Finite Group Databasebeta

← C3×#25470 #10644077

Subgroups — C3×#25470 #10644077

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#476C211yesmaximal, characteristic
2C2×C3×C8×Q8C211yesmaximal, characteristic, complemented
3C3×#2567C211yesmaximal, characteristic, complemented
4C3×#2538C211yesmaximal, characteristic
5C3×#2088C212yesmaximal, complemented
6C3×#2065C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1948C211yesmaximal, characteristic, complemented
9C3×#1941C212yesmaximal, complemented
10C3×#1934C212yesmaximal
11C3×C4×#102C211yesmaximal, characteristic
12#25470C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#476—24no—
18C3×C8×Q8—24no—
19C2×C3×#106C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C2×C3×C4×C8C2212yes—
28C3×#396—22no—
29C3×#395—22no—
30C3×#391—21no—
31C3×#390C2212yes—
32C3×#389C2212yes—
33C3×#389C2212yes—
34C3×#389C2212yes—
35C3×#384—21no—
36C3×#384C2211yescharacteristic
37C3×#380—21no—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2211yescharacteristic
42C3×#340C2211yescharacteristic
43C3×#340C2212yes—
44C2×#476C2×C311yescharacteristic
45C2×C8×Q8C2×C311yescharacteristic, complemented
46#2567C2×C311yescharacteristic, complemented
47#2538C2×C311yescharacteristic
48#2088C2×C312yescomplemented
49#2065C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1948C2×C311yescharacteristic, complemented
52#1941C2×C312yescomplemented
53#1934C2×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×C22⋊Q8—44no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C3×#106—24no—
89C3×#106—24no—
90C3×C4⋊C8—24no—
91C3×C4⋊C8D414yes—
92C3×#102—24no—
93C3×#102—22no—
94C3×#102Q814yescomplemented
95C3×C4×C8—24no—
96C3×C4×C8—22no—
97C3×#95—21no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C4×Q8C22×C311yescharacteristic
104C2×C4×D4C22×C311yescharacteristic
105#476—24no—
106C8×Q8—24no—
107C2×#106C22×C312yes—
108C2×#106C22×C312yes—
109C2×C4⋊C8C22×C312yes—
110C2×C4⋊C8C22×C311yescharacteristic
111C2×#102C22×C311yescharacteristic
112C2×#102C22×C312yes—
113C2×#102C22×C311yescharacteristic
114C2×C4×C8C22×C312yes—
115C2×C4×C8C22×C311yescharacteristic
116#396—22no—
117#395—22no—
118#391—21no—
119#390C22×C312yes—
120#389C22×C312yes—
121#389C22×C312yes—
122#389C22×C312yes—
123#384—21no—
124#384C22×C311yescharacteristic
125#380—21no—
126C4×C4⋊C4C22×C311yescharacteristic
127#340C22×C312yes—
128#340C22×C311yescharacteristic
129#340C22×C312yes—
130#340C22×C312yes—
131#340C22×C311yescharacteristic
132C24×C3—21no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—42no—
136C2×C3×Q8—81no—
137C2×C3×Q8—81no—
138C2×C3×Q8SD1614yes—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4C2×Q812yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×C3×C8—21no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8C2×D414yes—
166C2×C3×C8C2×Q812yes—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4—81no—
181C3×C4⋊C4—82no—
182C3×C4⋊C4—82no—
183C3×C4⋊C4C2×Q812yes—
184C3×C4⋊C4#4112yes—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—44no—
187C3×C22⋊C4—82no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—42no—
191C3×C42C2×D412yes—
192C3×C42#4112yes—
193C22×Q8—21no—
194C22×Q8C3×D411yescharacteristic
195C22×D4C23×C311yescharacteristic
196C23×C4—21no—
197C22×C8C23×C311yescharacteristic
198C22×C8C23×C312yes—
199C22×C8C23×C311yescharacteristic
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C22⋊Q8—44no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C3×D411yescharacteristic
214C2×C4⋊C4C3×D412yes—
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42C23×C312yes—
225C2×C42C23×C311yescharacteristic
226#106—24no—
227#106—24no—
228C4⋊C8—24no—
229C4⋊C8C3×D414yes—
230#102—24no—
231#102—22no—
232#102C3×Q814yescomplemented
233C4×C8—24no—
234C4×C8—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239C23×C3—22no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C22×D411yescharacteristic
247C3×Q8—28no—
248C3×Q8—84no—
249C3×D4—22no—
250C3×D4—22no—
251C3×D4—41no—
252C3×D4—41no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
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—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—84no—
276C2×C3×C4#12112yes—
277C2×C3×C4#12112yes—
278C2×C3×C4C2×SD1614yes—
279C2×C3×C4C2×SD1612yes—
280C2×C3×C4#13512yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C22×Q811yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287C3×C8—24no—
288C3×C8—22no—
289C3×C8—41no—
290C24—21no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—42no—
294C2×Q8—81no—
295C2×Q8—81no—
296C2×Q8C3×SD1614yes—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4C2×C3×Q812yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C24×C311yescharacteristic
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8C2×C3×D414yes—
324C2×C8C2×C3×Q812yes—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4C2×C3×Q812yes—
342C4⋊C4C3×#4112yes—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42C2×C3×D412yes—
350C42C3×#4112yes—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3C2×#12111yescharacteristic
359C22×C3#54811yescharacteristic
360C22×C3D4×Q811yescharacteristic
361C22×C3C22×SD1611yescharacteristic
362C22×C3C2×#13511yescharacteristic
363C22×C3#57611yescharacteristic
364C22×C3#57711yescharacteristic
365C3×C4—24no—
366C3×C4—22no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—82no—
371C3×C4—81no—
372C3×C4—82no—
373C3×C4#46514yes—
374C3×C4#53311yescharacteristic
375C3×C4#54111yescharacteristic
376C3×C4#54711yescharacteristic
377C3×C4D4×Q811yescharacteristic
378C23—22no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23C22×C3×D411yescharacteristic
386Q8—28no—
387Q8—84no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—84no—
415C2×C4C3×#12112yes—
416C2×C4C3×#12112yes—
417C2×C4C2×C3×SD1614yes—
418C2×C4C2×C3×SD1612yes—
419C2×C4C3×#13512yes—
420C2×C4C22×C3×D411yescharacteristic
421C2×C4C22×C3×Q811yescharacteristic
422C2×C4C2×C3×#4111yescharacteristic
423C2×C4C2×C3×#4111yescharacteristic
424C2×C4C3×2−1+411yescharacteristic
425C2×C4C3×2−1+411yescharacteristic
426C8—24no—
427C8—22no—
428C8—41no—
429C2×C3—82no—
430C2×C3#242911yescharacteristic
431C2×C3C2×#46511yescharacteristic
432C2×C3#306211yescharacteristic
433C2×C3#336011yescharacteristic
434C2×C3#336211yescharacteristic
435C2×C3Q8×SD1611yescharacteristic
436C2×C3#338011yescharacteristic
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22C2×C3×#12111yescharacteristic
445C22C3×#54811yescharacteristic
446C22C3×D4×Q811yescharacteristic
447C22C22×C3×SD1611yescharacteristic
448C22C2×C3×#13511yescharacteristic
449C22C3×#57611yescharacteristic
450C22C3×#57711yescharacteristic
451C4—24no—
452C4—22no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—82no—
459C4C3×#46514yes—
460C4C3×#53311yescharacteristic
461C4C3×#54111yescharacteristic
462C4C3×#54711yescharacteristic
463C4C3×D4×Q811yescharacteristic
464C3#2547011yescharacteristic, direct product factor, Sylow
465C2—82no—
466C2C3×#242911yescharacteristic
467C2C2×C3×#46511yescharacteristic
468C2C3×#306211yescharacteristic
469C2C3×#336011yescharacteristic
470C2C3×#336211yescharacteristic
471C2C3×Q8×SD1611yescharacteristic
472C2C3×#338011yescharacteristic
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×C8×Q8C2semidirect
3. C3×#2567C2semidirect
5. C3×#2088C2semidirect
7. C3×#1990C2semidirect
8. C3×#1948C2semidirect
9. C3×#1941C2semidirect
12. #25470C3direct (N × H)
45. C2×C8×Q8C2×C3semidirect
46. #2567C2×C3semidirect
48. #2088C2×C3semidirect
50. #1990C2×C3semidirect
51. #1948C2×C3semidirect
52. #1941C2×C3semidirect
94. C3×#102Q8semidirect
232. #102C3×Q8semidirect
464. C3#25470direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×C3×C8×Q82
C3×#25672
C3×#25382
C3×#20882
C3×#20652
C3×#19902
C3×#19482
C3×#19412
C3×#19342
C3×C4×#1022
#254703