Finite Group Databasebeta

← C3×#27790 #10646397

Subgroups — C3×#27790 #10646397

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#499C211yesmaximal, characteristic, complemented
2C2×C3×#466C211yesmaximal, characteristic
3C3×#2490C212yesmaximal, complemented
4C3×#2034C212yesmaximal, complemented
5C3×#2025C212yesmaximal, complemented
6C3×#1970C211yesmaximal, characteristic, complemented
7C3×#1970C211yesmaximal, characteristic, complemented
8C3×#1926C212yesmaximal, complemented
9C3×#1808C211yesmaximal, characteristic, complemented
10C3×C4×#107C211yesmaximal, characteristic, complemented
11C3×C8×C22⋊C4C211yesmaximal, characteristic, complemented
12#27790C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#125C2212yes—
16C2×C3×#121C2212yes—
17C3×#499—24no—
18C3×#466—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#397—22no—
30C3×#394—22no—
31C3×#390—22no—
32C3×#390C2212yes—
33C3×#390C2212yes—
34C3×#388—22no—
35C3×#380C2212yes—
36C3×C4×C4⋊C4C2212yes—
37C3×C4×C22⋊C4C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C2×#499C2×C311yescharacteristic, complemented
43C2×#466C2×C311yescharacteristic
44#2490C2×C312yescomplemented
45#2034C2×C312yescomplemented
46#2025C2×C312yescomplemented
47#1970C2×C311yescharacteristic, complemented
48#1970C2×C311yescharacteristic, complemented
49#1926C2×C312yescomplemented
50#1808C2×C311yescharacteristic, complemented
51C4×#107C2×C311yescharacteristic, complemented
52C8×C22⋊C4C2×C311yescharacteristic, complemented
53C22×C3×D4—22no—
54C22×C3×D4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C2×C3×D8—22no—
57C2×C3×D8—22no—
58C2×C3×D8—22no—
59C22×C3×C8—22no—
60C22×C3×C8—22no—
61C22×C3×C8—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#125—24no—
67C3×#125—42no—
68C3×#121—24no—
69C3×#121—42no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42—22no—
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#107—22no—
87C3×#107—22no—
88C3×#107D414yes—
89C3×#106—24no—
90C3×#102—24no—
91C3×#102—24no—
92C3×#102—42no—
93C3×C22⋊C8—22no—
94C3×C4×C8—22no—
95C3×C4×C8—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95D411yescharacteristic
99C3×#95D411yescharacteristic
100C22×D8C22×C311yescharacteristic
101C23×C8C22×C311yescharacteristic
102C2×#125C22×C312yes—
103C2×#121C22×C312yes—
104#499—24no—
105#466—24no—
106C2×#107C22×C311yescharacteristic
107C2×#107C22×C311yescharacteristic
108C2×#107C22×C311yescharacteristic
109C2×#106C22×C312yes—
110C2×#102C22×C312yes—
111C2×#102C22×C312yes—
112C2×#102C22×C312yes—
113C2×C22⋊C8C22×C311yescharacteristic
114C2×C4×C8C22×C311yescharacteristic
115C2×C4×C8C22×C311yescharacteristic
116#397—22no—
117#394—22no—
118#390—22no—
119#390C22×C312yes—
120#390C22×C312yes—
121#388—22no—
122#380C22×C312yes—
123C4×C4⋊C4C22×C312yes—
124C4×C22⋊C4C22×C311yescharacteristic
125#340C22×C312yes—
126#340C22×C312yes—
127#336C22×C311yescharacteristic
128#336C22×C311yescharacteristic
129C24×C3—22no—
130C24×C3C2×D411yescharacteristic
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4—82no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C3×D8—44no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—44no—
165C2×C3×C8C2×D412yes—
166C2×C3×C8#4112yes—
167C2×C3×C8#4112yes—
168C2×C3×C8#4112yes—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4C2×D414yes—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—82no—
182C3×C22⋊C4—82no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42—21no—
189C3×C42—82no—
190C22×D4—22no—
191C22×D4C23×C312yes—
192C23×C4C23×C311yescharacteristic
193C2×D8—22no—
194C2×D8—22no—
195C2×D8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C23×C311yescharacteristic
200C22×C8C23×C311yescharacteristic
201C22×C8C23×C311yescharacteristic
202C22×C8C23×C311yescharacteristic
203#125—24no—
204#125—42no—
205#121—24no—
206#121—42no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C312yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C3×D411yescharacteristic
219C2×C22⋊C4C3×D411yescharacteristic
220C2×C42—22no—
221C2×C42C23×C311yescharacteristic
222C2×C42C23×C311yescharacteristic
223#107—22no—
224#107—22no—
225#107C3×D414yes—
226#106—24no—
227#102—24no—
228#102—24no—
229#102—42no—
230C22⋊C8—22no—
231C4×C8—22no—
232C4×C8—22no—
233#95—22no—
234#95—22no—
235#95C3×D411yescharacteristic
236#95C3×D411yescharacteristic
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3#13512yes—
251C23×C3C22×D411yescharacteristic
252C3×D4—42no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—88no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4#13512yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4H(5,𝔽2)11yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C3×C8—22no—
292C3×C8—41no—
293C3×C8—41no—
294C3×C8#12414yes—
295C24—22no—
296C24C2×C3×D411yescharacteristic
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—82no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D411yescharacteristic
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C24×C311yescharacteristic
321D8—44no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—44no—
331C2×C8C2×C3×D412yes—
332C2×C8C3×#4112yes—
333C2×C8C3×#4112yes—
334C2×C8C3×#4112yes—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4C2×C3×D414yes—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—82no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3#53211yescharacteristic
371C22×C3D4211yescharacteristic
372C22×C3#54711yescharacteristic
373C22×C3C2×#13511yescharacteristic
374C22×C3C2×#13511yescharacteristic
375C22×C3#57611yescharacteristic
376C22×C3#57611yescharacteristic
377C3×C4—22no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—41no—
381C3×C4—44no—
382C3×C4—41no—
383C3×C4—82no—
384C3×C4—82no—
385C3×C4C2×#12411yescharacteristic
386C3×C4#54211yescharacteristic
387C3×C4#54811yescharacteristic
388C3×C4#55511yescharacteristic
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23C3×#13512yes—
403C23C22×C3×D411yescharacteristic
404D4—42no—
405D4—42no—
406D4—42no—
407D4—42no—
408D4—88no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4C3×#13512yes—
437C2×C4C22×C3×D411yescharacteristic
438C2×C4C2×C3×#4111yescharacteristic
439C2×C4C2×C3×#4111yescharacteristic
440C2×C4C2×C3×#4111yescharacteristic
441C2×C4C3×H(5,𝔽2)11yescharacteristic
442C2×C4C3×2−1+411yescharacteristic
443C8—22no—
444C8—41no—
445C8—41no—
446C8C3×#12414yes—
447C2×C3—42no—
448C2×C3—162no—
449C2×C3#249111yescharacteristic
450C2×C3#316811yescharacteristic
451C2×C3#317611yescharacteristic
452C2×C3#328011yescharacteristic
453C2×C3#328711yescharacteristic
454C2×C3#333911yescharacteristic
455C2×C3#333911yescharacteristic
456C22—22no—
457C22—22no—
458C22—22no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22C3×#53211yescharacteristic
471C22C3×D4211yescharacteristic
472C22C3×#54711yescharacteristic
473C22C2×C3×#13511yescharacteristic
474C22C2×C3×#13511yescharacteristic
475C22C3×#57611yescharacteristic
476C22C3×#57611yescharacteristic
477C4—22no—
478C4—41no—
479C4—41no—
480C4—44no—
481C4—41no—
482C4—41no—
483C4—82no—
484C4—82no—
485C4C2×C3×#12411yescharacteristic
486C4C3×#54211yescharacteristic
487C4C3×#54811yescharacteristic
488C4C3×#55511yescharacteristic
489C3#2779011yescharacteristic, direct product factor, Sylow
490C2—42no—
491C2—162no—
492C2C3×#249111yescharacteristic
493C2C3×#316811yescharacteristic
494C2C3×#317611yescharacteristic
495C2C3×#328011yescharacteristic
496C2C3×#328711yescharacteristic
497C2C3×#333911yescharacteristic
498C2C3×#333911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#466C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#499C2semidirect
3. C3×#2490C2semidirect
4. C3×#2034C2semidirect
5. C3×#2025C2semidirect
6. C3×#1970C2semidirect
7. C3×#1970C2semidirect
8. C3×#1926C2semidirect
9. C3×#1808C2semidirect
10. C3×C4×#107C2semidirect
11. C3×C8×C22⋊C4C2semidirect
12. #27790C3direct (N × H)
42. C2×#499C2×C3semidirect
44. #2490C2×C3semidirect
45. #2034C2×C3semidirect
46. #2025C2×C3semidirect
47. #1970C2×C3semidirect
48. #1970C2×C3semidirect
49. #1926C2×C3semidirect
50. #1808C2×C3semidirect
51. C4×#107C2×C3semidirect
52. C8×C22⋊C4C2×C3semidirect
489. C3#27790direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4992
C2×C3×#4662
C3×#24902
C3×#20342
C3×#20252
C3×#19702
C3×#19262
C3×#18082
C3×C4×#1072
C3×C8×C22⋊C42
#277903