Finite Group Databasebeta

← C3×#25828 #10644435

Subgroups — C3×#25828 #10644435

Order 768.

994 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic
2C3×#3166C212yesmaximal, complemented
3C3×#2566C211yesmaximal, characteristic
4C3×#2061C212yesmaximal, complemented
5C3×#2060C212yesmaximal, complemented
6C3×#1986C212yesmaximal, complemented
7C3×#1856C212yesmaximal
8C3×#1856C212yesmaximal
9C3×C4×#100C211yesmaximal, characteristic
10#25828C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—22no—
12C3×#554—24no—
13C3×D4×Q8—22no—
14C3×#544C2212yes—
15C3×#543—22no—
16C3×#533C2212yes—
17C2×C3×#125C2211yescharacteristic
18C2×C3×C22⋊Q8—22no—
19C3×#518C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C3×#501—22no—
22C3×#501—22no—
23C3×#499—24no—
24C3×#428C2214yes—
25C3×#428C2214yes—
26C2×C3×#100C2212yes—
27C3×C4×M16C2212yes—
28C3×#396—22no—
29C3×#390C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#385—22no—
32C3×#384C2211yescharacteristic
33C3×#382C2211yescharacteristic
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#343C2212yes—
36C3×#339C2214yes—
37C3×#339C2214yes—
38#3501C2×C311yescharacteristic
39#3166C2×C312yescomplemented
40#2566C2×C311yescharacteristic
41#2061C2×C312yescomplemented
42#2060C2×C312yescomplemented
43#1986C2×C312yescomplemented
44#1856C2×C312yes—
45#1856C2×C312yes—
46C4×#100C2×C311yescharacteristic
47C22×C3×Q8—22no—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4D412yes—
50C2×C3×M16C2314yes—
51C3×C4⋊Q8—24no—
52C3×C4⋊Q8D412yes—
53C3×C4⋊Q8D412yes—
54C3×#125—24no—
55C3×#125—22no—
56C3×#125—22no—
57C3×#125D414yes—
58C3×C22⋊Q8—24no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—28no—
61C3×C22⋊Q8—28no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×D4—24no—
68C3×C4×D4—24no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×#117C2314yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#107—24no—
89C3×#107—24no—
90C3×#106—24no—
91C3×#106—24no—
92C3×#100Q818yescomplemented
93C3×#97—22no—
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—22no—
97#554—22no—
98#554—24no—
99D4×Q8—22no—
100#544C22×C312yes—
101#543—22no—
102#533C22×C312yes—
103C2×#125C22×C311yescharacteristic
104C2×C22⋊Q8—22no—
105#518C22×C312yes—
106C2×C4×D4C22×C311yescharacteristic
107#501—22no—
108#501—22no—
109#499—24no—
110#428C22×C314yes—
111#428C22×C314yes—
112C2×#100C22×C312yes—
113C4×M16C22×C312yes—
114#396—22no—
115#390C22×C311yescharacteristic
116#390C22×C311yescharacteristic
117#385—22no—
118#384C22×C311yescharacteristic
119#382C22×C311yescharacteristic
120C4×C4⋊C4C22×C311yescharacteristic
121#343C22×C312yes—
122#339C22×C314yes—
123#339C22×C314yes—
124C24×C3C2×D412yes—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—84no—
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4C2×Q814yes—
132C22×C3×C4—21no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—82no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C3×M16—24no—
146C3×M16C2×Q818yescomplemented
147C2×C3×C8—24no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—42no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—82no—
162C3×C4⋊C4C2×D414yes—
163C3×C4⋊C4C2×D414yes—
164C3×C22⋊C4—28no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—44no—
168C3×C22⋊C4—82no—
169C3×C42—24no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42C2×D412yes—
173C3×C42C2×D412yes—
174C22×Q8—22no—
175C22×D4C23×C311yescharacteristic
176C23×C4C3×D412yes—
177C2×M16C23×C314yes—
178C4⋊Q8—24no—
179C4⋊Q8C3×D412yes—
180C4⋊Q8C3×D412yes—
181#125—24no—
182#125—22no—
183#125—22no—
184#125C3×D414yes—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—22no—
190C22⋊Q8—24no—
191C22⋊Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198#117C23×C314yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C311yescharacteristic
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C3×D412yes—
212C2×C42C23×C311yescharacteristic
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#107—24no—
216#107—24no—
217#106—24no—
218#106—24no—
219#100C3×Q818yescomplemented
220#97—22no—
221C4×C8—22no—
222#95—21no—
223#95—22no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3C4⋊Q814yes—
229C23×C3C22×D411yescharacteristic
230C3×Q8—44no—
231C3×Q8—82no—
232C3×Q8—84no—
233C3×D4—28no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—44no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—82no—
252C2×C3×C4—84no—
253C2×C3×C4—84no—
254C2×C3×C4#12012yes—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4C22×D412yes—
258C2×C3×C4C22×Q812yes—
259C2×C3×C42−1+412yes—
260C3×C8—28no—
261C24C2×C3×D412yes—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—84no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4C2×C3×Q814yes—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—82no—
278C22×C4C2×C3×D412yes—
279C22×C4C2×C3×D411yescharacteristic
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C24×C311yescharacteristic
282M16—24no—
283M16C2×C3×Q818yescomplemented
284C2×C8—24no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—82no—
299C4⋊C4C2×C3×D414yes—
300C4⋊C4C2×C3×D414yes—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—44no—
305C22⋊C4—82no—
306C42—24no—
307C42—22no—
308C42—22no—
309C42C2×C3×D412yes—
310C42C2×C3×D412yes—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—24no—
314C22×C3—44no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—82no—
318C22×C3C2×#12011yescharacteristic
319C22×C3C2×C4⋊Q811yescharacteristic
320C22×C3#53611yescharacteristic
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—44no—
326C3×C4—42no—
327C3×C4—44no—
328C3×C4—81no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4D4×Q814yes—
332C23—22no—
333C23—22no—
334C23—42no—
335C23—42no—
336C23C3×C4⋊Q814yes—
337C23C22×C3×D411yescharacteristic
338Q8—44no—
339Q8—82no—
340Q8—84no—
341D4—28no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4C3×#12012yes—
363C2×C4C3×#12011yescharacteristic
364C2×C4C3×#12011yescharacteristic
365C2×C4C22×C3×D412yes—
366C2×C4C22×C3×Q812yes—
367C2×C4C3×2−1+412yes—
368C8—28no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—44no—
372C2×C3#243411yescharacteristic
373C2×C3#301811yescharacteristic
374C2×C3#301911yescharacteristic
375C22—21no—
376C22—21no—
377C22—24no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22C2×C3×#12011yescharacteristic
383C22C2×C3×C4⋊Q811yescharacteristic
384C22C3×#53611yescharacteristic
385C4—22no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—44no—
390C4—42no—
391C4—44no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4C3×D4×Q814yes—
396C3#2582811yescharacteristic, direct product factor, Sylow
397C2—21no—
398C2—21no—
399C2—44no—
400C2C3×#243411yescharacteristic
401C2C3×#301811yescharacteristic
402C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3166C2semidirect
4. C3×#2061C2semidirect
5. C3×#2060C2semidirect
6. C3×#1986C2semidirect
10. #25828C3direct (N × H)
39. #3166C2×C3semidirect
41. #2061C2×C3semidirect
42. #2060C2×C3semidirect
43. #1986C2×C3semidirect
92. C3×#100Q8semidirect
146. C3×M16C2×Q8semidirect
219. #100C3×Q8semidirect
283. M16C2×C3×Q8semidirect
396. C3#25828direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35012
C3×#31662
C3×#25662
C3×#20612
C3×#20602
C3×#19862
C3×#18562
C3×C4×#1002
#258283