Finite Group Databasebeta

← C3×#27750 #10646357

Subgroups — C3×#27750 #10646357

Order 768.

762 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2571C211yesmaximal, characteristic, complemented
2C3×#2548C211yesmaximal, characteristic
3C3×#2073C212yesmaximal, complemented
4C3×#2042C212yesmaximal, complemented
5C3×#1982C211yesmaximal, characteristic
6C3×#1958C212yesmaximal, complemented
7C3×#1957C211yesmaximal, characteristic
8C3×#1934C212yesmaximal, complemented
9C3×#1803C211yesmaximal, characteristic, complemented
10C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×C4×#103C211yesmaximal, characteristic, complemented
12#27750C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#124C2212yes—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103C2211yescharacteristic
18C2×C3×#103C2211yescharacteristic
19C2×C3×#103C2212yes—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C2×C3×C4×C8C2212yes—
26C3×#400—22no—
27C3×#394—22no—
28C3×#390—21no—
29C3×#390—21no—
30C3×#390C2212yes—
31C3×#389—21no—
32C3×#389C2212yes—
33C3×#384C2211yescharacteristic
34C3×#382—21no—
35C3×#382C2212yes—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2212yes—
41C3×#336C2212yes—
42#2571C2×C311yescharacteristic, complemented
43#2548C2×C311yescharacteristic
44#2073C2×C312yescomplemented
45#2042C2×C312yescomplemented
46#1982C2×C311yescharacteristic
47#1958C2×C312yescomplemented
48#1957C2×C311yescharacteristic
49#1934C2×C312yescomplemented
50#1803C2×C311yescharacteristic, complemented
51C8×C4⋊C4C2×C311yescharacteristic, complemented
52C4×#103C2×C311yescharacteristic, complemented
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4C2312yes—
57C22×C3×C8C2312yes—
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×#127—22no—
61C3×#127—22no—
62C3×#124—24no—
63C3×#124—42no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—21no—
81C2×C3×C42—21no—
82C2×C3×C42C2312yes—
83C2×C3×C42C2311yescharacteristic
84C3×C4⋊C8—22no—
85C3×#103—22no—
86C3×#103—22no—
87C3×#103—42no—
88C3×#102—24no—
89C3×#102—42no—
90C3×#102—42no—
91C3×#102—42no—
92C3×C4×C8—22no—
93C3×C4×C8—42no—
94C3×#95—21no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95D412yes—
99C2×#127C22×C311yescharacteristic
100C2×#124C22×C312yes—
101C2×C4×Q8C22×C311yescharacteristic
102C2×C4⋊C8C22×C311yescharacteristic
103C2×#103C22×C311yescharacteristic
104C2×#103C22×C311yescharacteristic
105C2×#103C22×C312yes—
106C2×#102C22×C312yes—
107C2×#102C22×C312yes—
108C2×#102C22×C312yes—
109C2×#102C22×C312yes—
110C2×C4×C8C22×C311yescharacteristic
111C2×C4×C8C22×C312yes—
112#400—22no—
113#394—22no—
114#390—21no—
115#390—21no—
116#390C22×C312yes—
117#389—21no—
118#389C22×C312yes—
119#384C22×C311yescharacteristic
120#382—21no—
121#382C22×C312yes—
122C4×C4⋊C4C22×C311yescharacteristic
123C4×C4⋊C4C22×C311yescharacteristic
124#340C22×C312yes—
125#340C22×C311yescharacteristic
126#340C22×C311yescharacteristic
127#336C22×C312yes—
128C24×C3—22no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8#4112yes—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—42no—
145C2×C3×D4—81no—
146C2×C3×D4—81no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—21no—
164C2×C3×C8—22no—
165C2×C3×C8#4112yes—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—81no—
179C3×C4⋊C4—82no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—82no—
183C3×C22⋊C4—82no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—42no—
187C3×C42—42no—
188C3×C42#4112yes—
189C3×C42#4112yes—
190C22×Q8C23×C311yescharacteristic
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C23×C312yes—
194C22×C8C23×C311yescharacteristic
195C22×C8C23×C312yes—
196C22×C8C23×C311yescharacteristic
197#127—22no—
198#127—22no—
199#124—24no—
200#124—42no—
201C4×Q8—22no—
202C4×Q8—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C3×D411yescharacteristic
208C2×C4⋊C4C3×D411yescharacteristic
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C4⋊C4C23×C312yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C23×C312yes—
220C2×C42C23×C311yescharacteristic
221C4⋊C8—22no—
222#103—22no—
223#103—22no—
224#103—42no—
225#102—24no—
226#102—42no—
227#102—42no—
228#102—42no—
229C4×C8—22no—
230C4×C8—42no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95C3×D412yes—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3C22×D411yescharacteristic
244C3×Q8—22no—
245C3×Q8—22no—
246C3×Q8—41no—
247C3×Q8—41no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—84no—
253C3×D4—84no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4#12412yes—
278C2×C3×C4#12412yes—
279C2×C3×C4#13512yes—
280C2×C3×C4#13512yes—
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4H(5,𝔽2)11yescharacteristic
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C3×C8—22no—
288C3×C8—42no—
289C3×C8—41no—
290C24—22no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8C3×#4112yes—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—81no—
308C2×D4—81no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C24×C311yescharacteristic
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8C3×#4112yes—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—81no—
341C4⋊C4—82no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42C3×#4112yes—
351C42C3×#4112yes—
352C22×C3—82no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3C2×#12411yescharacteristic
360C22×C3#54611yescharacteristic
361C22×C3#55011yescharacteristic
362C22×C3C2×#13511yescharacteristic
363C22×C3C2×#13511yescharacteristic
364C22×C3#57611yescharacteristic
365C22×C3#57711yescharacteristic
366C3×C4—22no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—81no—
371C3×C4—81no—
372C3×C4—82no—
373C3×C4—81no—
374C3×C4#48814yes—
375C3×C4#52911yescharacteristic
376C3×C4#54011yescharacteristic
377C3×C4#55311yescharacteristic
378C3×C4#55311yescharacteristic
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23C22×C3×D411yescharacteristic
387Q8—22no—
388Q8—22no—
389Q8—41no—
390Q8—41no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—84no—
396D4—84no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4C3×#12412yes—
421C2×C4C3×#12412yes—
422C2×C4C3×#13512yes—
423C2×C4C3×#13512yes—
424C2×C4C2×C3×#4111yescharacteristic
425C2×C4C2×C3×#4111yescharacteristic
426C2×C4C2×C3×#4111yescharacteristic
427C2×C4C2×C3×#4111yescharacteristic
428C2×C4C3×H(5,𝔽2)11yescharacteristic
429C2×C4C3×H(5,𝔽2)11yescharacteristic
430C8—22no—
431C8—42no—
432C8—41no—
433C2×C3—162no—
434C2×C3#249011yescharacteristic
435C2×C3C2×#48811yescharacteristic
436C2×C3#313611yescharacteristic
437C2×C3#330411yescharacteristic
438C2×C3#330511yescharacteristic
439C2×C3#339411yescharacteristic
440C2×C3#339611yescharacteristic
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22C2×C3×#12411yescharacteristic
449C22C3×#54611yescharacteristic
450C22C3×#55011yescharacteristic
451C22C2×C3×#13511yescharacteristic
452C22C2×C3×#13511yescharacteristic
453C22C3×#57611yescharacteristic
454C22C3×#57711yescharacteristic
455C4—22no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—82no—
462C4—81no—
463C4C3×#48814yes—
464C4C3×#52911yescharacteristic
465C4C3×#54011yescharacteristic
466C4C3×#55311yescharacteristic
467C4C3×#55311yescharacteristic
468C3#2775011yescharacteristic, direct product factor, Sylow
469C2—162no—
470C2C3×#249011yescharacteristic
471C2C2×C3×#48811yescharacteristic
472C2C3×#313611yescharacteristic
473C2C3×#330411yescharacteristic
474C2C3×#330511yescharacteristic
475C2C3×#339411yescharacteristic
476C2C3×#339611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
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×#2073C2semidirect
4. C3×#2042C2semidirect
6. C3×#1958C2semidirect
8. C3×#1934C2semidirect
9. C3×#1803C2semidirect
10. C3×C8×C4⋊C4C2semidirect
11. C3×C4×#103C2semidirect
12. #27750C3direct (N × H)
42. #2571C2×C3semidirect
44. #2073C2×C3semidirect
45. #2042C2×C3semidirect
47. #1958C2×C3semidirect
49. #1934C2×C3semidirect
50. #1803C2×C3semidirect
51. C8×C4⋊C4C2×C3semidirect
52. C4×#103C2×C3semidirect
468. C3#27750direct (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×#25482
C3×#20732
C3×#20422
C3×#19822
C3×#19582
C3×#19572
C3×#19342
C3×#18032
C3×C8×C4⋊C42
C3×C4×#1032
#277503