Finite Group Databasebeta

← C3×#10915 #10629522

Subgroups — C3×#10915 #10629522

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2315C211yesmaximal, characteristic
2C3×#2273C211yesmaximal, characteristic
3C2×C3×#342C212yesmaximal
4C2×C3×#342C211yesmaximal, characteristic
5C2×C3×#342C212yesmaximal
6#10915C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C22×C3×C22⋊C4C2212yes—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C3×#385—22no—
13C3×#385C414yescomplemented
14C3×#380—22no—
15C3×C4×C22⋊C4—22no—
16C3×C4×C22⋊C4—24no—
17C3×#342—22no—
18C3×#342—24no—
19C3×#342C418yescomplemented
20#2315C2×C311yescharacteristic
21#2273C2×C311yescharacteristic
22C2×#342C2×C312yes—
23C2×#342C2×C311yescharacteristic
24C2×#342C2×C312yes—
25C25×C3C2311yescharacteristic
26C23×C3×C4—22no—
27C23×C3×C4—21no—
28C23×C3×C4—22no—
29C23×C3×C4C2×C412yes—
30C23×C3×C4D412yes—
31C2×C3×C4⋊C4—24no—
32C2×C3×C22⋊C4—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—21no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4C2×C414yes—
49C2×C3×C22⋊C4C2×C412yes—
50C2×C3×C22⋊C4C2×C412yes—
51C2×C3×C22⋊C4C2×C414yes—
52C2×C3×C22⋊C4C2×C414yes—
53C2×C3×C22⋊C4D412yes—
54C2×C3×C22⋊C4D414yes—
55C2×C3×C22⋊C4Q814yes—
56C2×C3×C42—24no—
57C3×#95—24no—
58C3×#95—24no—
59C22×C22⋊C4C22×C311yescharacteristic
60C22×C22⋊C4C22×C312yes—
61C22×C22⋊C4C22×C311yescharacteristic
62C22×C22⋊C4C22×C311yescharacteristic
63C22×C22⋊C4C22×C312yes—
64#385—22no—
65#385C3×C414yescomplemented
66#380—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—24no—
69#342—22no—
70#342—24no—
71#342C3×C418yescomplemented
72C24×C3—82no—
73C24×C3C22×C411yescharacteristic
74C24×C3C22×C412yes—
75C24×C3C2×D411yescharacteristic
76C24×C3C2×D411yescharacteristic
77C24×C3C2×D412yes—
78C24×C3C2×Q812yes—
79C24×C3#4112yes—
80C24×C3#4112yes—
81C24×C3#4112yes—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—24no—
85C22×C3×C4—24no—
86C22×C3×C4—24no—
87C22×C3×C4—24no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—82no—
95C22×C3×C4—84no—
96C22×C3×C4—82no—
97C22×C3×C4—82no—
98C22×C3×C4—84no—
99C22×C3×C4C22⋊C414yes—
100C3×C4⋊C4—84no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—24no—
103C3×C22⋊C4—24no—
104C3×C22⋊C4—24no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—24no—
107C3×C22⋊C4—24no—
108C3×C22⋊C4—24no—
109C3×C22⋊C4—24no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4C4214yescomplemented
116C3×C22⋊C4C4⋊C418yescomplemented
117C3×C42—84no—
118C25C23×C311yescharacteristic
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4C2×C3×C412yes—
123C23×C4C3×D412yes—
124C2×C4⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4C2×C3×C414yes—
142C2×C22⋊C4C2×C3×C412yes—
143C2×C22⋊C4C2×C3×C414yes—
144C2×C22⋊C4C2×C3×C412yes—
145C2×C22⋊C4C2×C3×C414yes—
146C2×C22⋊C4C3×D412yes—
147C2×C22⋊C4C3×D414yes—
148C2×C22⋊C4C3×Q814yes—
149C2×C42—24no—
150#95—24no—
151#95—24no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—44no—
156C23×C3—44no—
157C23×C3—82no—
158C23×C3—82no—
159C23×C3—82no—
160C23×C3—82no—
161C23×C3C2×C4211yescharacteristic
162C23×C3C2×C22⋊C411yescharacteristic
163C23×C3C2×C4⋊C412yes—
164C23×C3#11711yescharacteristic
165C23×C3#11712yes—
166C23×C3C4×D412yes—
167C23×C3C4×D412yes—
168C23×C3C4×D412yes—
169C23×C3C4×D412yes—
170C23×C3C4×Q812yes—
171C23×C3C4×Q812yes—
172C23×C3#12011yescharacteristic
173C23×C3#12112yes—
174C23×C3C22⋊Q812yes—
175C23×C3C22⋊Q812yes—
176C23×C3#12311yescharacteristic
177C23×C3#12312yes—
178C23×C3#12411yescharacteristic
179C23×C3#12512yes—
180C23×C3#12612yes—
181C23×C3C4⋊Q811yescharacteristic
182C2×C3×C4—24no—
183C2×C3×C4—28no—
184C2×C3×C4—24no—
185C2×C3×C4—48no—
186C2×C3×C4—44no—
187C2×C3×C4—44no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4—42no—
191C2×C3×C4—44no—
192C2×C3×C4—82no—
193C2×C3×C4—84no—
194C2×C3×C4—84no—
195C2×C3×C4—84no—
196C2×C3×C4—84no—
197C2×C3×C4—84no—
198C2×C3×C4—84no—
199C2×C3×C4—84no—
200C24—82no—
201C24C22×C3×C411yescharacteristic
202C24C22×C3×C412yes—
203C24C2×C3×D411yescharacteristic
204C24C2×C3×D411yescharacteristic
205C24C2×C3×D412yes—
206C24C2×C3×Q812yes—
207C24C3×#4112yes—
208C24C3×#4112yes—
209C24C3×#4112yes—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—84no—
227C22×C4C3×C22⋊C414yes—
228C4⋊C4—84no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4C3×C4214yescomplemented
244C22⋊C4C3×C4⋊C418yescomplemented
245C42—84no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—41no—
257C22×C3—42no—
258C22×C3—41no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—41no—
265C22×C3—42no—
266C22×C3—41no—
267C22×C3C4×C22⋊C411yescharacteristic
268C22×C3C4×C4⋊C412yes—
269C22×C3#38512yes—
270C22×C3#38611yescharacteristic
271C22×C3#38712yes—
272C22×C3#38712yes—
273C22×C3#38812yes—
274C22×C3#38912yes—
275C22×C3#39111yescharacteristic
276C3×C4—44no—
277C3×C4—48no—
278C3×C4—44no—
279C3×C4—84no—
280C3×C4—82no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—44no—
285C23—44no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23—82no—
290C23C2×C3×C4211yescharacteristic
291C23C2×C3×C22⋊C411yescharacteristic
292C23C2×C3×C4⋊C412yes—
293C23C3×#11711yescharacteristic
294C23C3×#11712yes—
295C23C3×C4×D412yes—
296C23C3×C4×D412yes—
297C23C3×C4×D412yes—
298C23C3×C4×D412yes—
299C23C3×C4×Q812yes—
300C23C3×C4×Q812yes—
301C23C3×#12011yescharacteristic
302C23C3×#12112yes—
303C23C3×C22⋊Q812yes—
304C23C3×C22⋊Q812yes—
305C23C3×#12311yescharacteristic
306C23C3×#12312yes—
307C23C3×#12411yescharacteristic
308C23C3×#12512yes—
309C23C3×#12612yes—
310C23C3×C4⋊Q811yescharacteristic
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3#143611yescharacteristic
338C2×C3#191312yes—
339C22—22no—
340C22—22no—
341C22—24no—
342C22—24no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22C3×C4×C22⋊C411yescharacteristic
361C22C3×C4×C4⋊C412yes—
362C22C3×#38512yes—
363C22C3×#38611yescharacteristic
364C22C3×#38712yes—
365C22C3×#38712yes—
366C22C3×#38812yes—
367C22C3×#38912yes—
368C22C3×#39111yescharacteristic
369C4—44no—
370C4—48no—
371C4—44no—
372C4—84no—
373C4—82no—
374C3#1091511yescharacteristic, direct product factor, Sylow
375C2—21no—
376C2—22no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—22no—
381C2—22no—
382C2—21no—
383C2C3×#143611yescharacteristic
384C2C3×#191312yes—
subgroupG/H
center, socleC22×C3#386
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
6. #10915C3direct (N × H)
13. C3×#385C4semidirect
19. C3×#342C4semidirect
65. #385C3×C4semidirect
71. #342C3×C4semidirect
115. C3×C22⋊C4C42semidirect
116. C3×C22⋊C4C4⋊C4semidirect
243. C22⋊C4C3×C42semidirect
244. C22⋊C4C3×C4⋊C4semidirect
374. C3#10915direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23152
C3×#22732
C2×C3×#3422
#109153