Finite Group Databasebeta

← C3×#35898 #10654505

Subgroups — C3×#35898 #10654505

Order 768.

1748 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2644C213yesmaximal, complemented
2C3×#2513C213yesmaximal, complemented
3C3×#2512C213yesmaximal, complemented
4C3×#2512C213yesmaximal, complemented
5C3×#2497C213yesmaximal, complemented
6C3×#2496C213yesmaximal, complemented
7C3×#2494C213yesmaximal, complemented
8C3×#2401C213yesmaximal, complemented
9C3×#2351C211yesmaximal, characteristic, complemented
10C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
12C3×#2341C211yesmaximal, characteristic
13C3×#2341C211yesmaximal, characteristic
14C3×#2341C211yesmaximal, characteristic
15C3×#2276C211yesmaximal, characteristic, complemented
16#35898C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×#41C2211yescharacteristic
18C2×C3×#126C2216yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C3×#401C2213yes—
26C3×#400C2211yescharacteristic
27C3×#400C2213yes—
28C3×#400C2213yes—
29C3×#397C2213yes—
30C3×#397C2216yes—
31C3×#397C2216yes—
32C3×#397C2213yes—
33C3×#397C2213yes—
34C3×#396C2216yes—
35C3×#396C2213yes—
36C3×#390C2213yes—
37C3×#390C2213yes—
38C3×#390C2213yes—
39C3×#390C2213yes—
40C3×#390C2213yes—
41C3×#390C2213yes—
42C3×#389C2213yes—
43C3×#389C2213yes—
44C3×#389C2213yes—
45C3×#389C2213yes—
46C3×#389C2213yes—
47C3×#389C2213yes—
48C3×#388C2216yes—
49C3×#388C2216yes—
50C3×#387C2216yes—
51C3×#384C2213yes—
52C3×#384C2213yes—
53C3×#382C2213yes—
54C3×#380C2213yes—
55C3×#380C2213yes—
56C3×#380C2213yes—
57C3×C4×C4⋊C4C2213yes—
58C3×C4×C4⋊C4C2213yes—
59C3×C4×C4⋊C4C2213yes—
60C3×C4×C4⋊C4C2213yes—
61C3×C4×C4⋊C4C2213yes—
62C3×C4×C4⋊C4C2213yes—
63C3×C4×C22⋊C4C2213yes—
64C3×C4×C22⋊C4C2213yes—
65C3×C4×C22⋊C4C2213yes—
66C3×#376C2213yes—
67#2644C2×C313yescomplemented
68#2513C2×C313yescomplemented
69#2512C2×C313yescomplemented
70#2512C2×C313yescomplemented
71#2497C2×C313yescomplemented
72#2496C2×C313yescomplemented
73#2494C2×C313yescomplemented
74#2401C2×C313yescomplemented
75#2351C2×C311yescharacteristic, complemented
76Q8×C4⋊C4C2×C311yescharacteristic, complemented
77Q8×C4⋊C4C2×C311yescharacteristic, complemented
78#2341C2×C311yescharacteristic
79#2341C2×C311yescharacteristic
80#2341C2×C311yescharacteristic
81#2276C2×C311yescharacteristic, complemented
82C2×C3×#41—24no—
83C2×C3×#41—24no—
84C2×C3×#41—24no—
85C22×C3×Q8C2311yescharacteristic
86C22×C3×D4C2313yes—
87C23×C3×C4C2313yes—
88C3×#126—412no—
89C3×C4×Q8—24no—
90C3×C4×Q8—24no—
91C3×C4×Q8—24no—
92C3×C4×Q8—24no—
93C3×C4×Q8—24no—
94C3×C4×Q8—24no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2313yes—
98C2×C3×C4⋊C4C2313yes—
99C2×C3×C4⋊C4C2313yes—
100C2×C3×C4⋊C4C2313yes—
101C2×C3×C4⋊C4C2313yes—
102C2×C3×C4⋊C4C2313yes—
103C2×C3×C4⋊C4C2313yes—
104C2×C3×C4⋊C4C2313yes—
105C2×C3×C4⋊C4C2313yes—
106C2×C3×C4⋊C4C2313yes—
107C2×C3×C4⋊C4C2316yes—
108C2×C3×C4⋊C4C2316yes—
109C2×C3×C22⋊C4C2313yes—
110C2×C3×C22⋊C4C2313yes—
111C2×C3×C22⋊C4C2313yes—
112C2×C3×C22⋊C4C2313yes—
113C2×C3×C22⋊C4C2313yes—
114C2×C3×C22⋊C4C2313yes—
115C2×C3×C22⋊C4C2316yes—
116C2×C3×C22⋊C4C2316yes—
117C2×C3×C22⋊C4C2316yes—
118C2×C3×C42C2313yes—
119C2×C3×C42C2313yes—
120C2×C3×C42C2313yes—
121C2×C3×C42C2313yes—
122C2×C3×C42C2313yes—
123C2×C3×C42C2313yes—
124C2×C3×C42C2316yes—
125C3×#95C2311yescharacteristic
126C3×#95C2313yes—
127C3×#95C2313yes—
128C3×#95C2313yes—
129C3×#95C2311yescharacteristic
130C3×#95C2311yescharacteristic
131C3×#95C2311yescharacteristic
132C3×#95C2313yes—
133C3×#95C2313yes—
134C3×#95C2313yes—
135C3×#95C2311yescharacteristic
136C3×#95C2313yes—
137C3×#95C2313yes—
138C3×#95C2313yes—
139C3×#95C2313yes—
140C3×#95C2313yes—
141C3×#95C2316yes—
142C22×#41C22×C311yescharacteristic
143C2×#126C22×C316yes—
144C2×C4×Q8C22×C311yescharacteristic
145C2×C4×Q8C22×C311yescharacteristic
146C2×C4×Q8C22×C311yescharacteristic
147C2×C4×Q8C22×C311yescharacteristic
148C2×C4×Q8C22×C311yescharacteristic
149C2×C4×Q8C22×C311yescharacteristic
150#401C22×C313yes—
151#400C22×C311yescharacteristic
152#400C22×C313yes—
153#400C22×C313yes—
154#397C22×C313yes—
155#397C22×C316yes—
156#397C22×C316yes—
157#397C22×C313yes—
158#397C22×C313yes—
159#396C22×C316yes—
160#396C22×C313yes—
161#390C22×C313yes—
162#390C22×C313yes—
163#390C22×C313yes—
164#390C22×C313yes—
165#390C22×C313yes—
166#390C22×C313yes—
167#389C22×C313yes—
168#389C22×C313yes—
169#389C22×C313yes—
170#389C22×C313yes—
171#389C22×C313yes—
172#389C22×C313yes—
173#388C22×C316yes—
174#388C22×C316yes—
175#387C22×C316yes—
176#384C22×C313yes—
177#384C22×C313yes—
178#382C22×C313yes—
179#380C22×C313yes—
180#380C22×C313yes—
181#380C22×C313yes—
182C4×C4⋊C4C22×C313yes—
183C4×C4⋊C4C22×C313yes—
184C4×C4⋊C4C22×C313yes—
185C4×C4⋊C4C22×C313yes—
186C4×C4⋊C4C22×C313yes—
187C4×C4⋊C4C22×C313yes—
188C4×C22⋊C4C22×C313yes—
189C4×C22⋊C4C22×C313yes—
190C4×C22⋊C4C22×C313yes—
191#376C22×C313yes—
192C24×C3C2413yes—
193C3×#41—416no—
194C2×C3×Q8#4114yes—
195C2×C3×Q8#4114yes—
196C2×C3×Q8#4114yes—
197C2×C3×D4—26no—
198C2×C3×D4—26no—
199C2×C3×D4—26no—
200C22×C3×C4—26no—
201C22×C3×C4—26no—
202C22×C3×C4—26no—
203C22×C3×C4C2411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×C3×C4C2413yes—
209C22×C3×C4C2413yes—
210C22×C3×C4C2413yes—
211C22×C3×C4C2413yes—
212C22×C3×C4C2413yes—
213C22×C3×C4C2413yes—
214C22×C3×C4C2411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×C3×C4C2413yes—
217C3×C4⋊C4—26no—
218C3×C4⋊C4—26no—
219C3×C4⋊C4—26no—
220C3×C4⋊C4—26no—
221C3×C4⋊C4—26no—
222C3×C4⋊C4—26no—
223C3×C4⋊C4—26no—
224C3×C4⋊C4—26no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—43no—
228C3×C4⋊C4—43no—
229C3×C4⋊C4—46no—
230C3×C4⋊C4—46no—
231C3×C22⋊C4—26no—
232C3×C22⋊C4—26no—
233C3×C22⋊C4—43no—
234C3×C22⋊C4—43no—
235C3×C22⋊C4—43no—
236C3×C22⋊C4—43no—
237C3×C22⋊C4—46no—
238C3×C22⋊C4—46no—
239C3×C22⋊C4—46no—
240C3×C42—26no—
241C3×C42—26no—
242C3×C42—26no—
243C3×C42—26no—
244C3×C42—26no—
245C3×C42—26no—
246C3×C42—46no—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—24no—
250C22×Q8C23×C311yescharacteristic
251C22×D4C23×C313yes—
252C23×C4C23×C313yes—
253#126—412no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C2×C4⋊C4C23×C313yes—
261C2×C4⋊C4C23×C313yes—
262C2×C4⋊C4C23×C313yes—
263C2×C4⋊C4C23×C313yes—
264C2×C4⋊C4C23×C313yes—
265C2×C4⋊C4C23×C313yes—
266C2×C4⋊C4C23×C313yes—
267C2×C4⋊C4C23×C313yes—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C316yes—
271C2×C4⋊C4C23×C313yes—
272C2×C4⋊C4C23×C313yes—
273C2×C4⋊C4C23×C316yes—
274C2×C22⋊C4C23×C313yes—
275C2×C22⋊C4C23×C313yes—
276C2×C22⋊C4C23×C316yes—
277C2×C22⋊C4C23×C316yes—
278C2×C22⋊C4C23×C313yes—
279C2×C22⋊C4C23×C313yes—
280C2×C22⋊C4C23×C313yes—
281C2×C22⋊C4C23×C313yes—
282C2×C22⋊C4C23×C316yes—
283C2×C42C23×C313yes—
284C2×C42C23×C313yes—
285C2×C42C23×C313yes—
286C2×C42C23×C313yes—
287C2×C42C23×C313yes—
288C2×C42C23×C313yes—
289C2×C42C23×C316yes—
290#95C23×C311yescharacteristic
291#95C23×C313yes—
292#95C23×C313yes—
293#95C23×C313yes—
294#95C23×C311yescharacteristic
295#95C23×C311yescharacteristic
296#95C23×C311yescharacteristic
297#95C23×C313yes—
298#95C23×C313yes—
299#95C23×C313yes—
300#95C23×C311yescharacteristic
301#95C23×C313yes—
302#95C23×C313yes—
303#95C23×C313yes—
304#95C23×C313yes—
305#95C23×C313yes—
306#95C23×C316yes—
307C23×C3—23no—
308C23×C3—23no—
309C23×C3—23no—
310C23×C3—23no—
311C23×C3—23no—
312C23×C3—23no—
313C23×C3—23no—
314C23×C3C2511yescharacteristic
315C3×Q8#126116yescomplemented
316C3×D4—412no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—23no—
326C2×C3×C4—23no—
327C2×C3×C4—23no—
328C2×C3×C4—23no—
329C2×C3×C4—23no—
330C2×C3×C4—23no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—23no—
335C2×C3×C4—23no—
336C2×C3×C4—23no—
337C2×C3×C4—23no—
338C2×C3×C4—23no—
339C2×C3×C4—23no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—23no—
348C2×C3×C4—23no—
349C2×C3×C4—23no—
350C2×C3×C4—23no—
351C2×C3×C4—23no—
352C2×C3×C4—23no—
353C2×C3×C4—21no—
354C2×C3×C4—412no—
355C2×C3×C4C2×#4116yes—
356C2×C3×C4C2×#4116yes—
357C2×C3×C4C2×#4116yes—
358C2×C3×C4H(5,𝔽2)12yes—
359C2×C3×C42−1+412yes—
360C24C24×C313yes—
361#41—416no—
362C2×Q8C3×#4114yes—
363C2×Q8C3×#4114yes—
364C2×Q8C3×#4114yes—
365C2×D4—26no—
366C2×D4—26no—
367C2×D4—26no—
368C22×C4—26no—
369C22×C4—26no—
370C22×C4—26no—
371C22×C4C24×C311yescharacteristic
372C22×C4C24×C311yescharacteristic
373C22×C4C24×C311yescharacteristic
374C22×C4C24×C311yescharacteristic
375C22×C4C24×C311yescharacteristic
376C22×C4C24×C313yes—
377C22×C4C24×C313yes—
378C22×C4C24×C313yes—
379C22×C4C24×C313yes—
380C22×C4C24×C313yes—
381C22×C4C24×C313yes—
382C22×C4C24×C311yescharacteristic
383C22×C4C24×C311yescharacteristic
384C22×C4C24×C313yes—
385C4⋊C4—26no—
386C4⋊C4—26no—
387C4⋊C4—26no—
388C4⋊C4—26no—
389C4⋊C4—26no—
390C4⋊C4—26no—
391C4⋊C4—26no—
392C4⋊C4—26no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—46no—
396C4⋊C4—43no—
397C4⋊C4—43no—
398C4⋊C4—46no—
399C22⋊C4—26no—
400C22⋊C4—26no—
401C22⋊C4—46no—
402C22⋊C4—46no—
403C22⋊C4—43no—
404C22⋊C4—43no—
405C22⋊C4—43no—
406C22⋊C4—43no—
407C22⋊C4—46no—
408C42—26no—
409C42—26no—
410C42—26no—
411C42—26no—
412C42—26no—
413C42—26no—
414C42—46no—
415C22×C3—43no—
416C22×C3—43no—
417C22×C3—43no—
418C22×C3—43no—
419C22×C3—43no—
420C22×C3—43no—
421C22×C3—43no—
422C22×C3C22×#4111yescharacteristic
423C22×C3C22×#4111yescharacteristic
424C22×C3C22×#4111yescharacteristic
425C22×C3C2×H(5,𝔽2)11yescharacteristic
426C22×C3C2×2−1+411yescharacteristic
427C22×C3#58511yescharacteristic
428C22×C3#58511yescharacteristic
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—43no—
434C3×C4—43no—
435C3×C4—43no—
436C3×C4—43no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—43no—
440C3×C4—41no—
441C3×C4—43no—
442C3×C4C2×#126112yes—
443C23—23no—
444C23—23no—
445C23—23no—
446C23—23no—
447C23—23no—
448C23—23no—
449C23—23no—
450C23C25×C311yescharacteristic
451Q8C3×#126116yescomplemented
452D4—412no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—23no—
487C2×C4—23no—
488C2×C4—23no—
489C2×C4—21no—
490C2×C4—412no—
491C2×C4C2×C3×#4116yes—
492C2×C4C2×C3×#4116yes—
493C2×C4C2×C3×#4116yes—
494C2×C4C3×H(5,𝔽2)12yes—
495C2×C4C3×2−1+412yes—
496C2×C3—83no—
497C2×C3C22×#12611yescharacteristic
498C2×C3#351611yescharacteristic
499C2×C3#351711yescharacteristic
500C2×C3#351711yescharacteristic
501C2×C3#351811yescharacteristic
502C2×C3#352211yescharacteristic
503C2×C3#352411yescharacteristic
504C22—43no—
505C22—43no—
506C22—43no—
507C22—43no—
508C22—43no—
509C22—43no—
510C22—43no—
511C22C22×C3×#4111yescharacteristic
512C22C22×C3×#4111yescharacteristic
513C22C22×C3×#4111yescharacteristic
514C22C2×C3×H(5,𝔽2)11yescharacteristic
515C22C2×C3×2−1+411yescharacteristic
516C22C3×#58511yescharacteristic
517C22C3×#58511yescharacteristic
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—43no—
523C4—43no—
524C4—43no—
525C4—43no—
526C4—41no—
527C4—41no—
528C4—43no—
529C4—41no—
530C4—43no—
531C4C2×C3×#126112yes—
532C3#3589811yescharacteristic, direct product factor, Sylow
533C2—83no—
534C2C22×C3×#12611yescharacteristic
535C2C3×#351611yescharacteristic
536C2C3×#351711yescharacteristic
537C2C3×#351711yescharacteristic
538C2C3×#351811yescharacteristic
539C2C3×#352211yescharacteristic
540C2C3×#352411yescharacteristic
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2644C2semidirect
2. C3×#2513C2semidirect
3. C3×#2512C2semidirect
4. C3×#2512C2semidirect
5. C3×#2497C2semidirect
6. C3×#2496C2semidirect
7. C3×#2494C2semidirect
8. C3×#2401C2semidirect
9. C3×#2351C2semidirect
10. C3×Q8×C4⋊C4C2semidirect
11. C3×Q8×C4⋊C4C2semidirect
15. C3×#2276C2semidirect
16. #35898C3direct (N × H)
67. #2644C2×C3semidirect
68. #2513C2×C3semidirect
69. #2512C2×C3semidirect
70. #2512C2×C3semidirect
71. #2497C2×C3semidirect
72. #2496C2×C3semidirect
73. #2494C2×C3semidirect
74. #2401C2×C3semidirect
75. #2351C2×C3semidirect
76. Q8×C4⋊C4C2×C3semidirect
77. Q8×C4⋊C4C2×C3semidirect
81. #2276C2×C3semidirect
315. C3×Q8#126semidirect
451. Q8C3×#126semidirect
532. C3#35898direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26442
C3×#25132
C3×#25122
C3×#24972
C3×#24962
C3×#24942
C3×#24012
C3×#23512
C3×Q8×C4⋊C42
C3×#23412
C3×#22762
#358983