Finite Group Databasebeta

← C3×#12985 #10631592

Subgroups — C3×#12985 #10631592

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×M32C211yesmaximal, characteristic, complemented
2C2×C3×#435C211yesmaximal, characteristic
3C2×C3×#349C212yesmaximal, complemented
4C3×#2112C211yesmaximal, characteristic, complemented
5C3×#1377C212yesmaximal, complemented
6#12985C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×M16C2211yescharacteristic
8C23×C3×C8C2211yescharacteristic
9C2×C3×C4×D4C411yescharacteristic
10C2×C3×M32—21no—
11C2×C3×M32—24no—
12C2×C3×M32C2212yes—
13C2×C3×M32C2212yes—
14C22×C3×C16—21no—
15C3×#435—21no—
16C3×#435—21no—
17C3×#435—21no—
18C3×#435—21no—
19C2×C3×C4⋊C8C411yescharacteristic
20C2×C3×C22⋊C8C411yescharacteristic
21C2×C3×C22⋊C8C411yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C3×#349—24no—
24C3×C22⋊C16—21no—
25C3×C22⋊C16—21no—
26C22×M32C2×C311yescharacteristic, complemented
27C2×#435C2×C311yescharacteristic
28C2×#349C2×C312yescomplemented
29#2112C2×C311yescharacteristic, complemented
30#1377C2×C312yescomplemented
31C22×C3×D4C811yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C2×C3×M16—21no—
35C2×C3×M16—21no—
36C2×C3×M16—22no—
37C2×C3×M16—22no—
38C22×C3×C8—22no—
39C22×C3×C8—22no—
40C22×C3×C8C2×C411yescharacteristic
41C22×C3×C8C2×C411yescharacteristic
42C22×C3×C8C2×C411yescharacteristic
43C22×C3×C8D411yescharacteristic
44C22×C3×C8D411yescharacteristic
45C22×C3×C8D411yescharacteristic
46C22×C3×C8D411yescharacteristic
47C22×C3×C8C2311yescharacteristic
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4C811yescharacteristic
53C2×C3×C22⋊C4C811yescharacteristic
54C2×C3×C22⋊C4C811yescharacteristic
55C2×C3×C42C2×C411yescharacteristic
56C3×M32—24no—
57C3×M32—22no—
58C3×M32—22no—
59C3×M32—22no—
60C3×M32D414yescomplemented
61C2×C3×C16—24no—
62C2×C3×C16—22no—
63C2×C3×C16—22no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97—21no—
72C22×M16C22×C311yescharacteristic
73C23×C8C22×C311yescharacteristic
74C2×C4×D4C3×C411yescharacteristic
75C2×M32—21no—
76C2×M32—24no—
77C2×M32C22×C312yes—
78C2×M32C22×C312yes—
79C22×C16—21no—
80#435—21no—
81#435—21no—
82#435—21no—
83#435—21no—
84C2×C4⋊C8C3×C411yescharacteristic
85C2×C22⋊C8C3×C411yescharacteristic
86C2×C22⋊C8C3×C411yescharacteristic
87C2×#97C22×C311yescharacteristic
88#349—24no—
89C22⋊C16—21no—
90C22⋊C16—21no—
91C24×C3C2×C811yescharacteristic
92C24×C3C2×C811yescharacteristic
93C2×C3×D4—21no—
94C2×C3×D4—21no—
95C2×C3×D4—42no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C2×C811yescharacteristic
105C22×C3×C4C2×C811yescharacteristic
106C22×C3×C4C2×C811yescharacteristic
107C22×C3×C4C2×C811yescharacteristic
108C22×C3×C4M1611yescharacteristic
109C22×C3×C4M1611yescharacteristic
110C22×C3×C4C22×C411yescharacteristic
111C3×M16—41no—
112C3×M16—42no—
113C3×M16—41no—
114C2×C3×C8—22no—
115C2×C3×C8—21no—
116C2×C3×C8—21no—
117C2×C3×C8—22no—
118C2×C3×C8—22no—
119C2×C3×C8—21no—
120C2×C3×C8—21no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8C22⋊C411yescharacteristic
126C2×C3×C8C22⋊C411yescharacteristic
127C2×C3×C8C2×D412yes—
128C2×C3×C8C2×D411yescharacteristic
129C2×C3×C8C2×D411yescharacteristic
130C2×C3×C8#4112yes—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C22⋊C4—21no—
134C3×C22⋊C4—21no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—21no—
137C3×C42—21no—
138C3×C42—21no—
139C3×C16—24no—
140C3×C16—42no—
141C22×D4C3×C811yescharacteristic
142C23×C4C2×C3×C411yescharacteristic
143C23×C4C2×C3×C411yescharacteristic
144C2×M16—22no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×C3×C411yescharacteristic
151C22×C8C2×C3×C411yescharacteristic
152C22×C8C2×C3×C411yescharacteristic
153C22×C8C3×D411yescharacteristic
154C22×C8C3×D411yescharacteristic
155C22×C8C3×D411yescharacteristic
156C22×C8C3×D411yescharacteristic
157C22×C8C23×C311yescharacteristic
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C2×C4⋊C4C3×C811yescharacteristic
163C2×C22⋊C4C3×C811yescharacteristic
164C2×C22⋊C4C3×C811yescharacteristic
165C2×C42C2×C3×C411yescharacteristic
166M32—24no—
167M32—22no—
168M32—22no—
169M32—22no—
170M32C3×D414yescomplemented
171C2×C16—24no—
172C2×C16—22no—
173C2×C16—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—21no—
180#97—21no—
181#97—21no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—21no—
185C23×C3—22no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3C22⋊C811yescharacteristic
190C23×C3C22⋊C811yescharacteristic
191C23×C3C22×C811yescharacteristic
192C3×D4—42no—
193C3×D4—42no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4C22⋊C811yescharacteristic
211C2×C3×C4C22⋊C811yescharacteristic
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D412yes—
214C2×C3×C4C2×M1611yescharacteristic
215C2×C3×C4#13112yes—
216C3×C8—21no—
217C3×C8—21no—
218C3×C8—41no—
219C3×C8—41no—
220C3×C8#12011yescharacteristic
221C3×C8#12112yes—
222C3×C8#12311yescharacteristic
223C24C2×C3×C811yescharacteristic
224C24C2×C3×C811yescharacteristic
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4C3×C22⋊C411yescharacteristic
235C22×C4C3×C22⋊C411yescharacteristic
236C22×C4C2×C3×C811yescharacteristic
237C22×C4C2×C3×C811yescharacteristic
238C22×C4C2×C3×C811yescharacteristic
239C22×C4C2×C3×C811yescharacteristic
240C22×C4C3×M1611yescharacteristic
241C22×C4C3×M1611yescharacteristic
242C22×C4C22×C3×C411yescharacteristic
243M16—41no—
244M16—42no—
245M16—41no—
246C2×C8—22no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8C3×C22⋊C411yescharacteristic
258C2×C8C3×C22⋊C411yescharacteristic
259C2×C8C2×C3×D412yes—
260C2×C8C2×C3×D411yescharacteristic
261C2×C8C2×C3×D411yescharacteristic
262C2×C8C3×#4112yes—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C42—21no—
270C42—21no—
271C16—24no—
272C16—42no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—44no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—82no—
285C22×C3#34912yescomplemented
286C22×C3C2×C22⋊C811yescharacteristic
287C22×C3C8×D412yes—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4#38611yescharacteristic
294C3×C4#40811yescharacteristic
295C3×C4#43512yes—
296C23—22no—
297C23—22no—
298C23—21no—
299C23—22no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23C3×C22⋊C811yescharacteristic
304C23C3×C22⋊C811yescharacteristic
305C23C22×C3×C811yescharacteristic
306D4—42no—
307D4—42no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4C3×C22⋊C811yescharacteristic
325C2×C4C3×C22⋊C811yescharacteristic
326C2×C4C2×C3×C22⋊C411yescharacteristic
327C2×C4C3×C4×D412yes—
328C2×C4C2×C3×M1611yescharacteristic
329C2×C4C3×#13112yes—
330C8—21no—
331C8—21no—
332C8—41no—
333C8—41no—
334C8C3×#12011yescharacteristic
335C8C3×#12112yes—
336C8C3×#12311yescharacteristic
337C2×C3—22no—
338C2×C3—22no—
339C2×C3—41no—
340C2×C3—81no—
341C2×C3#186911yescharacteristic
342C2×C3C2×#34911yescharacteristic
343C2×C3#211511yescharacteristic
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—82no—
356C22C3×#34912yescomplemented
357C22C2×C3×C22⋊C811yescharacteristic
358C22C3×C8×D412yes—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4C3×#38611yescharacteristic
365C4C3×#40811yescharacteristic
366C4C3×#43512yes—
367C3#1298511yescharacteristic, direct product factor, Sylow
368C2—22no—
369C2—22no—
370C2—41no—
371C2—81no—
372C2C3×#186911yescharacteristic
373C2C2×C3×#34911yescharacteristic
374C2C3×#211511yescharacteristic
subgroupG/H
centerC2×C3×C4C2×C22⋊C4
commutatorC23C22×C3×C8
Frattini, cosocleC22×C8C23×C3
socleC22×C3C2×C22⋊C8
prime radicalC22×C3×D4C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×M32C2semidirect
3. C2×C3×#349C2semidirect
4. C3×#2112C2semidirect
5. C3×#1377C2semidirect
6. #12985C3direct (N × H)
26. C22×M32C2×C3semidirect
28. C2×#349C2×C3semidirect
29. #2112C2×C3semidirect
30. #1377C2×C3semidirect
60. C3×M32D4semidirect
170. M32C3×D4semidirect
285. C22×C3#349semidirect
356. C22C3×#349semidirect
367. C3#12985direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×M322
C2×C3×#4352
C2×C3×#3492
C3×#21122
C3×#13772
#129853