Finite Group Databasebeta

← C3×#29655 #10648262

Subgroups — C3×#29655 #10648262

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3458C211yesmaximal, characteristic
2C3×#3095C212yesmaximal
3C3×#2499C211yesmaximal, characteristic
4C3×#2061C212yesmaximal, complemented
5C3×#2060C212yesmaximal, complemented
6C3×#2028C212yesmaximal, complemented
7C3×#2028C212yesmaximal, complemented
8C3×#1826C212yesmaximal
9C3×#1822C211yesmaximal, characteristic
10#29655C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#544—22no—
12C3×#544—22no—
13C3×#544C2212yes—
14C3×#544C2212yes—
15C3×#543—22no—
16C3×#533—24no—
17C3×#533—22no—
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#117C2212yes—
20C2×C3×#117C2211yescharacteristic
21C3×#485—24no—
22C3×#484—22no—
23C3×#483—22no—
24C3×#428C2214yes—
25C3×#419C2214yes—
26C3×#407C2212yes—
27C3×#397—22no—
28C3×#391C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#387—22no—
31C3×#384C2211yescharacteristic
32C3×#382C2211yescharacteristic
33C3×C4×C22⋊C4C2211yescharacteristic
34C3×#343C2212yes—
35C3×#343C2212yes—
36C3×#339C2214yes—
37C3×#339C2214yes—
38#3458C2×C311yescharacteristic
39#3095C2×C312yes—
40#2499C2×C311yescharacteristic
41#2061C2×C312yescomplemented
42#2060C2×C312yescomplemented
43#2028C2×C312yescomplemented
44#2028C2×C312yescomplemented
45#1826C2×C312yes—
46#1822C2×C311yescharacteristic
47C22×C3×Q8C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C2×C3×M16C2314yes—
50C3×C4⋊Q8—22no—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—24no—
53C3×C4⋊Q8—24no—
54C3×#125—24no—
55C3×#125—24no—
56C3×#125—24no—
57C3×C22⋊Q8—28no—
58C3×C22⋊Q8—28no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C4×Q8—24no—
64C3×C4×Q8—24no—
65C3×#117—22no—
66C3×#117—22no—
67C3×#117—22no—
68C3×#117—22no—
69C3×#117D414yes—
70C3×#117C2314yes—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4D412yes—
82C2×C3×C42—22no—
83C2×C3×C42D412yes—
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#106—24no—
88C3×#103—24no—
89C3×#103—24no—
90C3×C22⋊C8—22no—
91C3×C22⋊C8—22no—
92C3×#95—21no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96#544—22no—
97#544—22no—
98#544C22×C312yes—
99#544C22×C312yes—
100#543—22no—
101#533—24no—
102#533—22no—
103C2×C22⋊Q8C22×C311yescharacteristic
104C2×#117C22×C312yes—
105C2×#117C22×C311yescharacteristic
106#485—24no—
107#484—22no—
108#483—22no—
109#428C22×C314yes—
110#419C22×C314yes—
111#407C22×C312yes—
112#397—22no—
113#391C22×C311yescharacteristic
114#391C22×C311yescharacteristic
115#387—22no—
116#384C22×C311yescharacteristic
117#382C22×C311yescharacteristic
118C4×C22⋊C4C22×C311yescharacteristic
119#343C22×C312yes—
120#343C22×C312yes—
121#339C22×C314yes—
122#339C22×C314yes—
123C24×C3C2×D411yescharacteristic
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2411yescharacteristic
143C3×M16—24no—
144C3×M16—24no—
145C2×C3×C8—24no—
146C3×C4⋊C4—28no—
147C3×C4⋊C4—28no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—44no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4C2×D414yes—
159C3×C4⋊C4C2×D414yes—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—42no—
164C3×C22⋊C4—42no—
165C3×C42—24no—
166C3×C42—22no—
167C3×C42—22no—
168C3×C42—24no—
169C3×C42—22no—
170C3×C42—22no—
171C22×Q8C23×C311yescharacteristic
172C23×C4C23×C311yescharacteristic
173C2×M16C23×C314yes—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178#125—24no—
179#125—24no—
180#125—24no—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C4×Q8—24no—
188C4×Q8—24no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117C3×D414yes—
194#117C23×C314yes—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C3×D412yes—
198C2×C4⋊C4C3×D412yes—
199C2×C4⋊C4C23×C311yescharacteristic
200C2×C4⋊C4C23×C311yescharacteristic
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4C3×D412yes—
206C2×C42—22no—
207C2×C42C3×D412yes—
208C2×C42C23×C311yescharacteristic
209C2×C42C23×C311yescharacteristic
210#107—24no—
211#106—24no—
212#103—24no—
213#103—24no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3#12012yes—
225C23×C3C22×D411yescharacteristic
226C3×Q8—44no—
227C3×Q8—84no—
228C3×Q8—84no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4C22×D412yes—
255C2×C3×C42−1+412yes—
256C2×C3×C42−1+412yes—
257C3×C8—44no—
258C24C2×C3×D411yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4C2×C3×D411yescharacteristic
273C22×C4C2×C3×D412yes—
274C22×C4C2×C3×D412yes—
275C22×C4C2×C3×D412yes—
276C22×C4C2×C3×D412yes—
277C22×C4C24×C311yescharacteristic
278M16—24no—
279M16—24no—
280C2×C8—24no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—44no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4C2×C3×D414yes—
294C4⋊C4C2×C3×D414yes—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—24no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—22no—
305C42—22no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3C2×#12011yescharacteristic
314C22×C3#53611yescharacteristic
315C22×C3#53611yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—44no—
321C3×C4—42no—
322C3×C4—44no—
323C3×C4—81no—
324C3×C4—81no—
325C3×C4—81no—
326C3×C4#53614yes—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23C3×#12012yes—
332C23C22×C3×D411yescharacteristic
333Q8—44no—
334Q8—84no—
335Q8—84no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4C3×#12011yescharacteristic
360C2×C4C3×#12011yescharacteristic
361C2×C4C22×C3×D412yes—
362C2×C4C3×2−1+412yes—
363C2×C4C3×2−1+412yes—
364C8—44no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—42no—
368C2×C3#261411yescharacteristic
369C2×C3#301811yescharacteristic
370C2×C3#302211yescharacteristic
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—81no—
377C22—81no—
378C22C2×C3×#12011yescharacteristic
379C22C3×#53611yescharacteristic
380C22C3×#53611yescharacteristic
381C4—21no—
382C4—21no—
383C4—42no—
384C4—42no—
385C4—44no—
386C4—42no—
387C4—44no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4C3×#53614yes—
392C3#2965511yescharacteristic, direct product factor, Sylow
393C2—21no—
394C2—21no—
395C2—42no—
396C2C3×#261411yescharacteristic
397C2C3×#301811yescharacteristic
398C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
4. C3×#2061C2semidirect
5. C3×#2060C2semidirect
6. C3×#2028C2semidirect
7. C3×#2028C2semidirect
10. #29655C3direct (N × H)
41. #2061C2×C3semidirect
42. #2060C2×C3semidirect
43. #2028C2×C3semidirect
44. #2028C2×C3semidirect
392. C3#29655direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34582
C3×#30952
C3×#24992
C3×#20612
C3×#20602
C3×#20282
C3×#18262
C3×#18222
#296553