Finite Group Databasebeta

← C3×#13412 #10632019

Subgroups — C3×#13412 #10632019

Order 768.

430 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3156C211yesmaximal, characteristic
2C2×C3×#362C211yesmaximal, characteristic, complemented
3C2×C3×#359C211yesmaximal, characteristic
4C3×#2139C211yesmaximal, characteristic
5C2×C3×#358C211yesmaximal, characteristic, complemented
6C3×#1952C211yesmaximal, characteristic, complemented
7C3×#1379C211yesmaximal, characteristic, complemented
8#13412C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C2×C3×#135C2211yescharacteristic
11C22×C3×Q16C2211yescharacteristic
12C3×#536—21no—
13C2×C3×M32C2211yescharacteristic
14C22×C3×C16C2211yescharacteristic
15C3×#497—22no—
16C3×#495—21no—
17C3×#494—21no—
18C2×C3×#108C2211yescharacteristic
19C3×#428—21no—
20C2×C3×#107C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C3×#362—22no—
23C3×#359—22no—
24C3×#358—22no—
25C3×#358—21no—
26C3×#357—21no—
27C3×#343—21no—
28#3156C2×C311yescharacteristic
29C2×#362C2×C311yescharacteristic, complemented
30C2×#359C2×C311yescharacteristic
31#2139C2×C311yescharacteristic
32C2×#358C2×C311yescharacteristic, complemented
33#1952C2×C311yescharacteristic, complemented
34#1379C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C22×C3×Q8—21no—
37C3×#137—42no—
38C3×#137—42no—
39C3×#135—21no—
40C3×#135—21no—
41C3×#135—21no—
42C3×#135—21no—
43C2×C3×Q16—22no—
44C2×C3×Q16—21no—
45C2×C3×Q16—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16D412yes—
50C2×C3×Q16D411yescharacteristic
51C2×C3×SD16—22no—
52C2×C3×SD16—21no—
53C2×C3×D8D411yescharacteristic
54C2×C3×M16—21no—
55C2×C3×M16D411yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×C4⋊Q8—22no—
58C3×#124—22no—
59C3×#123—42no—
60C3×C22⋊Q8—41no—
61C3×C22⋊Q8—41no—
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×#108D412yes—
72C3×#107—21no—
73C3×#107—21no—
74C3×#107—41no—
75C3×#106—41no—
76C3×#97—21no—
77C3×C4×C8—21no—
78C2×#137—21no—
79C2×#135C22×C311yescharacteristic
80C22×Q16C22×C311yescharacteristic
81#536—21no—
82C2×M32C22×C311yescharacteristic
83C22×C16C22×C311yescharacteristic
84#497—22no—
85#495—21no—
86#494—21no—
87C2×#108C22×C311yescharacteristic
88#428—21no—
89C2×#107C22×C311yescharacteristic
90#405C22×C311yescharacteristic
91#362—22no—
92#359—22no—
93#358—22no—
94#358—21no—
95#357—21no—
96#343—21no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C2×C3×Q8—22no—
102C2×C3×Q8—21no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×D4—21no—
109C2×C3×D4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C3×Q16—22no—
115C3×Q16—21no—
116C3×Q16—22no—
117C3×Q16—21no—
118C3×Q16—44no—
119C3×Q16—41no—
120C3×Q16—41no—
121C3×SD16—44no—
122C3×SD16—41no—
123C3×SD16—41no—
124C3×D8—21no—
125C3×D8—21no—
126C3×M16—21no—
127C3×M16—22no—
128C3×M16—41no—
129C3×M16SD1612yes—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8#4111yescharacteristic
136C2×C3×C8#4111yescharacteristic
137C2×C3×C8#4111yescharacteristic
138C3×C4⋊C4—42no—
139C3×C4⋊C4—42no—
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×Q8—21no—
152#137—42no—
153#137—42no—
154#135—21no—
155#135—21no—
156#135—21no—
157#135—21no—
158C2×Q16—22no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16C3×D412yes—
165C2×Q16C3×D411yescharacteristic
166C2×SD16—22no—
167C2×SD16—21no—
168C2×D8C3×D411yescharacteristic
169C2×M16—21no—
170C2×M16C3×D411yescharacteristic
171C22×C8C23×C311yescharacteristic
172C4⋊Q8—22no—
173#124—22no—
174#123—42no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
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×D412yes—
187#107—21no—
188#107—21no—
189#107—41no—
190#106—41no—
191#97—21no—
192C4×C8—21no—
193C23×C3—41no—
194C23×C3AGL(1,ℤ/8)11yescharacteristic
195C3×Q8—41no—
196C3×Q8—42no—
197C3×Q8—42no—
198C3×Q8—41no—
199C3×Q8—81no—
200C3×Q8—81no—
201C3×D4—41no—
202C3×D4—41no—
203C3×D4—81no—
204C3×D4—81no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—161no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12411yescharacteristic
218C2×C3×C4C2×D811yescharacteristic
219C2×C3×C4C2×SD1611yescharacteristic
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C3×C8—22no—
222C3×C8—42no—
223C3×C8#12111yescharacteristic
224C3×C8#12111yescharacteristic
225C3×C8#12311yescharacteristic
226C3×C8#12311yescharacteristic
227#41—41no—
228#41—41no—
229#41—41no—
230#41—41no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×D4—21no—
239C2×D4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C2×C3×D411yescharacteristic
244Q16—22no—
245Q16—21no—
246Q16—22no—
247Q16—21no—
248Q16—44no—
249Q16—41no—
250Q16—41no—
251SD16—44no—
252SD16—41no—
253SD16—41no—
254D8—21no—
255D8—21no—
256M16—21no—
257M16—22no—
258M16—41no—
259M16C3×SD1612yes—
260C2×C8—21no—
261C2×C8—41no—
262C2×C8C2×C3×D411yescharacteristic
263C2×C8C2×C3×D411yescharacteristic
264C2×C8C2×C3×D411yescharacteristic
265C2×C8C3×#4111yescharacteristic
266C2×C8C3×#4111yescharacteristic
267C2×C8C3×#4111yescharacteristic
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4C3×D811yescharacteristic
273C4⋊C4C3×D811yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C42—21no—
278C16—41no—
279C16—41no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3#44711yescharacteristic
286C22×C3#46011yescharacteristic
287C22×C3#49011yescharacteristic
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—81no—
291C3×C4—82no—
292C3×C4—161no—
293C3×C4#39411yescharacteristic
294C3×C4#45011yescharacteristic
295C3×C4#45911yescharacteristic
296C3×C4#48611yescharacteristic
297C23—41no—
298C23C3×AGL(1,ℤ/8)11yescharacteristic
299Q8—41no—
300Q8—42no—
301Q8—42no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—161no—
319C2×C4C3×#12011yescharacteristic
320C2×C4C3×#12111yescharacteristic
321C2×C4C3×#12411yescharacteristic
322C2×C4C2×C3×D811yescharacteristic
323C2×C4C2×C3×SD1611yescharacteristic
324C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
325C8—22no—
326C8—42no—
327C8C3×#12111yescharacteristic
328C8C3×#12111yescharacteristic
329C8C3×#12311yescharacteristic
330C8C3×#12311yescharacteristic
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—161no—
334C2×C3#204111yescharacteristic
335C2×C3#219311yescharacteristic
336C2×C3#219511yescharacteristic
337C22—21no—
338C22—21no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C3×#44711yescharacteristic
343C22C3×#46011yescharacteristic
344C22C3×#49011yescharacteristic
345C4—41no—
346C4—41no—
347C4—81no—
348C4—82no—
349C4—161no—
350C4C3×#39411yescharacteristic
351C4C3×#45011yescharacteristic
352C4C3×#45911yescharacteristic
353C4C3×#48611yescharacteristic
354C3#1341211yescharacteristic, direct product factor, Sylow
355C2—21no—
356C2—21no—
357C2—161no—
358C2C3×#204111yescharacteristic
359C2C3×#219311yescharacteristic
360C2C3×#219511yescharacteristic
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#135C22
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×#362C2semidirect
5. C2×C3×#358C2semidirect
6. C3×#1952C2semidirect
7. C3×#1379C2semidirect
8. #13412C3direct (N × H)
29. C2×#362C2×C3semidirect
32. C2×#358C2×C3semidirect
33. #1952C2×C3semidirect
34. #1379C2×C3semidirect
354. C3#13412direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31562
C2×C3×#3622
C2×C3×#3592
C3×#21392
C2×C3×#3582
C3×#19522
C3×#13792
#134123