Finite Group Databasebeta

← C3×#27686 #10646293

Subgroups — C3×#27686 #10646293

Order 768.

798 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#382C211yesmaximal, characteristic
3C3×#2101C212yesmaximal, complemented
4C3×#2097C212yesmaximal, complemented
5C3×#1949C212yesmaximal, complemented
6C3×#1872C211yesmaximal, characteristic
7C3×#1872C211yesmaximal, characteristic
8C3×#1853C211yesmaximal, characteristic
9C3×#1852C211yesmaximal, characteristic
10C3×#1805C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic
12#27686C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#385—21no—
29C3×#382—21no—
30C3×#382—21no—
31C3×#382C2212yes—
32C3×#382C2212yes—
33C3×#380C2211yescharacteristic
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C22⋊C4—21no—
37C2×C3×#95—21no—
38C2×C3×#95—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C4×C22⋊Q8C2×C311yescharacteristic
45C2×#382C2×C311yescharacteristic
46#2101C2×C312yescomplemented
47#2097C2×C312yescomplemented
48#1949C2×C312yescomplemented
49#1872C2×C311yescharacteristic
50#1872C2×C311yescharacteristic
51#1853C2×C311yescharacteristic
52#1852C2×C311yescharacteristic
53#1805C2×C312yescomplemented
54#1797C2×C311yescharacteristic
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8Q814yes—
65C3×C4×Q8—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—24no—
81C2×C3×C42—21no—
82C2×C3×C42—21no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C3×#107—42no—
88C3×#106—42no—
89C3×C4⋊C8—24no—
90C3×#103—42no—
91C3×#103—42no—
92C3×C22⋊C8—22no—
93C3×C22⋊C8—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—21no—
103C2×C22⋊Q8C22×C311yescharacteristic
104C2×C4×Q8C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C42C22×C311yescharacteristic
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×C22⋊C8C22×C311yescharacteristic
114C2×C22⋊C8C22×C311yescharacteristic
115#391C22×C311yescharacteristic
116#389C22×C311yescharacteristic
117#387—21no—
118#385—21no—
119#382—21no—
120#382—21no—
121#382C22×C312yes—
122#382C22×C312yes—
123#380C22×C311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4C22×C311yescharacteristic
126C4×C22⋊C4—21no—
127C2×#95—21no—
128C2×#95—21no—
129#340C22×C312yes—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C312yes—
134C24×C3C2×D411yescharacteristic
135C2×C3×Q8—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8C2×Q812yes—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×Q812yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4C2×Q812yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—42no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—42no—
192C3×C42—42no—
193C3×C42#4114yes—
194C22×Q8C23×C311yescharacteristic
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C3×D411yescharacteristic
198C23×C4C3×D411yescharacteristic
199C23×C4C23×C311yescharacteristic
200C22×C8C23×C312yes—
201C22×C8C23×C312yes—
202C22⋊Q8—22no—
203C22⋊Q8C3×Q814yes—
204C4×Q8—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—24no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C3×D411yescharacteristic
223C2×C42C3×D411yescharacteristic
224C2×C42C23×C311yescharacteristic
225C2×C42C23×C311yescharacteristic
226#107—42no—
227#106—42no—
228C4⋊C8—24no—
229#103—42no—
230#103—42no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3C22⋊Q812yes—
247C23×C3#13512yes—
248C23×C3#13712yes—
249C23×C3C22×D411yescharacteristic
250C3×Q8—24no—
251C3×Q8—42no—
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—24no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
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—44no—
269C2×C3×C4—44no—
270C2×C3×C4—41no—
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—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—84no—
287C2×C3×C4—84no—
288C2×C3×C4C22⋊Q812yes—
289C2×C3×C4#12312yes—
290C2×C3×C4#12312yes—
291C2×C3×C4C22×Q811yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4H(5,𝔽2)12yes—
295C3×C8—42no—
296C3×C8—42no—
297C24C2×C3×D411yescharacteristic
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8C2×C3×Q812yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×Q812yes—
328C22×C4C3×#4112yes—
329C22×C4C24×C311yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4C2×C3×Q812yes—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42C3×#4114yes—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3#47914yes—
364C22×C3C2×C22⋊Q811yescharacteristic
365C22×C3C2×#12311yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3C2×#13511yescharacteristic
368C22×C3C2×#13711yescharacteristic
369C22×C3#57511yescharacteristic
370C22×C3#57511yescharacteristic
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—42no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—42no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4—82no—
382C3×C4#55312yes—
383C3×C4#55412yes—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23C3×C22⋊Q812yes—
389C23C3×#13512yes—
390C23C3×#13712yes—
391C23C22×C3×D411yescharacteristic
392Q8—24no—
393Q8—42no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4C3×C22⋊Q812yes—
431C2×C4C3×#12312yes—
432C2×C4C3×#12312yes—
433C2×C4C22×C3×Q811yescharacteristic
434C2×C4C2×C3×#4111yescharacteristic
435C2×C4C2×C3×#4112yes—
436C2×C4C3×H(5,𝔽2)12yes—
437C8—42no—
438C8—42no—
439C2×C3—24no—
440C2×C3#248711yescharacteristic
441C2×C3C2×#47911yescharacteristic
442C2×C3#308011yescharacteristic
443C2×C3#309111yescharacteristic
444C2×C3#309511yescharacteristic
445C2×C3#321111yescharacteristic
446C2×C3#321511yescharacteristic
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22C3×#47914yes—
454C22C2×C3×C22⋊Q811yescharacteristic
455C22C2×C3×#12311yescharacteristic
456C22C3×#53511yescharacteristic
457C22C2×C3×#13511yescharacteristic
458C22C2×C3×#13711yescharacteristic
459C22C3×#57511yescharacteristic
460C22C3×#57511yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—41no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4C3×#55312yes—
473C4C3×#55412yes—
474C3#2768611yescharacteristic, direct product factor, Sylow
475C2—24no—
476C2C3×#248711yescharacteristic
477C2C2×C3×#47911yescharacteristic
478C2C3×#308011yescharacteristic
479C2C3×#309111yescharacteristic
480C2C3×#309511yescharacteristic
481C2C3×#321111yescharacteristic
482C2C3×#321511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
3. C3×#2101C2semidirect
4. C3×#2097C2semidirect
5. C3×#1949C2semidirect
10. C3×#1805C2semidirect
12. #27686C3direct (N × H)
46. #2101C2×C3semidirect
47. #2097C2×C3semidirect
48. #1949C2×C3semidirect
53. #1805C2×C3semidirect
474. C3#27686direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×C3×#3822
C3×#21012
C3×#20972
C3×#19492
C3×#18722
C3×#18532
C3×#18522
C3×#18052
C3×#17972
#276863