162#137—44no—
163C2×Q16—24no—
164C2×SD16—24no—
165C2×SD16—22no—
166C2×M16C3×D412yes—
167C22×C8C3×D412yes—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C3×D412yes—
172C4⋊Q8C3×D411yescharacteristic
173C4⋊Q8C3×D411yescharacteristic
174C4⋊Q8C3×D411yescharacteristic
175#127C3×D411yescharacteristic
176#125—21no—
177#124—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180#121—21no—
181#121—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×D4—21no—
185C4×D4—21no—
186#117—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C42—21no—
191C2×C42C23×C311yescharacteristic
192#106—22no—
193#106—24no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197#103—22no—
198#103—24no—
199#102—22no—
200C22⋊C8C3×D412yes—
201#97—21no—
202C4×C8—21no—
203C4×C8—21no—
204#95—21no—
205C23×C3—41no—
206C23×C3#12711yescharacteristic
207C3×Q8—42no—
208C3×Q8—44no—
209C3×Q8—42no—
210C3×Q8—41no—
211C3×Q8—41no—
212C3×Q8—82no—
213C3×D4—42no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—81no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—82no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4C2×SD1612yes—
239C2×C3×C4AGL(1,ℤ/8)12yes—
240C3×C8—24no—
241C3×C8—42no—
242C3×C8—42no—
243#41—42no—
244#41—42no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C2×C3×D411yescharacteristic
260C22×C4C2×C3×D411yescharacteristic
261C22×C4C2×C3×D411yescharacteristic
262Q16—48no—
263SD16—48no—
264SD16—82no—
265M16—22no—
266M16C3×SD1614yescomplemented
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—42no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—41no—
289C42—42no—
290C42C2×C3×D411yescharacteristic
291C42C2×C3×D411yescharacteristic
292C42C2×C3×D411yescharacteristic
293C42C3×#4111yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3#39211yescharacteristic
300C22×C3#49211yescharacteristic
301C22×C3#49611yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4—82no—
310C3×C4#45012yes—
311C3×C4#45511yescharacteristic
312C3×C4#45611yescharacteristic
313C3×C4#46012yes—
314C23—41no—
315C23C3×#12711yescharacteristic
316Q8—42no—
317Q8—44no—
318Q8—42no—
319Q8—41no—
320Q8—41no—
321Q8—82no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4C3×#12011yescharacteristic
342C2×C4C3×#12011yescharacteristic
343C2×C4C3×#12011yescharacteristic
344C2×C4C3×#12111yescharacteristic
345C2×C4C3×#12111yescharacteristic
346C2×C4C3×#12111yescharacteristic
347C2×C4C2×C3×SD1612yes—
348C2×C4C3×AGL(1,ℤ/8)12yes—
349C8—24no—
350C8—42no—
351C8—42no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—161no—
355C2×C3#200711yescharacteristic
356C2×C3#200911yescharacteristic
357C2×C3#201411yescharacteristic
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C3×#39211yescharacteristic
364C22C3×#49211yescharacteristic
365C22C3×#49611yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—82no—
374C4C3×#45012yes—
375C4C3×#45511yescharacteristic
376C4C3×#45611yescharacteristic
377C4C3×#46012yes—
378C3#1020611yescharacteristic, direct product factor, Sylow
379C2—21no—
380C2—21no—
381C2—161no—
382C2C3×#200711yescharacteristic
383C2C3×#200911yescharacteristic
384C2C3×#201411yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3229C2semidirect
4. C3×#1988C2semidirect
5. C3×#1986C2semidirect
7. C3×#1283C2semidirect
8. #10206C3direct (N × H)
33. #3229C2×C3semidirect
35. #1988C2×C3semidirect
36. #1986C2×C3semidirect
38. #1283C2×C3semidirect
131. C3×M16SD16semidirect
266. M16C3×SD16semidirect
378. C3#10206direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32462
C3×#32292
C3×#31522
C3×#19882
C3×#19862
C3×#15022
C3×#12832
#102063