Finite Group Databasebeta

← C3×#13338 #10631945

Subgroups — C3×#13338 #10631945

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#510C211yesmaximal, characteristic, complemented
2C22×C3×SD32C211yesmaximal, characteristic, complemented
3C3×#3155C211yesmaximal, characteristic
4C2×C3×#361C211yesmaximal, characteristic, complemented
5C2×C3×#359C211yesmaximal, characteristic, complemented
6C3×#1979C211yesmaximal, characteristic, complemented
7C3×#1379C211yesmaximal, characteristic, complemented
8#13338C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C2×C3×#135C2211yescharacteristic
11C22×C3×Q16C2211yescharacteristic
12C22×C3×D8C2211yescharacteristic
13C3×#535—21no—
14C3×#510—22no—
15C3×#510—22no—
16C2×C3×Q32—21no—
17C2×C3×SD32—22no—
18C2×C3×SD32—22no—
19C2×C3×SD32—21no—
20C2×C3×SD32—22no—
21C2×C3×M32C2211yescharacteristic
22C22×C3×C16C2211yescharacteristic
23C3×#496—22no—
24C3×#495—21no—
25C3×#493—21no—
26C2×C3×#108C2211yescharacteristic
27C3×#419—21no—
28C2×C3×#101—21no—
29C3×#405C2211yescharacteristic
30C3×#361—22no—
31C3×#359—22no—
32C2×#510C2×C311yescharacteristic, complemented
33C22×SD32C2×C311yescharacteristic, complemented
34#3155C2×C311yescharacteristic
35C2×#361C2×C311yescharacteristic, complemented
36C2×#359C2×C311yescharacteristic, complemented
37#1979C2×C311yescharacteristic, complemented
38#1379C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×C3×D4—21no—
42C3×AGL(1,ℤ/8)—42no—
43C3×AGL(1,ℤ/8)—42no—
44C3×#135—21no—
45C3×#135—21no—
46C3×#135—21no—
47C3×#135—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16—21no—
50C2×C3×Q16—21no—
51C2×C3×Q16—21no—
52C2×C3×Q16—21no—
53C2×C3×Q16D411yescharacteristic
54C2×C3×Q16D412yescomplemented
55C2×C3×SD16—22no—
56C2×C3×SD16—21no—
57C2×C3×D8—22no—
58C2×C3×D8—21no—
59C2×C3×D8—21no—
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×D8—21no—
63C2×C3×D8D412yes—
64C2×C3×D8D411yescharacteristic
65C2×C3×M16—21no—
66C2×C3×M16D411yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×#127—22no—
69C3×#124—22no—
70C3×#121—42no—
71C3×#120—42no—
72C3×#117D411yescharacteristic
73C3×Q32—24no—
74C3×SD32—28no—
75C3×SD32—24no—
76C3×M32—21no—
77C3×M32D412yescomplemented
78C2×C3×C16—21no—
79C2×C3×C16—21no—
80C2×C3×C16—21no—
81C2×C3×C16D412yescomplemented
82C3×#108—21no—
83C3×#108—21no—
84C3×#103—42no—
85C3×#101—42no—
86C3×#97—21no—
87C3×C4×C8—21no—
88C2×AGL(1,ℤ/8)—21no—
89C2×#135C22×C311yescharacteristic
90C22×Q16C22×C311yescharacteristic
91C22×D8C22×C311yescharacteristic
92#535—21no—
93#510—22no—
94#510—22no—
95C2×Q32—21no—
96C2×SD32—22no—
97C2×SD32—22no—
98C2×SD32—21no—
99C2×SD32—22no—
100C2×M32C22×C311yescharacteristic
101C22×C16C22×C311yescharacteristic
102#496—22no—
103#495—21no—
104#493—21no—
105C2×#108C22×C311yescharacteristic
106#419—21no—
107C2×#101—21no—
108#405C22×C311yescharacteristic
109#361—22no—
110#359—22no—
111C24×C3—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C2×C3×Q8—21no—
117C2×C3×Q8—22no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—44no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C3×Q16—22no—
136C3×Q16—22no—
137C3×Q16—21no—
138C3×Q16—21no—
139C3×Q16—41no—
140C3×Q16—41no—
141C3×SD16—44no—
142C3×SD16—41no—
143C3×SD16—41no—
144C3×D8—22no—
145C3×D8—21no—
146C3×D8—22no—
147C3×D8—21no—
148C3×D8—44no—
149C3×D8—41no—
150C3×D8—41no—
151C3×M16—21no—
152C3×M16—41no—
153C3×M16—41no—
154C3×M16D812yes—
155C2×C3×C8—21no—
156C2×C3×C8—41no—
157C2×C3×C8C2×D411yescharacteristic
158C2×C3×C8C2×D411yescharacteristic
159C2×C3×C8C2×D411yescharacteristic
160C2×C3×C8C2×D411yescharacteristic
161C2×C3×C8C2×D411yescharacteristic
162C2×C3×C8#4111yescharacteristic
163C3×C4⋊C4D812yes—
164C3×C22⋊C4—21no—
165C3×C22⋊C4—44no—
166C3×C42—21no—
167C3×C16—22no—
168C3×C16—22no—
169C2×#41—21no—
170C22×Q8—21no—
171C22×D4—21no—
172AGL(1,ℤ/8)—42no—
173AGL(1,ℤ/8)—42no—
174#135—21no—
175#135—21no—
176#135—21no—
177#135—21no—
178C2×Q16—21no—
179C2×Q16—21no—
180C2×Q16—21no—
181C2×Q16—21no—
182C2×Q16—21no—
183C2×Q16C3×D411yescharacteristic
184C2×Q16C3×D412yescomplemented
185C2×SD16—22no—
186C2×SD16—21no—
187C2×D8—22no—
188C2×D8—21no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8—21no—
193C2×D8C3×D412yes—
194C2×D8C3×D411yescharacteristic
195C2×M16—21no—
196C2×M16C3×D411yescharacteristic
197C22×C8C23×C311yescharacteristic
198#127—22no—
199#124—22no—
200#121—42no—
201#120—42no—
202#117C3×D411yescharacteristic
203Q32—24no—
204SD32—28no—
205SD32—24no—
206M32—21no—
207M32C3×D412yescomplemented
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16C3×D412yescomplemented
212#108—21no—
213#108—21no—
214#103—42no—
215#101—42no—
216#97—21no—
217C4×C8—21no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—81no—
221C23×C3—81no—
222C23×C3—162no—
223C23×C3AGL(1,ℤ/8)11yescharacteristic
224C3×Q8—41no—
225C3×Q8—42no—
226C3×Q8—41no—
227C3×Q8—42no—
228C3×Q8—81no—
229C3×Q8—81no—
230C3×D4—42no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—42no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—164no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12711yescharacteristic
249C2×C3×C4C2×D811yescharacteristic
250C2×C3×C4C2×D811yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C3×C8—22no—
253C3×C8—81no—
254C3×C8#12011yescharacteristic
255C3×C8#12011yescharacteristic
256C3×C8#12111yescharacteristic
257C3×C8#12111yescharacteristic
258C24—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—44no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C2×C3×D411yescharacteristic
282Q16—22no—
283Q16—22no—
284Q16—21no—
285Q16—21no—
286Q16—41no—
287Q16—41no—
288SD16—44no—
289SD16—41no—
290SD16—41no—
291D8—22no—
292D8—21no—
293D8—22no—
294D8—21no—
295D8—44no—
296D8—41no—
297D8—41no—
298M16—21no—
299M16—41no—
300M16—41no—
301M16C3×D812yes—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8C2×C3×D411yescharacteristic
305C2×C8C2×C3×D411yescharacteristic
306C2×C8C2×C3×D411yescharacteristic
307C2×C8C2×C3×D411yescharacteristic
308C2×C8C2×C3×D411yescharacteristic
309C2×C8C3×#4111yescharacteristic
310C4⋊C4C3×D812yes—
311C22⋊C4—21no—
312C22⋊C4—44no—
313C42—21no—
314C16—22no—
315C16—22no—
316C22×C3—22no—
317C22×C3—82no—
318C22×C3—84no—
319C22×C3—81no—
320C22×C3—82no—
321C22×C3—162no—
322C22×C3—162no—
323C22×C3#44711yescharacteristic
324C22×C3#45911yescharacteristic
325C22×C3#49611yescharacteristic
326C3×C4—42no—
327C3×C4—81no—
328C3×C4—82no—
329C3×C4#39211yescharacteristic
330C3×C4#44711yescharacteristic
331C3×C4#45911yescharacteristic
332C3×C4#49311yescharacteristic
333C23—42no—
334C23—41no—
335C23—81no—
336C23—81no—
337C23—162no—
338C23C3×AGL(1,ℤ/8)11yescharacteristic
339Q8—41no—
340Q8—42no—
341Q8—41no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—42no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—164no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4C3×#12011yescharacteristic
362C2×C4C3×#12111yescharacteristic
363C2×C4C3×#12711yescharacteristic
364C2×C4C2×C3×D811yescharacteristic
365C2×C4C2×C3×D811yescharacteristic
366C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
367C8—22no—
368C8—81no—
369C8C3×#12011yescharacteristic
370C8C3×#12011yescharacteristic
371C8C3×#12111yescharacteristic
372C8C3×#12111yescharacteristic
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3—162no—
377C2×C3#201111yescharacteristic
378C2×C3#219412yes—
379C22—22no—
380C22—81no—
381C22—82no—
382C22—82no—
383C22—84no—
384C22—162no—
385C22—162no—
386C22C3×#44711yescharacteristic
387C22C3×#45911yescharacteristic
388C22C3×#49611yescharacteristic
389C4—42no—
390C4—81no—
391C4—82no—
392C4C3×#39211yescharacteristic
393C4C3×#44711yescharacteristic
394C4C3×#45911yescharacteristic
395C4C3×#49311yescharacteristic
396C3#1333811yescharacteristic, direct product factor, Sylow
397C2—21no—
398C2—21no—
399C2—162no—
400C2—161no—
401C2C3×#201111yescharacteristic
402C2C3×#219412yes—
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#510C2semidirect
2. C22×C3×SD32C2semidirect
4. C2×C3×#361C2semidirect
5. C2×C3×#359C2semidirect
6. C3×#1979C2semidirect
7. C3×#1379C2semidirect
8. #13338C3direct (N × H)
32. C2×#510C2×C3semidirect
33. C22×SD32C2×C3semidirect
35. C2×#361C2×C3semidirect
36. C2×#359C2×C3semidirect
37. #1979C2×C3semidirect
38. #1379C2×C3semidirect
54. C2×C3×Q16D4semidirect
77. C3×M32D4semidirect
81. C2×C3×C16D4semidirect
184. C2×Q16C3×D4semidirect
207. M32C3×D4semidirect
211. C2×C16C3×D4semidirect
396. C3#13338direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5102
C22×C3×SD322
C3×#31552
C2×C3×#3612
C2×C3×#3592
C3×#19792
C3×#13792
#133383