Finite Group Databasebeta

← C3×#13319 #10631926

Subgroups — C3×#13319 #10631926

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D16C211yesmaximal, characteristic, complemented
2C2×C3×#472C211yesmaximal, characteristic, complemented
3C3×#3058C211yesmaximal, characteristic, complemented
4C2×C3×#361C211yesmaximal, characteristic, complemented
5C3×#2140C211yesmaximal, characteristic, complemented
6C2×C3×#350C211yesmaximal, characteristic, complemented
7C3×#1382C211yesmaximal, characteristic, complemented
8#13319C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C22×C3×D8C2211yescharacteristic
11C22×C3×D8C2211yescharacteristic
12C2×C3×#131C2211yescharacteristic
13C3×#535—21no—
14C2×C3×D16—22no—
15C2×C3×D16—22no—
16C2×C3×D16—22no—
17C2×C3×M32C2211yescharacteristic
18C22×C3×C16C2211yescharacteristic
19C3×#472—22no—
20C3×#472—22no—
21C3×#469—22no—
22C3×#466—22no—
23C2×C3×#108C2211yescharacteristic
24C3×#427C2211yescharacteristic
25C3×#418—21no—
26C2×C3×#100—21no—
27C3×#361—22no—
28C3×#357—22no—
29C3×#350—22no—
30C22×D16C2×C311yescharacteristic, complemented
31C2×#472C2×C311yescharacteristic, complemented
32#3058C2×C311yescharacteristic, complemented
33C2×#361C2×C311yescharacteristic, complemented
34#2140C2×C311yescharacteristic, complemented
35C2×#350C2×C311yescharacteristic, complemented
36#1382C2×C311yescharacteristic, complemented
37C2×C3×#41D411yescharacteristic
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C3×AGL(1,ℤ/8)—24no—
41C3×AGL(1,ℤ/8)—42no—
42C2×C3×SD16—41no—
43C2×C3×D8—21no—
44C2×C3×D8—21no—
45C2×C3×D8—21no—
46C2×C3×D8—21no—
47C2×C3×D8—21no—
48C2×C3×D8—21no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8—21no—
53C2×C3×D8—41no—
54C2×C3×D8D412yes—
55C2×C3×D8D412yescomplemented
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×#127—41no—
66C3×#124—41no—
67C3×#121—24no—
68C3×#120—42no—
69C3×#117—21no—
70C3×D16—28no—
71C3×M32—21no—
72C3×M32—21no—
73C2×C3×C16—21no—
74C2×C3×C16—21no—
75C2×C3×C16—21no—
76C2×C3×C16D412yescomplemented
77C3×#108—21no—
78C3×#108D412yescomplemented
79C3×#107D412yes—
80C3×#106—21no—
81C3×#102—24no—
82C3×#100—24no—
83C2×AGL(1,ℤ/8)—21no—
84C22×D8C22×C311yescharacteristic
85C22×D8C22×C311yescharacteristic
86C2×#131C22×C311yescharacteristic
87#535—21no—
88C2×D16—22no—
89C2×D16—22no—
90C2×D16—22no—
91C2×M32C22×C311yescharacteristic
92C22×C16C22×C311yescharacteristic
93#472—22no—
94#472—22no—
95#469—22no—
96#466—22no—
97C2×#108C22×C311yescharacteristic
98#427C22×C311yescharacteristic
99#418—21no—
100C2×#100—21no—
101#361—22no—
102#357—22no—
103#350—22no—
104C24×C3—41no—
105C24×C3—41no—
106C3×#41—21no—
107C3×#41—21no—
108C3×#41—21no—
109C3×#41—21no—
110C2×C3×Q8D811yescharacteristic
111C2×C3×D4—21no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—44no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4D811yescharacteristic
126C22×C3×C4—21no—
127C22×C3×C4C2×D411yescharacteristic
128C3×SD16—44no—
129C3×D8—22no—
130C3×D8—22no—
131C3×D8—22no—
132C3×D8—22no—
133C3×D8—44no—
134C3×D8—41no—
135C3×D8—41no—
136C3×D8—41no—
137C3×D8—41no—
138C3×M16—21no—
139C3×M16—22no—
140C3×M16—21no—
141C3×M16—41no—
142C3×M16—41no—
143C3×M16—41no—
144C2×C3×C8—21no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8C2×D411yescharacteristic
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8C2×D411yescharacteristic
153C2×C3×C8#4111yescharacteristic
154C3×C4⋊C4—22no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—41no—
157C3×C42—41no—
158C3×C16—22no—
159C3×C16—41no—
160C2×#41C3×D411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163AGL(1,ℤ/8)—24no—
164AGL(1,ℤ/8)—42no—
165C2×SD16—41no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C2×D8—41no—
177C2×D8C3×D412yes—
178C2×D8C3×D412yescomplemented
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
188#127—41no—
189#124—41no—
190#121—24no—
191#120—42no—
192#117—21no—
193D16—28no—
194M32—21no—
195M32—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C3×D412yescomplemented
200#108—21no—
201#108C3×D412yescomplemented
202#107C3×D412yes—
203#106—21no—
204#102—24no—
205#100—24no—
206C23×C3—21no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3—162no—
214C23×C3—162no—
215C23×C3AGL(1,ℤ/8)11yescharacteristic
216C3×Q8—21no—
217C3×Q8—21no—
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—164no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4C2×D811yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C3×C8—41no—
242C3×C8—42no—
243C3×C8#12011yescharacteristic
244C3×C8#12111yescharacteristic
245C3×C8#12111yescharacteristic
246C3×C8#12711yescharacteristic
247C24—41no—
248C24—41no—
249#41—21no—
250#41—21no—
251#41—21no—
252#41—21no—
253C2×Q8C3×D811yescharacteristic
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4C3×D811yescharacteristic
269C22×C4—21no—
270C22×C4C2×C3×D411yescharacteristic
271SD16—44no—
272D8—22no—
273D8—22no—
274D8—22no—
275D8—22no—
276D8—44no—
277D8—41no—
278D8—41no—
279D8—41no—
280D8—41no—
281M16—21no—
282M16—22no—
283M16—21no—
284M16—41no—
285M16—41no—
286M16—41no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8C2×C3×D411yescharacteristic
292C2×C8C2×C3×D411yescharacteristic
293C2×C8C2×C3×D411yescharacteristic
294C2×C8C2×C3×D411yescharacteristic
295C2×C8C2×C3×D411yescharacteristic
296C2×C8C3×#4111yescharacteristic
297C4⋊C4—22no—
298C22⋊C4—44no—
299C22⋊C4—41no—
300C42—41no—
301C16—22no—
302C16—41no—
303C22×C3—22no—
304C22×C3—42no—
305C22×C3—41no—
306C22×C3—84no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—162no—
310C22×C3—162no—
311C22×C3—162no—
312C22×C3—162no—
313C22×C3—162no—
314C22×C3#44711yescharacteristic
315C22×C3#44911yescharacteristic
316C22×C3#46911yescharacteristic
317C3×C4—41no—
318C3×C4—82no—
319C3×C4#39211yescharacteristic
320C3×C4#44711yescharacteristic
321C3×C4#44911yescharacteristic
322C3×C4#46611yescharacteristic
323C23—21no—
324C23—42no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—162no—
331C23—162no—
332C23C3×AGL(1,ℤ/8)11yescharacteristic
333Q8—21no—
334Q8—21no—
335D4—21no—
336D4—21no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—42no—
341D4—42no—
342D4—42no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—164no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4C3×#12011yescharacteristic
353C2×C4C3×#12011yescharacteristic
354C2×C4C3×#12111yescharacteristic
355C2×C4C2×C3×D811yescharacteristic
356C2×C4C3×#13511yescharacteristic
357C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
358C8—41no—
359C8—42no—
360C8C3×#12011yescharacteristic
361C8C3×#12111yescharacteristic
362C8C3×#12111yescharacteristic
363C8C3×#12711yescharacteristic
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—81no—
367C2×C3—162no—
368C2×C3—162no—
369C2×C3#199911yescharacteristic
370C2×C3#219212yes—
371C22—22no—
372C22—41no—
373C22—42no—
374C22—82no—
375C22—82no—
376C22—84no—
377C22—162no—
378C22—162no—
379C22—162no—
380C22—162no—
381C22—162no—
382C22C3×#44711yescharacteristic
383C22C3×#44911yescharacteristic
384C22C3×#46911yescharacteristic
385C4—41no—
386C4—82no—
387C4C3×#39211yescharacteristic
388C4C3×#44711yescharacteristic
389C4C3×#44911yescharacteristic
390C4C3×#46611yescharacteristic
391C3#1331911yescharacteristic, direct product factor, Sylow
392C2—21no—
393C2—21no—
394C2—81no—
395C2—162no—
396C2—162no—
397C2C3×#199911yescharacteristic
398C2C3×#219212yes—
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
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×D16C2semidirect
2. C2×C3×#472C2semidirect
3. C3×#3058C2semidirect
4. C2×C3×#361C2semidirect
5. C3×#2140C2semidirect
6. C2×C3×#350C2semidirect
7. C3×#1382C2semidirect
8. #13319C3direct (N × H)
30. C22×D16C2×C3semidirect
31. C2×#472C2×C3semidirect
32. #3058C2×C3semidirect
33. C2×#361C2×C3semidirect
34. #2140C2×C3semidirect
35. C2×#350C2×C3semidirect
36. #1382C2×C3semidirect
55. C2×C3×D8D4semidirect
76. C2×C3×C16D4semidirect
78. C3×#108D4semidirect
178. C2×D8C3×D4semidirect
199. C2×C16C3×D4semidirect
201. #108C3×D4semidirect
391. C3#13319direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D162
C2×C3×#4722
C3×#30582
C2×C3×#3612
C3×#21402
C2×C3×#3502
C3×#13822
#133193