Finite Group Databasebeta

← C3×#2034 #72749

Subgroups — C3×#2034 #72749

Order 384.

368 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×C3×#121C211yesmaximal, characteristic
2C2×C3×#106C211yesmaximal, characteristic, complemented
3C2×C3×#102C211yesmaximal, characteristic, complemented
4C2×C3×#102C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
6C3×#380C211yesmaximal, characteristic, complemented
7C3×#340C211yesmaximal, characteristic, complemented
8#2034C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×D4—21no—
10C22×C3×D4C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C22×C3×C8C2211yescharacteristic
14C3×#121—22no—
15C3×#121—22no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4—21no—
20C2×C3×C22⋊C4—21no—
21C3×#106—22no—
22C3×#102—22no—
23C3×#102—22no—
24C3×C22⋊C8—22no—
25C3×#95—21no—
26C2×#121C2×C311yescharacteristic
27C2×#106C2×C311yescharacteristic, complemented
28C2×#102C2×C311yescharacteristic, complemented
29C2×#102C2×C311yescharacteristic, complemented
30C2×C22⋊C8C2×C311yescharacteristic, complemented
31#380C2×C311yescharacteristic, complemented
32#340C2×C311yescharacteristic, complemented
33C24×C3—21no—
34C24×C3D411yescharacteristic
35C2×C3×D4—24no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4—21no—
41C2×C3×D4—42no—
42C2×C3×D4—42no—
43C2×C3×D4D412yes—
44C22×C3×C4—21no—
45C22×C3×C4—21no—
46C22×C3×C4—21no—
47C22×C3×C4—21no—
48C22×C3×C4—21no—
49C22×C3×C4—21no—
50C22×C3×C4—21no—
51C22×C3×C4—21no—
52C22×C3×C4—21no—
53C22×C3×C4D411yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C2×C3×C8—21no—
56C2×C3×C8—21no—
57C2×C3×C8—21no—
58C2×C3×C8—21no—
59C2×C3×C8—21no—
60C2×C3×C8D412yescomplemented
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4D412yes—
67C3×C22⋊C4—24no—
68C3×C22⋊C4—42no—
69C22×D4—21no—
70C22×D4C22×C311yescharacteristic
71C23×C4C22×C311yescharacteristic
72C22×C8C22×C311yescharacteristic
73C22×C8C22×C311yescharacteristic
74#121—22no—
75#121—22no—
76C2×C4⋊C4C22×C311yescharacteristic
77C2×C4⋊C4C22×C311yescharacteristic
78C2×C4⋊C4C22×C311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81#106—22no—
82#102—22no—
83#102—22no—
84C22⋊C8—22no—
85#95—21no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3—21no—
89C23×C3—21no—
90C23×C3—21no—
91C23×C3—21no—
92C23×C3—22no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3—41no—
97C23×C3—41no—
98C23×C3—41no—
99C23×C3SD1612yes—
100C23×C3C2×D411yescharacteristic
101C3×D4—22no—
102C3×D4—22no—
103C3×D4—48no—
104C3×D4—41no—
105C3×D4—41no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×C3×C4—41no—
113C2×C3×C4—41no—
114C2×C3×C4—41no—
115C2×C3×C4—41no—
116C2×C3×C4—41no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4#4111yescharacteristic
126C2×C3×C4#4111yescharacteristic
127C2×C3×C4#4111yescharacteristic
128C3×C8—22no—
129C3×C8—41no—
130C24—21no—
131C24C3×D411yescharacteristic
132C2×D4—24no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4C3×D412yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C3×D411yescharacteristic
151C22×C4C23×C311yescharacteristic
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8C3×D412yescomplemented
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4C3×D412yes—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C22×C3—22no—
167C22×C3—22no—
168C22×C3—22no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—42no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—81no—
177C22×C3—81no—
178C22×C3—81no—
179C22×C3—81no—
180C22×C3#12011yescharacteristic
181C22×C3#12111yescharacteristic
182C22×C3#12311yescharacteristic
183C22×C3C2×SD1611yescharacteristic
184C22×C3#13511yescharacteristic
185C22×C3AGL(1,ℤ/8)11yescharacteristic
186C22×C3AGL(1,ℤ/8)11yescharacteristic
187C3×C4—41no—
188C3×C4—42no—
189C3×C4—81no—
190C3×C4—81no—
191C3×C4#12111yescharacteristic
192C3×C4#12111yescharacteristic
193C3×C4#12311yescharacteristic
194C3×C4#12411yescharacteristic
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—22no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23C3×SD1612yes—
209C23C2×C3×D411yescharacteristic
210D4—22no—
211D4—22no—
212D4—48no—
213D4—41no—
214D4—41no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4C2×C3×D411yescharacteristic
232C2×C4C2×C3×D411yescharacteristic
233C2×C4C2×C3×D411yescharacteristic
234C2×C4C3×#4111yescharacteristic
235C2×C4C3×#4111yescharacteristic
236C2×C4C3×#4111yescharacteristic
237C8—22no—
238C8—41no—
239C2×C3—42no—
240C2×C3—82no—
241C2×C3#39411yescharacteristic
242C2×C3#44911yescharacteristic
243C2×C3#45011yescharacteristic
244C2×C3#46511yescharacteristic
245C2×C3#46911yescharacteristic
246C2×C3#48111yescharacteristic
247C2×C3#48211yescharacteristic
248C22—22no—
249C22—22no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22C3×#12011yescharacteristic
263C22C3×#12111yescharacteristic
264C22C3×#12311yescharacteristic
265C22C2×C3×SD1611yescharacteristic
266C22C3×#13511yescharacteristic
267C22C3×AGL(1,ℤ/8)11yescharacteristic
268C22C3×AGL(1,ℤ/8)11yescharacteristic
269C4—41no—
270C4—42no—
271C4—81no—
272C4—81no—
273C4C3×#12111yescharacteristic
274C4C3×#12111yescharacteristic
275C4C3×#12311yescharacteristic
276C4C3×#12411yescharacteristic
277C3#203411yescharacteristic, direct product factor, Sylow
278C2—42no—
279C2—82no—
280C2C3×#39411yescharacteristic
281C2C3×#44911yescharacteristic
282C2C3×#45011yescharacteristic
283C2C3×#46511yescharacteristic
284C2C3×#46911yescharacteristic
285C2C3×#48111yescharacteristic
286C2C3×#48211yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×#121C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#106C2semidirect
3. C2×C3×#102C2semidirect
4. C2×C3×#102C2semidirect
5. C2×C3×C22⋊C8C2semidirect
6. C3×#380C2semidirect
7. C3×#340C2semidirect
8. #2034C3direct (N × H)
27. C2×#106C2×C3semidirect
28. C2×#102C2×C3semidirect
29. C2×#102C2×C3semidirect
30. C2×C22⋊C8C2×C3semidirect
31. #380C2×C3semidirect
32. #340C2×C3semidirect
60. C2×C3×C8D4semidirect
157. C2×C8C3×D4semidirect
277. C3#2034direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×C3×#1062
C2×C3×#1022
C2×C3×C22⋊C82
C3×#3802
C3×#3402
#20343