Finite Group Databasebeta

← C3×#29807 #10648414

Subgroups — C3×#29807 #10648414

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#492C211yesmaximal, characteristic, complemented
2C2×C3×#431C211yesmaximal, characteristic, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#2856C211yesmaximal, characteristic
5C3×#2055C214yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1957C211yesmaximal, characteristic, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1834C211yesmaximal, characteristic, complemented
10#29807C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×#121—21no—
16C2×C3×#121—21no—
17C2×C3×#121C2212yes—
18C22×C3×C42C2211yescharacteristic
19C3×#492—24no—
20C3×#431—24no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#103C2214yes—
23C2×C3×#102C2214yes—
24C2×C3×C22⋊C8C2212yes—
25C2×C3×C4×C8C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#391C2212yes—
28C3×#390—21no—
29C3×#390—21no—
30C3×#390C2212yes—
31C3×#389—21no—
32C3×#389—21no—
33C3×#384C2211yescharacteristic
34C3×#384C2211yescharacteristic
35C3×#380—21no—
36C3×#380—21no—
37C3×#340C2214yes—
38C2×#492C2×C311yescharacteristic, complemented
39C2×#431C2×C311yescharacteristic, complemented
40#2857C2×C311yescharacteristic, complemented
41#2856C2×C311yescharacteristic
42#2055C2×C314yescomplemented
43#2000C2×C314yescomplemented
44#1957C2×C311yescharacteristic, complemented
45#1956C2×C311yescharacteristic, complemented
46#1834C2×C311yescharacteristic, complemented
47C22×C3×Q8C2312yes—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C2×C3×SD16—28no—
56C2×C3×SD16—44no—
57C2×C3×SD16—44no—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×#127—22no—
63C3×#127—22no—
64C3×C22⋊Q8—44no—
65C3×#121—44no—
66C3×#121—44no—
67C3×#121—44no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C3×C4⋊C8—24no—
88C3×#103—44no—
89C3×#102—44no—
90C3×C22⋊C8—24no—
91C3×C4×C8—22no—
92C3×C4×C8D414yescomplemented
93C3×#95—22no—
94C3×#95—22no—
95C22×SD16C22×C314yes—
96C2×C4⋊Q8C22×C311yescharacteristic
97C2×#127C22×C311yescharacteristic
98C2×C22⋊Q8C22×C312yes—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C22×C312yes—
102C22×C42C22×C311yescharacteristic
103#492—24no—
104#431—24no—
105C2×C4⋊C8C22×C312yes—
106C2×#103C22×C314yes—
107C2×#102C22×C314yes—
108C2×C22⋊C8C22×C312yes—
109C2×C4×C8C22×C311yescharacteristic
110C2×C4×C8C22×C311yescharacteristic
111#391C22×C312yes—
112#390—21no—
113#390—21no—
114#390C22×C312yes—
115#389—21no—
116#389—21no—
117#384C22×C311yescharacteristic
118#384C22×C311yescharacteristic
119#380—21no—
120#380—21no—
121#340C22×C314yes—
122C24×C3—22no—
123C24×C3C2×D411yescharacteristic
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—24no—
127C2×C3×Q8—24no—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4SD1614yescomplemented
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C3×SD16—416no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8C2×D414yes—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—44no—
178C3×C22⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42C2×D412yes—
185C3×C42#4112yes—
186C22×Q8C23×C312yes—
187C22×D4—22no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C23×C312yes—
191C23×C4C3×D411yescharacteristic
192C23×C4C3×D411yescharacteristic
193C23×C4C23×C311yescharacteristic
194C2×SD16—28no—
195C2×SD16—44no—
196C2×SD16—44no—
197C22×C8C23×C312yes—
198C22×C8C23×C312yes—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201#127—22no—
202#127—22no—
203C22⋊Q8—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C312yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C3×D411yescharacteristic
223C2×C42C3×D411yescharacteristic
224C2×C42C23×C311yescharacteristic
225C2×C42C23×C311yescharacteristic
226C4⋊C8—24no—
227#103—44no—
228#102—44no—
229C22⋊C8—24no—
230C4×C8—22no—
231C4×C8C3×D414yescomplemented
232#95—22no—
233#95—22no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3C2×SD1612yes—
248C23×C3C22×D411yescharacteristic
249C3×Q8—44no—
250C3×Q8—44no—
251C3×D4—44no—
252C3×D4—44no—
253C3×D4—88no—
254C3×D4—84no—
255C3×D4—84no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—28no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4#12112yes—
278C2×C3×C4#12112yes—
279C2×C3×C4C2×SD1612yes—
280C2×C3×C4C2×SD1612yes—
281C2×C3×C4#13512yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4#13712yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)12yes—
287C2×C3×C4H(5,𝔽2)12yes—
288C3×C8—24no—
289C3×C8—42no—
290C24—22no—
291C24C2×C3×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4C3×SD1614yescomplemented
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C24×C311yescharacteristic
328SD16—416no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8C2×C3×D414yes—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C42—22no—
349C42—22no—
350C42—21no—
351C42—21no—
352C42C2×C3×D412yes—
353C42C3×#4112yes—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3C2×#12111yescharacteristic
369C22×C3#53511yescharacteristic
370C22×C3#53511yescharacteristic
371C22×C3C22×SD1611yescharacteristic
372C22×C3C2×#13511yescharacteristic
373C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C3C2×#13711yescharacteristic
375C3×C4—22no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—82no—
379C3×C4—82no—
380C3×C4—82no—
381C3×C4#46514yes—
382C3×C4#46814yes—
383C3×C4#53512yes—
384C3×C4#54012yes—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23C2×C3×SD1612yes—
399C23C22×C3×D411yescharacteristic
400Q8—44no—
401Q8—44no—
402D4—44no—
403D4—44no—
404D4—88no—
405D4—84no—
406D4—84no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—28no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4C3×#12112yes—
429C2×C4C3×#12112yes—
430C2×C4C2×C3×SD1612yes—
431C2×C4C2×C3×SD1612yes—
432C2×C4C3×#13512yes—
433C2×C4C3×AGL(1,ℤ/8)12yes—
434C2×C4C3×#13712yes—
435C2×C4C22×C3×D411yescharacteristic
436C2×C4C2×C3×#4111yescharacteristic
437C2×C4C3×H(5,𝔽2)12yes—
438C2×C4C3×H(5,𝔽2)12yes—
439C8—24no—
440C8—42no—
441C2×C3—42no—
442C2×C3—162no—
443C2×C3#261911yescharacteristic
444C2×C3C2×#46511yescharacteristic
445C2×C3C2×#46811yescharacteristic
446C2×C3#320611yescharacteristic
447C2×C3#320711yescharacteristic
448C2×C3#320811yescharacteristic
449C2×C3#321011yescharacteristic
450C22—22no—
451C22—22no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22C2×C3×#12111yescharacteristic
465C22C3×#53511yescharacteristic
466C22C3×#53511yescharacteristic
467C22C22×C3×SD1611yescharacteristic
468C22C2×C3×#13511yescharacteristic
469C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
470C22C2×C3×#13711yescharacteristic
471C4—22no—
472C4—41no—
473C4—41no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4C3×#46514yes—
478C4C3×#46814yes—
479C4C3×#53512yes—
480C4C3×#54012yes—
481C3#2980711yescharacteristic, direct product factor, Sylow
482C2—42no—
483C2—162no—
484C2C3×#261911yescharacteristic
485C2C2×C3×#46511yescharacteristic
486C2C2×C3×#46811yescharacteristic
487C2C3×#320611yescharacteristic
488C2C3×#320711yescharacteristic
489C2C3×#320811yescharacteristic
490C2C3×#321011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#492C2semidirect
2. C2×C3×#431C2semidirect
3. C3×#2857C2semidirect
5. C3×#2055C2semidirect
6. C3×#2000C2semidirect
7. C3×#1957C2semidirect
8. C3×#1956C2semidirect
9. C3×#1834C2semidirect
10. #29807C3direct (N × H)
38. C2×#492C2×C3semidirect
39. C2×#431C2×C3semidirect
40. #2857C2×C3semidirect
42. #2055C2×C3semidirect
43. #2000C2×C3semidirect
44. #1957C2×C3semidirect
45. #1956C2×C3semidirect
46. #1834C2×C3semidirect
92. C3×C4×C8D4semidirect
153. C22×C3×C4SD16semidirect
231. C4×C8C3×D4semidirect
321. C22×C4C3×SD16semidirect
481. C3#29807direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4922
C2×C3×#4312
C3×#28572
C3×#28562
C3×#20552
C3×#20002
C3×#19572
C3×#19562
C3×#18342
#298073