Finite Group Databasebeta

← C3×#9517 #10628124

Subgroups — C3×#9517 #10628124

Order 768.

520 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2973C211yesmaximal, characteristic
2C3×#2579C211yesmaximal, characteristic
3C3×#1465C211yesmaximal, characteristic
4C3×#1463C212yesmaximal, complemented
5C3×#1305C212yesmaximal, complemented
6#9517C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#435—24no—
10C3×#432—21no—
11C3×#407C2212yes—
12C3×#407C2212yes—
13C2×C3×C22⋊C8—21no—
14C3×#398—21no—
15C3×#396—21no—
16C3×#387—21no—
17C3×#386C411yescharacteristic
18C3×#386C411yescharacteristic
19C3×#385—21no—
20C3×#385C411yescharacteristic
21C3×#385C411yescharacteristic
22C3×#380C2211yescharacteristic
23C3×#324—21no—
24C3×#324—24no—
25C3×#323—21no—
26#2973C2×C311yescharacteristic
27#2579C2×C311yescharacteristic
28#1465C2×C311yescharacteristic
29#1463C2×C312yescomplemented
30#1305C2×C312yescomplemented
31C22×C3×D4C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C2×C3×M16—22no—
37C2×C3×M16—22no—
38C22×C3×C8—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C3×C4⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8D414yes—
62C3×#97—22no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—21no—
68C2×C4×D4C22×C311yescharacteristic
69C22×C4⋊C4C22×C311yescharacteristic
70#435—24no—
71#432—21no—
72#407C22×C312yes—
73#407C22×C312yes—
74C2×C22⋊C8—21no—
75#398—21no—
76#396—21no—
77#387—21no—
78#386C3×C411yescharacteristic
79#386C3×C411yescharacteristic
80#385—21no—
81#385C3×C411yescharacteristic
82#385C3×C411yescharacteristic
83#380C22×C311yescharacteristic
84#324—21no—
85#324—24no—
86#323—21no—
87C24×C3C22⋊C411yescharacteristic
88C24×C3C22×C411yescharacteristic
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—41no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—42no—
101C22×C3×C4—81no—
102C22×C3×C4—81no—
103C22×C3×C4—81no—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4112yes—
111C3×M16—82no—
112C3×M16—82no—
113C2×C3×C8—24no—
114C2×C3×C8—42no—
115C2×C3×C8—82no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—42no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—81no—
122C3×C4⋊C4—81no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—42no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—81no—
129C3×C42—21no—
130C3×C42—21no—
131C22×D4C2×C3×C411yescharacteristic
132C23×C4C2×C3×C411yescharacteristic
133C23×C4C2×C3×C411yescharacteristic
134C23×C4C3×D411yescharacteristic
135C23×C4C23×C311yescharacteristic
136C2×M16—22no—
137C2×M16—22no—
138C22×C8—22no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4C3×D411yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4C2×C3×C411yescharacteristic
152C2×C22⋊C4C2×C3×C411yescharacteristic
153C2×C22⋊C4C2×C3×C411yescharacteristic
154C2×C22⋊C4C3×D411yescharacteristic
155C2×C42C3×D411yescharacteristic
156C4⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8C3×D414yes—
162#97—22no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3#10112yes—
177C23×C3C2×C22⋊C411yescharacteristic
178C23×C3C4×D412yes—
179C3×D4—44no—
180C2×C3×C4—22no—
181C2×C3×C4—24no—
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—42no—
189C2×C3×C4—42no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4—82no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12112yes—
210C2×C3×C4#12311yescharacteristic
211C3×C8—44no—
212C3×C8—82no—
213C24C3×C22⋊C411yescharacteristic
214C24C22×C3×C411yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4C3×C22⋊C411yescharacteristic
231C22×C4C3×C22⋊C411yescharacteristic
232C22×C4C3×C22⋊C411yescharacteristic
233C22×C4C2×C3×D412yes—
234C22×C4C2×C3×D411yescharacteristic
235C22×C4C2×C3×D411yescharacteristic
236C22×C4C3×#4112yes—
237M16—82no—
238M16—82no—
239C2×C8—24no—
240C2×C8—42no—
241C2×C8—82no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—42no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—81no—
255C42—21no—
256C42—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—81no—
269C22×C3#38611yescharacteristic
270C22×C3C2×#10111yescharacteristic
271C22×C3#41311yescharacteristic
272C3×C4—21no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—42no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—81no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C3×#10112yes—
291C23C2×C3×C22⋊C411yescharacteristic
292C23C3×C4×D412yes—
293D4—44no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4C3×#12011yescharacteristic
323C2×C4C3×#12112yes—
324C2×C4C3×#12311yescharacteristic
325C8—44no—
326C8—82no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—42no—
331C2×C3#189111yescharacteristic
332C2×C3#190111yescharacteristic
333C2×C3#191311yescharacteristic
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22C3×#38611yescharacteristic
347C22C2×C3×#10111yescharacteristic
348C22C3×#41311yescharacteristic
349C4—21no—
350C4—22no—
351C4—21no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C4—81no—
359C3#951711yescharacteristic, direct product factor, Sylow
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2C3×#189111yescharacteristic
365C2C3×#190111yescharacteristic
366C2C3×#191311yescharacteristic
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
4. C3×#1463C2semidirect
5. C3×#1305C2semidirect
6. #9517C3direct (N × H)
29. #1463C2×C3semidirect
30. #1305C2×C3semidirect
359. C3#9517direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#29732
C3×#25792
C3×#14652
C3×#14632
C3×#13052
#95173