Finite Group Databasebeta

← C3×#27735 #10646342

Subgroups — C3×#27735 #10646342

Order 768.

698 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2563C211yesmaximal, characteristic
2C3×#2552C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2071C212yesmaximal
5C3×#1990C211yesmaximal, characteristic, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1936C212yesmaximal
8C3×#1836C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×#1770C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27735C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#102C2211yescharacteristic
18C2×C3×#102C2211yescharacteristic
19C2×C3×#102C2212yes—
20C2×C3×#97C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22C3×#398—22no—
23C3×#393—22no—
24C3×#390C2212yes—
25C3×#389—21no—
26C3×#389C2212yes—
27C3×#384C2211yescharacteristic
28C3×#384C2212yes—
29C3×#382C2212yes—
30C3×#380—21no—
31C3×#380—21no—
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#2563C2×C311yescharacteristic
43#2552C2×C311yescharacteristic, complemented
44#2101C2×C312yescomplemented
45#2071C2×C312yes—
46#1990C2×C311yescharacteristic, complemented
47#1987C2×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×#125—44no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—21no—
81C2×C3×C42C2312yes—
82C2×C3×C42C2311yescharacteristic
83C3×C4⋊C8—22no—
84C3×#102—22no—
85C3×#102—24no—
86C3×#102—22no—
87C3×#97—24no—
88C3×C4×C8—22no—
89C3×#95—21no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95D412yes—
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×#125—21no—
96C2×C4×D4C22×C311yescharacteristic
97C2×C4⋊C8C22×C311yescharacteristic
98C2×#102C22×C312yes—
99C2×#102C22×C311yescharacteristic
100C2×#102C22×C311yescharacteristic
101C2×#97C22×C312yes—
102C2×C4×C8C22×C311yescharacteristic
103#398—22no—
104#393—22no—
105#390C22×C312yes—
106#389—21no—
107#389C22×C312yes—
108#384C22×C311yescharacteristic
109#384C22×C312yes—
110#382C22×C312yes—
111#380—21no—
112#380—21no—
113C4×C4⋊C4C22×C311yescharacteristic
114C4×C4⋊C4C22×C311yescharacteristic
115#340C22×C312yes—
116#340C22×C312yes—
117#340C22×C312yes—
118#340C22×C311yescharacteristic
119#340C22×C312yes—
120#340C22×C311yescharacteristic
121#340C22×C312yes—
122#336C22×C312yes—
123C24×C3—21no—
124C2×C3×Q8—42no—
125C2×C3×Q8—81no—
126C2×C3×Q8—81no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4#4112yes—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C2×C3×C8—21no—
154C2×C3×C8#4112yes—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—44no—
168C3×C4⋊C4—44no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—81no—
172C3×C4⋊C4—82no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—82no—
175C3×C22⋊C4—82no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—42no—
179C3×C42#4112yes—
180C3×C42#4112yes—
181C22×Q8—21no—
182C22×D4C23×C311yescharacteristic
183C23×C4—21no—
184C22×C8C23×C311yescharacteristic
185C22×C8C23×C312yes—
186C22×C8C23×C311yescharacteristic
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189#125—44no—
190C4×D4—22no—
191C4×D4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C3×D411yescharacteristic
199C2×C4⋊C4C3×D411yescharacteristic
200C2×C4⋊C4C23×C311yescharacteristic
201C2×C4⋊C4C23×C312yes—
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C312yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C42—21no—
209C2×C42C23×C312yes—
210C2×C42C23×C311yescharacteristic
211C4⋊C8—22no—
212#102—24no—
213#102—22no—
214#102—22no—
215#97—24no—
216C4×C8—22no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95C3×D412yes—
222C23×C3—22no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3C22×D411yescharacteristic
230C3×Q8—84no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—84no—
259C2×C3×C4#12412yes—
260C2×C3×C4#12412yes—
261C2×C3×C4AGL(1,ℤ/8)12yes—
262C2×C3×C4#13712yes—
263C2×C3×C4C2×#4111yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C42−1+411yescharacteristic
268C2×C3×C42−1+411yescharacteristic
269C3×C8—22no—
270C3×C8—42no—
271C3×C8—41no—
272C24—21no—
273C2×Q8—42no—
274C2×Q8—81no—
275C2×Q8—81no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4C3×#4112yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C2×C3×D412yes—
291C22×C4C2×C3×D412yes—
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C24×C311yescharacteristic
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8C3×#4112yes—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C22⋊C4—42no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—22no—
326C42—22no—
327C42—42no—
328C42C3×#4112yes—
329C42C3×#4112yes—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3C2×#12411yescharacteristic
338C22×C3#54711yescharacteristic
339C22×C3#54811yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3C2×#13711yescharacteristic
342C22×C3#57611yescharacteristic
343C22×C3#57711yescharacteristic
344C3×C4—22no—
345C3×C4—42no—
346C3×C4—42no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4—82no—
350C3×C4—81no—
351C3×C4—82no—
352C3×C4#48914yes—
353C3×C4#52911yescharacteristic
354C3×C4#54111yescharacteristic
355C3×C4#55211yescharacteristic
356C3×C4#55211yescharacteristic
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C22×C3×D411yescharacteristic
365Q8—84no—
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—84no—
394C2×C4C3×#12412yes—
395C2×C4C3×#12412yes—
396C2×C4C3×AGL(1,ℤ/8)12yes—
397C2×C4C3×#13712yes—
398C2×C4C2×C3×#4111yescharacteristic
399C2×C4C2×C3×#4111yescharacteristic
400C2×C4C2×C3×#4111yescharacteristic
401C2×C4C2×C3×#4111yescharacteristic
402C2×C4C3×2−1+411yescharacteristic
403C2×C4C3×2−1+411yescharacteristic
404C8—22no—
405C8—42no—
406C8—41no—
407C2×C3—82no—
408C2×C3#248911yescharacteristic
409C2×C3C2×#48911yescharacteristic
410C2×C3#313611yescharacteristic
411C2×C3#335411yescharacteristic
412C2×C3#335711yescharacteristic
413C2×C3#336611yescharacteristic
414C2×C3#337211yescharacteristic
415C22—42no—
416C22—42no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22C2×C3×#12411yescharacteristic
423C22C3×#54711yescharacteristic
424C22C3×#54811yescharacteristic
425C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
426C22C2×C3×#13711yescharacteristic
427C22C3×#57611yescharacteristic
428C22C3×#57711yescharacteristic
429C4—22no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—82no—
434C4—81no—
435C4—81no—
436C4—82no—
437C4C3×#48914yes—
438C4C3×#52911yescharacteristic
439C4C3×#54111yescharacteristic
440C4C3×#55211yescharacteristic
441C4C3×#55211yescharacteristic
442C3#2773511yescharacteristic, direct product factor, Sylow
443C2—82no—
444C2C3×#248911yescharacteristic
445C2C2×C3×#48911yescharacteristic
446C2C3×#313611yescharacteristic
447C2C3×#335411yescharacteristic
448C2C3×#335711yescharacteristic
449C2C3×#336611yescharacteristic
450C2C3×#337211yescharacteristic
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
2. C3×#2552C2semidirect
3. C3×#2101C2semidirect
5. C3×#1990C2semidirect
6. C3×#1987C2semidirect
8. C3×#1836C2semidirect
10. C3×#1770C2semidirect
12. #27735C3direct (N × H)
43. #2552C2×C3semidirect
44. #2101C2×C3semidirect
46. #1990C2×C3semidirect
47. #1987C2×C3semidirect
49. #1836C2×C3semidirect
51. #1770C2×C3semidirect
442. C3#27735direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25632
C3×#25522
C3×#21012
C3×#20712
C3×#19902
C3×#19872
C3×#19362
C3×#18362
C3×#18022
C3×#17702
C3×C4×#1022
#277353