Finite Group Databasebeta

← C3×#19078 #10637685

Subgroups — C3×#19078 #10637685

Order 768.

944 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3550C211yesmaximal, characteristic
2C3×#3165C212yesmaximal, complemented
3C3×#3152C211yesmaximal, characteristic
4C3×#1660C218yesmaximal, complemented
5C3×#1526C212yesmaximal, complemented
6C3×#1525C211yesmaximal, characteristic
7#19078C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×#137—22no—
9C3×#563C2214yes—
10C3×#559—21no—
11C3×Q82—21no—
12C3×#556—22no—
13C3×D4×Q8—22no—
14C3×#542—24no—
15C3×#536—22no—
16C3×#533C2212yes—
17C3×#532C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C3×#501C2218yes—
20C3×#500—22no—
21C3×#498—22no—
22C3×#497C2214yes—
23C3×#494—21no—
24C3×#492—21no—
25C3×#428—24no—
26C3×C4×M16C2211yescharacteristic
27C3×C4×M16C2212yes—
28C3×#330C2218yes—
29C3×#329C2214yes—
30#3550C2×C311yescharacteristic
31#3165C2×C312yescomplemented
32#3152C2×C311yescharacteristic
33#1660C2×C318yescomplemented
34#1526C2×C312yescomplemented
35#1525C2×C311yescharacteristic
36C2×C3×#41D412yes—
37C22×C3×Q8—22no—
38C22×C3×Q8—21no—
39C3×#137—48no—
40C2×C3×Q16—24no—
41C2×C3×SD16—24no—
42C2×C3×M16—21no—
43C2×C3×M16—22no—
44C3×C4⋊Q8—24no—
45C3×C4⋊Q8—24no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×C4⋊Q8D412yes—
50C3×C4⋊Q8D412yes—
51C3×C4⋊Q8C2312yes—
52C3×#127D411yescharacteristic
53C3×#126—28no—
54C3×#125C2314yes—
55C3×#124—24no—
56C3×#124—24no—
57C3×#124C2312yes—
58C3×#123—24no—
59C3×#123—24no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—28no—
64C3×#121—22no—
65C3×C4×Q8—24no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×D4—24no—
69C3×C4×D4—22no—
70C3×#117—22no—
71C3×#117D414yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C42C2311yescharacteristic
74C3×#107—28no—
75C3×#106—28no—
76C3×#97C2314yes—
77C3×#97C2312yes—
78C3×C4×C8—21no—
79C3×C4×C8—22no—
80C2×#137—22no—
81#563C22×C314yes—
82#559—21no—
83Q82—21no—
84#556—22no—
85D4×Q8—22no—
86#542—24no—
87#536—22no—
88#533C22×C312yes—
89#532C22×C311yescharacteristic
90C2×C4⋊Q8C22×C311yescharacteristic
91#501C22×C318yes—
92#500—22no—
93#498—22no—
94#497C22×C314yes—
95#494—21no—
96#492—21no—
97#428—24no—
98C4×M16C22×C311yescharacteristic
99C4×M16C22×C312yes—
100#330C22×C318yes—
101#329C22×C314yes—
102C3×#41—44no—
103C2×C3×Q8—24no—
104C2×C3×Q8—22no—
105C2×C3×Q8—22no—
106C2×C3×Q8—22no—
107C2×C3×Q8—44no—
108C2×C3×Q8—41no—
109C2×C3×Q8—84no—
110C2×C3×Q8C2×D412yes—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4C2×D412yes—
114C22×C3×C4—24no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C3×Q16—48no—
120C3×SD16—48no—
121C3×M16—28no—
122C3×M16—24no—
123C2×C3×C8—24no—
124C2×C3×C8—22no—
125C3×C4⋊C4—28no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—42no—
133C3×C4⋊C4C2×D414yes—
134C3×C4⋊C4C2×D414yes—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—28no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C42—24no—
140C3×C42—24no—
141C3×C42C2×D412yes—
142C3×C42C2×D411yescharacteristic
143C3×C42C2411yescharacteristic
144C2×#41C3×D412yes—
145C22×Q8—22no—
146C22×Q8—21no—
147#137—48no—
148C2×Q16—24no—
149C2×SD16—24no—
150C2×M16—21no—
151C2×M16—22no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8C3×D411yescharacteristic
157C4⋊Q8C3×D412yes—
158C4⋊Q8C3×D412yes—
159C4⋊Q8C23×C312yes—
160#127C3×D411yescharacteristic
161#126—28no—
162#125C23×C314yes—
163#124—24no—
164#124—24no—
165#124C23×C312yes—
166#123—24no—
167#123—24no—
168C22⋊Q8—24no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—28no—
172#121—22no—
173C4×Q8—24no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×D4—24no—
177C4×D4—22no—
178#117—22no—
179#117C3×D414yes—
180C2×C4⋊C4—22no—
181C2×C42C23×C311yescharacteristic
182#107—28no—
183#106—28no—
184#97C23×C314yes—
185#97C23×C312yes—
186C4×C8—21no—
187C4×C8—22no—
188C23×C3—22no—
189C23×C3#12011yescharacteristic
190C3×Q8—42no—
191C3×Q8—44no—
192C3×Q8—44no—
193C3×Q8—44no—
194C3×Q8—84no—
195C3×Q8—82no—
196C3×D4—42no—
197C3×D4—44no—
198C3×D4—82no—
199C2×C3×C4—22no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—22no—
203C2×C3×C4—24no—
204C2×C3×C4—44no—
205C2×C3×C4—42no—
206C2×C3×C4—84no—
207C2×C3×C4—82no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12012yes—
210C2×C3×C4C22×D412yes—
211C2×C3×C4C22×D411yescharacteristic
212C3×C8—28no—
213C3×C8—24no—
214#41—44no—
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—84no—
222C2×Q8C2×C3×D412yes—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4C2×C3×D412yes—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4C2×C3×D412yes—
230C22×C4C2×C3×D411yescharacteristic
231Q16—48no—
232SD16—48no—
233M16—24no—
234M16—28no—
235C2×C8—24no—
236C2×C8—22no—
237C4⋊C4—28no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4C2×C3×D414yes—
246C4⋊C4C2×C3×D414yes—
247C22⋊C4—24no—
248C22⋊C4—28no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C42—24no—
252C42—24no—
253C42C2×C3×D412yes—
254C42C2×C3×D411yescharacteristic
255C42C24×C311yescharacteristic
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—42no—
259C22×C3—44no—
260C22×C3C2×#12011yescharacteristic
261C3×C4—21no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—44no—
265C3×C4—44no—
266C3×C4—82no—
267C3×C4#45312yes—
268C3×C4#45614yes—
269C23—22no—
270C23C3×#12011yescharacteristic
271Q8—42no—
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—84no—
276Q8—82no—
277D4—42no—
278D4—44no—
279D4—82no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—84no—
288C2×C4—82no—
289C2×C4C3×#12011yescharacteristic
290C2×C4C3×#12012yes—
291C2×C4C22×C3×D412yes—
292C2×C4C22×C3×D411yescharacteristic
293C8—24no—
294C8—28no—
295C2×C3—41no—
296C2×C3—82no—
297C2×C3C2×#45311yescharacteristic
298C2×C3C2×#45612yes—
299C22—21no—
300C22—22no—
301C22—42no—
302C22—44no—
303C22C2×C3×#12011yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—44no—
308C4—44no—
309C4—82no—
310C4C3×#45312yes—
311C4C3×#45614yes—
312C3#1907811yescharacteristic, direct product factor, Sylow
313C2—41no—
314C2—82no—
315C2C2×C3×#45311yescharacteristic
316C2C2×C3×#45612yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#532C22
LCSG ▷ C42 ▷ C22 ▷ 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×#1660C2semidirect
5. C3×#1526C2semidirect
7. #19078C3direct (N × H)
31. #3165C2×C3semidirect
33. #1660C2×C3semidirect
34. #1526C2×C3semidirect
312. C3#19078direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35502
C3×#31652
C3×#31522
C3×#16602
C3×#15262
C3×#15252
#190783