Finite Group Databasebeta

← C3×#32969 #10651576

Subgroups — C3×#32969 #10651576

Order 768.

1350 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#3079C212yesmaximal, complemented
3C3×#2542C211yesmaximal, characteristic, complemented
4C3×#1859C214yesmaximal, complemented
5C3×#1856C212yesmaximal, complemented
6C3×#1806C212yesmaximal, complemented
7C3×#1800C211yesmaximal, characteristic, complemented
8C3×#1792C212yesmaximal
9#32969C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic, complemented
12C2×C3×C22⋊Q8—22no—
13C3×#520—22no—
14C3×#520C2212yes—
15C3×#518—22no—
16C3×#518—22no—
17C3×#518C2211yescharacteristic
18C3×C4×#41—22no—
19C3×C4×#41C2212yes—
20C2×C3×C4×D4—22no—
21C2×C3×C4×D4—22no—
22C2×C3×C4×D4C2211yescharacteristic
23C3×#477—22no—
24C3×#476—22no—
25C3×#475—22no—
26C3×#474—22no—
27C3×#428C2214yescomplemented
28C3×#423C2212yescomplemented
29C2×C3×#104C2214yes—
30C3×#417C2214yes—
31C2×C3×#100C2212yes—
32C3×#399—22no—
33C3×#390C2212yes—
34C3×#384C2212yescomplemented
35C3×C4×C4⋊C4C2211yescharacteristic, complemented
36C3×#343C2212yescomplemented
37C3×#339C2214yescomplemented
38C4×H(5,𝔽2)C2×C311yescharacteristic
39#3079C2×C312yescomplemented
40#2542C2×C311yescharacteristic, complemented
41#1859C2×C314yescomplemented
42#1856C2×C312yescomplemented
43#1806C2×C312yescomplemented
44#1800C2×C311yescharacteristic, complemented
45#1792C2×C312yes—
46C3×H(5,𝔽2)—24no—
47C3×H(5,𝔽2)—22no—
48C3×H(5,𝔽2)Q814yescomplemented
49C2×C3×#41—22no—
50C2×C3×#41C2312yes—
51C22×C3×Q8—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4D412yes—
57C2×C3×M16C2314yes—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C22⋊Q8—48no—
61C3×C4×Q8—22no—
62C3×C4×Q8D412yes—
63C3×C4×Q8D412yes—
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C3×C4×D4—24no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—24no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4D412yes—
77C3×C4×D4D412yes—
78C3×#117—24no—
79C3×#117—22no—
80C3×#117—22no—
81C3×#117C2312yes—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—24no—
93C2×C3×C22⋊C4D412yes—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42C2312yes—
97C2×C3×C42C2311yescharacteristic
98C3×#107—24no—
99C3×#106—24no—
100C3×C4⋊C8—22no—
101C3×C4⋊C8—22no—
102C3×#104—28no—
103C3×#103—24no—
104C3×#102—24no—
105C3×#100—24no—
106C3×#95—21no—
107C2×H(5,𝔽2)C22×C311yescharacteristic
108C2×C4⋊Q8C22×C311yescharacteristic, complemented
109C2×C22⋊Q8—22no—
110#520—22no—
111#520C22×C312yes—
112#518—22no—
113#518—22no—
114#518C22×C311yescharacteristic
115C4×#41—22no—
116C4×#41C22×C312yes—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4C22×C311yescharacteristic
120#477—22no—
121#476—22no—
122#475—22no—
123#474—22no—
124#428C22×C314yescomplemented
125#423C22×C312yescomplemented
126C2×#104C22×C314yes—
127#417C22×C314yes—
128C2×#100C22×C312yes—
129#399—22no—
130#390C22×C312yes—
131#384C22×C312yescomplemented
132C4×C4⋊C4C22×C311yescharacteristic, complemented
133#343C22×C312yescomplemented
134#339C22×C314yescomplemented
135C24×C3—22no—
136C24×C3C2×D412yes—
137C3×#41—28no—
138C3×#41—28no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41C2×Q814yes—
142C3×#41#4114yes—
143C2×C3×Q8—44no—
144C2×C3×Q8—41no—
145C2×C3×Q8—41no—
146C2×C3×Q8C2×D412yes—
147C2×C3×D4—24no—
148C2×C3×D4—28no—
149C2×C3×D4—28no—
150C2×C3×D4—28no—
151C2×C3×D4—22no—
152C2×C3×D4—28no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4C2×D412yes—
156C2×C3×D4C2×Q812yes—
157C2×C3×D4#4112yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—44no—
170C22×C3×C4—84no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×M16—24no—
176C3×M16—24no—
177C2×C3×C8—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—48no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—48no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—24no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202H(5,𝔽2)—24no—
203H(5,𝔽2)—22no—
204H(5,𝔽2)C3×Q814yescomplemented
205C2×#41—22no—
206C2×#41C23×C312yes—
207C22×Q8—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C23×C311yescharacteristic
211C23×C4—22no—
212C23×C4C3×D412yes—
213C2×M16C23×C314yes—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C22⋊Q8—48no—
217C4×Q8—22no—
218C4×Q8C3×D412yes—
219C4×Q8C3×D412yes—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—24no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C3×D412yes—
233C4×D4C3×D412yes—
234#117—24no—
235#117—22no—
236#117—22no—
237#117C23×C312yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C23×C312yes—
244C2×C4⋊C4C23×C311yescharacteristic
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4C3×D412yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42C23×C312yes—
253C2×C42C23×C311yescharacteristic
254#107—24no—
255#106—24no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258#104—28no—
259#103—24no—
260#102—24no—
261#100—24no—
262#95—21no—
263C23×C3—28no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—82no—
270C23×C3—82no—
271C23×C3C22⋊Q814yes—
272C23×C3C22×D411yescharacteristic
273C3×Q8—22no—
274C3×Q8—82no—
275C3×Q8—82no—
276C3×Q8C22⋊Q814yescomplemented
277C3×D4—216no—
278C3×D4—24no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—22no—
282C3×D4—24no—
283C3×D4C22⋊Q814yescomplemented
284C2×C3×C4—28no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—42no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—82no—
302C2×C3×C4—82no—
303C2×C3×C4—82no—
304C2×C3×C4—84no—
305C2×C3×C4—82no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12011yescharacteristic
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4C22×D412yes—
310C2×C3×C4C22×Q811yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C4C2×#4111yescharacteristic
313C3×C8—44no—
314C24—22no—
315C24C2×C3×D412yes—
316#41—28no—
317#41—28no—
318#41—22no—
319#41—22no—
320#41C2×C3×Q814yes—
321#41C3×#4114yes—
322C2×Q8—44no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8C2×C3×D412yes—
326C2×D4—24no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—22no—
331C2×D4—28no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4C2×C3×D412yes—
335C2×D4C2×C3×Q812yes—
336C2×D4C3×#4112yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—44no—
349C22×C4—84no—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C24×C311yescharacteristic
354M16—24no—
355M16—24no—
356C2×C8—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4C2×C3×D412yes—
366C4⋊C4C2×C3×D412yes—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—24no—
373C22⋊C4—48no—
374C42—24no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—24no—
379C42C2×C3×D412yes—
380C42C2×C3×D412yes—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—44no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—82no—
391C22×C3—88no—
392C22×C3C2×#12011yescharacteristic
393C22×C3C2×C22⋊Q811yescharacteristic
394C22×C3#52511yescharacteristic
395C3×C4—24no—
396C3×C4—21no—
397C3×C4—22no—
398C3×C4—44no—
399C3×C4—42no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—82no—
403C3×C4#45312yes—
404C3×C4C2×C22⋊Q812yes—
405C3×C4#52512yes—
406C23—28no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—42no—
411C23—42no—
412C23—82no—
413C23—82no—
414C23C3×C22⋊Q814yes—
415C23C22×C3×D411yescharacteristic
416Q8—22no—
417Q8—82no—
418Q8—82no—
419Q8C3×C22⋊Q814yescomplemented
420D4—216no—
421D4—24no—
422D4—24no—
423D4—24no—
424D4—22no—
425D4—24no—
426D4C3×C22⋊Q814yescomplemented
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4C3×#12012yes—
450C2×C4C3×#12011yescharacteristic
451C2×C4C3×#12011yescharacteristic
452C2×C4C22×C3×D412yes—
453C2×C4C22×C3×Q811yescharacteristic
454C2×C4C2×C3×#4112yes—
455C2×C4C2×C3×#4111yescharacteristic
456C8—44no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—44no—
460C2×C3—44no—
461C2×C3#284811yescharacteristic
462C2×C3C2×#45311yescharacteristic
463C2×C3#302211yescharacteristic
464C22—21no—
465C22—21no—
466C22—24no—
467C22—24no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—82no—
474C22—88no—
475C22C2×C3×#12011yescharacteristic
476C22C2×C3×C22⋊Q811yescharacteristic
477C22C3×#52511yescharacteristic
478C4—21no—
479C4—22no—
480C4—24no—
481C4—44no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—82no—
486C4C3×#45312yes—
487C4C2×C3×C22⋊Q812yes—
488C4C3×#52512yes—
489C3#3296911yescharacteristic, direct product factor, Sylow
490C2—21no—
491C2—21no—
492C2—44no—
493C2—44no—
494C2C3×#284811yescharacteristic
495C2C2×C3×#45311yescharacteristic
496C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3079C2semidirect
3. C3×#2542C2semidirect
4. C3×#1859C2semidirect
5. C3×#1856C2semidirect
6. C3×#1806C2semidirect
7. C3×#1800C2semidirect
9. #32969C3direct (N × H)
11. C2×C3×C4⋊Q8C22semidirect
27. C3×#428C22semidirect
28. C3×#423C22semidirect
34. C3×#384C22semidirect
35. C3×C4×C4⋊C4C22semidirect
36. C3×#343C22semidirect
37. C3×#339C22semidirect
39. #3079C2×C3semidirect
40. #2542C2×C3semidirect
41. #1859C2×C3semidirect
42. #1856C2×C3semidirect
43. #1806C2×C3semidirect
44. #1800C2×C3semidirect
48. C3×H(5,𝔽2)Q8semidirect
108. C2×C4⋊Q8C22×C3semidirect
124. #428C22×C3semidirect
125. #423C22×C3semidirect
131. #384C22×C3semidirect
132. C4×C4⋊C4C22×C3semidirect
133. #343C22×C3semidirect
134. #339C22×C3semidirect
204. H(5,𝔽2)C3×Q8semidirect
276. C3×Q8C22⋊Q8semidirect
283. C3×D4C22⋊Q8semidirect
419. Q8C3×C22⋊Q8semidirect
426. D4C3×C22⋊Q8semidirect
489. C3#32969direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C3×#30792
C3×#25422
C3×#18592
C3×#18562
C3×#18062
C3×#18002
C3×#17922
#329693