Finite Group Databasebeta

← C3×#14836 #10633443

Subgroups — C3×#14836 #10633443

Order 768.

2254 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C211yesmaximal, characteristic, complemented
2C2×C3×#386C213yesmaximal
3C2×C3×#380C213yesmaximal, complemented
4C22×C3×#95C211yesmaximal, characteristic, complemented
5C3×#1445C213yesmaximal, complemented
6C3×#1444C211yesmaximal, characteristic, complemented
7C3×#1439C213yesmaximal, complemented
8#14836C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C24×C3×C4C2213yes—
11C22×C3×C4⋊C4C2213yes—
12C22×C3×C4⋊C4C2213yes—
13C22×C3×C22⋊C4C2213yes—
14C22×C3×C22⋊C4C2213yes—
15C22×C3×C22⋊C4C2213yes—
16C3×#399—24no—
17C3×#386—212no—
18C3×#380—212no—
19C2×C3×#95—24no—
20C2×C3×#95C2213yes—
21C2×C3×#95C2216yes—
22C2×C3×#95C2213yes—
23C2×C3×#95C2213yes—
24C2×C3×#95C2211yescharacteristic
25C2×#399C2×C311yescharacteristic, complemented
26C2×#386C2×C313yes—
27C2×#380C2×C313yescomplemented
28C22×#95C2×C311yescharacteristic, complemented
29#1445C2×C313yescomplemented
30#1444C2×C311yescharacteristic, complemented
31#1439C2×C313yescomplemented
32C25×C3C2311yescharacteristic
33C25×C3C2311yescharacteristic
34C22×C3×D4—26no—
35C22×C3×D4—26no—
36C22×C3×D4Q814yes—
37C23×C3×C4—212no—
38C23×C3×C4—26no—
39C23×C3×C4—26no—
40C23×C3×C4—46no—
41C23×C3×C4D4112yes—
42C23×C3×C4C2313yes—
43C23×C3×C4C2313yes—
44C23×C3×C4C2313yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2313yes—
47C2×C3×C4⋊C4—26no—
48C2×C3×C4⋊C4—26no—
49C2×C3×C4⋊C4—26no—
50C2×C3×C4⋊C4—26no—
51C2×C3×C4⋊C4—46no—
52C2×C3×C22⋊C4—212no—
53C2×C3×C22⋊C4—26no—
54C2×C3×C22⋊C4—26no—
55C2×C3×C22⋊C4—26no—
56C2×C3×C22⋊C4—26no—
57C2×C3×C22⋊C4—26no—
58C2×C3×C22⋊C4—46no—
59C3×#95—26no—
60C3×#95—212no—
61C3×#95—26no—
62C3×#95—26no—
63C3×#95D414yescomplemented
64C23×D4C22×C311yescharacteristic
65C24×C4C22×C313yes—
66C22×C4⋊C4C22×C313yes—
67C22×C4⋊C4C22×C313yes—
68C22×C22⋊C4C22×C313yes—
69C22×C22⋊C4C22×C313yes—
70C22×C22⋊C4C22×C313yes—
71#399—24no—
72#386—212no—
73#380—212no—
74C2×#95—24no—
75C2×#95C22×C313yes—
76C2×#95C22×C316yes—
77C2×#95C22×C313yes—
78C2×#95C22×C313yes—
79C2×#95C22×C311yescharacteristic
80C24×C3—21no—
81C24×C3—23no—
82C24×C3—23no—
83C24×C3—21no—
84C24×C3—21no—
85C24×C3—23no—
86C24×C3—23no—
87C24×C3—23no—
88C24×C3—23no—
89C24×C3—21no—
90C24×C3C2×D416yes—
91C24×C3C2×Q812yes—
92C24×C3C2×Q812yes—
93C24×C3#4116yes—
94C24×C3C2411yescharacteristic
95C2×C3×D4—224no—
96C2×C3×D4—412no—
97C2×C3×D4—44no—
98C22×C3×C4—26no—
99C22×C3×C4—26no—
100C22×C3×C4—224no—
101C22×C3×C4—23no—
102C22×C3×C4—23no—
103C22×C3×C4—23no—
104C22×C3×C4—26no—
105C22×C3×C4—23no—
106C22×C3×C4—26no—
107C22×C3×C4—224no—
108C22×C3×C4—23no—
109C22×C3×C4—23no—
110C22×C3×C4—26no—
111C22×C3×C4—26no—
112C22×C3×C4—23no—
113C22×C3×C4—23no—
114C22×C3×C4—26no—
115C22×C3×C4—26no—
116C22×C3×C4—23no—
117C22×C3×C4—23no—
118C22×C3×C4—412no—
119C22×C3×C4—424no—
120C22×C3×C4—424no—
121C22×C3×C4C2×D416yes—
122C22×C3×C4C2×D416yes—
123C22×C3×C4C2×D416yes—
124C22×C3×C4C2×Q812yes—
125C22×C3×C4#4116yes—
126C3×C4⋊C4—412no—
127C3×C4⋊C4—412no—
128C3×C22⋊C4—412no—
129C3×C22⋊C4—412no—
130C3×C22⋊C4—412no—
131C25C23×C311yescharacteristic
132C25C23×C311yescharacteristic
133C22×D4—26no—
134C22×D4—26no—
135C22×D4C3×Q814yes—
136C23×C4—212no—
137C23×C4—26no—
138C23×C4—26no—
139C23×C4—46no—
140C23×C4C3×D4112yes—
141C23×C4C23×C313yes—
142C23×C4C23×C313yes—
143C23×C4C23×C313yes—
144C23×C4C23×C311yescharacteristic
145C23×C4C23×C313yes—
146C2×C4⋊C4—26no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—46no—
151C2×C22⋊C4—212no—
152C2×C22⋊C4—26no—
153C2×C22⋊C4—26no—
154C2×C22⋊C4—26no—
155C2×C22⋊C4—26no—
156C2×C22⋊C4—26no—
157C2×C22⋊C4—46no—
158#95—26no—
159#95—212no—
160#95—26no—
161#95—26no—
162#95C3×D414yescomplemented
163C23×C3—26no—
164C23×C3—26no—
165C23×C3—26no—
166C23×C3—26no—
167C23×C3—26no—
168C23×C3—22no—
169C23×C3—212no—
170C23×C3—26no—
171C23×C3—26no—
172C23×C3—26no—
173C23×C3—22no—
174C23×C3—26no—
175C23×C3—43no—
176C23×C3—43no—
177C23×C3—41no—
178C23×C3—43no—
179C23×C3—43no—
180C23×C3—43no—
181C23×C3—43no—
182C23×C3—46no—
183C23×C3—43no—
184C23×C3C22⋊Q8112yes—
185C23×C3#123112yes—
186C23×C3#12714yes—
187C23×C3C22×D411yescharacteristic
188C23×C3C22×D413yes—
189C23×C3C22×Q811yescharacteristic
190C23×C3C2×#4113yes—
191C23×C3C2×#4113yes—
192C23×C3H(5,𝔽2)13yes—
193C23×C32−1+411yescharacteristic
194C3×D4—416no—
195C2×C3×C4—26no—
196C2×C3×C4—26no—
197C2×C3×C4—212no—
198C2×C3×C4—212no—
199C2×C3×C4—26no—
200C2×C3×C4—26no—
201C2×C3×C4—26no—
202C2×C3×C4—26no—
203C2×C3×C4—448no—
204C2×C3×C4—43no—
205C2×C3×C4—41no—
206C2×C3×C4—46no—
207C2×C3×C4—46no—
208C2×C3×C4—43no—
209C2×C3×C4—46no—
210C2×C3×C4—43no—
211C2×C3×C4—43no—
212C2×C3×C4—43no—
213C2×C3×C4—46no—
214C2×C3×C4—43no—
215C2×C3×C4—46no—
216C2×C3×C4—43no—
217C2×C3×C4—46no—
218C2×C3×C4#121112yes—
219C24—21no—
220C24—23no—
221C24—23no—
222C24—21no—
223C24—21no—
224C24—23no—
225C24—23no—
226C24—23no—
227C24—23no—
228C24—21no—
229C24C2×C3×D416yes—
230C24C2×C3×Q812yes—
231C24C2×C3×Q812yes—
232C24C3×#4116yes—
233C24C24×C311yescharacteristic
234C2×D4—224no—
235C2×D4—412no—
236C2×D4—44no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—224no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—26no—
244C22×C4—23no—
245C22×C4—26no—
246C22×C4—224no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—412no—
258C22×C4—424no—
259C22×C4—424no—
260C22×C4C2×C3×D416yes—
261C22×C4C2×C3×D416yes—
262C22×C4C2×C3×D416yes—
263C22×C4C2×C3×Q812yes—
264C22×C4C3×#4116yes—
265C4⋊C4—412no—
266C4⋊C4—412no—
267C22⋊C4—412no—
268C22⋊C4—412no—
269C22⋊C4—412no—
270C22×C3—212no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—212no—
274C22×C3—212no—
275C22×C3—212no—
276C22×C3—46no—
277C22×C3—42no—
278C22×C3—46no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—83no—
282C22×C3—83no—
283C22×C3#39918yescomplemented
284C22×C3C2×#12113yes—
285C22×C3C2×C22⋊Q813yes—
286C22×C3C2×#12313yes—
287C22×C3C2×#12711yescharacteristic
288C22×C3#53213yes—
289C22×C3#53413yes—
290C22×C3#53713yes—
291C22×C3#54613yes—
292C22×C3D4×Q811yescharacteristic
293C22×C3#55116yes—
294C22×C3#55213yes—
295C22×C3#55413yes—
296C3×C4—212no—
297C3×C4—42no—
298C3×C4—46no—
299C3×C4—83no—
300C3×C4—83no—
301C23—26no—
302C23—26no—
303C23—26no—
304C23—26no—
305C23—26no—
306C23—22no—
307C23—212no—
308C23—26no—
309C23—26no—
310C23—26no—
311C23—22no—
312C23—26no—
313C23—43no—
314C23—43no—
315C23—41no—
316C23—43no—
317C23—43no—
318C23—43no—
319C23—43no—
320C23—46no—
321C23—43no—
322C23C3×C22⋊Q8112yes—
323C23C3×#123112yes—
324C23C3×#12714yes—
325C23C22×C3×D411yescharacteristic
326C23C22×C3×D413yes—
327C23C22×C3×Q811yescharacteristic
328C23C2×C3×#4113yes—
329C23C2×C3×#4113yes—
330C23C3×H(5,𝔽2)13yes—
331C23C3×2−1+411yescharacteristic
332D4—416no—
333C2×C4—26no—
334C2×C4—26no—
335C2×C4—212no—
336C2×C4—212no—
337C2×C4—26no—
338C2×C4—26no—
339C2×C4—26no—
340C2×C4—26no—
341C2×C4—448no—
342C2×C4—43no—
343C2×C4—41no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—43no—
347C2×C4—46no—
348C2×C4—43no—
349C2×C4—43no—
350C2×C4—43no—
351C2×C4—46no—
352C2×C4—43no—
353C2×C4—46no—
354C2×C4—43no—
355C2×C4—46no—
356C2×C4C3×#121112yes—
357C2×C3—28no—
358C2×C3—82no—
359C2×C3C2×#39911yescharacteristic
360C2×C3#241213yes—
361C2×C3#241713yes—
362C2×C3#249813yes—
363C2×C3#256413yes—
364C2×C3#265611yescharacteristic
365C2×C3#285611yescharacteristic
366C22—24no—
367C22—212no—
368C22—212no—
369C22—212no—
370C22—212no—
371C22—24no—
372C22—42no—
373C22—46no—
374C22—46no—
375C22—81no—
376C22—83no—
377C22—83no—
378C22—81no—
379C22C3×#39918yescomplemented
380C22C2×C3×#12113yes—
381C22C2×C3×C22⋊Q813yes—
382C22C2×C3×#12313yes—
383C22C2×C3×#12711yescharacteristic
384C22C3×#53213yes—
385C22C3×#53413yes—
386C22C3×#53713yes—
387C22C3×#54613yes—
388C22C3×D4×Q811yescharacteristic
389C22C3×#55116yes—
390C22C3×#55213yes—
391C22C3×#55413yes—
392C4—212no—
393C4—42no—
394C4—46no—
395C4—83no—
396C4—83no—
397C3#1483611yescharacteristic, direct product factor, Sylow
398C2—28no—
399C2—82no—
400C2C2×C3×#39911yescharacteristic
401C2C3×#241213yes—
402C2C3×#241713yes—
403C2C3×#249813yes—
404C2C3×#256413yes—
405C2C3×#265611yescharacteristic
406C2C3×#285611yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#399C2semidirect
3. C2×C3×#380C2semidirect
4. C22×C3×#95C2semidirect
5. C3×#1445C2semidirect
6. C3×#1444C2semidirect
7. C3×#1439C2semidirect
8. #14836C3direct (N × H)
25. C2×#399C2×C3semidirect
27. C2×#380C2×C3semidirect
28. C22×#95C2×C3semidirect
29. #1445C2×C3semidirect
30. #1444C2×C3semidirect
31. #1439C2×C3semidirect
63. C3×#95D4semidirect
162. #95C3×D4semidirect
283. C22×C3#399semidirect
379. C22C3×#399semidirect
397. C3#14836direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3862
C2×C3×#3802
C22×C3×#952
C3×#14452
C3×#14442
C3×#14392
#148363