Finite Group Databasebeta

← C3×#29774 #10648381

Subgroups — C3×#29774 #10648381

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic
2C22×C3×C4⋊C8C211yesmaximal, characteristic, complemented
3C3×#2855C211yesmaximal, characteristic, complemented
4C3×#2854C211yesmaximal, characteristic
5C3×#2088C212yesmaximal, complemented
6C3×#2066C212yesmaximal, complemented
7C3×#2065C212yesmaximal, complemented
8C3×#2055C212yesmaximal, complemented
9C3×#1893C211yesmaximal, characteristic
10C3×#1892C211yesmaximal, characteristic, complemented
11C3×#1807C211yesmaximal, characteristic, complemented
12#29774C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#123—21no—
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#465—24no—
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8—24no—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×C4⋊C8C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C3×#391C2212yes—
31C3×#390C2212yes—
32C3×#389C2212yes—
33C3×#389C2212yes—
34C3×#388—22no—
35C3×#387—22no—
36C3×#386—21no—
37C3×#385—21no—
38C3×#381—21no—
39C3×#380C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#465C2×C311yescharacteristic
45C22×C4⋊C8C2×C311yescharacteristic, complemented
46#2855C2×C311yescharacteristic, complemented
47#2854C2×C311yescharacteristic
48#2088C2×C312yescomplemented
49#2066C2×C312yescomplemented
50#2065C2×C312yescomplemented
51#2055C2×C312yescomplemented
52#1893C2×C311yescharacteristic
53#1892C2×C311yescharacteristic, complemented
54#1807C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C2×C3×SD16—22no—
62C2×C3×SD16—42no—
63C22×C3×C8—22no—
64C22×C3×C8—42no—
65C22×C3×C8—42no—
66C22×C3×C8D414yes—
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#123—44no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×#121—22no—
74C3×#121—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42D412yes—
91C2×C3×C42C2312yes—
92C3×#106—22no—
93C3×#106—42no—
94C3×C4⋊C8—24no—
95C3×C4⋊C8—24no—
96C3×#103—22no—
97C3×#103—42no—
98C3×#102—22no—
99C3×#102—42no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—22no—
106C22×SD16C22×C311yescharacteristic
107C23×C8C22×C311yescharacteristic
108C23×C8C22×C311yescharacteristic
109C2×#123—21no—
110C2×C22⋊Q8C22×C311yescharacteristic
111C2×#121C22×C311yescharacteristic
112C22×C42C22×C311yescharacteristic
113#465—24no—
114C2×#106C22×C311yescharacteristic
115C2×#106C22×C312yes—
116C2×C4⋊C8—24no—
117C2×C4⋊C8C22×C312yes—
118C2×C4⋊C8C22×C312yes—
119C2×#103C22×C311yescharacteristic
120C2×#103C22×C312yes—
121C2×#102C22×C311yescharacteristic
122C2×#102C22×C312yes—
123#391C22×C312yes—
124#390C22×C312yes—
125#389C22×C312yes—
126#389C22×C312yes—
127#388—22no—
128#387—22no—
129#386—21no—
130#385—21no—
131#381—21no—
132#380C22×C311yescharacteristic
133#380C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136#340C22×C312yes—
137C24×C3—21no—
138C24×C3C2×D411yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—42no—
161C22×C3×C4SD1614yescomplemented
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C3×SD16—44no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—44no—
175C2×C3×C8—44no—
176C2×C3×C8C2×D412yes—
177C2×C3×C8C2×D412yes—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—44no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—42no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—44no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C22×Q8C23×C311yescharacteristic
199C22×D4—21no—
200C22×D4C23×C311yescharacteristic
201C23×C4C3×D411yescharacteristic
202C23×C4C3×D411yescharacteristic
203C23×C4C23×C311yescharacteristic
204C2×SD16—22no—
205C2×SD16—42no—
206C22×C8—22no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8C3×D414yes—
210C22×C8C23×C312yes—
211C22×C8C23×C311yescharacteristic
212C22×C8C23×C311yescharacteristic
213#123—44no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216#121—22no—
217#121—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C312yes—
224C2×C4⋊C4C23×C312yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C3×D412yes—
234C2×C42C23×C312yes—
235#106—22no—
236#106—42no—
237C4⋊C8—24no—
238C4⋊C8—24no—
239#103—22no—
240#103—42no—
241#102—22no—
242#102—42no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3#12112yes—
257C23×C3C2×SD1612yes—
258C23×C3#13512yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—41no—
261C3×Q8—41no—
262C3×Q8—42no—
263C3×D4—41no—
264C3×D4—41no—
265C3×D4—42no—
266C3×D4—84no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—28no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—44no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4#12112yes—
292C2×C3×C4C2×SD1614yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4H(5,𝔽2)12yes—
296C2×C3×C42−1+412yes—
297C3×C8—22no—
298C3×C8—22no—
299C3×C8—42no—
300C24—21no—
301C24C2×C3×D411yescharacteristic
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4C3×SD1614yescomplemented
325C22×C4C2×C3×D412yes—
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D412yes—
328C22×C4C2×C3×D412yes—
329C22×C4C3×#4112yes—
330C22×C4C24×C311yescharacteristic
331SD16—44no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8C2×C3×D412yes—
340C2×C8C2×C3×D412yes—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C42—24no—
359C42—22no—
360C42—22no—
361C22×C3—22no—
362C22×C3—24no—
363C22×C3—24no—
364C22×C3—22no—
365C22×C3—81no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—81no—
369C22×C3—81no—
370C22×C3#46514yes—
371C22×C3C2×#12111yescharacteristic
372C22×C3#53712yes—
373C22×C3C22×SD1611yescharacteristic
374C22×C3C2×#13511yescharacteristic
375C22×C3#57512yes—
376C3×C4—22no—
377C3×C4—24no—
378C3×C4—42no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4—82no—
382C3×C4—81no—
383C3×C4—81no—
384C3×C4#53411yescharacteristic
385C3×C4#53611yescharacteristic
386C3×C4#53912yes—
387C23—22no—
388C23—22no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23C3×#12112yes—
395C23C2×C3×SD1612yes—
396C23C3×#13512yes—
397C23C22×C3×D411yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—42no—
401D4—41no—
402D4—41no—
403D4—42no—
404D4—84no—
405C2×C4—22no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4C3×#12112yes—
430C2×C4C2×C3×SD1614yes—
431C2×C4C22×C3×D411yescharacteristic
432C2×C4C2×C3×#4111yescharacteristic
433C2×C4C3×H(5,𝔽2)12yes—
434C2×C4C3×2−1+412yes—
435C8—22no—
436C8—22no—
437C8—42no—
438C2×C3—24no—
439C2×C3—161no—
440C2×C3#261811yescharacteristic
441C2×C3C2×#46511yescharacteristic
442C2×C3#305411yescharacteristic
443C2×C3#324612yes—
444C2×C3#325012yes—
445C22—22no—
446C22—24no—
447C22—24no—
448C22—22no—
449C22—81no—
450C22—82no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22C3×#46514yes—
455C22C2×C3×#12111yescharacteristic
456C22C3×#53712yes—
457C22C22×C3×SD1611yescharacteristic
458C22C2×C3×#13511yescharacteristic
459C22C3×#57512yes—
460C4—22no—
461C4—24no—
462C4—42no—
463C4—81no—
464C4—82no—
465C4—82no—
466C4—81no—
467C4—81no—
468C4C3×#53411yescharacteristic
469C4C3×#53611yescharacteristic
470C4C3×#53912yes—
471C3#2977411yescharacteristic, direct product factor, Sylow
472C2—24no—
473C2—161no—
474C2C3×#261811yescharacteristic
475C2C2×C3×#46511yescharacteristic
476C2C3×#305411yescharacteristic
477C2C3×#324612yes—
478C2C3×#325012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
2. C22×C3×C4⋊C8C2semidirect
3. C3×#2855C2semidirect
5. C3×#2088C2semidirect
6. C3×#2066C2semidirect
7. C3×#2065C2semidirect
8. C3×#2055C2semidirect
10. C3×#1892C2semidirect
11. C3×#1807C2semidirect
12. #29774C3direct (N × H)
45. C22×C4⋊C8C2×C3semidirect
46. #2855C2×C3semidirect
48. #2088C2×C3semidirect
49. #2066C2×C3semidirect
50. #2065C2×C3semidirect
51. #2055C2×C3semidirect
53. #1892C2×C3semidirect
54. #1807C2×C3semidirect
161. C22×C3×C4SD16semidirect
324. C22×C4C3×SD16semidirect
471. C3#29774direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4652
C22×C3×C4⋊C82
C3×#28552
C3×#28542
C3×#20882
C3×#20662
C3×#20652
C3×#20552
C3×#18932
C3×#18922
C3×#18072
#297743