Finite Group Databasebeta

← C3×#25823 #10644430

Subgroups — C3×#25823 #10644430

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic
2C3×#3165C212yesmaximal, complemented
3C3×#2563C211yesmaximal, characteristic
4C3×#2061C214yesmaximal, complemented
5C3×#1986C212yesmaximal, complemented
6C3×#1856C214yesmaximal
7C3×C4×#100C211yesmaximal, characteristic
8#25823C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#554—22no—
10C3×#554—24no—
11C3×D4×Q8—22no—
12C3×#544C2212yes—
13C3×#543—22no—
14C3×#533C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C22⋊Q8—22no—
17C3×#518C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C3×#501—24no—
20C3×#500—22no—
21C3×#498—22no—
22C3×#428C2218yes—
23C2×C3×#100C2212yes—
24C3×C4×M16C2212yes—
25C3×#393—22no—
26C3×#390C2212yes—
27C3×#382C2212yes—
28C3×#380—22no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×#343C2212yes—
31C3×#339C2218yes—
32#3501C2×C311yescharacteristic
33#3165C2×C312yescomplemented
34#2563C2×C311yescharacteristic
35#2061C2×C314yescomplemented
36#1986C2×C312yescomplemented
37#1856C2×C314yes—
38C4×#100C2×C311yescharacteristic
39C22×C3×Q8—22no—
40C22×C3×Q8—21no—
41C22×C3×D4C2311yescharacteristic
42C23×C3×C4D412yes—
43C2×C3×M16C2314yes—
44C3×C4⋊Q8—24no—
45C3×C4⋊Q8—22no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8D412yes—
48C3×C4⋊Q8D412yes—
49C3×#125—24no—
50C3×#125D414yes—
51C3×C22⋊Q8—24no—
52C3×C22⋊Q8—24no—
53C3×C22⋊Q8—28no—
54C3×C22⋊Q8—28no—
55C3×C22⋊Q8—22no—
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—22no—
58C3×C4×Q8—22no—
59C3×C4×Q8—22no—
60C3×C4×D4—24no—
61C3×C4×D4—24no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C3×#117C2314yes—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C3×#107—28no—
77C3×#106—28no—
78C3×#100Q818yescomplemented
79C3×#97—22no—
80C3×C4×C8—22no—
81C3×#95—21no—
82C3×#95—24no—
83#554—22no—
84#554—24no—
85D4×Q8—22no—
86#544C22×C312yes—
87#543—22no—
88#533C22×C312yes—
89C2×C4⋊Q8C22×C311yescharacteristic
90C2×C22⋊Q8—22no—
91#518C22×C312yes—
92C2×C4×D4C22×C311yescharacteristic
93#501—24no—
94#500—22no—
95#498—22no—
96#428C22×C318yes—
97C2×#100C22×C312yes—
98C4×M16C22×C312yes—
99#393—22no—
100#390C22×C312yes—
101#382C22×C312yes—
102#380—22no—
103C4×C4⋊C4C22×C311yescharacteristic
104#343C22×C312yes—
105#339C22×C318yes—
106C24×C3C2×D412yes—
107C2×C3×Q8—24no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—82no—
112C2×C3×Q8—84no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4C2×Q814yes—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—44no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2411yescharacteristic
127C3×M16—24no—
128C3×M16C2×Q818yescomplemented
129C2×C3×C8—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—28no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4C2×D414yes—
142C3×C4⋊C4C2×D414yes—
143C3×C22⋊C4—28no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—84no—
147C3×C42—24no—
148C3×C42—24no—
149C3×C42C2×D412yes—
150C3×C42C2×D412yes—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×D4C23×C311yescharacteristic
154C23×C4C3×D412yes—
155C2×M16C23×C314yes—
156C4⋊Q8—24no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8C3×D412yes—
160C4⋊Q8C3×D412yes—
161#125—24no—
162#125C3×D414yes—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—28no—
166C22⋊Q8—28no—
167C22⋊Q8—22no—
168C22⋊Q8—24no—
169C22⋊Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176#117C23×C314yes—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C23×C312yes—
182C2×C4⋊C4C23×C311yescharacteristic
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4C3×D412yes—
186C2×C42C23×C312yes—
187C2×C42C23×C311yescharacteristic
188#107—28no—
189#106—28no—
190#100C3×Q818yescomplemented
191#97—22no—
192C4×C8—22no—
193#95—21no—
194#95—24no—
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3C4⋊Q814yes—
200C23×C3C22×D411yescharacteristic
201C3×Q8—44no—
202C3×Q8—82no—
203C3×Q8—84no—
204C3×Q8—84no—
205C3×D4—28no—
206C2×C3×C4—22no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—44no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—82no—
219C2×C3×C4—84no—
220C2×C3×C4—84no—
221C2×C3×C4#12012yes—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4C22×D412yes—
225C2×C3×C4C22×Q812yes—
226C2×C3×C42−1+412yes—
227C3×C8—28no—
228C24C2×C3×D412yes—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—42no—
233C2×Q8—82no—
234C2×Q8—84no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4C2×C3×Q814yes—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4C2×C3×D411yescharacteristic
246C22×C4C2×C3×D412yes—
247C22×C4C2×C3×D411yescharacteristic
248C22×C4C24×C311yescharacteristic
249M16—24no—
250M16C2×C3×Q818yescomplemented
251C2×C8—24no—
252C4⋊C4—24no—
253C4⋊C4—28no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4C2×C3×D414yes—
264C4⋊C4C2×C3×D414yes—
265C22⋊C4—28no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—84no—
269C42—24no—
270C42—24no—
271C42C2×C3×D412yes—
272C42C2×C3×D412yes—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—24no—
276C22×C3—44no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—82no—
280C22×C3C2×#12011yescharacteristic
281C22×C3C2×C4⋊Q811yescharacteristic
282C22×C3#53611yescharacteristic
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—42no—
286C3×C4—44no—
287C3×C4—44no—
288C3×C4—82no—
289C3×C4—82no—
290C3×C4#45512yes—
291C3×C4D4×Q814yes—
292C23—22no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23C3×C4⋊Q814yes—
297C23C22×C3×D411yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—84no—
301Q8—84no—
302D4—28no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4C3×#12012yes—
319C2×C4C3×#12011yescharacteristic
320C2×C4C3×#12011yescharacteristic
321C2×C4C22×C3×D412yes—
322C2×C4C22×C3×Q812yes—
323C2×C4C3×2−1+412yes—
324C8—28no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—44no—
328C2×C3#243411yescharacteristic
329C2×C3C2×#45511yescharacteristic
330C2×C3#302111yescharacteristic
331C22—21no—
332C22—21no—
333C22—24no—
334C22—44no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22C2×C3×#12011yescharacteristic
339C22C2×C3×C4⋊Q811yescharacteristic
340C22C3×#53611yescharacteristic
341C4—21no—
342C4—22no—
343C4—42no—
344C4—44no—
345C4—44no—
346C4—82no—
347C4—82no—
348C4C3×#45512yes—
349C4C3×D4×Q814yes—
350C3#2582311yescharacteristic, direct product factor, Sylow
351C2—21no—
352C2—21no—
353C2—44no—
354C2C3×#243411yescharacteristic
355C2C2×C3×#45511yescharacteristic
356C2C3×#302111yescharacteristic
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×#3165C2semidirect
4. C3×#2061C2semidirect
5. C3×#1986C2semidirect
8. #25823C3direct (N × H)
33. #3165C2×C3semidirect
35. #2061C2×C3semidirect
36. #1986C2×C3semidirect
78. C3×#100Q8semidirect
128. C3×M16C2×Q8semidirect
190. #100C3×Q8semidirect
250. M16C2×C3×Q8semidirect
350. C3#25823direct (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×#31652
C3×#25632
C3×#20612
C3×#19862
C3×#18562
C3×C4×#1002
#258233