Finite Group Databasebeta

← C3×#33767 #10652374

Subgroups — C3×#33767 #10652374

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3061C211yesmaximal, characteristic
2C3×#3059C211yesmaximal, characteristic, complemented
3C3×#2237C211yesmaximal, characteristic, complemented
4C3×#2236C211yesmaximal, characteristic, complemented
5C3×#2235C211yesmaximal, characteristic, complemented
6C3×#2234C211yesmaximal, characteristic, complemented
7C3×#2220C211yesmaximal, characteristic, complemented
8C3×#2219C211yesmaximal, characteristic, complemented
9C3×#2217C211yesmaximal, characteristic, complemented
10C3×#2216C211yesmaximal, characteristic, complemented
11C3×#2189C211yesmaximal, characteristic, complemented
12C3×#2189C211yesmaximal, characteristic, complemented
13C3×#2188C211yesmaximal, characteristic, complemented
14C3×#2185C211yesmaximal, characteristic, complemented
15C3×#2113C211yesmaximal, characteristic, complemented
16#33767C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#135C2211yescharacteristic
18C22×C3×Q16C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#537—21no—
21C3×#536—21no—
22C2×C3×Q32C2211yescharacteristic
23C2×C3×Q32C2211yescharacteristic
24C2×C3×SD32C2211yescharacteristic
25C2×C3×SD32C2211yescharacteristic
26C2×C3×M32C2211yescharacteristic
27C22×C3×C16C2211yescharacteristic
28C3×#470—21no—
29C3×#470—21no—
30C3×#468—21no—
31C3×#468—21no—
32C3×#467C2211yescharacteristic
33C3×#467C2211yescharacteristic
34C3×#467C2211yescharacteristic
35C3×#467C2211yescharacteristic
36C3×#467C2211yescharacteristic
37C3×#467C2211yescharacteristic
38C3×#466C2211yescharacteristic
39C3×#466C2211yescharacteristic
40C3×#427C2211yescharacteristic
41C2×C3×#107C2211yescharacteristic
42C3×#419—21no—
43C3×#417—21no—
44C3×#367C2211yescharacteristic
45C3×#367C2211yescharacteristic
46C3×#366C2211yescharacteristic
47C3×#366C2211yescharacteristic
48C3×#358C2211yescharacteristic
49C3×#358C2211yescharacteristic
50C3×#358C2211yescharacteristic
51C3×#358C2211yescharacteristic
52C3×#358C2211yescharacteristic
53C3×#358C2211yescharacteristic
54C3×#357C2211yescharacteristic
55C3×#357C2211yescharacteristic
56C3×C22⋊C16C2211yescharacteristic
57C3×C22⋊C16C2211yescharacteristic
58C3×C22⋊C16C2211yescharacteristic
59C3×C22⋊C16C2211yescharacteristic
60#3061C2×C311yescharacteristic
61#3059C2×C311yescharacteristic, complemented
62#2237C2×C311yescharacteristic, complemented
63#2236C2×C311yescharacteristic, complemented
64#2235C2×C311yescharacteristic, complemented
65#2234C2×C311yescharacteristic, complemented
66#2220C2×C311yescharacteristic, complemented
67#2219C2×C311yescharacteristic, complemented
68#2217C2×C311yescharacteristic, complemented
69#2216C2×C311yescharacteristic, complemented
70#2189C2×C311yescharacteristic, complemented
71#2189C2×C311yescharacteristic, complemented
72#2188C2×C311yescharacteristic, complemented
73#2185C2×C311yescharacteristic, complemented
74#2113C2×C311yescharacteristic, complemented
75C2×C3×#41—21no—
76C2×C3×#41D411yescharacteristic
77C22×C3×Q8—21no—
78C3×#135—41no—
79C3×#135—41no—
80C2×C3×Q16—21no—
81C2×C3×Q16—41no—
82C2×C3×Q16—41no—
83C2×C3×Q16C2311yescharacteristic
84C2×C3×Q16C2311yescharacteristic
85C2×C3×Q16C2311yescharacteristic
86C2×C3×SD16—21no—
87C2×C3×D8C2311yescharacteristic
88C3×#131—41no—
89C3×#131—41no—
90C2×C3×M16D411yescharacteristic
91C2×C3×M16D411yescharacteristic
92C2×C3×M16D411yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2311yescharacteristic
95C22×C3×C8C2311yescharacteristic
96C3×C4⋊Q8—41no—
97C3×#124—41no—
98C3×#123—41no—
99C3×#123—41no—
100C3×C22⋊Q8—21no—
101C3×C22⋊Q8—21no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8—41no—
107C3×#121—21no—
108C3×#121—21no—
109C3×#121—41no—
110C3×#121—41no—
111C3×#121—41no—
112C3×#117—21no—
113C2×C3×C4⋊C4—21no—
114C3×Q32—41no—
115C3×Q32—41no—
116C3×SD32—41no—
117C3×SD32—41no—
118C3×M32—41no—
119C2×C3×C16—41no—
120C2×C3×C16C2311yescharacteristic
121C2×C3×C16C2311yescharacteristic
122C2×C3×C16C2311yescharacteristic
123C2×C3×C16C2311yescharacteristic
124C3×#107—21no—
125C3×#107C2311yescharacteristic
126C3×#107C2311yescharacteristic
127C3×#107C2311yescharacteristic
128C3×#107C2311yescharacteristic
129C3×#106—21no—
130C3×#103—21no—
131C3×#103—21no—
132C3×#103—21no—
133C3×#103—21no—
134C3×#103—21no—
135C3×#103—21no—
136C3×#102—21no—
137C3×#102—21no—
138C2×#135C22×C311yescharacteristic
139C22×Q16C22×C311yescharacteristic
140C2×#131C22×C311yescharacteristic
141#537—21no—
142#536—21no—
143C2×Q32C22×C311yescharacteristic
144C2×Q32C22×C311yescharacteristic
145C2×SD32C22×C311yescharacteristic
146C2×SD32C22×C311yescharacteristic
147C2×M32C22×C311yescharacteristic
148C22×C16C22×C311yescharacteristic
149#470—21no—
150#470—21no—
151#468—21no—
152#468—21no—
153#467C22×C311yescharacteristic
154#467C22×C311yescharacteristic
155#467C22×C311yescharacteristic
156#467C22×C311yescharacteristic
157#467C22×C311yescharacteristic
158#467C22×C311yescharacteristic
159#466C22×C311yescharacteristic
160#466C22×C311yescharacteristic
161#427C22×C311yescharacteristic
162C2×#107C22×C311yescharacteristic
163#419—21no—
164#417—21no—
165#367C22×C311yescharacteristic
166#367C22×C311yescharacteristic
167#366C22×C311yescharacteristic
168#366C22×C311yescharacteristic
169#358C22×C311yescharacteristic
170#358C22×C311yescharacteristic
171#358C22×C311yescharacteristic
172#358C22×C311yescharacteristic
173#358C22×C311yescharacteristic
174#358C22×C311yescharacteristic
175#357C22×C311yescharacteristic
176#357C22×C311yescharacteristic
177C22⋊C16C22×C311yescharacteristic
178C22⋊C16C22×C311yescharacteristic
179C22⋊C16C22×C311yescharacteristic
180C22⋊C16C22×C311yescharacteristic
181C3×#41—41no—
182C3×#41—41no—
183C3×#41—42no—
184C3×#41—42no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—42no—
189C2×C3×Q8—42no—
190C2×C3×Q8—41no—
191C2×C3×Q8D811yescharacteristic
192C2×C3×D4—21no—
193C2×C3×D4—41no—
194C2×C3×D4—41no—
195C2×C3×D4—41no—
196C2×C3×D4D811yescharacteristic
197C2×C3×D4D811yescharacteristic
198C2×C3×D4D811yescharacteristic
199C22×C3×C4—41no—
200C22×C3×C4—41no—
201C22×C3×C4—41no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C3×Q16—21no—
206C3×Q16—21no—
207C3×Q16—21no—
208C3×Q16—21no—
209C3×Q16—21no—
210C3×Q16—21no—
211C3×Q16—42no—
212C3×SD16—42no—
213C3×D8—21no—
214C3×D8—21no—
215C3×M16—41no—
216C3×M16—41no—
217C3×M16—41no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8C2×D411yescharacteristic
225C2×C3×C8C2×D411yescharacteristic
226C2×C3×C8C2×D411yescharacteristic
227C2×C3×C8C2411yescharacteristic
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C22⋊C4—41no—
240C3×C22⋊C4—41no—
241C3×C22⋊C4—41no—
242C3×C22⋊C4—41no—
243C3×C22⋊C4—41no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C42—41no—
249C3×C16—21no—
250C3×C16—21no—
251C3×C16—21no—
252C3×C16—21no—
253C2×#41—21no—
254C2×#41C3×D411yescharacteristic
255C22×Q8—21no—
256#135—41no—
257#135—41no—
258C2×Q16—21no—
259C2×Q16—41no—
260C2×Q16—41no—
261C2×Q16C23×C311yescharacteristic
262C2×Q16C23×C311yescharacteristic
263C2×Q16C23×C311yescharacteristic
264C2×SD16—21no—
265C2×D8C23×C311yescharacteristic
266#131—41no—
267#131—41no—
268C2×M16C3×D411yescharacteristic
269C2×M16C3×D411yescharacteristic
270C2×M16C3×D411yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C4⋊Q8—41no—
275#124—41no—
276#123—41no—
277#123—41no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—41no—
285#121—21no—
286#121—21no—
287#121—41no—
288#121—41no—
289#121—41no—
290#117—21no—
291C2×C4⋊C4—21no—
292Q32—41no—
293Q32—41no—
294SD32—41no—
295SD32—41no—
296M32—41no—
297C2×C16—41no—
298C2×C16C23×C311yescharacteristic
299C2×C16C23×C311yescharacteristic
300C2×C16C23×C311yescharacteristic
301C2×C16C23×C311yescharacteristic
302#107—21no—
303#107C23×C311yescharacteristic
304#107C23×C311yescharacteristic
305#107C23×C311yescharacteristic
306#107C23×C311yescharacteristic
307#106—21no—
308#103—21no—
309#103—21no—
310#103—21no—
311#103—21no—
312#103—21no—
313#103—21no—
314#102—21no—
315#102—21no—
316C23×C3—41no—
317C23×C3C2×D811yescharacteristic
318C23×C3C2×D811yescharacteristic
319C23×C3C2×D811yescharacteristic
320C3×Q8—41no—
321C3×Q8—41no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—82no—
328C3×D4—41no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—82no—
334C3×D4—161no—
335C3×D4—161no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—82no—
350C2×C3×C4—82no—
351C2×C3×C4—161no—
352C2×C3×C4C2×D811yescharacteristic
353C2×C3×C4C2×D811yescharacteristic
354C2×C3×C4C2×D811yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C3×C8—21no—
357C3×C8—21no—
358C3×C8—21no—
359C3×C8H(5,𝔽2)11yescharacteristic
360C3×C8H(5,𝔽2)11yescharacteristic
361#41—41no—
362#41—41no—
363#41—42no—
364#41—42no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—41no—
371C2×Q8C3×D811yescharacteristic
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4C3×D811yescharacteristic
377C2×D4C3×D811yescharacteristic
378C2×D4C3×D811yescharacteristic
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385Q16—21no—
386Q16—21no—
387Q16—21no—
388Q16—21no—
389Q16—21no—
390Q16—21no—
391Q16—42no—
392SD16—42no—
393D8—21no—
394D8—21no—
395M16—41no—
396M16—41no—
397M16—41no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8C2×C3×D411yescharacteristic
405C2×C8C2×C3×D411yescharacteristic
406C2×C8C2×C3×D411yescharacteristic
407C2×C8C24×C311yescharacteristic
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C42—41no—
429C16—21no—
430C16—21no—
431C16—21no—
432C16—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3C22×D811yescharacteristic
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—81no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4—81no—
453C3×C4—81no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4#53411yescharacteristic
457C3×C4#57511yescharacteristic
458C23—41no—
459C23C2×C3×D811yescharacteristic
460C23C2×C3×D811yescharacteristic
461C23C2×C3×D811yescharacteristic
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—82no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—161no—
477D4—161no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—161no—
494C2×C4C2×C3×D811yescharacteristic
495C2×C4C2×C3×D811yescharacteristic
496C2×C4C2×C3×D811yescharacteristic
497C2×C4C22×C3×D411yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8C3×H(5,𝔽2)11yescharacteristic
502C8C3×H(5,𝔽2)11yescharacteristic
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—161no—
507C2×C3#318611yescharacteristic
508C2×C3#341611yescharacteristic
509C2×C3#341711yescharacteristic
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C22×C3×D811yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4C3×#53411yescharacteristic
534C4C3×#57511yescharacteristic
535C3#3376711yescharacteristic, direct product factor, Sylow
536C2—41no—
537C2—41no—
538C2—41no—
539C2—161no—
540C2C3×#318611yescharacteristic
541C2C3×#341611yescharacteristic
542C2C3×#341711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC3×#3061C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3059C2semidirect
3. C3×#2237C2semidirect
4. C3×#2236C2semidirect
5. C3×#2235C2semidirect
6. C3×#2234C2semidirect
7. C3×#2220C2semidirect
8. C3×#2219C2semidirect
9. C3×#2217C2semidirect
10. C3×#2216C2semidirect
11. C3×#2189C2semidirect
12. C3×#2189C2semidirect
13. C3×#2188C2semidirect
14. C3×#2185C2semidirect
15. C3×#2113C2semidirect
16. #33767C3direct (N × H)
61. #3059C2×C3semidirect
62. #2237C2×C3semidirect
63. #2236C2×C3semidirect
64. #2235C2×C3semidirect
65. #2234C2×C3semidirect
66. #2220C2×C3semidirect
67. #2219C2×C3semidirect
68. #2217C2×C3semidirect
69. #2216C2×C3semidirect
70. #2189C2×C3semidirect
71. #2189C2×C3semidirect
72. #2188C2×C3semidirect
73. #2185C2×C3semidirect
74. #2113C2×C3semidirect
535. C3#33767direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30612
C3×#30592
C3×#22372
C3×#22362
C3×#22352
C3×#22342
C3×#22202
C3×#22192
C3×#22172
C3×#22162
C3×#21892
C3×#21882
C3×#21852
C3×#21132
#337673