Finite Group Databasebeta

← C3×#10246 #10628853

Subgroups — C3×#10246 #10628853

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3208C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic
4C3×#1545C211yesmaximal, characteristic, complemented
5C3×#1503C211yesmaximal, characteristic, complemented
6C3×#1478C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10246C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127—21no—
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121—22no—
13C2×C3×C4×D4C2211yescharacteristic
14C3×#492—21no—
15C3×#486—21no—
16C3×#478—22no—
17C3×#476—21no—
18C3×#474—21no—
19C3×#460—22no—
20C3×#450—22no—
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#428—21no—
24C3×#423C2211yescharacteristic
25C3×#418—21no—
26C3×#392—22no—
27C3×#390—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#331—22no—
30C3×#328—22no—
31C3×D4⋊C8—22no—
32#3208C2×C311yescharacteristic, complemented
33#3077C2×C311yescharacteristic, complemented
34#2541C2×C311yescharacteristic
35#1545C2×C311yescharacteristic, complemented
36#1503C2×C311yescharacteristic, complemented
37#1478C2×C311yescharacteristic, complemented
38#1303C2×C311yescharacteristic, complemented
39C22×C3×D4—24no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D412yes—
44C23×C3×C4—21no—
45C2×C3×SD16—24no—
46C2×C3×M16—21no—
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×#127—21no—
53C3×#127—42no—
54C3×#127—42no—
55C3×#125—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×#121—48no—
59C3×C4×Q8—21no—
60C3×C4×Q8—21no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×#117—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72C3×#107—22no—
73C3×#106—22no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yescomplemented
77C3×C4⋊C8D412yescomplemented
78C3×#102—22no—
79C3×#102—22no—
80C3×#102—24no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C4×C8—21no—
84C3×C4×C8—21no—
85C3×#95D412yes—
86#533C22×C311yescharacteristic
87C2×#127—21no—
88C2×#127C22×C311yescharacteristic
89C2×#121—22no—
90C2×C4×D4C22×C311yescharacteristic
91#492—21no—
92#486—21no—
93#478—22no—
94#476—21no—
95#474—21no—
96#460—22no—
97#450—22no—
98#431C22×C311yescharacteristic
99#431C22×C311yescharacteristic
100#428—21no—
101#423C22×C311yescharacteristic
102#418—21no—
103#392—22no—
104#390—21no—
105C4×C4⋊C4C22×C311yescharacteristic
106#331—22no—
107#328—22no—
108D4⋊C8—22no—
109C24×C3—21no—
110C24×C3—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×D4—21no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—44no—
121C2×C3×D4—48no—
122C2×C3×D4—48no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—82no—
132C2×C3×D4—84no—
133C2×C3×D4SD1612yescomplemented
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—44no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C3×SD16—84no—
143C3×M16—41no—
144C3×M16—41no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—41no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—48no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C42—41no—
162C3×C42—41no—
163C3×C42—41no—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42#4111yescharacteristic
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C3×D411yescharacteristic
172C22×D4C3×D412yes—
173C23×C4—21no—
174C2×SD16—24no—
175C2×M16—21no—
176C4⋊Q8C3×D411yescharacteristic, complemented
177C4⋊Q8C3×D411yescharacteristic, complemented
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—42no—
183#127—42no—
184#125—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187#121—48no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194#117—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C3×D411yescharacteristic
197C2×C22⋊C4—24no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42C23×C311yescharacteristic
201#107—22no—
202#106—22no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8C3×D412yescomplemented
206C4⋊C8C3×D412yescomplemented
207#102—22no—
208#102—22no—
209#102—24no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214#95C3×D412yes—
215C23×C3—22no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—44no—
219C23×C3—44no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3#12011yescharacteristic
226C3×Q8—42no—
227C3×Q8—81no—
228C3×D4—22no—
229C3×D4—22no—
230C3×D4—41no—
231C3×D4—44no—
232C3×D4—41no—
233C3×D4—82no—
234C3×D4—82no—
235C3×D4—82no—
236C3×D4—816no—
237C3×D4—81no—
238C3×D4—81no—
239C3×D4—81no—
240C3×D4—81no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×C3×C4C2×SD1611yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4#13711yescharacteristic
262C3×C8—42no—
263C3×C8—42no—
264C3×C8—81no—
265C24—21no—
266C24—22no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×D4—21no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4C3×SD1612yescomplemented
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298SD16—84no—
299M16—41no—
300M16—41no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C22⋊C4—48no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42C2×C3×D411yescharacteristic
321C42C2×C3×D411yescharacteristic
322C42C2×C3×D411yescharacteristic
323C42C3×#4111yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3—84no—
330C22×C3—84no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3—82no—
336C22×C3#39211yescharacteristic
337C22×C3#45011yescharacteristic
338C22×C3#45211yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4#44811yescharacteristic
347C3×C4#44911yescharacteristic
348C3×C4#45312yes—
349C3×C4#46511yescharacteristic
350C3×C4#46811yescharacteristic
351C23—22no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—44no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23C3×#12011yescharacteristic
362Q8—42no—
363Q8—81no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—44no—
368D4—41no—
369D4—82no—
370D4—82no—
371D4—82no—
372D4—816no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4C3×#12011yescharacteristic
389C2×C4C3×#12011yescharacteristic
390C2×C4C3×#12111yescharacteristic
391C2×C4C3×#12111yescharacteristic
392C2×C4C3×#12111yescharacteristic
393C2×C4C3×#12711yescharacteristic
394C2×C4C2×C3×SD1611yescharacteristic
395C2×C4C3×#13511yescharacteristic
396C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
397C2×C4C3×#13711yescharacteristic
398C8—42no—
399C8—42no—
400C8—81no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—82no—
404C2×C3—162no—
405C2×C3#200011yescharacteristic
406C2×C3#200211yescharacteristic
407C2×C3#200811yescharacteristic
408C22—21no—
409C22—21no—
410C22—44no—
411C22—42no—
412C22—82no—
413C22—84no—
414C22—84no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22C3×#39211yescharacteristic
421C22C3×#45011yescharacteristic
422C22C3×#45211yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—44no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4C3×#44811yescharacteristic
431C4C3×#44911yescharacteristic
432C4C3×#45312yes—
433C4C3×#46511yescharacteristic
434C4C3×#46811yescharacteristic
435C3#1024611yescharacteristic, direct product factor, Sylow
436C2—21no—
437C2—21no—
438C2—82no—
439C2—162no—
440C2C3×#200011yescharacteristic
441C2C3×#200211yescharacteristic
442C2C3×#200811yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#2541C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3208C2semidirect
2. C3×#3077C2semidirect
4. C3×#1545C2semidirect
5. C3×#1503C2semidirect
6. C3×#1478C2semidirect
7. C3×#1303C2semidirect
8. #10246C3direct (N × H)
32. #3208C2×C3semidirect
33. #3077C2×C3semidirect
35. #1545C2×C3semidirect
36. #1503C2×C3semidirect
37. #1478C2×C3semidirect
38. #1303C2×C3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×C4⋊Q8D4semidirect
76. C3×C4⋊C8D4semidirect
77. C3×C4⋊C8D4semidirect
133. C2×C3×D4SD16semidirect
176. C4⋊Q8C3×D4semidirect
177. C4⋊Q8C3×D4semidirect
205. C4⋊C8C3×D4semidirect
206. C4⋊C8C3×D4semidirect
289. C2×D4C3×SD16semidirect
435. C3#10246direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32082
C3×#30772
C3×#25412
C3×#15452
C3×#15032
C3×#14782
C3×#13032
#102463