Finite Group Databasebeta

← C3×#27736 #10646343

Subgroups — C3×#27736 #10646343

Order 768.

718 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2571C211yesmaximal, characteristic, complemented
2C3×#2542C211yesmaximal, characteristic
3C3×#2100C212yesmaximal, complemented
4C3×#2073C212yesmaximal
5C3×#1990C211yesmaximal, characteristic, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1936C212yesmaximal
8C3×#1836C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×#1770C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27736C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102C2211yescharacteristic
19C2×C3×#102C2211yescharacteristic
20C2×C3×#102C2212yes—
21C2×C3×#97C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#400—22no—
24C3×#399—22no—
25C3×#390C2212yes—
26C3×#389—21no—
27C3×#389C2212yes—
28C3×#384C2211yescharacteristic
29C3×#384C2212yes—
30C3×#382—21no—
31C3×#382C2212yes—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34C3×#340C2212yes—
35C3×#340C2212yes—
36C3×#340C2211yescharacteristic
37C3×#340C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42#2571C2×C311yescharacteristic, complemented
43#2542C2×C311yescharacteristic
44#2100C2×C312yescomplemented
45#2073C2×C312yes—
46#1990C2×C311yescharacteristic, complemented
47#1988C2×C311yescharacteristic, complemented
48#1936C2×C312yes—
49#1836C2×C312yescomplemented
50#1802C2×C311yescharacteristic
51#1770C2×C311yescharacteristic, complemented
52C4×#102C2×C311yescharacteristic
53C22×C3×Q8—21no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2312yes—
58C22×C3×C8C2311yescharacteristic
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C22⋊Q8—44no—
62C3×C22⋊Q8—44no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—21no—
82C2×C3×C42C2312yes—
83C2×C3×C42C2311yescharacteristic
84C3×C4⋊C8—22no—
85C3×#102—22no—
86C3×#102—24no—
87C3×#102—22no—
88C3×#97—24no—
89C3×C4×C8—22no—
90C3×#95—21no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95D412yes—
95C2×C4⋊Q8C22×C311yescharacteristic
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C4×D4C22×C311yescharacteristic
99C2×C4⋊C8C22×C311yescharacteristic
100C2×#102C22×C312yes—
101C2×#102C22×C311yescharacteristic
102C2×#102C22×C311yescharacteristic
103C2×#97C22×C312yes—
104C2×C4×C8C22×C311yescharacteristic
105#400—22no—
106#399—22no—
107#390C22×C312yes—
108#389—21no—
109#389C22×C312yes—
110#384C22×C311yescharacteristic
111#384C22×C312yes—
112#382—21no—
113#382C22×C312yes—
114C4×C4⋊C4C22×C311yescharacteristic
115C4×C4⋊C4C22×C311yescharacteristic
116#340C22×C312yes—
117#340C22×C312yes—
118#340C22×C312yes—
119#340C22×C311yescharacteristic
120#340C22×C312yes—
121#340C22×C311yescharacteristic
122#340C22×C312yes—
123#336C22×C312yes—
124C24×C3—21no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C2×C3×Q8—42no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4#4112yes—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—21no—
153C2×C3×C8—22no—
154C2×C3×C8—21no—
155C2×C3×C8#4112yes—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—44no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—81no—
173C3×C4⋊C4—82no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—42no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—42no—
180C3×C42#4112yes—
181C3×C42#4112yes—
182C22×Q8—21no—
183C22×D4C23×C311yescharacteristic
184C23×C4—21no—
185C22×C8C23×C311yescharacteristic
186C22×C8C23×C312yes—
187C22×C8C23×C311yescharacteristic
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C3×D411yescharacteristic
201C2×C4⋊C4C3×D411yescharacteristic
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C312yes—
204C2×C4⋊C4C23×C311yescharacteristic
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C312yes—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C42—21no—
211C2×C42C23×C312yes—
212C2×C42C23×C311yescharacteristic
213C4⋊C8—22no—
214#102—24no—
215#102—22no—
216#102—22no—
217#97—24no—
218C4×C8—22no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95C3×D412yes—
224C23×C3—22no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3C22×D411yescharacteristic
232C3×Q8—84no—
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—41no—
236C3×D4—41no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—84no—
261C2×C3×C4#12412yes—
262C2×C3×C4#12412yes—
263C2×C3×C4AGL(1,ℤ/8)12yes—
264C2×C3×C4AGL(1,ℤ/8)12yes—
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C42−1+411yescharacteristic
270C2×C3×C42−1+411yescharacteristic
271C3×C8—22no—
272C3×C8—42no—
273C3×C8—41no—
274C24—21no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4C3×#4112yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C2×C3×D412yes—
293C22×C4C2×C3×D412yes—
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C24×C311yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8C3×#4112yes—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42C3×#4112yes—
331C42C3×#4112yes—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3C2×#12411yescharacteristic
340C22×C3#54711yescharacteristic
341C22×C3#54811yescharacteristic
342C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
343C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
344C22×C3#57711yescharacteristic
345C22×C3#57811yescharacteristic
346C3×C4—22no—
347C3×C4—42no—
348C3×C4—42no—
349C3×C4—81no—
350C3×C4—81no—
351C3×C4—82no—
352C3×C4—81no—
353C3×C4—82no—
354C3×C4#49014yes—
355C3×C4#52911yescharacteristic
356C3×C4#54111yescharacteristic
357C3×C4#55211yescharacteristic
358C3×C4#55211yescharacteristic
359C23—22no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23C22×C3×D411yescharacteristic
367Q8—84no—
368D4—22no—
369D4—22no—
370D4—41no—
371D4—41no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—84no—
396C2×C4C3×#12412yes—
397C2×C4C3×#12412yes—
398C2×C4C3×AGL(1,ℤ/8)12yes—
399C2×C4C3×AGL(1,ℤ/8)12yes—
400C2×C4C2×C3×#4111yescharacteristic
401C2×C4C2×C3×#4111yescharacteristic
402C2×C4C2×C3×#4111yescharacteristic
403C2×C4C2×C3×#4111yescharacteristic
404C2×C4C3×2−1+411yescharacteristic
405C2×C4C3×2−1+411yescharacteristic
406C8—22no—
407C8—42no—
408C8—41no—
409C2×C3—82no—
410C2×C3#248911yescharacteristic
411C2×C3C2×#49011yescharacteristic
412C2×C3#313711yescharacteristic
413C2×C3#335311yescharacteristic
414C2×C3#335411yescharacteristic
415C2×C3#336711yescharacteristic
416C2×C3#337311yescharacteristic
417C22—42no—
418C22—42no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C2×C3×#12411yescharacteristic
425C22C3×#54711yescharacteristic
426C22C3×#54811yescharacteristic
427C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
428C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
429C22C3×#57711yescharacteristic
430C22C3×#57811yescharacteristic
431C4—22no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4—81no—
438C4—82no—
439C4C3×#49014yes—
440C4C3×#52911yescharacteristic
441C4C3×#54111yescharacteristic
442C4C3×#55211yescharacteristic
443C4C3×#55211yescharacteristic
444C3#2773611yescharacteristic, direct product factor, Sylow
445C2—82no—
446C2C3×#248911yescharacteristic
447C2C2×C3×#49011yescharacteristic
448C2C3×#313711yescharacteristic
449C2C3×#335311yescharacteristic
450C2C3×#335411yescharacteristic
451C2C3×#336711yescharacteristic
452C2C3×#337311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2571C2semidirect
3. C3×#2100C2semidirect
5. C3×#1990C2semidirect
6. C3×#1988C2semidirect
8. C3×#1836C2semidirect
10. C3×#1770C2semidirect
12. #27736C3direct (N × H)
42. #2571C2×C3semidirect
44. #2100C2×C3semidirect
46. #1990C2×C3semidirect
47. #1988C2×C3semidirect
49. #1836C2×C3semidirect
51. #1770C2×C3semidirect
444. C3#27736direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25712
C3×#25422
C3×#21002
C3×#20732
C3×#19902
C3×#19882
C3×#19362
C3×#18362
C3×#18022
C3×#17702
C3×C4×#1022
#277363