Finite Group Databasebeta

← C3×#13411 #10632018

Subgroups — C3×#13411 #10632018

Order 768.

462 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3155C211yesmaximal, characteristic
2C2×C3×#361C211yesmaximal, characteristic, complemented
3C2×C3×#359C211yesmaximal, characteristic, complemented
4C3×#2139C211yesmaximal, characteristic, complemented
5C2×C3×#357C211yesmaximal, characteristic, complemented
6C3×#1952C211yesmaximal, characteristic, complemented
7C3×#1379C211yesmaximal, characteristic, complemented
8#13411C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C2×C3×#135C2211yescharacteristic
11C22×C3×D8C2211yescharacteristic
12C3×#535—21no—
13C2×C3×M32C2211yescharacteristic
14C22×C3×C16C2211yescharacteristic
15C3×#496—22no—
16C3×#495—21no—
17C3×#493—21no—
18C2×C3×#108C2211yescharacteristic
19C3×#428—21no—
20C2×C3×#107C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C3×#361—22no—
23C3×#359—22no—
24C3×#358—21no—
25C3×#357—22no—
26C3×#357—21no—
27C3×#343—21no—
28#3155C2×C311yescharacteristic
29C2×#361C2×C311yescharacteristic, complemented
30C2×#359C2×C311yescharacteristic, complemented
31#2139C2×C311yescharacteristic, complemented
32C2×#357C2×C311yescharacteristic, complemented
33#1952C2×C311yescharacteristic, complemented
34#1379C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C22×C3×D4—21no—
37C3×AGL(1,ℤ/8)—42no—
38C3×AGL(1,ℤ/8)—42no—
39C3×#135—21no—
40C3×#135—21no—
41C3×#135—21no—
42C3×#135—21no—
43C2×C3×Q16D411yescharacteristic
44C2×C3×SD16—22no—
45C2×C3×SD16—21no—
46C2×C3×D8—22no—
47C2×C3×D8—21no—
48C2×C3×D8—21no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8D412yes—
53C2×C3×D8D411yescharacteristic
54C2×C3×M16—21no—
55C2×C3×M16D411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×#127—22no—
58C3×#124—22no—
59C3×#121—41no—
60C3×#121—41no—
61C3×#120—42no—
62C3×#117D411yescharacteristic
63C2×C3×C4⋊C4—21no—
64C3×M32—21no—
65C3×M32—21no—
66C2×C3×C16—21no—
67C2×C3×C16—21no—
68C2×C3×C16—21no—
69C2×C3×C16—21no—
70C3×#108—21no—
71C3×#108D412yescomplemented
72C3×#107—21no—
73C3×#107—21no—
74C3×#107—41no—
75C3×#106—41no—
76C3×#97—21no—
77C3×C4×C8—21no—
78C2×AGL(1,ℤ/8)—21no—
79C2×#135C22×C311yescharacteristic
80C22×D8C22×C311yescharacteristic
81#535—21no—
82C2×M32C22×C311yescharacteristic
83C22×C16C22×C311yescharacteristic
84#496—22no—
85#495—21no—
86#493—21no—
87C2×#108C22×C311yescharacteristic
88#428—21no—
89C2×#107C22×C311yescharacteristic
90#405C22×C311yescharacteristic
91#361—22no—
92#359—22no—
93#358—21no—
94#357—22no—
95#357—21no—
96#343—21no—
97C24×C3—41no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C3×#41—41no—
102C2×C3×Q8—21no—
103C2×C3×D4—22no—
104C2×C3×D4—21no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C3×Q16—21no—
117C3×Q16—21no—
118C3×SD16—44no—
119C3×SD16—41no—
120C3×SD16—41no—
121C3×D8—22no—
122C3×D8—21no—
123C3×D8—22no—
124C3×D8—21no—
125C3×D8—44no—
126C3×D8—41no—
127C3×D8—41no—
128C3×M16—21no—
129C3×M16—22no—
130C3×M16—41no—
131C3×M16SD1612yes—
132C2×C3×C8—21no—
133C2×C3×C8—41no—
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8C2×D411yescharacteristic
136C2×C3×C8C2×D411yescharacteristic
137C2×C3×C8#4111yescharacteristic
138C2×C3×C8#4111yescharacteristic
139C2×C3×C8#4111yescharacteristic
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4D811yescharacteristic
143C3×C4⋊C4D811yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147C3×C42—21no—
148C3×C16—41no—
149C3×C16—41no—
150C2×#41—21no—
151C22×D4—21no—
152AGL(1,ℤ/8)—42no—
153AGL(1,ℤ/8)—42no—
154#135—21no—
155#135—21no—
156#135—21no—
157#135—21no—
158C2×Q16C3×D411yescharacteristic
159C2×SD16—22no—
160C2×SD16—21no—
161C2×D8—22no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8C3×D412yes—
168C2×D8C3×D411yescharacteristic
169C2×M16—21no—
170C2×M16C3×D411yescharacteristic
171C22×C8C23×C311yescharacteristic
172#127—22no—
173#124—22no—
174#121—41no—
175#121—41no—
176#120—42no—
177#117C3×D411yescharacteristic
178C2×C4⋊C4—21no—
179M32—21no—
180M32—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185#108—21no—
186#108C3×D412yescomplemented
187#107—21no—
188#107—21no—
189#107—41no—
190#106—41no—
191#97—21no—
192C4×C8—21no—
193C23×C3—42no—
194C23×C3—41no—
195C23×C3—81no—
196C23×C3—81no—
197C23×C3—161no—
198C23×C3—161no—
199C23×C3AGL(1,ℤ/8)11yescharacteristic
200C3×Q8—41no—
201C3×Q8—41no—
202C3×D4—42no—
203C3×D4—41no—
204C3×D4—41no—
205C3×D4—42no—
206C3×D4—81no—
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C3×D4—162no—
211C3×D4—162no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—161no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12411yescharacteristic
222C2×C3×C4C2×D811yescharacteristic
223C2×C3×C4C2×SD1611yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C3×C8—22no—
226C3×C8—42no—
227C3×C8#12111yescharacteristic
228C3×C8#12111yescharacteristic
229C3×C8#12311yescharacteristic
230C3×C8#12311yescharacteristic
231C24—41no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—41no—
236C2×Q8—21no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C3×D411yescharacteristic
250Q16—21no—
251Q16—21no—
252SD16—44no—
253SD16—41no—
254SD16—41no—
255D8—22no—
256D8—21no—
257D8—22no—
258D8—21no—
259D8—44no—
260D8—41no—
261D8—41no—
262M16—21no—
263M16—22no—
264M16—41no—
265M16C3×SD1612yes—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8C2×C3×D411yescharacteristic
269C2×C8C2×C3×D411yescharacteristic
270C2×C8C2×C3×D411yescharacteristic
271C2×C8C3×#4111yescharacteristic
272C2×C8C3×#4111yescharacteristic
273C2×C8C3×#4111yescharacteristic
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4C3×D811yescharacteristic
277C4⋊C4C3×D811yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C42—21no—
282C16—41no—
283C16—41no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—81no—
287C22×C3—82no—
288C22×C3—81no—
289C22×C3—82no—
290C22×C3—81no—
291C22×C3—82no—
292C22×C3—162no—
293C22×C3—162no—
294C22×C3#44711yescharacteristic
295C22×C3#46011yescharacteristic
296C22×C3#49011yescharacteristic
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—81no—
300C3×C4—161no—
301C3×C4#39411yescharacteristic
302C3×C4#45011yescharacteristic
303C3×C4#45911yescharacteristic
304C3×C4#48611yescharacteristic
305C23—42no—
306C23—41no—
307C23—81no—
308C23—81no—
309C23—161no—
310C23—161no—
311C23C3×AGL(1,ℤ/8)11yescharacteristic
312Q8—41no—
313Q8—41no—
314D4—42no—
315D4—41no—
316D4—41no—
317D4—42no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—162no—
323D4—162no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—161no—
331C2×C4C3×#12011yescharacteristic
332C2×C4C3×#12111yescharacteristic
333C2×C4C3×#12411yescharacteristic
334C2×C4C2×C3×D811yescharacteristic
335C2×C4C2×C3×SD1611yescharacteristic
336C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
337C8—22no—
338C8—42no—
339C8C3×#12111yescharacteristic
340C8C3×#12111yescharacteristic
341C8C3×#12311yescharacteristic
342C8C3×#12311yescharacteristic
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—161no—
346C2×C3—162no—
347C2×C3#204111yescharacteristic
348C2×C3#219211yescharacteristic
349C2×C3#219411yescharacteristic
350C22—21no—
351C22—21no—
352C22—81no—
353C22—82no—
354C22—81no—
355C22—82no—
356C22—81no—
357C22—82no—
358C22—162no—
359C22—162no—
360C22C3×#44711yescharacteristic
361C22C3×#46011yescharacteristic
362C22C3×#49011yescharacteristic
363C4—41no—
364C4—41no—
365C4—81no—
366C4—161no—
367C4C3×#39411yescharacteristic
368C4C3×#45011yescharacteristic
369C4C3×#45911yescharacteristic
370C4C3×#48611yescharacteristic
371C3#1341111yescharacteristic, direct product factor, Sylow
372C2—21no—
373C2—21no—
374C2—162no—
375C2—161no—
376C2C3×#204111yescharacteristic
377C2C3×#219211yescharacteristic
378C2C3×#219411yescharacteristic
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
2. C2×C3×#361C2semidirect
3. C2×C3×#359C2semidirect
4. C3×#2139C2semidirect
5. C2×C3×#357C2semidirect
6. C3×#1952C2semidirect
7. C3×#1379C2semidirect
8. #13411C3direct (N × H)
29. C2×#361C2×C3semidirect
30. C2×#359C2×C3semidirect
31. #2139C2×C3semidirect
32. C2×#357C2×C3semidirect
33. #1952C2×C3semidirect
34. #1379C2×C3semidirect
71. C3×#108D4semidirect
186. #108C3×D4semidirect
371. C3#13411direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31552
C2×C3×#3612
C2×C3×#3592
C3×#21392
C2×C3×#3572
C3×#19522
C3×#13792
#134113