Finite Group Databasebeta

← C3×#12986 #10631593

Subgroups — C3×#12986 #10631593

Order 768.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3407C211yesmaximal, characteristic, complemented
2C2×C3×#444C211yesmaximal, characteristic
3C3×#2117C211yesmaximal, characteristic, complemented
4C3×#2117C211yesmaximal, characteristic, complemented
5C3×#2115C211yesmaximal, characteristic, complemented
6C3×#1377C211yesmaximal, characteristic, complemented
7C3×#1375C211yesmaximal, characteristic, complemented
8#12986C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C411yescharacteristic
10C2×C3×#131C2211yescharacteristic
11C2×C3×#131C2211yescharacteristic
12C3×#504—22no—
13C3×#504—22no—
14C2×C3×M32—22no—
15C2×C3×M32C2211yescharacteristic
16C2×C3×M32C2211yescharacteristic
17C2×C3×M32C2211yescharacteristic
18C2×C3×M32C2211yescharacteristic
19C3×#444—22no—
20C3×#444—22no—
21C2×C3×#108C411yescharacteristic
22C2×C3×#104C411yescharacteristic
23C2×C3×#104C411yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×#350—22no—
26C3×#350—22no—
27C3×#349—22no—
28#3407C2×C311yescharacteristic, complemented
29C2×#444C2×C311yescharacteristic
30#2117C2×C311yescharacteristic, complemented
31#2117C2×C311yescharacteristic, complemented
32#2115C2×C311yescharacteristic, complemented
33#1377C2×C311yescharacteristic, complemented
34#1375C2×C311yescharacteristic, complemented
35C2×C3×#41C2×C411yescharacteristic
36C2×C3×#41C2×C411yescharacteristic
37C3×#135—22no—
38C3×#135—22no—
39C2×C3×Q16C811yescharacteristic
40C2×C3×SD16C811yescharacteristic
41C2×C3×SD16C811yescharacteristic
42C2×C3×D8C811yescharacteristic
43C3×#131—21no—
44C3×#131—21no—
45C3×#131—21no—
46C3×#131—21no—
47C3×#131—21no—
48C3×#131—21no—
49C3×#131D412yes—
50C3×#131D412yes—
51C2×C3×M16—21no—
52C2×C3×M16—21no—
53C2×C3×M16C2×C411yescharacteristic
54C2×C3×M16C2×C411yescharacteristic
55C22×C3×C8—21no—
56C22×C3×C8—21no—
57C22×C3×C8C2×C411yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C42C2×C411yescharacteristic
60C3×M32—24no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32—21no—
64C3×M32—21no—
65C3×M32—21no—
66C3×M32—21no—
67C3×M32—21no—
68C3×M32D412yes—
69C2×C3×C16—24no—
70C2×C3×C16—21no—
71C2×C3×C16—21no—
72C2×C3×C16—21no—
73C2×C3×C16D412yes—
74C3×#108—22no—
75C3×#104—22no—
76C3×#104—22no—
77C3×#97—21no—
78C3×#97—21no—
79C2×#135C3×C411yescharacteristic
80C2×#131C22×C311yescharacteristic
81C2×#131C22×C311yescharacteristic
82#504—22no—
83#504—22no—
84C2×M32—22no—
85C2×M32C22×C311yescharacteristic
86C2×M32C22×C311yescharacteristic
87C2×M32C22×C311yescharacteristic
88C2×M32C22×C311yescharacteristic
89#444—22no—
90#444—22no—
91C2×#108C3×C411yescharacteristic
92C2×#104C3×C411yescharacteristic
93C2×#104C3×C411yescharacteristic
94C2×#97C22×C311yescharacteristic
95#350—22no—
96#350—22no—
97#349—22no—
98C3×#41—21no—
99C3×#41—21no—
100C3×#41—21no—
101C3×#41—21no—
102C3×#41—21no—
103C3×#41—21no—
104C3×#41C22⋊C412yes—
105C3×#41M1612yes—
106C2×C3×Q8C2×C811yescharacteristic
107C2×C3×Q8C2×C811yescharacteristic
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4C2×C811yescharacteristic
111C2×C3×D4C2×C811yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C3×Q16—22no—
117C3×SD16—22no—
118C3×SD16—22no—
119C3×D8—22no—
120C3×M16—21no—
121C3×M16—22no—
122C3×M16—22no—
123C3×M16—21no—
124C3×M16—21no—
125C3×M16—41no—
126C3×M16—41no—
127C3×M16C22⋊C412yes—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—21no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8C2×C811yescharacteristic
137C2×C3×C8C2×C811yescharacteristic
138C2×C3×C8C2×D411yescharacteristic
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8#4111yescharacteristic
143C2×C3×C8#4111yescharacteristic
144C3×C42—21no—
145C3×C42—21no—
146C3×C16—22no—
147C3×C16—22no—
148C3×C16—41no—
149C3×C16—41no—
150C2×#41C2×C3×C411yescharacteristic
151C2×#41C2×C3×C411yescharacteristic
152#135—22no—
153#135—22no—
154C2×Q16C3×C811yescharacteristic
155C2×SD16C3×C811yescharacteristic
156C2×SD16C3×C811yescharacteristic
157C2×D8C3×C811yescharacteristic
158#131—21no—
159#131—21no—
160#131—21no—
161#131—21no—
162#131—21no—
163#131—21no—
164#131C3×D412yes—
165#131C3×D412yes—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16C2×C3×C411yescharacteristic
169C2×M16C2×C3×C411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×C3×C411yescharacteristic
173C22×C8C23×C311yescharacteristic
174C2×C42C2×C3×C411yescharacteristic
175M32—24no—
176M32—21no—
177M32—21no—
178M32—21no—
179M32—21no—
180M32—21no—
181M32—21no—
182M32—21no—
183M32C3×D412yes—
184C2×C16—24no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16C3×D412yes—
189#108—22no—
190#104—22no—
191#104—22no—
192#97—21no—
193#97—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3#13111yescharacteristic
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—21no—
200C3×Q8C22⋊C812yes—
201C3×D4—22no—
202C3×D4—22no—
203C3×D4—21no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—41no—
207C3×D4—41no—
208C3×D4C22⋊C812yes—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4C22×C811yescharacteristic
221C2×C3×C4C2×M1611yescharacteristic
222C2×C3×C4#13111yescharacteristic
223C3×C8—22no—
224C3×C8—21no—
225C3×C8—21no—
226C3×C8—41no—
227C3×C8#12011yescharacteristic
228C3×C8#12111yescharacteristic
229C3×C8#12111yescharacteristic
230C3×C8#12311yescharacteristic
231#41—21no—
232#41—21no—
233#41—21no—
234#41—21no—
235#41—21no—
236#41—21no—
237#41C3×C22⋊C412yes—
238#41C3×M1612yes—
239C2×Q8C2×C3×C811yescharacteristic
240C2×Q8C2×C3×C811yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4C2×C3×C811yescharacteristic
244C2×D4C2×C3×C811yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C22×C3×C411yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—22no—
252D8—22no—
253M16—21no—
254M16—22no—
255M16—22no—
256M16—21no—
257M16—21no—
258M16—41no—
259M16—41no—
260M16C3×C22⋊C412yes—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8C2×C3×C811yescharacteristic
270C2×C8C2×C3×C811yescharacteristic
271C2×C8C2×C3×D411yescharacteristic
272C2×C8C2×C3×D411yescharacteristic
273C2×C8C2×C3×D411yescharacteristic
274C2×C8C2×C3×D411yescharacteristic
275C2×C8C3×#4111yescharacteristic
276C2×C8C3×#4111yescharacteristic
277C42—21no—
278C42—21no—
279C16—22no—
280C16—22no—
281C16—41no—
282C16—41no—
283C22×C3—22no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#40811yescharacteristic
289C22×C3C8×D411yescharacteristic
290C22×C3#43511yescharacteristic
291C3×C4—22no—
292C3×C4—41no—
293C3×C4—81no—
294C3×C4#38611yescharacteristic
295C3×C4C2×C22⋊C811yescharacteristic
296C3×C4C8×D411yescharacteristic
297C3×C4#43511yescharacteristic
298C23—21no—
299C23—21no—
300C23C3×#13111yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8C3×C22⋊C812yes—
305D4—22no—
306D4—22no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4C3×C22⋊C812yes—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4C2×C3×C22⋊C411yescharacteristic
322C2×C4C3×C4×D411yescharacteristic
323C2×C4C3×C4×D411yescharacteristic
324C2×C4C22×C3×C811yescharacteristic
325C2×C4C2×C3×M1611yescharacteristic
326C2×C4C3×#13111yescharacteristic
327C8—22no—
328C8—21no—
329C8—21no—
330C8—41no—
331C8C3×#12011yescharacteristic
332C8C3×#12111yescharacteristic
333C8C3×#12111yescharacteristic
334C8C3×#12311yescharacteristic
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—42no—
339C2×C3—81no—
340C2×C3#186911yescharacteristic
341C22—22no—
342C22—41no—
343C22—41no—
344C22—81no—
345C22—81no—
346C22C3×#40811yescharacteristic
347C22C3×C8×D411yescharacteristic
348C22C3×#43511yescharacteristic
349C4—22no—
350C4—41no—
351C4—81no—
352C4C3×#38611yescharacteristic
353C4C2×C3×C22⋊C811yescharacteristic
354C4C3×C8×D411yescharacteristic
355C4C3×#43511yescharacteristic
356C3#1298611yescharacteristic, direct product factor, Sylow
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2C3×#186911yescharacteristic
subgroupG/H
centerC3×C4#386
commutatorC2×C4C22×C3×C8
Frattini, cosocleC22×C8C23×C3
socleC2×C3#1869
prime radicalC2×C3×#135C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3407C2semidirect
3. C3×#2117C2semidirect
4. C3×#2117C2semidirect
5. C3×#2115C2semidirect
6. C3×#1377C2semidirect
7. C3×#1375C2semidirect
8. #12986C3direct (N × H)
28. #3407C2×C3semidirect
30. #2117C2×C3semidirect
31. #2117C2×C3semidirect
32. #2115C2×C3semidirect
33. #1377C2×C3semidirect
34. #1375C2×C3semidirect
356. C3#12986direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34072
C2×C3×#4442
C3×#21172
C3×#21152
C3×#13772
C3×#13752
#129863