Finite Group Databasebeta

← C3×#13320 #10631927

Subgroups — C3×#13320 #10631927

Order 768.

540 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×SD32C211yesmaximal, characteristic, complemented
2C2×C3×#472C211yesmaximal, characteristic
3C3×#3059C211yesmaximal, characteristic, complemented
4C2×C3×#362C211yesmaximal, characteristic, complemented
5C3×#2140C211yesmaximal, characteristic, complemented
6C2×C3×#350C211yesmaximal, characteristic, complemented
7C3×#1382C211yesmaximal, characteristic, complemented
8#13320C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C22×C3×Q16C2211yescharacteristic
11C22×C3×D8C2211yescharacteristic
12C2×C3×#131C2211yescharacteristic
13C3×#536—21no—
14C2×C3×SD32—22no—
15C2×C3×SD32—22no—
16C2×C3×SD32—22no—
17C2×C3×M32C2211yescharacteristic
18C22×C3×C16C2211yescharacteristic
19C3×#472—22no—
20C3×#472—22no—
21C3×#470—22no—
22C3×#467—22no—
23C2×C3×#108C2211yescharacteristic
24C3×#427C2211yescharacteristic
25C3×#419—21no—
26C2×C3×#100—21no—
27C3×#362—22no—
28C3×#357—22no—
29C3×#350—22no—
30C22×SD32C2×C311yescharacteristic, complemented
31C2×#472C2×C311yescharacteristic
32#3059C2×C311yescharacteristic, complemented
33C2×#362C2×C311yescharacteristic, complemented
34#2140C2×C311yescharacteristic, complemented
35C2×#350C2×C311yescharacteristic, complemented
36#1382C2×C311yescharacteristic, complemented
37C2×C3×#41D411yescharacteristic
38C22×C3×Q8—21no—
39C22×C3×D4—21no—
40C3×AGL(1,ℤ/8)—24no—
41C3×AGL(1,ℤ/8)—42no—
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C2×C3×Q16—21no—
45C2×C3×Q16—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16D412yes—
48C2×C3×SD16—41no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8—21no—
53C2×C3×D8—21no—
54C2×C3×D8—41no—
55C2×C3×D8D412yes—
56C3×#131—21no—
57C3×#131—21no—
58C3×#131—21no—
59C3×#131—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D411yescharacteristic
63C22×C3×C8—21no—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—41no—
66C3×#124—41no—
67C3×#123—42no—
68C3×C22⋊Q8—24no—
69C3×#117—21no—
70C3×SD32—28no—
71C3×M32—21no—
72C3×M32—21no—
73C2×C3×C16—21no—
74C2×C3×C16—21no—
75C2×C3×C16—21no—
76C2×C3×C16D412yes—
77C3×#108—21no—
78C3×#108D412yes—
79C3×#107D412yes—
80C3×#106—21no—
81C3×#103—24no—
82C3×#100—24no—
83C2×AGL(1,ℤ/8)—21no—
84C22×Q16C22×C311yescharacteristic
85C22×D8C22×C311yescharacteristic
86C2×#131C22×C311yescharacteristic
87#536—21no—
88C2×SD32—22no—
89C2×SD32—22no—
90C2×SD32—22no—
91C2×M32C22×C311yescharacteristic
92C22×C16C22×C311yescharacteristic
93#472—22no—
94#472—22no—
95#470—22no—
96#467—22no—
97C2×#108C22×C311yescharacteristic
98#427C22×C311yescharacteristic
99#419—21no—
100C2×#100—21no—
101#362—22no—
102#357—22no—
103#350—22no—
104C24×C3—41no—
105C3×#41—21no—
106C3×#41—21no—
107C3×#41—21no—
108C3×#41—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8D811yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—22no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4D811yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C3×Q16—22no—
128C3×Q16—22no—
129C3×Q16—41no—
130C3×Q16—41no—
131C3×SD16—44no—
132C3×D8—22no—
133C3×D8—22no—
134C3×D8—44no—
135C3×D8—41no—
136C3×D8—41no—
137C3×M16—21no—
138C3×M16—22no—
139C3×M16—21no—
140C3×M16—41no—
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—21no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2×D411yescharacteristic
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8#4111yescharacteristic
153C3×C4⋊C4—22no—
154C3×C4⋊C4—44no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—41no—
157C3×C42—41no—
158C3×C16—22no—
159C3×C16—41no—
160C2×#41C3×D411yescharacteristic
161C22×Q8—21no—
162C22×D4—21no—
163AGL(1,ℤ/8)—24no—
164AGL(1,ℤ/8)—42no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16C3×D412yes—
171C2×SD16—41no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—21no—
177C2×D8—41no—
178C2×D8C3×D412yes—
179#131—21no—
180#131—21no—
181#131—21no—
182#131—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16C3×D411yescharacteristic
186C22×C8—21no—
187C22×C8C23×C311yescharacteristic
188C4⋊Q8—41no—
189#124—41no—
190#123—42no—
191C22⋊Q8—24no—
192#117—21no—
193SD32—28no—
194M32—21no—
195M32—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C3×D412yes—
200#108—21no—
201#108C3×D412yes—
202#107C3×D412yes—
203#106—21no—
204#103—24no—
205#100—24no—
206C23×C3—21no—
207C23×C3—42no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—162no—
211C23×C3AGL(1,ℤ/8)11yescharacteristic
212C3×Q8—21no—
213C3×Q8—21no—
214C3×Q8—42no—
215C3×Q8—42no—
216C3×Q8—81no—
217C3×Q8—81no—
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4#13511yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C3×C8—41no—
240C3×C8—42no—
241C3×C8#12011yescharacteristic
242C3×C8#12111yescharacteristic
243C3×C8#12111yescharacteristic
244C3×C8#12711yescharacteristic
245C24—41no—
246#41—21no—
247#41—21no—
248#41—21no—
249#41—21no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8C3×D811yescharacteristic
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4C3×D811yescharacteristic
265C22×C4—21no—
266C22×C4—41no—
267C22×C4C2×C3×D411yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272SD16—44no—
273D8—22no—
274D8—22no—
275D8—44no—
276D8—41no—
277D8—41no—
278M16—21no—
279M16—22no—
280M16—21no—
281M16—41no—
282M16—41no—
283M16—41no—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8C2×C3×D411yescharacteristic
289C2×C8C2×C3×D411yescharacteristic
290C2×C8C2×C3×D411yescharacteristic
291C2×C8C2×C3×D411yescharacteristic
292C2×C8C2×C3×D411yescharacteristic
293C2×C8C3×#4111yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—41no—
298C42—41no—
299C16—22no—
300C16—41no—
301C22×C3—22no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—82no—
305C22×C3—162no—
306C22×C3—162no—
307C22×C3—162no—
308C22×C3#44711yescharacteristic
309C22×C3#44911yescharacteristic
310C22×C3#46911yescharacteristic
311C3×C4—41no—
312C3×C4—82no—
313C3×C4—82no—
314C3×C4#39211yescharacteristic
315C3×C4#44711yescharacteristic
316C3×C4#44911yescharacteristic
317C3×C4#46611yescharacteristic
318C23—21no—
319C23—42no—
320C23—81no—
321C23—81no—
322C23—162no—
323C23C3×AGL(1,ℤ/8)11yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—42no—
327Q8—42no—
328Q8—81no—
329Q8—81no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—42no—
335D4—42no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C3×#12011yescharacteristic
346C2×C4C3×#12011yescharacteristic
347C2×C4C3×#12111yescharacteristic
348C2×C4C2×C3×D811yescharacteristic
349C2×C4C3×#13511yescharacteristic
350C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
351C8—41no—
352C8—42no—
353C8C3×#12011yescharacteristic
354C8C3×#12111yescharacteristic
355C8C3×#12111yescharacteristic
356C8C3×#12711yescharacteristic
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—81no—
360C2×C3—162no—
361C2×C3#199911yescharacteristic
362C2×C3#219312yes—
363C22—22no—
364C22—41no—
365C22—42no—
366C22—82no—
367C22—162no—
368C22—162no—
369C22—162no—
370C22C3×#44711yescharacteristic
371C22C3×#44911yescharacteristic
372C22C3×#46911yescharacteristic
373C4—41no—
374C4—82no—
375C4—82no—
376C4C3×#39211yescharacteristic
377C4C3×#44711yescharacteristic
378C4C3×#44911yescharacteristic
379C4C3×#46611yescharacteristic
380C3#1332011yescharacteristic, direct product factor, Sylow
381C2—21no—
382C2—21no—
383C2—81no—
384C2—162no—
385C2C3×#199911yescharacteristic
386C2C3×#219312yes—
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#472C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×SD32C2semidirect
3. C3×#3059C2semidirect
4. C2×C3×#362C2semidirect
5. C3×#2140C2semidirect
6. C2×C3×#350C2semidirect
7. C3×#1382C2semidirect
8. #13320C3direct (N × H)
30. C22×SD32C2×C3semidirect
32. #3059C2×C3semidirect
33. C2×#362C2×C3semidirect
34. #2140C2×C3semidirect
35. C2×#350C2×C3semidirect
36. #1382C2×C3semidirect
380. C3#13320direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×SD322
C2×C3×#4722
C3×#30592
C2×C3×#3622
C3×#21402
C2×C3×#3502
C3×#13822
#133203