Finite Group Databasebeta

← C3×#33744 #10652351

Subgroups — C3×#33744 #10652351

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3176C211yesmaximal, characteristic, complemented
2C3×#3175C211yesmaximal, characteristic, complemented
3C3×#2255C211yesmaximal, characteristic, complemented
4C3×#2250C211yesmaximal, characteristic, complemented
5C3×#2242C211yesmaximal, characteristic, complemented
6C3×#2240C211yesmaximal, characteristic, complemented
7C3×#2220C211yesmaximal, characteristic, complemented
8C3×#2218C211yesmaximal, characteristic, complemented
9C3×#2217C211yesmaximal, characteristic, complemented
10C3×#2215C211yesmaximal, characteristic, complemented
11C3×#2177C211yesmaximal, characteristic, complemented
12C3×#2175C211yesmaximal, characteristic, complemented
13C3×C4×SD32C211yesmaximal, characteristic, complemented
14C3×C4×D16C211yesmaximal, characteristic, complemented
15C3×#2108C211yesmaximal, characteristic, complemented
16#33744C311yesmaximal, characteristic, direct product factor, Sylow
17C3×#542—21no—
18C3×#540—21no—
19C2×C3×SD32C2211yescharacteristic
20C2×C3×SD32C2211yescharacteristic
21C2×C3×D16C2211yescharacteristic
22C2×C3×D16C2211yescharacteristic
23C2×C3×M32C2211yescharacteristic
24C22×C3×C16C2211yescharacteristic
25C3×#500C2211yescharacteristic
26C3×#499C2211yescharacteristic
27C3×#495C2211yescharacteristic
28C3×#493C2211yescharacteristic
29C3×#490—21no—
30C3×#489—21no—
31C3×#469—21no—
32C3×#468—21no—
33C3×#467C2211yescharacteristic
34C3×#466C2211yescharacteristic
35C3×#466C2211yescharacteristic
36C3×#466C2211yescharacteristic
37C3×#442—21no—
38C3×#440—21no—
39C3×C4×Q16C2211yescharacteristic
40C3×C4×D8C2211yescharacteristic
41C3×C4×D8C2211yescharacteristic
42C3×C4×D8C2211yescharacteristic
43C3×#405C2211yescharacteristic
44C3×#367C2211yescharacteristic
45C3×#367C2211yescharacteristic
46C3×#366C2211yescharacteristic
47C3×#366C2211yescharacteristic
48C3×#358C2211yescharacteristic
49C3×#358C2211yescharacteristic
50C3×#357C2211yescharacteristic
51C3×#357C2211yescharacteristic
52C3×#357C2211yescharacteristic
53C3×#357C2211yescharacteristic
54C3×#357C2211yescharacteristic
55C3×#357C2211yescharacteristic
56C3×#346C2211yescharacteristic
57C3×#346C2211yescharacteristic
58C3×C4×C16C2211yescharacteristic
59C3×C4×C16C2211yescharacteristic
60#3176C2×C311yescharacteristic, complemented
61#3175C2×C311yescharacteristic, complemented
62#2255C2×C311yescharacteristic, complemented
63#2250C2×C311yescharacteristic, complemented
64#2242C2×C311yescharacteristic, complemented
65#2240C2×C311yescharacteristic, complemented
66#2220C2×C311yescharacteristic, complemented
67#2218C2×C311yescharacteristic, complemented
68#2217C2×C311yescharacteristic, complemented
69#2215C2×C311yescharacteristic, complemented
70#2177C2×C311yescharacteristic, complemented
71#2175C2×C311yescharacteristic, complemented
72C4×SD32C2×C311yescharacteristic, complemented
73C4×D16C2×C311yescharacteristic, complemented
74#2108C2×C311yescharacteristic, complemented
75C2×C3×Q16C2311yescharacteristic
76C2×C3×SD16—21no—
77C2×C3×D8—21no—
78C2×C3×D8C2311yescharacteristic
79C2×C3×D8C2311yescharacteristic
80C2×C3×D8C2311yescharacteristic
81C2×C3×M16D411yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—21no—
84C3×#127—21no—
85C3×#126—41no—
86C3×#125—21no—
87C3×#124—21no—
88C3×#124—41no—
89C3×#123—41no—
90C3×#123—41no—
91C3×#123—41no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—41no—
94C3×#121—21no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—41no—
98C3×#121—41no—
99C3×C4×Q8—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×#117D411yescharacteristic
104C3×SD32—22no—
105C3×SD32—22no—
106C3×D16—22no—
107C3×D16—22no—
108C3×M32—22no—
109C2×C3×C16—22no—
110C2×C3×C16C2311yescharacteristic
111C2×C3×C16C2311yescharacteristic
112C2×C3×C16C2311yescharacteristic
113C2×C3×C16C2311yescharacteristic
114C3×#107—21no—
115C3×#107C2311yescharacteristic
116C3×#107C2311yescharacteristic
117C3×#107C2311yescharacteristic
118C3×#107C2311yescharacteristic
119C3×#106—21no—
120C3×#103—21no—
121C3×#103—21no—
122C3×#102—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×#97D411yescharacteristic
129C3×#97D411yescharacteristic
130C3×C4×C8C2311yescharacteristic
131C3×C4×C8C2311yescharacteristic
132#542—21no—
133#540—21no—
134C2×SD32C22×C311yescharacteristic
135C2×SD32C22×C311yescharacteristic
136C2×D16C22×C311yescharacteristic
137C2×D16C22×C311yescharacteristic
138C2×M32C22×C311yescharacteristic
139C22×C16C22×C311yescharacteristic
140#500C22×C311yescharacteristic
141#499C22×C311yescharacteristic
142#495C22×C311yescharacteristic
143#493C22×C311yescharacteristic
144#490—21no—
145#489—21no—
146#469—21no—
147#468—21no—
148#467C22×C311yescharacteristic
149#466C22×C311yescharacteristic
150#466C22×C311yescharacteristic
151#466C22×C311yescharacteristic
152#442—21no—
153#440—21no—
154C4×Q16C22×C311yescharacteristic
155C4×D8C22×C311yescharacteristic
156C4×D8C22×C311yescharacteristic
157C4×D8C22×C311yescharacteristic
158#405C22×C311yescharacteristic
159#367C22×C311yescharacteristic
160#367C22×C311yescharacteristic
161#366C22×C311yescharacteristic
162#366C22×C311yescharacteristic
163#358C22×C311yescharacteristic
164#358C22×C311yescharacteristic
165#357C22×C311yescharacteristic
166#357C22×C311yescharacteristic
167#357C22×C311yescharacteristic
168#357C22×C311yescharacteristic
169#357C22×C311yescharacteristic
170#357C22×C311yescharacteristic
171#346C22×C311yescharacteristic
172#346C22×C311yescharacteristic
173C4×C16C22×C311yescharacteristic
174C4×C16C22×C311yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—41no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—41no—
181C2×C3×D4—41no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C2×C3×D4—41no—
185C22×C3×C4—41no—
186C22×C3×C4—41no—
187C22×C3×C4—41no—
188C22×C3×C4C2×D411yescharacteristic
189C3×Q16—21no—
190C3×Q16—21no—
191C3×SD16—42no—
192C3×D8—21no—
193C3×D8—21no—
194C3×D8—21no—
195C3×D8—21no—
196C3×D8—21no—
197C3×D8—21no—
198C3×D8—42no—
199C3×M16—22no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8C2×D411yescharacteristic
203C2×C3×C8C2×D411yescharacteristic
204C2×C3×C8C2×D411yescharacteristic
205C2×C3×C8C2411yescharacteristic
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4D811yescharacteristic
216C3×C4⋊C4D811yescharacteristic
217C3×C22⋊C4—41no—
218C3×C22⋊C4—41no—
219C3×C22⋊C4—41no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—41no—
222C3×C22⋊C4—41no—
223C3×C22⋊C4—41no—
224C3×C22⋊C4—41no—
225C3×C22⋊C4—41no—
226C3×C22⋊C4D811yescharacteristic
227C3×C22⋊C4D811yescharacteristic
228C3×C42—41no—
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C16—21no—
232C3×C16—21no—
233C3×C16#4112yes—
234C3×C16#4112yes—
235C2×Q16C23×C311yescharacteristic
236C2×SD16—21no—
237C2×D8—21no—
238C2×D8C23×C311yescharacteristic
239C2×D8C23×C311yescharacteristic
240C2×D8C23×C311yescharacteristic
241C2×M16C3×D411yescharacteristic
242C22×C8C23×C311yescharacteristic
243C4⋊Q8—21no—
244#127—21no—
245#126—41no—
246#125—21no—
247#124—21no—
248#124—41no—
249#123—41no—
250#123—41no—
251#123—41no—
252C22⋊Q8—21no—
253C22⋊Q8—41no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—41no—
258#121—41no—
259C4×Q8—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263#117C3×D411yescharacteristic
264SD32—22no—
265SD32—22no—
266D16—22no—
267D16—22no—
268M32—22no—
269C2×C16—22no—
270C2×C16C23×C311yescharacteristic
271C2×C16C23×C311yescharacteristic
272C2×C16C23×C311yescharacteristic
273C2×C16C23×C311yescharacteristic
274#107—21no—
275#107C23×C311yescharacteristic
276#107C23×C311yescharacteristic
277#107C23×C311yescharacteristic
278#107C23×C311yescharacteristic
279#106—21no—
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#97C3×D411yescharacteristic
289#97C3×D411yescharacteristic
290C4×C8C23×C311yescharacteristic
291C4×C8C23×C311yescharacteristic
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3C2×D811yescharacteristic
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—161no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—41no—
303C3×D4—41no—
304C3×D4—41no—
305C3×D4—161no—
306C3×D4—161no—
307C3×D4—161no—
308C3×D4—161no—
309C3×D4—161no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—161no—
318C2×C3×C4—161no—
319C2×C3×C4—161no—
320C2×C3×C4C2×D811yescharacteristic
321C2×C3×C4C2×D811yescharacteristic
322C2×C3×C4C2×D811yescharacteristic
323C2×C3×C4C2×D811yescharacteristic
324C2×C3×C4C2×D811yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C3×C8—21no—
327C3×C8—21no—
328C3×C8#13512yes—
329C3×C8C2×#4111yescharacteristic
330C3×C8C2×#4111yescharacteristic
331C2×Q8—21no—
332C2×Q8—41no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C2×C3×D411yescharacteristic
345Q16—21no—
346Q16—21no—
347SD16—42no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—21no—
352D8—21no—
353D8—21no—
354D8—42no—
355M16—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×C3×D411yescharacteristic
359C2×C8C2×C3×D411yescharacteristic
360C2×C8C2×C3×D411yescharacteristic
361C2×C8C24×C311yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4C3×D811yescharacteristic
372C4⋊C4C3×D811yescharacteristic
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4C3×D811yescharacteristic
383C22⋊C4C3×D811yescharacteristic
384C42—41no—
385C42C2×C3×D411yescharacteristic
386C42C2×C3×D411yescharacteristic
387C16—21no—
388C16—21no—
389C16C3×#4112yes—
390C16C3×#4112yes—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3C22×D811yescharacteristic
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4#53211yescharacteristic
415C3×C4C2×#13511yescharacteristic
416C23—41no—
417C23—41no—
418C23—41no—
419C23C2×C3×D811yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—161no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—161no—
430D4—161no—
431D4—161no—
432D4—161no—
433D4—161no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—161no—
442C2×C4—161no—
443C2×C4—161no—
444C2×C4C2×C3×D811yescharacteristic
445C2×C4C2×C3×D811yescharacteristic
446C2×C4C2×C3×D811yescharacteristic
447C2×C4C2×C3×D811yescharacteristic
448C2×C4C2×C3×D811yescharacteristic
449C2×C4C22×C3×D411yescharacteristic
450C8—21no—
451C8—21no—
452C8C3×#13512yes—
453C8C2×C3×#4111yescharacteristic
454C8C2×C3×#4111yescharacteristic
455C2×C3—41no—
456C2×C3—161no—
457C2×C3—161no—
458C2×C3—161no—
459C2×C3#316911yescharacteristic
460C2×C3#341511yescharacteristic
461C2×C3#341611yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22C22×C3×D811yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4C3×#53211yescharacteristic
486C4C2×C3×#13511yescharacteristic
487C3#3374411yescharacteristic, direct product factor, Sylow
488C2—41no—
489C2—161no—
490C2—161no—
491C2—161no—
492C2C3×#316911yescharacteristic
493C2C3×#341511yescharacteristic
494C2C3×#341611yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3176C2semidirect
2. C3×#3175C2semidirect
3. C3×#2255C2semidirect
4. C3×#2250C2semidirect
5. C3×#2242C2semidirect
6. C3×#2240C2semidirect
7. C3×#2220C2semidirect
8. C3×#2218C2semidirect
9. C3×#2217C2semidirect
10. C3×#2215C2semidirect
11. C3×#2177C2semidirect
12. C3×#2175C2semidirect
13. C3×C4×SD32C2semidirect
14. C3×C4×D16C2semidirect
15. C3×#2108C2semidirect
16. #33744C3direct (N × H)
60. #3176C2×C3semidirect
61. #3175C2×C3semidirect
62. #2255C2×C3semidirect
63. #2250C2×C3semidirect
64. #2242C2×C3semidirect
65. #2240C2×C3semidirect
66. #2220C2×C3semidirect
67. #2218C2×C3semidirect
68. #2217C2×C3semidirect
69. #2215C2×C3semidirect
70. #2177C2×C3semidirect
71. #2175C2×C3semidirect
72. C4×SD32C2×C3semidirect
73. C4×D16C2×C3semidirect
74. #2108C2×C3semidirect
487. C3#33744direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31762
C3×#31752
C3×#22552
C3×#22502
C3×#22422
C3×#22402
C3×#22202
C3×#22182
C3×#22172
C3×#22152
C3×#21772
C3×#21752
C3×C4×SD322
C3×C4×D162
C3×#21082
#337443