Finite Group Databasebeta

← C3×#15171 #10633778

Subgroups — C3×#15171 #10633778

Order 768.

1302 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#401C211yesmaximal, characteristic, complemented
2C2×C3×#396C211yesmaximal, characteristic
3C3×#1442C213yesmaximal, complemented
4C3×#1440C213yesmaximal
5C3×#1437C213yesmaximal
6C3×#1436C213yesmaximal, complemented
7C3×C4×#95C211yesmaximal, characteristic, complemented
8#15171C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×C4⋊C4C2213yes—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2213yes—
12C22×C3×C22⋊C4C2213yes—
13C22×C3×C42C2213yes—
14C3×#401—24no—
15C3×#396—24no—
16C2×C3×#95C2213yes—
17C2×C3×#95C2213yes—
18C2×C3×#95C2213yes—
19C2×C3×#95C2213yes—
20C2×C3×#95C2213yes—
21C2×C3×#95C2213yes—
22C2×C3×#95C2213yes—
23C2×C3×#95C2211yescharacteristic
24C2×#401C2×C311yescharacteristic, complemented
25C2×#396C2×C311yescharacteristic
26#1442C2×C313yescomplemented
27#1440C2×C313yes—
28#1437C2×C313yes—
29#1436C2×C313yescomplemented
30C4×#95C2×C311yescharacteristic, complemented
31C25×C3C2311yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×C3×C4C2313yes—
34C23×C3×C4C2313yes—
35C23×C3×C4C2313yes—
36C23×C3×C4C2313yes—
37C23×C3×C4C2311yescharacteristic
38C2×C3×C4⋊C4—26no—
39C2×C3×C4⋊C4—26no—
40C2×C3×C4⋊C4—43no—
41C2×C3×C22⋊C4—26no—
42C2×C3×C22⋊C4—26no—
43C2×C3×C22⋊C4—26no—
44C2×C3×C22⋊C4—26no—
45C2×C3×C22⋊C4—26no—
46C2×C3×C22⋊C4—43no—
47C2×C3×C22⋊C4—43no—
48C2×C3×C42—26no—
49C2×C3×C42—43no—
50C2×C3×C42—43no—
51C3×#95—26no—
52C3×#95—26no—
53C3×#95—26no—
54C3×#95—26no—
55C3×#95—26no—
56C3×#95—26no—
57C3×#95—43no—
58C3×#95D414yes—
59C22×C4⋊C4C22×C313yes—
60C22×C22⋊C4C22×C311yescharacteristic
61C22×C22⋊C4C22×C313yes—
62C22×C22⋊C4C22×C313yes—
63C22×C42C22×C313yes—
64#401—24no—
65#396—24no—
66C2×#95C22×C313yes—
67C2×#95C22×C313yes—
68C2×#95C22×C313yes—
69C2×#95C22×C313yes—
70C2×#95C22×C313yes—
71C2×#95C22×C313yes—
72C2×#95C22×C313yes—
73C2×#95C22×C311yescharacteristic
74C24×C3—21no—
75C24×C3—23no—
76C24×C3—23no—
77C24×C3—23no—
78C24×C3—23no—
79C24×C3—21no—
80C24×C3#4112yes—
81C24×C3C2411yescharacteristic
82C22×C3×C4—23no—
83C22×C3×C4—21no—
84C22×C3×C4—23no—
85C22×C3×C4—23no—
86C22×C3×C4—23no—
87C22×C3×C4—23no—
88C22×C3×C4—23no—
89C22×C3×C4—23no—
90C22×C3×C4—23no—
91C22×C3×C4—23no—
92C22×C3×C4—23no—
93C22×C3×C4—23no—
94C22×C3×C4—23no—
95C22×C3×C4—23no—
96C22×C3×C4—23no—
97C22×C3×C4—23no—
98C22×C3×C4—23no—
99C22×C3×C4—23no—
100C22×C3×C4—23no—
101C22×C3×C4—23no—
102C22×C3×C4—21no—
103C22×C3×C4—23no—
104C22×C3×C4—23no—
105C22×C3×C4—23no—
106C22×C3×C4—23no—
107C22×C3×C4—23no—
108C22×C3×C4—23no—
109C22×C3×C4—23no—
110C22×C3×C4—23no—
111C22×C3×C4—23no—
112C22×C3×C4C2×D416yes—
113C22×C3×C4#4116yes—
114C22×C3×C4#4116yes—
115C22×C3×C4#4116yes—
116C3×C4⋊C4—412no—
117C3×C22⋊C4—412no—
118C3×C22⋊C4—412no—
119C3×C22⋊C4—44no—
120C3×C42—412no—
121C25C23×C311yescharacteristic
122C23×C4C23×C311yescharacteristic
123C23×C4C23×C313yes—
124C23×C4C23×C313yes—
125C23×C4C23×C313yes—
126C23×C4C23×C313yes—
127C23×C4C23×C311yescharacteristic
128C2×C4⋊C4—26no—
129C2×C4⋊C4—26no—
130C2×C4⋊C4—43no—
131C2×C22⋊C4—26no—
132C2×C22⋊C4—26no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4—26no—
136C2×C22⋊C4—43no—
137C2×C22⋊C4—43no—
138C2×C42—26no—
139C2×C42—43no—
140C2×C42—43no—
141#95—26no—
142#95—26no—
143#95—26no—
144#95—26no—
145#95—26no—
146#95—26no—
147#95—43no—
148#95C3×D414yes—
149C23×C3—26no—
150C23×C3—22no—
151C23×C3—26no—
152C23×C3—41no—
153C23×C3—43no—
154C23×C3—43no—
155C23×C3—43no—
156C23×C3—43no—
157C23×C3—43no—
158C23×C3—43no—
159C23×C3—43no—
160C23×C3—43no—
161C23×C3—43no—
162C23×C3C22×D411yescharacteristic
163C23×C3C2×#4113yes—
164C23×C3C2×#4113yes—
165C23×C3C2×#4111yescharacteristic
166C23×C3C2×#4113yes—
167C23×C3H(5,𝔽2)13yes—
168C23×C32−1+411yescharacteristic
169C2×C3×C4—26no—
170C2×C3×C4—26no—
171C2×C3×C4—26no—
172C2×C3×C4—26no—
173C2×C3×C4—26no—
174C2×C3×C4—26no—
175C2×C3×C4—26no—
176C2×C3×C4—26no—
177C2×C3×C4—26no—
178C2×C3×C4—26no—
179C2×C3×C4—26no—
180C2×C3×C4—43no—
181C2×C3×C4—43no—
182C2×C3×C4—43no—
183C2×C3×C4—43no—
184C2×C3×C4—43no—
185C2×C3×C4—43no—
186C2×C3×C4—43no—
187C2×C3×C4—43no—
188C2×C3×C4—43no—
189C2×C3×C4—43no—
190C2×C3×C4—43no—
191C2×C3×C4—43no—
192C2×C3×C4—41no—
193C2×C3×C4—43no—
194C2×C3×C4—43no—
195C2×C3×C4—43no—
196C2×C3×C4—43no—
197C2×C3×C4—43no—
198C2×C3×C4—43no—
199C2×C3×C4—43no—
200C2×C3×C4—43no—
201C2×C3×C4—43no—
202C2×C3×C4#12614yes—
203C24—21no—
204C24—23no—
205C24—23no—
206C24—23no—
207C24—23no—
208C24—21no—
209C24C3×#4112yes—
210C24C24×C311yescharacteristic
211C22×C4—23no—
212C22×C4—21no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—23no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—21no—
232C22×C4—23no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4C2×C3×D416yes—
242C22×C4C3×#4116yes—
243C22×C4C3×#4116yes—
244C22×C4C3×#4116yes—
245C4⋊C4—412no—
246C22⋊C4—412no—
247C22⋊C4—412no—
248C22⋊C4—44no—
249C42—412no—
250C22×C3—46no—
251C22×C3—46no—
252C22×C3—42no—
253C22×C3—83no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3—83no—
257C22×C3C2×#12611yescharacteristic
258C22×C3#52913yes—
259C22×C3#52913yes—
260C22×C3#52913yes—
261C22×C3#53213yes—
262C22×C3#53813yes—
263C22×C3#53913yes—
264C22×C3#54613yes—
265C22×C3#54811yescharacteristic
266C22×C3#55113yes—
267C22×C3#55313yes—
268C22×C3#55313yes—
269C22×C3#55613yes—
270C3×C4—24no—
271C3×C4—46no—
272C3×C4—46no—
273C3×C4—46no—
274C3×C4—83no—
275C3×C4—81no—
276C23—26no—
277C23—22no—
278C23—26no—
279C23—41no—
280C23—43no—
281C23—43no—
282C23—43no—
283C23—43no—
284C23—43no—
285C23—43no—
286C23—43no—
287C23—43no—
288C23—43no—
289C23C22×C3×D411yescharacteristic
290C23C2×C3×#4113yes—
291C23C2×C3×#4113yes—
292C23C2×C3×#4111yescharacteristic
293C23C2×C3×#4113yes—
294C23C3×H(5,𝔽2)13yes—
295C23C3×2−1+411yescharacteristic
296C2×C4—26no—
297C2×C4—26no—
298C2×C4—26no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—26no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—43no—
308C2×C4—43no—
309C2×C4—43no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—43no—
313C2×C4—43no—
314C2×C4—43no—
315C2×C4—43no—
316C2×C4—43no—
317C2×C4—43no—
318C2×C4—43no—
319C2×C4—41no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—43no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—43no—
328C2×C4—43no—
329C2×C4C3×#12614yes—
330C2×C3—82no—
331C2×C3#240111yescharacteristic
332C2×C3#255913yes—
333C2×C3#256013yes—
334C2×C3#257613yes—
335C2×C3#259413yes—
336C2×C3#264311yescharacteristic
337C2×C3#286011yescharacteristic
338C22—42no—
339C22—46no—
340C22—46no—
341C22—81no—
342C22—83no—
343C22—83no—
344C22—81no—
345C22C2×C3×#12611yescharacteristic
346C22C3×#52913yes—
347C22C3×#52913yes—
348C22C3×#52913yes—
349C22C3×#53213yes—
350C22C3×#53813yes—
351C22C3×#53913yes—
352C22C3×#54613yes—
353C22C3×#54811yescharacteristic
354C22C3×#55113yes—
355C22C3×#55313yes—
356C22C3×#55313yes—
357C22C3×#55613yes—
358C4—24no—
359C4—46no—
360C4—46no—
361C4—46no—
362C4—83no—
363C4—81no—
364C3#1517111yescharacteristic, direct product factor, Sylow
365C2—82no—
366C2C3×#240111yescharacteristic
367C2C3×#255913yes—
368C2C3×#256013yes—
369C2C3×#257613yes—
370C2C3×#259413yes—
371C2C3×#264311yescharacteristic
372C2C3×#286011yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#401C2semidirect
3. C3×#1442C2semidirect
6. C3×#1436C2semidirect
7. C3×C4×#95C2semidirect
8. #15171C3direct (N × H)
24. C2×#401C2×C3semidirect
26. #1442C2×C3semidirect
29. #1436C2×C3semidirect
30. C4×#95C2×C3semidirect
364. C3#15171direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4012
C2×C3×#3962
C3×#14422
C3×#14402
C3×#14372
C3×#14362
C3×C4×#952
#151713