Finite Group Databasebeta

← C3×#25825 #10644432

Subgroups — C3×#25825 #10644432

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3502C211yesmaximal, characteristic
2C3×#3165C212yesmaximal
3C3×#2567C211yesmaximal, characteristic
4C3×#2060C214yesmaximal
5C3×#1986C212yesmaximal
6C3×#1857C214yesmaximal
7C3×C4×#101C211yesmaximal, characteristic
8#25825C311yesmaximal, characteristic, direct product factor, Sylow
9C3×Q82—22no—
10C3×#557—22no—
11C3×#557—24no—
12C3×#544—22no—
13C3×#544C2212yes—
14C3×#533—22no—
15C3×#533C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C3×#519C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C3×#501—24no—
20C3×#500—22no—
21C3×#498—22no—
22C3×#428C2218yes—
23C2×C3×#101C2212yes—
24C3×C4×M16C2212yes—
25C3×#395—22no—
26C3×#391C2212yes—
27C3×#389—22no—
28C3×#384C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×#343C2212yes—
31C3×#339C2218yes—
32#3502C2×C311yescharacteristic
33#3165C2×C312yes—
34#2567C2×C311yescharacteristic
35#2060C2×C314yes—
36#1986C2×C312yes—
37#1857C2×C314yes—
38C4×#101C2×C311yescharacteristic
39C22×C3×Q8—21no—
40C22×C3×Q8C2311yescharacteristic
41C2×C3×M16C2314yes—
42C3×C4⋊Q8—22no—
43C3×C4⋊Q8—28no—
44C3×C4⋊Q8—22no—
45C3×C4⋊Q8—24no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8—22no—
48C3×C4⋊Q8D412yes—
49C3×C4⋊Q8D412yes—
50C3×#125—24no—
51C3×#125—28no—
52C3×#125—24no—
53C3×#125D414yes—
54C3×C22⋊Q8—22no—
55C3×C22⋊Q8—24no—
56C3×C22⋊Q8—22no—
57C3×C4×Q8—24no—
58C3×C4×Q8—24no—
59C3×C4×Q8—24no—
60C3×C4×Q8—24no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×#117—24no—
66C3×#117C2314yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42D412yes—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C3×#107—28no—
77C3×#106—28no—
78C3×#101Q818yescomplemented
79C3×#97—22no—
80C3×C4×C8—22no—
81C3×#95—21no—
82C3×#95—24no—
83Q82—22no—
84#557—22no—
85#557—24no—
86#544—22no—
87#544C22×C312yes—
88#533—22no—
89#533C22×C312yes—
90C2×C4⋊Q8C22×C311yescharacteristic
91#519C22×C312yes—
92C2×C4×Q8C22×C311yescharacteristic
93#501—24no—
94#500—22no—
95#498—22no—
96#428C22×C318yes—
97C2×#101C22×C312yes—
98C4×M16C22×C312yes—
99#395—22no—
100#391C22×C312yes—
101#389—22no—
102#384C22×C312yes—
103C4×C4⋊C4C22×C311yescharacteristic
104#343C22×C312yes—
105#339C22×C318yes—
106C2×C3×Q8—24no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—42no—
112C2×C3×Q8—82no—
113C2×C3×Q8C2×Q814yes—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2411yescharacteristic
121C3×M16—24no—
122C3×M16C2×Q818yes—
123C2×C3×C8—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—28no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—84no—
139C3×C4⋊C4C2×D414yes—
140C3×C4⋊C4C2×D414yes—
141C3×C22⋊C4—24no—
142C3×C42—28no—
143C3×C42—22no—
144C3×C42—22no—
145C3×C42—24no—
146C3×C42—24no—
147C3×C42C2×D412yes—
148C3×C42C2×D412yes—
149C22×Q8—21no—
150C22×Q8C23×C311yescharacteristic
151C2×M16C23×C314yes—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—28no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8C3×D412yes—
159C4⋊Q8C3×D412yes—
160#125—24no—
161#125—28no—
162#125—24no—
163#125C3×D414yes—
164C22⋊Q8—22no—
165C22⋊Q8—24no—
166C22⋊Q8—22no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175#117—24no—
176#117C23×C314yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4C3×D412yes—
181C2×C4⋊C4C23×C312yes—
182C2×C4⋊C4C23×C311yescharacteristic
183C2×C42C3×D412yes—
184C2×C42C23×C312yes—
185C2×C42C23×C311yescharacteristic
186#107—28no—
187#106—28no—
188#101C3×Q818yescomplemented
189#97—22no—
190C4×C8—22no—
191#95—21no—
192#95—24no—
193C23×C3C22×D411yescharacteristic
194C3×Q8—28no—
195C3×Q8—44no—
196C3×Q8—82no—
197C3×Q8—84no—
198C3×Q8—84no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—42no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12012yes—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4C4⋊Q814yes—
216C2×C3×C4C22×D412yes—
217C2×C3×C4C22×Q812yes—
218C2×C3×C42−1+412yes—
219C3×C8—28no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—82no—
227C2×Q8C2×C3×Q814yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4C2×C3×D411yescharacteristic
231C22×C4C2×C3×D412yes—
232C22×C4C2×C3×D412yes—
233C22×C4C2×C3×D411yescharacteristic
234C22×C4C24×C311yescharacteristic
235M16—24no—
236M16C2×C3×Q818yes—
237C2×C8—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—84no—
253C4⋊C4C2×C3×D414yes—
254C4⋊C4C2×C3×D414yes—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42—24no—
260C42—24no—
261C42C2×C3×D412yes—
262C42C2×C3×D412yes—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3C2×#12011yescharacteristic
266C22×C3C2×C4⋊Q811yescharacteristic
267C22×C3#53611yescharacteristic
268C3×C4—24no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—44no—
272C3×C4—44no—
273C3×C4—42no—
274C3×C4—44no—
275C3×C4—82no—
276C3×C4#45612yes—
277C3×C4D4×Q814yes—
278C23C22×C3×D411yescharacteristic
279Q8—28no—
280Q8—44no—
281Q8—82no—
282Q8—84no—
283Q8—84no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4C3×#12011yescharacteristic
298C2×C4C3×#12012yes—
299C2×C4C3×#12011yescharacteristic
300C2×C4C3×C4⋊Q814yes—
301C2×C4C22×C3×D412yes—
302C2×C4C22×C3×Q812yes—
303C2×C4C3×2−1+412yes—
304C8—28no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3#243411yescharacteristic
308C2×C3C2×#45611yescharacteristic
309C2×C3#302111yescharacteristic
310C22—21no—
311C22—21no—
312C22C2×C3×#12011yescharacteristic
313C22C2×C3×C4⋊Q811yescharacteristic
314C22C3×#53611yescharacteristic
315C4—24no—
316C4—21no—
317C4—22no—
318C4—44no—
319C4—44no—
320C4—42no—
321C4—44no—
322C4—82no—
323C4C3×#45612yes—
324C4C3×D4×Q814yes—
325C3#2582511yescharacteristic, direct product factor, Sylow
326C2—21no—
327C2—21no—
328C2C3×#243411yescharacteristic
329C2C2×C3×#45611yescharacteristic
330C2C3×#302111yescharacteristic
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
8. #25825C3direct (N × H)
78. C3×#101Q8semidirect
188. #101C3×Q8semidirect
325. C3#25825direct (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×#31652
C3×#25672
C3×#20602
C3×#19862
C3×#18572
C3×C4×#1012
#258253