Finite Group Databasebeta

← C3×#60492 #10679099

Subgroups — C3×#60492 #10679099

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3458C211yesmaximal, characteristic, complemented
2C2×C3×#520C211yesmaximal, characteristic
3C3×#3084C218yesmaximal, complemented
4C3×#3083C218yesmaximal, complemented
5C3×#3078C214yesmaximal, complemented
6C3×#3077C214yesmaximal, complemented
7C2×C3×#428C212yesmaximal, complemented
8C3×#2905C211yesmaximal, characteristic, complemented
9C3×#2896C212yesmaximal
10#60492C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×M16C2212yes—
13C3×#544—21no—
14C3×#544—21no—
15C3×#544C2214yescomplemented
16C3×#544C2214yescomplemented
17C3×#543—22no—
18C3×#533—24no—
19C3×#533C2214yes—
20C3×#520—24no—
21C3×#520C2218yes—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×D4—22no—
24C2×C3×C4×D4C2212yes—
25C2×C3×#117—21no—
26C2×C3×#117C2212yes—
27C2×C3×#117C2211yescharacteristic
28C22×C3×C4⋊C4—21no—
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#479C22116yescomplemented
31C3×#478C22116yescomplemented
32C3×#477C2218yescomplemented
33C3×#476C2218yescomplemented
34C3×#475C2218yescomplemented
35C3×#474C2218yescomplemented
36C3×#428C2218yes—
37C3×#428C2218yescomplemented
38C2×C3×#107C2214yescomplemented
39C2×C3×#106C2214yescomplemented
40C3×#424C2218yescomplemented
41C3×#423C2214yes—
42C3×#419C2214yes—
43C3×#418C2214yes—
44C3×#417C2214yes—
45C3×#417C2214yes—
46C3×#416C2218yes—
47#3458C2×C311yescharacteristic, complemented
48C2×#520C2×C311yescharacteristic
49#3084C2×C318yescomplemented
50#3083C2×C318yescomplemented
51#3078C2×C314yescomplemented
52#3077C2×C314yescomplemented
53C2×#428C2×C312yescomplemented
54#2905C2×C311yescharacteristic, complemented
55#2896C2×C312yes—
56C2×C3×#41—22no—
57C2×C3×#41—24no—
58C2×C3×#41Q818yescomplemented
59C2×C3×#41C2314yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4—22no—
66C23×C3×C4C2311yescharacteristic
67C2×C3×M16—28no—
68C2×C3×M16C2314yes—
69C2×C3×M16C2314yes—
70C22×C3×C8C2314yes—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8C2314yes—
74C3×C4⋊Q8C2314yes—
75C3×#125—24no—
76C3×#125—24no—
77C3×#125C2318yes—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—28no—
81C3×C22⋊Q8—28no—
82C3×C4×Q8—24no—
83C3×C4×Q8—24no—
84C3×C4×Q8—24no—
85C3×C4×Q8C2318yes—
86C3×C4×D4—216no—
87C3×C4×D4—24no—
88C3×C4×D4C2318yes—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117—28no—
92C3×#117D414yes—
93C3×#117C2314yes—
94C3×#117C2314yes—
95C3×#117C2314yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—28no—
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4D411yescharacteristic
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4D412yes—
109C2×C3×C42—22no—
110C2×C3×C42—22no—
111C2×C3×C42C2312yes—
112C3×#107C23116yes—
113C3×#106C23116yes—
114C3×C4⋊C8C2318yes—
115C3×C4⋊C8C2318yes—
116C3×#103C2318yes—
117C3×#103C2318yes—
118C3×#102C2318yes—
119C3×#102C2318yes—
120C22×#41C22×C311yescharacteristic
121C22×M16C22×C312yes—
122#544—21no—
123#544—21no—
124#544C22×C314yescomplemented
125#544C22×C314yescomplemented
126#543—22no—
127#533—24no—
128#533C22×C314yes—
129#520—24no—
130#520C22×C318yes—
131C2×C4×Q8C22×C312yes—
132C2×C4×D4—22no—
133C2×C4×D4C22×C312yes—
134C2×#117—21no—
135C2×#117C22×C312yes—
136C2×#117C22×C311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4C22×C311yescharacteristic
139#479C22×C3116yescomplemented
140#478C22×C3116yescomplemented
141#477C22×C318yescomplemented
142#476C22×C318yescomplemented
143#475C22×C318yescomplemented
144#474C22×C318yescomplemented
145#428C22×C318yes—
146#428C22×C318yescomplemented
147C2×#107C22×C314yescomplemented
148C2×#106C22×C314yescomplemented
149#424C22×C318yescomplemented
150#423C22×C314yes—
151#419C22×C314yes—
152#418C22×C314yes—
153#417C22×C314yes—
154#417C22×C314yes—
155#416C22×C318yes—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C3×#41—216no—
159C3×#41—28no—
160C3×#41C2×Q8116yes—
161C2×C3×Q8—21no—
162C2×C3×Q8—24no—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8C2×Q814yes—
166C2×C3×Q8C2412yes—
167C2×C3×D4—28no—
168C2×C3×D4—24no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4C2×Q814yes—
172C2×C3×D4C2412yes—
173C22×C3×C4—21no—
174C22×C3×C4—28no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—28no—
181C22×C3×C4—84no—
182C22×C3×C4—82no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×Q814yes—
188C22×C3×C4#4114yes—
189C22×C3×C4C2411yescharacteristic
190C22×C3×C4C2412yes—
191C3×M16—28no—
192C3×M16—28no—
193C2×C3×C8—28no—
194C2×C3×C8C2418yes—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4C2×D414yes—
205C3×C4⋊C4C2×D414yes—
206C3×C4⋊C4C2414yes—
207C3×C4⋊C4C2414yes—
208C3×C4⋊C4C2414yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4C2×D418yes—
213C3×C42—28no—
214C3×C42—24no—
215C3×C42—24no—
216C3×C42C2×D414yes—
217C3×C42C2414yes—
218C2×#41—22no—
219C2×#41—24no—
220C2×#41C3×Q818yescomplemented
221C2×#41C23×C314yes—
222C22×Q8C23×C311yescharacteristic
223C22×D4—21no—
224C22×D4C23×C311yescharacteristic
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4C23×C311yescharacteristic
229C2×M16—28no—
230C2×M16C23×C314yes—
231C2×M16C23×C314yes—
232C22×C8C23×C314yes—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8C23×C314yes—
236C4⋊Q8C23×C314yes—
237#125—24no—
238#125—24no—
239#125C23×C318yes—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—28no—
243C22⋊Q8—28no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8C23×C318yes—
248C4×D4—216no—
249C4×D4—24no—
250C4×D4C23×C318yes—
251#117—22no—
252#117—22no—
253#117—28no—
254#117C3×D414yes—
255#117C23×C314yes—
256#117C23×C314yes—
257#117C23×C314yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4C3×D411yescharacteristic
263C2×C4⋊C4C3×D411yescharacteristic
264C2×C4⋊C4C23×C312yes—
265C2×C4⋊C4C23×C314yes—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C3×D412yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C23×C312yes—
274#107C23×C3116yes—
275#106C23×C3116yes—
276C4⋊C8C23×C318yes—
277C4⋊C8C23×C318yes—
278#103C23×C318yes—
279#103C23×C318yes—
280#102C23×C318yes—
281#102C23×C318yes—
282C23×C3—24no—
283C23×C3—22no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3C22⋊Q814yes—
289C23×C3C22×D411yescharacteristic
290C23×C3C22×D412yes—
291C3×Q8—24no—
292C3×Q8—22no—
293C3×Q8—84no—
294C3×Q8—84no—
295C3×Q8C22×Q814yes—
296C3×D4—216no—
297C3×D4—24no—
298C3×D4—22no—
299C3×D4C22×Q814yes—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—216no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—44no—
309C2×C3×C4—88no—
310C2×C3×C4—82no—
311C2×C3×C4—82no—
312C2×C3×C4—84no—
313C2×C3×C4C22⋊Q814yes—
314C2×C3×C4C22⋊Q818yes—
315C2×C3×C4C22×D414yes—
316C2×C3×C4C22×D414yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D412yes—
319C2×C3×C4C22×Q812yes—
320C2×C3×C4C22×Q814yes—
321C2×C3×C4C2×#4114yes—
322C2×C3×C4C2×#4112yes—
323C2×C3×C4C2511yescharacteristic
324C3×C8—28no—
325C24—21no—
326C24C2×C3×D411yescharacteristic
327#41—216no—
328#41—28no—
329#41C2×C3×Q8116yes—
330C2×Q8—21no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8C2×C3×Q814yes—
335C2×Q8C24×C312yes—
336C2×D4—28no—
337C2×D4—24no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4C2×C3×Q814yes—
341C2×D4C24×C312yes—
342C22×C4—21no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—84no—
351C22×C4—82no—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×D412yes—
354C22×C4C2×C3×D412yes—
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×Q814yes—
357C22×C4C3×#4114yes—
358C22×C4C24×C311yescharacteristic
359C22×C4C24×C312yes—
360M16—28no—
361M16—28no—
362C2×C8—28no—
363C2×C8C24×C318yes—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4C2×C3×D414yes—
374C4⋊C4C2×C3×D414yes—
375C4⋊C4C24×C314yes—
376C4⋊C4C24×C314yes—
377C4⋊C4C24×C314yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—28no—
381C22⋊C4C2×C3×D418yes—
382C42—28no—
383C42—24no—
384C42—24no—
385C42C2×C3×D414yes—
386C42C24×C314yes—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—22no—
391C22×C3—24no—
392C22×C3—44no—
393C22×C3—48no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3C2×C22⋊Q812yes—
397C22×C3C2×C22⋊Q814yes—
398C22×C3C23×D411yescharacteristic
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—24no—
402C3×C4—44no—
403C3×C4—44no—
404C3×C4—44no—
405C3×C4C2×C22⋊Q814yes—
406C3×C4C2×C22⋊Q812yes—
407C3×C4C23×Q811yescharacteristic
408C3×C4C22×#4111yescharacteristic
409C23—22no—
410C23—24no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23C3×C22⋊Q814yes—
416C23C22×C3×D411yescharacteristic
417C23C22×C3×D412yes—
418Q8—22no—
419Q8—24no—
420Q8—84no—
421Q8—84no—
422Q8C22×C3×Q814yes—
423D4—216no—
424D4—22no—
425D4—24no—
426D4C22×C3×Q814yes—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—216no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—88no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4C3×C22⋊Q814yes—
441C2×C4C3×C22⋊Q818yes—
442C2×C4C22×C3×D414yes—
443C2×C4C22×C3×D414yes—
444C2×C4C22×C3×D411yescharacteristic
445C2×C4C22×C3×D412yes—
446C2×C4C22×C3×Q812yes—
447C2×C4C22×C3×Q814yes—
448C2×C4C2×C3×#4114yes—
449C2×C4C2×C3×#4112yes—
450C2×C4C25×C311yescharacteristic
451C8—28no—
452C2×C3—22no—
453C2×C3—24no—
454C2×C3—44no—
455C2×C3C22×C22⋊Q811yescharacteristic
456C2×C3#358511yescharacteristic
457C2×C3#358611yescharacteristic
458C22—21no—
459C22—22no—
460C22—21no—
461C22—22no—
462C22—24no—
463C22—44no—
464C22—48no—
465C22—42no—
466C22—42no—
467C22C2×C3×C22⋊Q812yes—
468C22C2×C3×C22⋊Q814yes—
469C22C23×C3×D411yescharacteristic
470C4—24no—
471C4—24no—
472C4—24no—
473C4—44no—
474C4—44no—
475C4—44no—
476C4C2×C3×C22⋊Q814yes—
477C4C2×C3×C22⋊Q812yes—
478C4C23×C3×Q811yescharacteristic
479C4C22×C3×#4111yescharacteristic
480C3#6049211yescharacteristic, direct product factor, Sylow
481C2—22no—
482C2—24no—
483C2—44no—
484C2C22×C3×C22⋊Q811yescharacteristic
485C2C3×#358511yescharacteristic
486C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3458C2semidirect
3. C3×#3084C2semidirect
4. C3×#3083C2semidirect
5. C3×#3078C2semidirect
6. C3×#3077C2semidirect
7. C2×C3×#428C2semidirect
8. C3×#2905C2semidirect
10. #60492C3direct (N × H)
15. C3×#544C22semidirect
16. C3×#544C22semidirect
30. C3×#479C22semidirect
31. C3×#478C22semidirect
32. C3×#477C22semidirect
33. C3×#476C22semidirect
34. C3×#475C22semidirect
35. C3×#474C22semidirect
37. C3×#428C22semidirect
38. C2×C3×#107C22semidirect
39. C2×C3×#106C22semidirect
40. C3×#424C22semidirect
47. #3458C2×C3semidirect
49. #3084C2×C3semidirect
50. #3083C2×C3semidirect
51. #3078C2×C3semidirect
52. #3077C2×C3semidirect
53. C2×#428C2×C3semidirect
54. #2905C2×C3semidirect
58. C2×C3×#41Q8semidirect
124. #544C22×C3semidirect
125. #544C22×C3semidirect
139. #479C22×C3semidirect
140. #478C22×C3semidirect
141. #477C22×C3semidirect
142. #476C22×C3semidirect
143. #475C22×C3semidirect
144. #474C22×C3semidirect
146. #428C22×C3semidirect
147. C2×#107C22×C3semidirect
148. C2×#106C22×C3semidirect
149. #424C22×C3semidirect
220. C2×#41C3×Q8semidirect
480. C3#60492direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34582
C2×C3×#5202
C3×#30842
C3×#30832
C3×#30782
C3×#30772
C2×C3×#4282
C3×#29052
C3×#28962
#604923