Finite Group Databasebeta

← C3×#19067 #10637674

Subgroups — C3×#19067 #10637674

Order 768.

784 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 222 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×#3556C211yesmaximal, characteristic
2C3×#3237C212yesmaximal
3C3×#3162C211yesmaximal, characteristic
4C3×#1663C218yesmaximal, complemented
5C3×#1537C211yesmaximal, characteristic
6C3×#1531C212yesmaximal
7#19067C311yesmaximal, characteristic, direct product factor, Sylow
8C3×#564C2214yes—
9C3×#557—22no—
10C3×#557—22no—
11C3×#557—22no—
12C3×#541—22no—
13C3×#541—24no—
14C3×#533C2211yescharacteristic
15C3×#533C2211yescharacteristic
16C3×#533C2212yes—
17C3×#501—22no—
18C3×#501C2214yes—
19C3×#501C2214yes—
20C3×#491C2214yes—
21C3×#485—24no—
22C3×#477—22no—
23C3×#475—22no—
24C3×#432C2212yes—
25C3×#427—22no—
26C2×C3×#97C2211yescharacteristic
27C3×#330C2218yes—
28C3×#330C2214yes—
29#3556C2×C311yescharacteristic
30#3237C2×C312yes—
31#3162C2×C311yescharacteristic
32#1663C2×C318yescomplemented
33#1537C2×C311yescharacteristic
34#1531C2×C312yes—
35C22×C3×C8—21no—
36C3×C4⋊Q8—24no—
37C3×C4⋊Q8—22no—
38C3×C4⋊Q8D412yes—
39C3×C4⋊Q8D412yes—
40C3×C4⋊Q8D412yes—
41C3×C4⋊Q8C2312yes—
42C3×#126—28no—
43C3×#125—24no—
44C3×#125—22no—
45C3×#125—28no—
46C3×#125—28no—
47C3×#125—28no—
48C3×#125—21no—
49C3×#125C2314yes—
50C3×#125C2312yes—
51C3×C22⋊Q8—22no—
52C3×C22⋊Q8—22no—
53C3×C22⋊Q8—22no—
54C3×C22⋊Q8—22no—
55C3×C4×Q8—24no—
56C3×C4×Q8—24no—
57C3×C4×Q8—24no—
58C3×C4×Q8—22no—
59C3×C4×Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×#117—21no—
63C3×#117D412yes—
64C3×#117D414yes—
65C2×C3×C42C2311yescharacteristic
66C3×#107—24no—
67C3×#107—24no—
68C3×#106—24no—
69C3×#106—24no—
70C3×C4⋊C8—22no—
71C3×#103—24no—
72C3×#103—24no—
73C3×C22⋊C8—22no—
74C3×#97—21no—
75C3×#97C2312yes—
76C3×#97C2314yes—
77#564C22×C314yes—
78#557—22no—
79#557—22no—
80#557—22no—
81#541—22no—
82#541—24no—
83#533C22×C311yescharacteristic
84#533C22×C311yescharacteristic
85#533C22×C312yes—
86#501—22no—
87#501C22×C314yes—
88#501C22×C314yes—
89#491C22×C314yes—
90#485—24no—
91#477—22no—
92#475—22no—
93#432C22×C312yes—
94#427—22no—
95C2×#97C22×C311yescharacteristic
96#330C22×C314yes—
97#330C22×C318yes—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D412yes—
104C2×C3×C8—24no—
105C2×C3×C8—22no—
106C2×C3×C8—24no—
107C3×C4⋊C4—28no—
108C3×C4⋊C4—28no—
109C3×C4⋊C4—28no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4—24no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—42no—
119C3×C4⋊C4C2×D414yes—
120C3×C4⋊C4C2×D414yes—
121C3×C4⋊C4C2×D414yes—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—24no—
124C3×C22⋊C4—41no—
125C3×C42—28no—
126C3×C42—24no—
127C3×C42—22no—
128C3×C42—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D412yes—
131C3×C42C2411yescharacteristic
132C22×C8—21no—
133C4⋊Q8—24no—
134C4⋊Q8—22no—
135C4⋊Q8C3×D412yes—
136C4⋊Q8C3×D412yes—
137C4⋊Q8C3×D412yes—
138C4⋊Q8C23×C312yes—
139#126—28no—
140#125—24no—
141#125—22no—
142#125—28no—
143#125—28no—
144#125—28no—
145#125—21no—
146#125C23×C314yes—
147#125C23×C312yes—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159#117—21no—
160#117C3×D412yes—
161#117C3×D414yes—
162C2×C42C23×C311yescharacteristic
163#107—24no—
164#107—24no—
165#106—24no—
166#106—24no—
167C4⋊C8—22no—
168#103—24no—
169#103—24no—
170C22⋊C8—22no—
171#97—21no—
172#97C23×C312yes—
173#97C23×C314yes—
174C23×C3#12011yescharacteristic
175C3×Q8—44no—
176C3×Q8—44no—
177C3×Q8—44no—
178C3×Q8—44no—
179C2×C3×C4—22no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4#12012yes—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4C22×D411yescharacteristic
188C2×C3×C4C22×D412yes—
189C3×C8—24no—
190C3×C8—44no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C22×C4C2×C3×D411yescharacteristic
196C22×C4C2×C3×D412yes—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—24no—
200C4⋊C4—28no—
201C4⋊C4—28no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4C2×C3×D414yes—
213C4⋊C4C2×C3×D414yes—
214C4⋊C4C2×C3×D414yes—
215C22⋊C4—22no—
216C22⋊C4—24no—
217C22⋊C4—41no—
218C42—28no—
219C42—24no—
220C42—22no—
221C42—41no—
222C42C2×C3×D411yescharacteristic
223C42C2×C3×D412yes—
224C42C24×C311yescharacteristic
225C22×C3—21no—
226C22×C3—22no—
227C22×C3C2×#12011yescharacteristic
228C3×C4—21no—
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—44no—
232C3×C4—44no—
233C3×C4—44no—
234C3×C4—82no—
235C3×C4#45612yes—
236C23C3×#12011yescharacteristic
237Q8—44no—
238Q8—44no—
239Q8—44no—
240Q8—44no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4C3×#12012yes—
248C2×C4C3×#12011yescharacteristic
249C2×C4C22×C3×D411yescharacteristic
250C2×C4C22×C3×D412yes—
251C8—24no—
252C8—44no—
253C2×C3—41no—
254C2×C3C2×#45611yescharacteristic
255C2×C3#302012yes—
256C22—21no—
257C22—22no—
258C22C2×C3×#12011yescharacteristic
259C4—21no—
260C4—22no—
261C4—22no—
262C4—44no—
263C4—44no—
264C4—44no—
265C4—82no—
266C4C3×#45612yes—
267C3#1906711yescharacteristic, direct product factor, Sylow
268C2—41no—
269C2C2×C3×#45611yescharacteristic
270C2C3×#302012yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC23×C3#120
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
4. C3×#1663C2semidirect
7. #19067C3direct (N × H)
32. #1663C2×C3semidirect
267. C3#19067direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35562
C3×#32372
C3×#31622
C3×#16632
C3×#15372
C3×#15312
#190673