Finite Group Databasebeta

← C3×#29835 #10648442

Subgroups — C3×#29835 #10648442

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#479C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic
3C3×#2641C211yesmaximal, characteristic
4C3×#2320C211yesmaximal, characteristic, complemented
5C3×#2067C212yesmaximal, complemented
6C3×#2065C212yesmaximal, complemented
7C3×#2055C212yesmaximal, complemented
8C3×#2055C212yesmaximal, complemented
9C3×#1989C211yesmaximal, characteristic, complemented
10C3×#1982C211yesmaximal, characteristic
11C3×#1843C211yesmaximal, characteristic, complemented
12#29835C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C3×#479—24no—
18C3×#460—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C3×#394—21no—
31C3×#394—21no—
32C3×#394—22no—
33C3×#391C2212yes—
34C3×#390C2212yes—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#387—22no—
38C3×#383—21no—
39C3×#382—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#479C2×C311yescharacteristic, complemented
45C2×#460C2×C311yescharacteristic
46#2641C2×C311yescharacteristic
47#2320C2×C311yescharacteristic, complemented
48#2067C2×C312yescomplemented
49#2065C2×C312yescomplemented
50#2055C2×C312yescomplemented
51#2055C2×C312yescomplemented
52#1989C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic
54#1843C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C2×C3×SD16—24no—
60C2×C3×SD16—42no—
61C2×C3×SD16—42no—
62C22×C3×C8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#127—22no—
66C3×#127—22no—
67C3×#125—22no—
68C3×#125—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8D414yes—
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⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#107—22no—
88C3×#106—22no—
89C3×C4⋊C8—22no—
90C3×C4⋊C8—22no—
91C3×C4⋊C8—42no—
92C3×C4⋊C8—42no—
93C3×#103—22no—
94C3×#103—22no—
95C3×#102—24no—
96C3×#102—42no—
97C3×#102—42no—
98C3×#95—21no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×SD16C22×C312yes—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×Q8C22×C311yescharacteristic
109#479—24no—
110#460—24no—
111C2×#107C22×C311yescharacteristic
112C2×#106C22×C311yescharacteristic
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C311yescharacteristic
118C2×#103C22×C311yescharacteristic
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C312yes—
122#394—21no—
123#394—21no—
124#394—22no—
125#391C22×C312yes—
126#390C22×C312yes—
127#390C22×C312yes—
128#389C22×C312yes—
129#387—22no—
130#383—21no—
131#382—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133#376C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136C24×C3—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×D412yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—81no—
150C2×C3×D4—82no—
151C2×C3×D4—81no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C3×SD16—48no—
163C2×C3×C8—22no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—82no—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—82no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C22⋊C4—82no—
186C3×C42—24no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—81no—
190C3×C42C2×D412yes—
191C3×C42#4112yes—
192C22×Q8C23×C311yescharacteristic
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C23×C312yes—
196C2×SD16—24no—
197C2×SD16—42no—
198C2×SD16—42no—
199C22×C8C23×C312yes—
200C22×C8C23×C311yescharacteristic
201C22×C8C23×C311yescharacteristic
202#127—22no—
203#127—22no—
204#125—22no—
205#125—22no—
206C4×Q8—22no—
207C4×Q8C3×D414yes—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C23×C312yes—
223C2×C42C23×C311yescharacteristic
224#107—22no—
225#106—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230#103—22no—
231#103—22no—
232#102—24no—
233#102—42no—
234#102—42no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95C3×D411yescharacteristic
240#95C3×D411yescharacteristic
241#95C3×D412yes—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3C22×D411yescharacteristic
250C3×Q8—22no—
251C3×Q8—22no—
252C3×Q8—42no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—84no—
258C3×D4—84no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4#12112yes—
282C2×C3×C4#12112yes—
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4H(5,𝔽2)12yes—
286C2×C3×C42−1+412yes—
287C3×C8—41no—
288C3×C8—41no—
289C3×C8—42no—
290C24—22no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8C2×C3×D412yes—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—81no—
304C2×D4—82no—
305C2×D4—81no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D412yes—
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C24×C311yescharacteristic
316SD16—48no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C4⋊C4—24no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—82no—
335C4⋊C4C2×C3×D412yes—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—24no—
341C42—22no—
342C42—22no—
343C42—81no—
344C42C2×C3×D412yes—
345C42C3×#4112yes—
346C22×C3—82no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3C2×#12111yescharacteristic
354C22×C3#53712yes—
355C22×C3#57611yescharacteristic
356C22×C3#57712yes—
357C22×C3#57711yescharacteristic
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—42no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—82no—
365C3×C4—81no—
366C3×C4—81no—
367C3×C4#53511yescharacteristic
368C3×C4#53611yescharacteristic
369C3×C4#54212yes—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23C22×C3×D411yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—42no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—84no—
386D4—84no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4C3×#12112yes—
410C2×C4C3×#12112yes—
411C2×C4C22×C3×D411yescharacteristic
412C2×C4C2×C3×#4111yescharacteristic
413C2×C4C3×H(5,𝔽2)12yes—
414C2×C4C3×2−1+412yes—
415C8—42no—
416C8—41no—
417C8—41no—
418C2×C3—162no—
419C2×C3#262011yescharacteristic
420C2×C3#304311yescharacteristic
421C2×C3#304511yescharacteristic
422C2×C3#325712yes—
423C2×C3#326012yes—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22C2×C3×#12111yescharacteristic
432C22C3×#53712yes—
433C22C3×#57611yescharacteristic
434C22C3×#57712yes—
435C22C3×#57711yescharacteristic
436C4—22no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4C3×#53511yescharacteristic
446C4C3×#53611yescharacteristic
447C4C3×#54212yes—
448C3#2983511yescharacteristic, direct product factor, Sylow
449C2—162no—
450C2C3×#262011yescharacteristic
451C2C3×#304311yescharacteristic
452C2C3×#304511yescharacteristic
453C2C3×#325712yes—
454C2C3×#326012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#479C2semidirect
4. C3×#2320C2semidirect
5. C3×#2067C2semidirect
6. C3×#2065C2semidirect
7. C3×#2055C2semidirect
8. C3×#2055C2semidirect
9. C3×#1989C2semidirect
11. C3×#1843C2semidirect
12. #29835C3direct (N × H)
44. C2×#479C2×C3semidirect
47. #2320C2×C3semidirect
48. #2067C2×C3semidirect
49. #2065C2×C3semidirect
50. #2055C2×C3semidirect
51. #2055C2×C3semidirect
52. #1989C2×C3semidirect
54. #1843C2×C3semidirect
448. C3#29835direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4792
C2×C3×#4602
C3×#26412
C3×#23202
C3×#20672
C3×#20652
C3×#20552
C3×#19892
C3×#19822
C3×#18432
#298353