Finite Group Databasebeta

← C3×#29234 #10647841

Subgroups — C3×#29234 #10647841

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C3×#2086C212yesmaximal, complemented
4C3×#2067C212yesmaximal, complemented
5C3×#1918C212yesmaximal, complemented
6C3×#1874C211yesmaximal, characteristic
7C3×#1874C211yesmaximal, characteristic
8C3×#1862C212yesmaximal, complemented
9C3×#1852C211yesmaximal, characteristic, complemented
10C3×#1852C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29234C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×C3×#107C2212yes—
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389—21no—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#389C2211yescharacteristic
32C3×#389C2212yes—
33C3×#388—21no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C2×C3×#95—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C4×#121C2×C311yescharacteristic
45C2×#389C2×C311yescharacteristic, complemented
46#2086C2×C312yescomplemented
47#2067C2×C312yescomplemented
48#1918C2×C312yescomplemented
49#1874C2×C311yescharacteristic
50#1874C2×C311yescharacteristic
51#1862C2×C312yescomplemented
52#1852C2×C311yescharacteristic, complemented
53#1852C2×C311yescharacteristic, complemented
54#1797C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×#121—22no—
66C3×#121Q814yes—
67C3×C4×D4—42no—
68C3×C4×D4—44no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—42no—
76C2×C3×C4⋊C4—44no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42—42no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C3×#107—24no—
94C3×#107—42no—
95C3×C4⋊C8—42no—
96C3×#102—24no—
97C3×#102—42no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C2×#121C22×C311yescharacteristic
106C2×C4×D4—21no—
107C2×C4×D4C22×C311yescharacteristic
108C22×C4⋊C4—21no—
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C42C22×C311yescharacteristic
112C2×#107C22×C312yes—
113C2×#107C22×C312yes—
114C2×C4⋊C8C22×C312yes—
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×C22⋊C8C22×C311yescharacteristic
118C2×C22⋊C8C22×C311yescharacteristic
119#390C22×C311yescharacteristic
120#389—21no—
121#389—21no—
122#389C22×C312yes—
123#389C22×C311yescharacteristic
124#389C22×C312yes—
125#388—21no—
126#386—21no—
127#380C22×C311yescharacteristic
128C4×C4⋊C4C22×C311yescharacteristic
129C4×C22⋊C4—21no—
130C2×#95—21no—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#340C22×C312yes—
135#336C22×C312yes—
136C24×C3—21no—
137C24×C3C2×D411yescharacteristic
138C2×C3×D4—24no—
139C2×C3×D4—22no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4C2×Q812yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4D814yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×Q812yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C2×Q812yes—
193C3×C4⋊C4#4114yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—42no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—41no—
202C3×C42—41no—
203C3×C42—42no—
204C22×D4—21no—
205C22×D4C23×C311yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C3×D411yescharacteristic
210C23×C4C3×D411yescharacteristic
211C23×C4C23×C311yescharacteristic
212C22×C8C23×C312yes—
213C22×C8C23×C312yes—
214#121—22no—
215#121C3×Q814yes—
216C4×D4—42no—
217C4×D4—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C23×C312yes—
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C4⋊C4C23×C312yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42—21no—
237C2×C42—42no—
238C2×C42C3×D411yescharacteristic
239C2×C42C3×D411yescharacteristic
240C2×C42C23×C311yescharacteristic
241C2×C42C23×C311yescharacteristic
242#107—24no—
243#107—42no—
244C4⋊C8—42no—
245#102—24no—
246#102—42no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3C22⋊Q812yes—
266C23×C3C2×D812yes—
267C23×C3#13712yes—
268C23×C3C22×D411yescharacteristic
269C3×D4—24no—
270C3×D4—48no—
271C3×D4—42no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—44no—
289C2×C3×C4—48no—
290C2×C3×C4—44no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—84no—
305C2×C3×C4—84no—
306C2×C3×C4C22⋊Q812yes—
307C2×C3×C4C2×D812yes—
308C2×C3×C4C2×D812yes—
309C2×C3×C4C22×Q811yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C4H(5,𝔽2)12yes—
313C3×C8—42no—
314C3×C8—42no—
315C24—21no—
316C24C2×C3×D411yescharacteristic
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4C2×C3×Q812yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4C3×D814yes—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×Q812yes—
352C22×C4C3×#4112yes—
353C22×C4C24×C311yescharacteristic
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4C2×C3×Q812yes—
372C4⋊C4C3×#4114yes—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C42—22no—
379C42—22no—
380C42—41no—
381C42—41no—
382C42—42no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3#47414yes—
397C22×C3C2×C22⋊Q811yescharacteristic
398C22×C3#52511yescharacteristic
399C22×C3#53411yescharacteristic
400C22×C3C22×D811yescharacteristic
401C22×C3C2×#13711yescharacteristic
402C22×C3#57511yescharacteristic
403C22×C3#57511yescharacteristic
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—44no—
408C3×C4—41no—
409C3×C4—41no—
410C3×C4—42no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4—82no—
414C3×C4#55312yes—
415C3×C4#55412yes—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C3×C22⋊Q812yes—
428C23C2×C3×D812yes—
429C23C3×#13712yes—
430C23C22×C3×D411yescharacteristic
431D4—24no—
432D4—48no—
433D4—42no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4C3×C22⋊Q812yes—
469C2×C4C2×C3×D812yes—
470C2×C4C2×C3×D812yes—
471C2×C4C22×C3×Q811yescharacteristic
472C2×C4C2×C3×#4111yescharacteristic
473C2×C4C2×C3×#4112yes—
474C2×C4C3×H(5,𝔽2)12yes—
475C8—42no—
476C8—42no—
477C2×C3—24no—
478C2×C3—82no—
479C2×C3#257911yescharacteristic
480C2×C3C2×#47411yescharacteristic
481C2×C3#308011yescharacteristic
482C2×C3#310011yescharacteristic
483C2×C3#311711yescharacteristic
484C2×C3#318611yescharacteristic
485C2×C3#319611yescharacteristic
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C3×#47414yes—
500C22C2×C3×C22⋊Q811yescharacteristic
501C22C3×#52511yescharacteristic
502C22C3×#53411yescharacteristic
503C22C22×C3×D811yescharacteristic
504C22C2×C3×#13711yescharacteristic
505C22C3×#57511yescharacteristic
506C22C3×#57511yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—42no—
513C4—44no—
514C4—81no—
515C4—82no—
516C4—81no—
517C4C3×#55312yes—
518C4C3×#55412yes—
519C3#2923411yescharacteristic, direct product factor, Sylow
520C2—24no—
521C2—82no—
522C2C3×#257911yescharacteristic
523C2C2×C3×#47411yescharacteristic
524C2C3×#308011yescharacteristic
525C2C3×#310011yescharacteristic
526C2C3×#311711yescharacteristic
527C2C3×#318611yescharacteristic
528C2C3×#319611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
2. C2×C3×#389C2semidirect
3. C3×#2086C2semidirect
4. C3×#2067C2semidirect
5. C3×#1918C2semidirect
8. C3×#1862C2semidirect
9. C3×#1852C2semidirect
10. C3×#1852C2semidirect
11. C3×#1797C2semidirect
12. #29234C3direct (N × H)
45. C2×#389C2×C3semidirect
46. #2086C2×C3semidirect
47. #2067C2×C3semidirect
48. #1918C2×C3semidirect
51. #1862C2×C3semidirect
52. #1852C2×C3semidirect
53. #1852C2×C3semidirect
54. #1797C2×C3semidirect
519. C3#29234direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C2×C3×#3892
C3×#20862
C3×#20672
C3×#19182
C3×#18742
C3×#18622
C3×#18522
C3×#17972
#292343