Finite Group Databasebeta

← C3×#25478 #10644085

Subgroups — C3×#25478 #10644085

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic
2C2×C3×#446C211yesmaximal, characteristic, complemented
3C3×#2546C211yesmaximal, characteristic, complemented
4C3×#2538C211yesmaximal, characteristic
5C3×#2087C212yesmaximal, complemented
6C3×#2065C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1948C211yesmaximal, characteristic, complemented
9C3×#1944C212yesmaximal, complemented
10C3×#1936C212yesmaximal
11C3×C4×#102C211yesmaximal, characteristic
12#25478C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C3×#474—24no—
19C3×#446—24no—
20C2×C3×#107C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2212yes—
28C2×C3×C4×C8C2211yescharacteristic
29C3×#398—22no—
30C3×#396—22no—
31C3×#391—21no—
32C3×#390C2212yes—
33C3×#389C2212yes—
34C3×#389C2212yes—
35C3×#389C2212yes—
36C3×#384C2211yescharacteristic
37C3×#380—21no—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2211yescharacteristic
42C3×#340C2211yescharacteristic
43C3×#340C2212yes—
44C2×#474C2×C311yescharacteristic
45C2×#446C2×C311yescharacteristic, complemented
46#2546C2×C311yescharacteristic, complemented
47#2538C2×C311yescharacteristic
48#2087C2×C312yescomplemented
49#2065C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1948C2×C311yescharacteristic, complemented
52#1944C2×C312yescomplemented
53#1936C2×C312yes—
54C4×#102C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8D411yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2311yescharacteristic
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—44no—
65C3×C22⋊Q8—44no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D412yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×#106—24no—
91C3×C4⋊C8—24no—
92C3×C4⋊C8D414yes—
93C3×#102—24no—
94C3×#102—22no—
95C3×#102Q814yescomplemented
96C3×#97—24no—
97C3×C4×C8—22no—
98C3×#95—21no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C22×C311yescharacteristic
104C2×C22⋊Q8—21no—
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107#474—24no—
108#446—24no—
109C2×#107C22×C312yes—
110C2×#106C22×C312yes—
111C2×C4⋊C8C22×C312yes—
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#102C22×C311yescharacteristic
114C2×#102C22×C312yes—
115C2×#102C22×C311yescharacteristic
116C2×#97C22×C312yes—
117C2×C4×C8C22×C311yescharacteristic
118#398—22no—
119#396—22no—
120#391—21no—
121#390C22×C312yes—
122#389C22×C312yes—
123#389C22×C312yes—
124#389C22×C312yes—
125#384C22×C311yescharacteristic
126#380—21no—
127C4×C4⋊C4C22×C311yescharacteristic
128#340C22×C312yes—
129#340C22×C311yescharacteristic
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C311yescharacteristic
133C24×C3—21no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4C2×Q812yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C2×C3×C8—21no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8C2×D414yes—
167C2×C3×C8C2×Q812yes—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—81no—
184C3×C4⋊C4C2×Q812yes—
185C3×C4⋊C4#4112yes—
186C3×C22⋊C4—44no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—82no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—42no—
192C3×C42C2×D412yes—
193C3×C42#4112yes—
194C22×Q8—21no—
195C22×Q8C3×D411yescharacteristic
196C22×D4C23×C311yescharacteristic
197C23×C4—21no—
198C22×C8C23×C311yescharacteristic
199C22×C8C23×C312yes—
200C22×C8C23×C311yescharacteristic
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—44no—
204C22⋊Q8—44no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C3×D411yescharacteristic
216C2×C4⋊C4C3×D412yes—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42—21no—
226C2×C42C23×C312yes—
227C2×C42C23×C311yescharacteristic
228#107—24no—
229#106—24no—
230C4⋊C8—24no—
231C4⋊C8C3×D414yes—
232#102—24no—
233#102—22no—
234#102C3×Q814yescomplemented
235#97—24no—
236C4×C8—22no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—22no—
241C23×C3—22no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3C22×D411yescharacteristic
249C3×Q8—44no—
250C3×Q8—84no—
251C3×D4—22no—
252C3×D4—22no—
253C3×D4—41no—
254C3×D4—41no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—84no—
279C2×C3×C4#12112yes—
280C2×C3×C4#12112yes—
281C2×C3×C4AGL(1,ℤ/8)12yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C22×Q811yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×C3×C42−1+411yescharacteristic
289C3×C8—24no—
290C3×C8—22no—
291C3×C8—41no—
292C24—21no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C2×C3×Q812yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C24×C311yescharacteristic
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8C2×C3×D414yes—
326C2×C8C2×C3×Q812yes—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—81no—
343C4⋊C4C2×C3×Q812yes—
344C4⋊C4C3×#4112yes—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—82no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42C2×C3×D412yes—
352C42C3×#4112yes—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3C2×#12111yescharacteristic
361C22×C3#54811yescharacteristic
362C22×C3D4×Q811yescharacteristic
363C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
364C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
365C22×C3#57711yescharacteristic
366C22×C3#57811yescharacteristic
367C3×C4—22no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—82no—
373C3×C4—81no—
374C3×C4—82no—
375C3×C4#46914yes—
376C3×C4#53311yescharacteristic
377C3×C4#54111yescharacteristic
378C3×C4#54711yescharacteristic
379C3×C4D4×Q811yescharacteristic
380C23—22no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C22×C3×D411yescharacteristic
388Q8—44no—
389Q8—84no—
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4C3×#12112yes—
419C2×C4C3×#12112yes—
420C2×C4C3×AGL(1,ℤ/8)12yes—
421C2×C4C3×AGL(1,ℤ/8)12yes—
422C2×C4C22×C3×D411yescharacteristic
423C2×C4C22×C3×Q811yescharacteristic
424C2×C4C2×C3×#4111yescharacteristic
425C2×C4C2×C3×#4111yescharacteristic
426C2×C4C3×2−1+411yescharacteristic
427C2×C4C3×2−1+411yescharacteristic
428C8—24no—
429C8—22no—
430C8—41no—
431C2×C3—82no—
432C2×C3#242911yescharacteristic
433C2×C3C2×#46911yescharacteristic
434C2×C3#306311yescharacteristic
435C2×C3#336711yescharacteristic
436C2×C3#336811yescharacteristic
437C2×C3#338411yescharacteristic
438C2×C3#338611yescharacteristic
439C22—42no—
440C22—42no—
441C22—42no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22C2×C3×#12111yescharacteristic
447C22C3×#54811yescharacteristic
448C22C3×D4×Q811yescharacteristic
449C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
450C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
451C22C3×#57711yescharacteristic
452C22C3×#57811yescharacteristic
453C4—22no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—82no—
460C4—82no—
461C4C3×#46914yes—
462C4C3×#53311yescharacteristic
463C4C3×#54111yescharacteristic
464C4C3×#54711yescharacteristic
465C4C3×D4×Q811yescharacteristic
466C3#2547811yescharacteristic, direct product factor, Sylow
467C2—82no—
468C2C3×#242911yescharacteristic
469C2C2×C3×#46911yescharacteristic
470C2C3×#306311yescharacteristic
471C2C3×#336711yescharacteristic
472C2C3×#336811yescharacteristic
473C2C3×#338411yescharacteristic
474C2C3×#338611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#446C2semidirect
3. C3×#2546C2semidirect
5. C3×#2087C2semidirect
7. C3×#1990C2semidirect
8. C3×#1948C2semidirect
9. C3×#1944C2semidirect
12. #25478C3direct (N × H)
45. C2×#446C2×C3semidirect
46. #2546C2×C3semidirect
48. #2087C2×C3semidirect
50. #1990C2×C3semidirect
51. #1948C2×C3semidirect
52. #1944C2×C3semidirect
95. C3×#102Q8semidirect
234. #102C3×Q8semidirect
466. C3#25478direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×#4462
C3×#25462
C3×#25382
C3×#20872
C3×#20652
C3×#19902
C3×#19482
C3×#19442
C3×#19362
C3×C4×#1022
#254783