Finite Group Databasebeta

← C3×#25827 #10644434

Subgroups — C3×#25827 #10644434

Order 768.

910 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×#3502C211yesmaximal, characteristic
2C3×#3166C212yesmaximal
3C3×#2565C211yesmaximal, characteristic
4C3×#2061C212yesmaximal
5C3×#2060C212yesmaximal
6C3×#1986C212yesmaximal
7C3×#1857C212yesmaximal
8C3×#1857C212yesmaximal
9C3×C4×#101C211yesmaximal, characteristic
10#25827C311yesmaximal, characteristic, direct product factor, Sylow
11C3×Q82—22no—
12C3×#557—22no—
13C3×#557—24no—
14C3×#544—22no—
15C3×#544C2212yes—
16C3×#533—22no—
17C3×#533C2212yes—
18C2×C3×#125C2211yescharacteristic
19C3×#519C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C3×#501—22no—
22C3×#501—22no—
23C3×#499—24no—
24C3×#428C2214yes—
25C3×#428C2214yes—
26C2×C3×#101C2212yes—
27C3×C4×M16C2212yes—
28C3×#398—22no—
29C3×#391C2211yescharacteristic
30C3×#391C2211yescharacteristic
31C3×#387—22no—
32C3×#384C2211yescharacteristic
33C3×#382C2211yescharacteristic
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#343C2212yes—
36C3×#339C2214yes—
37C3×#339C2214yes—
38#3502C2×C311yescharacteristic
39#3166C2×C312yes—
40#2565C2×C311yescharacteristic
41#2061C2×C312yes—
42#2060C2×C312yes—
43#1986C2×C312yes—
44#1857C2×C312yes—
45#1857C2×C312yes—
46C4×#101C2×C311yescharacteristic
47C22×C3×Q8C2311yescharacteristic
48C2×C3×M16C2314yes—
49C3×C4⋊Q8—22no—
50C3×C4⋊Q8—28no—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—24no—
53C3×C4⋊Q8D412yes—
54C3×C4⋊Q8D412yes—
55C3×#125—24no—
56C3×#125—28no—
57C3×#125—24no—
58C3×#125—22no—
59C3×#125—22no—
60C3×#125D414yes—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—22no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×C4×Q8—24no—
67C3×C4×Q8—24no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×#117—24no—
73C3×#117C2314yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D412yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C42D412yes—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#107—24no—
88C3×#106—24no—
89C3×#106—24no—
90C3×#101Q818yescomplemented
91C3×#97—22no—
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97Q82—22no—
98#557—22no—
99#557—24no—
100#544—22no—
101#544C22×C312yes—
102#533—22no—
103#533C22×C312yes—
104C2×#125C22×C311yescharacteristic
105#519C22×C312yes—
106C2×C4×Q8C22×C311yescharacteristic
107#501—22no—
108#501—22no—
109#499—24no—
110#428C22×C314yes—
111#428C22×C314yes—
112C2×#101C22×C312yes—
113C4×M16C22×C312yes—
114#398—22no—
115#391C22×C311yescharacteristic
116#391C22×C311yescharacteristic
117#387—22no—
118#384C22×C311yescharacteristic
119#382C22×C311yescharacteristic
120C4×C4⋊C4C22×C311yescharacteristic
121#343C22×C312yes—
122#339C22×C314yes—
123#339C22×C314yes—
124C2×C3×Q8—24no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8C2×Q814yes—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2411yescharacteristic
139C3×M16—24no—
140C3×M16C2×Q818yes—
141C2×C3×C8—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—28no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—44no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—42no—
159C3×C4⋊C4C2×D414yes—
160C3×C4⋊C4C2×D414yes—
161C3×C22⋊C4—24no—
162C3×C42—28no—
163C3×C42—22no—
164C3×C42—22no—
165C3×C42—24no—
166C3×C42—22no—
167C3×C42—22no—
168C3×C42C2×D412yes—
169C3×C42C2×D412yes—
170C22×Q8C23×C311yescharacteristic
171C2×M16C23×C314yes—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—28no—
175C4⋊Q8—24no—
176C4⋊Q8C3×D412yes—
177C4⋊Q8C3×D412yes—
178#125—24no—
179#125—28no—
180#125—24no—
181#125—22no—
182#125—22no—
183#125C3×D414yes—
184C22⋊Q8—22no—
185C22⋊Q8—24no—
186C22⋊Q8—22no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195#117—24no—
196#117C23×C314yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C3×D412yes—
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C311yescharacteristic
205C2×C42C3×D412yes—
206C2×C42C23×C311yescharacteristic
207C2×C42C23×C311yescharacteristic
208C2×C42C23×C311yescharacteristic
209#107—24no—
210#107—24no—
211#106—24no—
212#106—24no—
213#101C3×Q818yescomplemented
214#97—22no—
215C4×C8—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220C23×C3C22×D411yescharacteristic
221C3×Q8—28no—
222C3×Q8—44no—
223C3×Q8—82no—
224C3×Q8—84no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12012yes—
246C2×C3×C4C4⋊Q814yes—
247C2×C3×C4C22×D412yes—
248C2×C3×C4C22×Q812yes—
249C2×C3×C42−1+412yes—
250C3×C8—28no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8C2×C3×Q814yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C2×C3×D412yes—
262C22×C4C2×C3×D412yes—
263C22×C4C2×C3×D411yescharacteristic
264C22×C4C2×C3×D411yescharacteristic
265C22×C4C24×C311yescharacteristic
266M16—24no—
267M16C2×C3×Q818yes—
268C2×C8—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—28no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4C2×C3×D414yes—
287C4⋊C4C2×C3×D414yes—
288C22⋊C4—24no—
289C42—28no—
290C42—22no—
291C42—22no—
292C42—24no—
293C42—22no—
294C42—22no—
295C42C2×C3×D412yes—
296C42C2×C3×D412yes—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3C2×#12011yescharacteristic
300C22×C3C2×C4⋊Q811yescharacteristic
301C22×C3#53611yescharacteristic
302C3×C4—24no—
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—42no—
307C3×C4—42no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4D4×Q814yes—
314C23C22×C3×D411yescharacteristic
315Q8—28no—
316Q8—44no—
317Q8—82no—
318Q8—84no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4C3×#12011yescharacteristic
338C2×C4C3×#12011yescharacteristic
339C2×C4C3×#12012yes—
340C2×C4C3×C4⋊Q814yes—
341C2×C4C22×C3×D412yes—
342C2×C4C22×C3×Q812yes—
343C2×C4C3×2−1+412yes—
344C8—28no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#243411yescharacteristic
348C2×C3#301811yescharacteristic
349C2×C3#302011yescharacteristic
350C22—21no—
351C22—21no—
352C22C2×C3×#12011yescharacteristic
353C22C2×C3×C4⋊Q811yescharacteristic
354C22C3×#53611yescharacteristic
355C4—24no—
356C4—22no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4—44no—
362C4—44no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4C3×D4×Q814yes—
367C3#2582711yescharacteristic, direct product factor, Sylow
368C2—21no—
369C2—21no—
370C2C3×#243411yescharacteristic
371C2C3×#301811yescharacteristic
372C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3C22×D4
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
10. #25827C3direct (N × H)
90. C3×#101Q8semidirect
213. #101C3×Q8semidirect
367. C3#25827direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35022
C3×#31662
C3×#25652
C3×#20612
C3×#20602
C3×#19862
C3×#18572
C3×C4×#1012
#258273