Finite Group Databasebeta

← C3×#29198 #10647805

Subgroups — C3×#29198 #10647805

Order 768.

758 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
1C3×#2859C211yesmaximal, characteristic, complemented
2C3×C4×#124C211yesmaximal, characteristic
3C3×#2067C212yesmaximal
4C3×#2066C212yesmaximal, complemented
5C3×#1959C211yesmaximal, characteristic
6C3×#1959C211yesmaximal, characteristic
7C3×#1864C212yesmaximal
8C3×#1863C212yesmaximal, complemented
9C3×#1847C211yesmaximal, characteristic, complemented
10C3×#1836C211yesmaximal, characteristic, complemented
11C3×#1767C211yesmaximal, characteristic, complemented
12#29198C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125—21no—
14C2×C3×#124C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#97C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389—21no—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#389C2212yes—
32C3×#388—21no—
33C3×#388—21no—
34C3×#384C2211yescharacteristic
35C3×#384C2211yescharacteristic
36C3×#382—21no—
37C3×#382C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C3×C4×C22⋊C4—21no—
40C3×C43C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44#2859C2×C311yescharacteristic, complemented
45C4×#124C2×C311yescharacteristic
46#2067C2×C312yes—
47#2066C2×C312yescomplemented
48#1959C2×C311yescharacteristic
49#1959C2×C311yescharacteristic
50#1864C2×C312yes—
51#1863C2×C312yescomplemented
52#1847C2×C311yescharacteristic, complemented
53#1836C2×C311yescharacteristic, complemented
54#1767C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×#125—44no—
61C3×#124—22no—
62C3×#124Q814yes—
63C3×C4×Q8—42no—
64C3×C4×D4—42no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—21no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C3×#107—42no—
90C3×#106—42no—
91C3×C4⋊C8—24no—
92C3×#103—24no—
93C3×#103—42no—
94C3×#102—24no—
95C3×#102—42no—
96C3×#97—22no—
97C3×#97—22no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—22no—
103C3×#95—22no—
104C2×#125—21no—
105C2×#124C22×C311yescharacteristic
106C2×C4×Q8C22×C311yescharacteristic
107C2×C4×D4C22×C311yescharacteristic
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×#97C22×C311yescharacteristic
116C2×#97C22×C311yescharacteristic
117#391C22×C311yescharacteristic
118#390C22×C311yescharacteristic
119#389—21no—
120#389—21no—
121#389C22×C312yes—
122#389C22×C312yes—
123#388—21no—
124#388—21no—
125#384C22×C311yescharacteristic
126#384C22×C311yescharacteristic
127#382—21no—
128#382C22×C311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C43C22×C311yescharacteristic
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8C2×Q812yes—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—22no—
143C2×C3×D4C2×Q812yes—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8#4114yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—44no—
177C3×C4⋊C4—44no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—41no—
188C3×C42—41no—
189C3×C42—42no—
190C3×C42C2×Q812yes—
191C3×C42#4112yes—
192C22×Q8C23×C311yescharacteristic
193C22×D4C23×C311yescharacteristic
194C23×C4—21no—
195C22×C8C23×C312yes—
196C22×C8C23×C312yes—
197#125—44no—
198#124—22no—
199#124C3×Q814yes—
200C4×Q8—42no—
201C4×D4—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C42—21no—
219C2×C42C3×D411yescharacteristic
220C2×C42C3×D411yescharacteristic
221C2×C42C3×D411yescharacteristic
222C2×C42C3×D411yescharacteristic
223C2×C42C23×C311yescharacteristic
224C2×C42C23×C311yescharacteristic
225C2×C42C23×C311yescharacteristic
226#107—42no—
227#106—42no—
228C4⋊C8—24no—
229#103—24no—
230#103—42no—
231#102—24no—
232#102—42no—
233#97—22no—
234#97—22no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
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—24no—
250C3×Q8—42no—
251C3×D4—24no—
252C3×D4—42no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—84no—
283C2×C3×C4C22⋊Q812yes—
284C2×C3×C4C22⋊Q812yes—
285C2×C3×C4AGL(1,ℤ/8)12yes—
286C2×C3×C4#13712yes—
287C2×C3×C4C22×Q811yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4112yes—
290C2×C3×C4H(5,𝔽2)12yes—
291C3×C8—24no—
292C3×C8—42no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8C2×C3×Q812yes—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—22no—
301C2×D4C2×C3×Q812yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C24×C311yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8C3×#4114yes—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C42—42no—
348C42C2×C3×Q812yes—
349C42C3×#4112yes—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3C2×C22⋊Q811yescharacteristic
358C22×C3#53211yescharacteristic
359C22×C3#53511yescharacteristic
360C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
361C22×C3C2×#13711yescharacteristic
362C22×C3#57511yescharacteristic
363C22×C3#57511yescharacteristic
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—42no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—82no—
372C3×C4—81no—
373C3×C4—82no—
374C3×C4—81no—
375C3×C4#55312yes—
376C3×C4#55412yes—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C22×C3×D411yescharacteristic
385Q8—24no—
386Q8—42no—
387D4—24no—
388D4—42no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4C3×C22⋊Q812yes—
420C2×C4C3×C22⋊Q812yes—
421C2×C4C3×AGL(1,ℤ/8)12yes—
422C2×C4C3×#13712yes—
423C2×C4C22×C3×Q811yescharacteristic
424C2×C4C2×C3×#4111yescharacteristic
425C2×C4C2×C3×#4112yes—
426C2×C4C3×H(5,𝔽2)12yes—
427C8—24no—
428C8—42no—
429C2×C3—82no—
430C2×C3#256611yescharacteristic
431C2×C3#307911yescharacteristic
432C2×C3#308011yescharacteristic
433C2×C3#317211yescharacteristic
434C2×C3#318011yescharacteristic
435C2×C3#321411yescharacteristic
436C2×C3#321511yescharacteristic
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22C2×C3×C22⋊Q811yescharacteristic
445C22C3×#53211yescharacteristic
446C22C3×#53511yescharacteristic
447C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
448C22C2×C3×#13711yescharacteristic
449C22C3×#57511yescharacteristic
450C22C3×#57511yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—42no—
458C4—82no—
459C4—81no—
460C4—82no—
461C4—81no—
462C4C3×#55312yes—
463C4C3×#55412yes—
464C3#2919811yescharacteristic, direct product factor, Sylow
465C2—82no—
466C2C3×#256611yescharacteristic
467C2C3×#307911yescharacteristic
468C2C3×#308011yescharacteristic
469C2C3×#317211yescharacteristic
470C2C3×#318011yescharacteristic
471C2C3×#321411yescharacteristic
472C2C3×#321511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2859C2semidirect
4. C3×#2066C2semidirect
8. C3×#1863C2semidirect
9. C3×#1847C2semidirect
10. C3×#1836C2semidirect
11. C3×#1767C2semidirect
12. #29198C3direct (N × H)
44. #2859C2×C3semidirect
47. #2066C2×C3semidirect
51. #1863C2×C3semidirect
52. #1847C2×C3semidirect
53. #1836C2×C3semidirect
54. #1767C2×C3semidirect
464. C3#29198direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28592
C3×C4×#1242
C3×#20672
C3×#20662
C3×#19592
C3×#18642
C3×#18632
C3×#18472
C3×#18362
C3×#17672
#291983