Finite Group Databasebeta

← C3×#9506 #10628113

Subgroups — C3×#9506 #10628113

Order 768.

682 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2975C211yesmaximal, characteristic
2C3×#2409C211yesmaximal, characteristic
3C3×#1784C212yesmaximal, complemented
4C3×#1465C211yesmaximal, characteristic
5C3×#1305C212yesmaximal, complemented
6#9506C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#436—24no—
11C3×#431—21no—
12C2×C3×#101—24no—
13C3×#407—21no—
14C3×#407C2212yes—
15C3×#407C2212yes—
16C3×#395—21no—
17C3×#393—21no—
18C3×#391—21no—
19C3×#386C412yes—
20C3×#385C412yes—
21C3×#380C2211yescharacteristic
22C3×#324—21no—
23C3×#323—21no—
24#2975C2×C311yescharacteristic
25#2409C2×C311yescharacteristic
26#1784C2×C312yescomplemented
27#1465C2×C311yescharacteristic
28#1305C2×C312yescomplemented
29C22×C3×Q8—21no—
30C22×C3×Q8—22no—
31C22×C3×Q8—21no—
32C22×C3×Q8—22no—
33C22×C3×Q8—44no—
34C22×C3×D4C2×C411yescharacteristic
35C23×C3×C4C2×C412yes—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C2×C3×M16—22no—
39C2×C3×M16—22no—
40C2×C3×M16D414yes—
41C3×C22⋊Q8—44no—
42C3×C4×D4—22no—
43C3×C4×D4—22no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4C2×C412yes—
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C3×C4⋊C8—22no—
53C3×#101—48no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×C4×C8—22no—
60C3×#95—22no—
61C3×#95—22no—
62C3×#95—21no—
63C23×Q8C22×C311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C4×D4C22×C311yescharacteristic
66#436—24no—
67#431—21no—
68C2×#101—24no—
69#407—21no—
70#407C22×C312yes—
71#407C22×C312yes—
72#395—21no—
73#393—21no—
74#391—21no—
75#386C3×C412yes—
76#385C3×C412yes—
77#380C22×C311yescharacteristic
78#324—21no—
79#323—21no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22×C411yescharacteristic
82C2×C3×Q8—28no—
83C2×C3×Q8—24no—
84C2×C3×Q8—42no—
85C2×C3×Q8—44no—
86C2×C3×Q8—44no—
87C2×C3×Q8—48no—
88C2×C3×Q8—41no—
89C2×C3×Q8—42no—
90C2×C3×Q8—41no—
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—41no—
94C22×C3×C4—22no—
95C22×C3×C4—24no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4—44no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4112yes—
109C3×M16—44no—
110C3×M16—44no—
111C3×M16—82no—
112C2×C3×C8—24no—
113C2×C3×C8—42no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—42no—
116C3×C4⋊C4—44no—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—44no—
120C3×C22⋊C4—42no—
121C3×C42—21no—
122C3×C42—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—44no—
128C22×D4C2×C3×C411yescharacteristic
129C23×C4C2×C3×C412yes—
130C23×C4C3×D411yescharacteristic
131C23×C4C23×C311yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16C3×D414yes—
135C22⋊Q8—44no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C3×D411yescharacteristic
141C2×C22⋊C4—22no—
142C2×C22⋊C4C2×C3×C411yescharacteristic
143C2×C22⋊C4C2×C3×C412yes—
144C2×C22⋊C4C3×D411yescharacteristic
145C2×C42C3×D411yescharacteristic
146C4⋊C8—22no—
147#101—48no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C4×C8—22no—
154#95—22no—
155#95—22no—
156#95—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3C2×C22⋊C411yescharacteristic
167C23×C3C4×D412yes—
168C3×Q8—48no—
169C3×Q8—44no—
170C3×Q8—44no—
171C3×D4—44no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—24no—
177C2×C3×C4—28no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—44no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—81no—
192C2×C3×C4#10112yescomplemented
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C3×C8—44no—
197C3×C8—82no—
198C24C3×C22⋊C411yescharacteristic
199C24C22×C3×C411yescharacteristic
200C2×Q8—28no—
201C2×Q8—24no—
202C2×Q8—42no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—48no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C3×C22⋊C411yescharacteristic
221C22×C4C3×C22⋊C411yescharacteristic
222C22×C4C3×C22⋊C411yescharacteristic
223C22×C4C2×C3×D412yes—
224C22×C4C2×C3×D411yescharacteristic
225C22×C4C2×C3×D411yescharacteristic
226C22×C4C3×#4112yes—
227M16—44no—
228M16—44no—
229M16—82no—
230C2×C8—24no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C42—21no—
240C42—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—81no—
253C22×C3#38611yescharacteristic
254C22×C3C2×#10111yescharacteristic
255C22×C3#41311yescharacteristic
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—44no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—82no—
263C3×C4B(4,𝔽2)12yes—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C2×C3×C22⋊C411yescharacteristic
274C23C3×C4×D412yes—
275Q8—48no—
276Q8—44no—
277Q8—44no—
278D4—44no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4C3×#10112yescomplemented
300C2×C4C3×#12011yescharacteristic
301C2×C4C3×#12112yes—
302C2×C4C3×#12311yescharacteristic
303C8—44no—
304C8—82no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—81no—
309C2×C3#188911yescharacteristic
310C2×C3#190111yescharacteristic
311C2×C3#191511yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—81no—
324C22C3×#38611yescharacteristic
325C22C2×C3×#10111yescharacteristic
326C22C3×#41311yescharacteristic
327C4—22no—
328C4—21no—
329C4—22no—
330C4—44no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4C3×B(4,𝔽2)12yes—
335C3#950611yescharacteristic, direct product factor, Sylow
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2C3×#188911yescharacteristic
341C2C3×#190111yescharacteristic
342C2C3×#191511yescharacteristic
subgroupG/H
center, socleC22×C3#386
commutatorC24C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC22×C3×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1784C2semidirect
5. C3×#1305C2semidirect
6. #9506C3direct (N × H)
26. #1784C2×C3semidirect
28. #1305C2×C3semidirect
192. C2×C3×C4#101semidirect
299. C2×C4C3×#101semidirect
335. C3#9506direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#29752
C3×#24092
C3×#17842
C3×#14652
C3×#13052
#95063