Finite Group Databasebeta

← C3×#9032 #10627639

Subgroups — C3×#9032 #10627639

Order 768.

516 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2983C211yesmaximal, characteristic, complemented
2C3×#2398C211yesmaximal, characteristic
3C3×C4×#101C211yesmaximal, characteristic, complemented
4C3×#1490C212yesmaximal
5C3×#1300C212yesmaximal, complemented
6#9032C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#446—24no—
10C3×#432—21no—
11C3×#432C2212yes—
12C3×#431—21no—
13C2×C3×#101—21no—
14C3×C4×M16C2212yes—
15C3×#400C411yescharacteristic
16C3×#400C411yescharacteristic
17C3×#397—21no—
18C3×#394C411yescharacteristic
19C3×#394C411yescharacteristic
20C3×#393—21no—
21C3×C4×C22⋊C4—21no—
22C3×C4×C22⋊C4—21no—
23C3×#376C2211yescharacteristic
24C3×#343—21no—
25C3×D4⋊C8—24no—
26#2983C2×C311yescharacteristic, complemented
27#2398C2×C311yescharacteristic
28C4×#101C2×C311yescharacteristic, complemented
29#1490C2×C312yes—
30#1300C2×C312yescomplemented
31C22×C3×Q8D411yescharacteristic
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4—21no—
34C2×C3×M16—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C3×C4×D4—22no—
38C3×C4×D4—22no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C3×C4⋊C8—22no—
53C3×C4⋊C8—22no—
54C3×C4⋊C8D414yes—
55C3×#101—24no—
56C3×C22⋊C8—22no—
57C3×#97—22no—
58C3×#97—22no—
59C3×#97—22no—
60C3×C4×C8—22no—
61C3×C4×C8—22no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95C2×C411yescharacteristic
68C3×#95C2×C411yescharacteristic
69C2×C4×Q8C22×C311yescharacteristic
70C2×C4×D4C22×C311yescharacteristic
71#446—24no—
72#432—21no—
73#432C22×C312yes—
74#431—21no—
75C2×#101—21no—
76C4×M16C22×C312yes—
77#400C3×C411yescharacteristic
78#400C3×C411yescharacteristic
79#397—21no—
80#394C3×C411yescharacteristic
81#394C3×C411yescharacteristic
82#393—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85#376C22×C311yescharacteristic
86#343—21no—
87D4⋊C8—24no—
88C24×C3—21no—
89C2×C3×Q8—22no—
90C2×C3×Q8—22no—
91C2×C3×Q8—41no—
92C2×C3×D4—22no—
93C2×C3×D4—42no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C3×M16—24no—
109C3×M16—42no—
110C2×C3×C8—24no—
111C2×C3×C8—42no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—42no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—42no—
126C3×C42C2×D412yes—
127C3×C42#4112yes—
128C22×Q8C3×D411yescharacteristic
129C22×D4C2×C3×C411yescharacteristic
130C23×C4—21no—
131C2×M16—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×D4—22no—
135C4×D4—22no—
136C2×C4⋊C4C2×C3×C411yescharacteristic
137C2×C4⋊C4C2×C3×C411yescharacteristic
138C2×C4⋊C4C2×C3×C411yescharacteristic
139C2×C4⋊C4C3×D411yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C42—21no—
146C2×C42C3×D411yescharacteristic
147C2×C42C3×D411yescharacteristic
148C2×C42C23×C311yescharacteristic
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8C3×D414yes—
152#101—24no—
153C22⋊C8—22no—
154#97—22no—
155#97—22no—
156#97—22no—
157C4×C8—22no—
158C4×C8—22no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95C2×C3×C411yescharacteristic
165#95C2×C3×C411yescharacteristic
166C23×C3—22no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3C2×C22⋊C411yescharacteristic
174C3×Q8—44no—
175C3×D4—24no—
176C3×D4—42no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—82no—
200C2×C3×C4—82no—
201C2×C3×C4#10412yes—
202C2×C3×C4C4×D412yes—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12112yes—
205C2×C3×C4#12311yescharacteristic
206C3×C8—44no—
207C3×C8—44no—
208C24—21no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×D4—22no—
213C2×D4—42no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4C3×C22⋊C411yescharacteristic
222C22×C4C3×C22⋊C411yescharacteristic
223C22×C4C3×C22⋊C411yescharacteristic
224C22×C4C3×C22⋊C411yescharacteristic
225C22×C4C22×C3×C411yescharacteristic
226C22×C4C2×C3×D411yescharacteristic
227C22×C4C2×C3×D411yescharacteristic
228M16—24no—
229M16—42no—
230C2×C8—24no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—42no—
246C42C2×C3×D412yes—
247C42C3×#4112yes—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3—81no—
257C22×C3#38611yescharacteristic
258C22×C3C2×#10411yescharacteristic
259C22×C3#42111yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4#45412yes—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C3×C22⋊C411yescharacteristic
279Q8—44no—
280D4—24no—
281D4—42no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4C3×#10412yes—
307C2×C4C3×C4×D412yes—
308C2×C4C3×#12011yescharacteristic
309C2×C4C3×#12112yes—
310C2×C4C3×#12311yescharacteristic
311C8—44no—
312C8—44no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—82no—
316C2×C3#187111yescharacteristic
317C2×C3#188811yescharacteristic
318C2×C3#189111yescharacteristic
319C22—21no—
320C22—21no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C3×#38611yescharacteristic
329C22C2×C3×#10411yescharacteristic
330C22C3×#42111yescharacteristic
331C4—22no—
332C4—22no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4C3×#45412yes—
342C3#903211yescharacteristic, direct product factor, Sylow
343C2—21no—
344C2—21no—
345C2—82no—
346C2C3×#187111yescharacteristic
347C2C3×#188811yescharacteristic
348C2C3×#189111yescharacteristic
subgroupG/H
centerC2×C3×C4#120
commutatorC22×C4C22×C3×C4
Frattini, cosocleC2×C42C23×C3
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2983C2semidirect
3. C3×C4×#101C2semidirect
5. C3×#1300C2semidirect
6. #9032C3direct (N × H)
26. #2983C2×C3semidirect
28. C4×#101C2×C3semidirect
30. #1300C2×C3semidirect
342. C3#9032direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#29832
C3×#23982
C3×C4×#1012
C3×#14902
C3×#13002
#90323