Finite Group Databasebeta

← C3×#27789 #10646396

Subgroups — C3×#27789 #10646396

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#499C211yesmaximal, characteristic, complemented
2C2×C3×#467C211yesmaximal, characteristic
3C3×#2489C212yesmaximal
4C3×#2035C212yesmaximal
5C3×#2026C212yesmaximal, complemented
6C3×#1969C211yesmaximal, characteristic, complemented
7C3×#1969C211yesmaximal, characteristic, complemented
8C3×#1927C212yesmaximal, complemented
9C3×#1808C211yesmaximal, characteristic
10C3×C4×#107C211yesmaximal, characteristic, complemented
11C3×C8×C22⋊C4C211yesmaximal, characteristic
12#27789C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#125C2212yes—
16C2×C3×C22⋊Q8C2212yes—
17C3×#499—24no—
18C3×#467—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#400—22no—
30C3×#397—22no—
31C3×#391C2212yes—
32C3×#391C2212yes—
33C3×#388—22no—
34C3×#382—22no—
35C3×#380C2212yes—
36C3×C4×C4⋊C4C2212yes—
37C3×C4×C22⋊C4C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C2×#499C2×C311yescharacteristic, complemented
43C2×#467C2×C311yescharacteristic
44#2489C2×C312yes—
45#2035C2×C312yes—
46#2026C2×C312yescomplemented
47#1969C2×C311yescharacteristic, complemented
48#1969C2×C311yescharacteristic, complemented
49#1927C2×C312yescomplemented
50#1808C2×C311yescharacteristic
51C4×#107C2×C311yescharacteristic, complemented
52C8×C22⋊C4C2×C311yescharacteristic
53C22×C3×Q8C2312yes—
54C23×C3×C4C2311yescharacteristic
55C2×C3×Q16—22no—
56C2×C3×Q16—22no—
57C2×C3×Q16—22no—
58C22×C3×C8—22no—
59C22×C3×C8—22no—
60C22×C3×C8—22no—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#125—24no—
66C3×#125—42no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—42no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D411yescharacteristic
78C2×C3×C22⋊C4D411yescharacteristic
79C2×C3×C42—22no—
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C3×#107—22no—
83C3×#107—22no—
84C3×#107D414yes—
85C3×#106—24no—
86C3×#103—24no—
87C3×#103—24no—
88C3×#103—42no—
89C3×C22⋊C8—22no—
90C3×C4×C8—22no—
91C3×C4×C8—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95D411yescharacteristic
99C3×#95D411yescharacteristic
100C22×Q16C22×C311yescharacteristic
101C23×C8C22×C311yescharacteristic
102C2×#125C22×C312yes—
103C2×C22⋊Q8C22×C312yes—
104#499—24no—
105#467—24no—
106C2×#107C22×C311yescharacteristic
107C2×#107C22×C311yescharacteristic
108C2×#107C22×C311yescharacteristic
109C2×#106C22×C312yes—
110C2×#103C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×C22⋊C8C22×C311yescharacteristic
114C2×C4×C8C22×C311yescharacteristic
115C2×C4×C8C22×C311yescharacteristic
116#400—22no—
117#397—22no—
118#391C22×C312yes—
119#391C22×C312yes—
120#388—22no—
121#382—22no—
122#380C22×C312yes—
123C4×C4⋊C4C22×C312yes—
124C4×C22⋊C4C22×C311yescharacteristic
125#340C22×C312yes—
126#340C22×C312yes—
127#336C22×C311yescharacteristic
128#336C22×C311yescharacteristic
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C3×Q16—44no—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—44no—
162C2×C3×C8C2×D412yes—
163C2×C3×C8#4112yes—
164C2×C3×C8#4112yes—
165C2×C3×C8#4112yes—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4C2×D414yes—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—22no—
177C3×C22⋊C4—44no—
178C3×C22⋊C4—44no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—44no—
184C22×Q8C23×C312yes—
185C23×C4C23×C311yescharacteristic
186C2×Q16—22no—
187C2×Q16—22no—
188C2×Q16—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C23×C311yescharacteristic
193C22×C8C23×C311yescharacteristic
194C22×C8C23×C311yescharacteristic
195C22×C8C23×C311yescharacteristic
196#125—24no—
197#125—42no—
198C22⋊Q8—24no—
199C22⋊Q8—42no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4C23×C312yes—
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C3×D411yescharacteristic
209C2×C22⋊C4C3×D411yescharacteristic
210C2×C42—22no—
211C2×C42C23×C311yescharacteristic
212C2×C42C23×C311yescharacteristic
213#107—22no—
214#107—22no—
215#107C3×D414yes—
216#106—24no—
217#103—24no—
218#103—24no—
219#103—42no—
220C22⋊C8—22no—
221C4×C8—22no—
222C4×C8—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95C3×D411yescharacteristic
230#95C3×D411yescharacteristic
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3#13512yes—
238C23×C3C22×D411yescharacteristic
239C3×Q8—42no—
240C3×Q8—42no—
241C3×Q8—42no—
242C3×Q8—42no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4#13512yes—
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4H(5,𝔽2)11yescharacteristic
279C2×C3×C42−1+411yescharacteristic
280C3×C8—22no—
281C3×C8—41no—
282C3×C8—41no—
283C3×C8#12414yes—
284C24C2×C3×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4C2×C3×D411yescharacteristic
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D411yescharacteristic
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C24×C311yescharacteristic
307Q16—44no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—44no—
317C2×C8C2×C3×D412yes—
318C2×C8C3×#4112yes—
319C2×C8C3×#4112yes—
320C2×C8C3×#4112yes—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4C2×C3×D414yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—44no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3#53211yescharacteristic
347C22×C3D4211yescharacteristic
348C22×C3#54711yescharacteristic
349C22×C3C2×#13511yescharacteristic
350C22×C3C2×#13511yescharacteristic
351C22×C3#57811yescharacteristic
352C22×C3#57811yescharacteristic
353C3×C4—22no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—41no—
357C3×C4—44no—
358C3×C4—41no—
359C3×C4—82no—
360C3×C4—82no—
361C3×C4—82no—
362C3×C4C2×#12411yescharacteristic
363C3×C4#54211yescharacteristic
364C3×C4#54811yescharacteristic
365C3×C4#55511yescharacteristic
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23C3×#13512yes—
373C23C22×C3×D411yescharacteristic
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4C3×#13512yes—
409C2×C4C22×C3×D411yescharacteristic
410C2×C4C2×C3×#4111yescharacteristic
411C2×C4C2×C3×#4111yescharacteristic
412C2×C4C2×C3×#4111yescharacteristic
413C2×C4C3×H(5,𝔽2)11yescharacteristic
414C2×C4C3×2−1+411yescharacteristic
415C8—22no—
416C8—41no—
417C8—41no—
418C8C3×#12414yes—
419C2×C3—42no—
420C2×C3#249111yescharacteristic
421C2×C3#316811yescharacteristic
422C2×C3#317711yescharacteristic
423C2×C3#328311yescharacteristic
424C2×C3#328511yescharacteristic
425C2×C3#333611yescharacteristic
426C2×C3#333611yescharacteristic
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22C3×#53211yescharacteristic
435C22C3×D4211yescharacteristic
436C22C3×#54711yescharacteristic
437C22C2×C3×#13511yescharacteristic
438C22C2×C3×#13511yescharacteristic
439C22C3×#57811yescharacteristic
440C22C3×#57811yescharacteristic
441C4—22no—
442C4—41no—
443C4—41no—
444C4—44no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4C2×C3×#12411yescharacteristic
451C4C3×#54211yescharacteristic
452C4C3×#54811yescharacteristic
453C4C3×#55511yescharacteristic
454C3#2778911yescharacteristic, direct product factor, Sylow
455C2—42no—
456C2C3×#249111yescharacteristic
457C2C3×#316811yescharacteristic
458C2C3×#317711yescharacteristic
459C2C3×#328311yescharacteristic
460C2C3×#328511yescharacteristic
461C2C3×#333611yescharacteristic
462C2C3×#333611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#499C2semidirect
5. C3×#2026C2semidirect
6. C3×#1969C2semidirect
7. C3×#1969C2semidirect
8. C3×#1927C2semidirect
10. C3×C4×#107C2semidirect
12. #27789C3direct (N × H)
42. C2×#499C2×C3semidirect
46. #2026C2×C3semidirect
47. #1969C2×C3semidirect
48. #1969C2×C3semidirect
49. #1927C2×C3semidirect
51. C4×#107C2×C3semidirect
454. C3#27789direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4992
C2×C3×#4672
C3×#24892
C3×#20352
C3×#20262
C3×#19692
C3×#19272
C3×#18082
C3×C4×#1072
C3×C8×C22⋊C42
#277893