Finite Group Databasebeta

← C3×#29711 #10648318

Subgroups — C3×#29711 #10648318

Order 768.

1116 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
1C3×#3459C211yesmaximal, characteristic
2C3×#2579C213yesmaximal, complemented
3C3×#2062C211yesmaximal, characteristic, complemented
4C3×#2062C211yesmaximal, characteristic, complemented
5C3×#2044C213yesmaximal, complemented
6C3×#2044C213yesmaximal, complemented
7C3×#1799C213yesmaximal, complemented
8#29711C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543C2211yescharacteristic
10C3×#539—23no—
11C3×#538—23no—
12C3×#534C2213yes—
13C3×#529—23no—
14C3×#525—23no—
15C2×C3×C4×D4C2213yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×C3×#99C2216yes—
18C3×#398—23no—
19C3×#396—23no—
20C3×#387—23no—
21C3×#386C2213yes—
22C3×#386C2213yes—
23C3×#385—23no—
24C3×#385C2213yes—
25C3×#385C2213yes—
26C3×#380C2213yes—
27C3×#342C2213yes—
28C3×#342C2213yes—
29#3459C2×C311yescharacteristic
30#2579C2×C313yescomplemented
31#2062C2×C311yescharacteristic, complemented
32#2062C2×C311yescharacteristic, complemented
33#2044C2×C313yescomplemented
34#2044C2×C313yescomplemented
35#1799C2×C313yescomplemented
36C2×C3×#41—21no—
37C22×C3×D4C2313yes—
38C23×C3×C4D413yes—
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C3×#126—26no—
43C3×#125—23no—
44C3×#124—23no—
45C3×#123—26no—
46C3×#123—26no—
47C3×C22⋊Q8—23no—
48C3×C22⋊Q8—23no—
49C3×C22⋊Q8—23no—
50C3×#121—23no—
51C3×#121—23no—
52C3×#121—23no—
53C3×#120—26no—
54C3×C4×Q8—21no—
55C3×C4×D4—23no—
56C3×C4×D4—26no—
57C3×C4×D4—26no—
58C3×#117—23no—
59C2×C3×C4⋊C4—23no—
60C2×C3×C4⋊C4—23no—
61C2×C3×C4⋊C4—23no—
62C2×C3×C4⋊C4—23no—
63C2×C3×C4⋊C4—23no—
64C2×C3×C4⋊C4D413yes—
65C2×C3×C22⋊C4—23no—
66C2×C3×C22⋊C4—23no—
67C2×C3×C22⋊C4D413yes—
68C2×C3×C22⋊C4C2313yes—
69C2×C3×C22⋊C4C2313yes—
70C2×C3×C22⋊C4C2313yes—
71C2×C3×C42D413yes—
72C3×#99—46no—
73C3×#95—23no—
74C3×#95—23no—
75C3×#95—23no—
76C3×#95—23no—
77C3×#95—23no—
78#543C22×C311yescharacteristic
79#539—23no—
80#538—23no—
81#534C22×C313yes—
82#529—23no—
83#525—23no—
84C2×C4×D4C22×C313yes—
85C22×C4⋊C4C22×C311yescharacteristic
86C2×#99C22×C316yes—
87#398—23no—
88#396—23no—
89#387—23no—
90#386C22×C313yes—
91#386C22×C313yes—
92#385—23no—
93#385C22×C313yes—
94#385C22×C313yes—
95#380C22×C313yes—
96#342C22×C313yes—
97#342C22×C313yes—
98C24×C3C2×D413yes—
99C24×C3C2411yescharacteristic
100C3×#41—82no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×D4—26no—
104C2×C3×D4—23no—
105C2×C3×D4—26no—
106C2×C3×D4—23no—
107C2×C3×D4—23no—
108C2×C3×D4—23no—
109C22×C3×C4—23no—
110C22×C3×C4—23no—
111C22×C3×C4—23no—
112C22×C3×C4—23no—
113C22×C3×C4—23no—
114C22×C3×C4—23no—
115C22×C3×C4—42no—
116C22×C3×C4—81no—
117C22×C3×C4—81no—
118C22×C3×C4—83no—
119C22×C3×C4C2×D413yes—
120C22×C3×C4C2×D413yes—
121C22×C3×C4C2×D413yes—
122C22×C3×C4C2×D413yes—
123C22×C3×C4C2×D413yes—
124C3×C4⋊C4—23no—
125C3×C4⋊C4—23no—
126C3×C4⋊C4—23no—
127C3×C4⋊C4—23no—
128C3×C4⋊C4—26no—
129C3×C4⋊C4—46no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—43no—
132C3×C4⋊C4—83no—
133C3×C4⋊C4—83no—
134C3×C22⋊C4—26no—
135C3×C22⋊C4—26no—
136C3×C22⋊C4—26no—
137C3×C22⋊C4—26no—
138C3×C22⋊C4—23no—
139C3×C22⋊C4—23no—
140C3×C22⋊C4—23no—
141C3×C22⋊C4—23no—
142C3×C22⋊C4—46no—
143C3×C22⋊C4—83no—
144C3×C42—23no—
145C3×C42—23no—
146C2×#41—21no—
147C22×D4C23×C313yes—
148C23×C4C3×D413yes—
149C23×C4C23×C311yescharacteristic
150C23×C4C23×C311yescharacteristic
151C23×C4C23×C311yescharacteristic
152#126—26no—
153#125—23no—
154#124—23no—
155#123—26no—
156#123—26no—
157C22⋊Q8—23no—
158C22⋊Q8—23no—
159C22⋊Q8—23no—
160#121—23no—
161#121—23no—
162#121—23no—
163#120—26no—
164C4×Q8—21no—
165C4×D4—23no—
166C4×D4—26no—
167C4×D4—26no—
168#117—23no—
169C2×C4⋊C4—23no—
170C2×C4⋊C4—23no—
171C2×C4⋊C4—23no—
172C2×C4⋊C4—23no—
173C2×C4⋊C4—23no—
174C2×C4⋊C4C3×D413yes—
175C2×C22⋊C4—23no—
176C2×C22⋊C4—23no—
177C2×C22⋊C4C3×D413yes—
178C2×C22⋊C4C23×C313yes—
179C2×C22⋊C4C23×C313yes—
180C2×C22⋊C4C23×C313yes—
181C2×C42C3×D413yes—
182#99—46no—
183#95—23no—
184#95—23no—
185#95—23no—
186#95—23no—
187#95—23no—
188C23×C3—23no—
189C23×C3—23no—
190C23×C3—23no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—43no—
194C23×C3—43no—
195C23×C3C22×D413yes—
196C23×C3H(5,𝔽2)11yescharacteristic
197C23×C32−1+413yes—
198C3×Q8—81no—
199C3×Q8—81no—
200C3×D4—46no—
201C3×D4—46no—
202C3×D4—83no—
203C3×D4—83no—
204C2×C3×C4—26no—
205C2×C3×C4—23no—
206C2×C3×C4—23no—
207C2×C3×C4—23no—
208C2×C3×C4—23no—
209C2×C3×C4—23no—
210C2×C3×C4—43no—
211C2×C3×C4—43no—
212C2×C3×C4—46no—
213C2×C3×C4—43no—
214C2×C3×C4—41no—
215C2×C3×C4—43no—
216C2×C3×C4—43no—
217C2×C3×C4—43no—
218C2×C3×C4—43no—
219C2×C3×C4—43no—
220C2×C3×C4—43no—
221C2×C3×C4—43no—
222C2×C3×C4—43no—
223C2×C3×C4—41no—
224C2×C3×C4—43no—
225C2×C3×C4—43no—
226C2×C3×C4—83no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12013yes—
229C24C2×C3×D413yes—
230C24C24×C311yescharacteristic
231#41—82no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×D4—26no—
235C2×D4—23no—
236C2×D4—26no—
237C2×D4—23no—
238C2×D4—23no—
239C2×D4—23no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—23no—
246C22×C4—42no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—83no—
250C22×C4C2×C3×D413yes—
251C22×C4C2×C3×D413yes—
252C22×C4C2×C3×D413yes—
253C22×C4C2×C3×D413yes—
254C22×C4C2×C3×D413yes—
255C4⋊C4—23no—
256C4⋊C4—23no—
257C4⋊C4—23no—
258C4⋊C4—23no—
259C4⋊C4—26no—
260C4⋊C4—46no—
261C4⋊C4—41no—
262C4⋊C4—43no—
263C4⋊C4—83no—
264C4⋊C4—83no—
265C22⋊C4—26no—
266C22⋊C4—26no—
267C22⋊C4—26no—
268C22⋊C4—26no—
269C22⋊C4—23no—
270C22⋊C4—23no—
271C22⋊C4—23no—
272C22⋊C4—23no—
273C22⋊C4—46no—
274C22⋊C4—83no—
275C42—23no—
276C42—23no—
277C22×C3—23no—
278C22×C3—23no—
279C22×C3—41no—
280C22×C3—43no—
281C22×C3—46no—
282C22×C3—43no—
283C22×C3—43no—
284C22×C3—43no—
285C22×C3—83no—
286C22×C3C2×#12011yescharacteristic
287C22×C3#53613yes—
288C22×C3#53713yes—
289C3×C4—21no—
290C3×C4—23no—
291C3×C4—43no—
292C3×C4—42no—
293C3×C4—43no—
294C3×C4—43no—
295C3×C4—83no—
296C3×C4—83no—
297C3×C4—81no—
298C23—23no—
299C23—23no—
300C23—23no—
301C23—41no—
302C23—41no—
303C23—43no—
304C23—43no—
305C23C22×C3×D413yes—
306C23C3×H(5,𝔽2)11yescharacteristic
307C23C3×2−1+413yes—
308Q8—81no—
309Q8—81no—
310D4—46no—
311D4—46no—
312D4—83no—
313D4—83no—
314C2×C4—26no—
315C2×C4—23no—
316C2×C4—23no—
317C2×C4—23no—
318C2×C4—23no—
319C2×C4—23no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—46no—
323C2×C4—43no—
324C2×C4—41no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—43no—
328C2×C4—43no—
329C2×C4—43no—
330C2×C4—43no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—41no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—83no—
337C2×C4C3×#12011yescharacteristic
338C2×C4C3×#12013yes—
339C2×C3—23no—
340C2×C3—23no—
341C2×C3—83no—
342C2×C3#261611yescharacteristic
343C2×C3#302511yescharacteristic
344C2×C3#302611yescharacteristic
345C22—23no—
346C22—23no—
347C22—41no—
348C22—43no—
349C22—46no—
350C22—43no—
351C22—43no—
352C22—43no—
353C22—83no—
354C22C2×C3×#12011yescharacteristic
355C22C3×#53613yes—
356C22C3×#53713yes—
357C4—21no—
358C4—23no—
359C4—43no—
360C4—42no—
361C4—43no—
362C4—43no—
363C4—83no—
364C4—83no—
365C4—81no—
366C3#2971111yescharacteristic, direct product factor, Sylow
367C2—23no—
368C2—23no—
369C2—83no—
370C2C3×#261611yescharacteristic
371C2C3×#302511yescharacteristic
372C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#3459C2
LCSG ▷ C24 ▷ 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×#2579C2semidirect
3. C3×#2062C2semidirect
4. C3×#2062C2semidirect
5. C3×#2044C2semidirect
6. C3×#2044C2semidirect
7. C3×#1799C2semidirect
8. #29711C3direct (N × H)
30. #2579C2×C3semidirect
31. #2062C2×C3semidirect
32. #2062C2×C3semidirect
33. #2044C2×C3semidirect
34. #2044C2×C3semidirect
35. #1799C2×C3semidirect
366. C3#29711direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34592
C3×#25792
C3×#20622
C3×#20442
C3×#17992
#297113