Finite Group Databasebeta

← C3×#23557 #10642164

Subgroups — C3×#23557 #10642164

Order 768.

1110 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic
3C2×C3×#427C211yesmaximal, characteristic
4C3×#2316C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2059C212yesmaximal, complemented
7C3×#2058C212yesmaximal, complemented
8C3×#2057C212yesmaximal, complemented
9C3×#1736C211yesmaximal, characteristic
10C2×C3×#340C211yesmaximal, characteristic
11C3×#1729C211yesmaximal, characteristic
12#23557C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C3×#544—24no—
16C3×#544—24no—
17C2×C3×C4⋊Q8—21no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×#125—21no—
20C2×C3×#125C2212yes—
21C2×C3×C22⋊Q8—22no—
22C2×C3×#117C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#428—24no—
28C3×#427—24no—
29C2×C3×#107C2211yescharacteristic
30C2×C3×#107C2211yescharacteristic
31C2×C3×#107C2212yes—
32C2×C3×#106C2211yescharacteristic
33C2×C3×#106C2211yescharacteristic
34C2×C3×#106C2212yes—
35C3×#389—21no—
36C3×#389—21no—
37C3×#389C2212yes—
38C3×#389C2212yes—
39C3×#380—22no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#544C2×C311yescharacteristic
47C2×#428C2×C311yescharacteristic
48C2×#427C2×C311yescharacteristic
49#2316C2×C311yescharacteristic
50#2059C2×C312yescomplemented
51#2059C2×C312yescomplemented
52#2058C2×C312yescomplemented
53#2057C2×C312yescomplemented
54#1736C2×C311yescharacteristic
55C2×#340C2×C311yescharacteristic
56#1729C2×C311yescharacteristic
57C22×C3×Q8—22no—
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C2×C3×M16—22no—
62C2×C3×M16—22no—
63C2×C3×M16Q814yescomplemented
64C22×C3×C8—22no—
65C22×C3×C8—22no—
66C22×C3×C8Q814yescomplemented
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—28no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×#125—28no—
74C3×#125—24no—
75C3×#125—24no—
76C3×C22⋊Q8—216no—
77C3×#117—22no—
78C3×#117—22no—
79C3×#117—22no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117D414yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D414yes—
95C2×C3×C4⋊C4Q814yescomplemented
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4D412yes—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42C2312yes—
108C3×#107—22no—
109C3×#107—22no—
110C3×#107—24no—
111C3×#106—22no—
112C3×#106—22no—
113C3×#106—24no—
114C3×#95—22no—
115C3×#95—22no—
116C22×M16C22×C311yescharacteristic
117C23×C8C22×C311yescharacteristic
118#544—24no—
119#544—24no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8C22×C312yes—
122C2×#125—21no—
123C2×#125C22×C312yes—
124C2×C22⋊Q8—22no—
125C2×#117C22×C311yescharacteristic
126C2×#117C22×C311yescharacteristic
127C2×#117C22×C311yescharacteristic
128C22×C4⋊C4C22×C311yescharacteristic
129C22×C4⋊C4C22×C311yescharacteristic
130#428—24no—
131#427—24no—
132C2×#107C22×C311yescharacteristic
133C2×#107C22×C312yes—
134C2×#107C22×C311yescharacteristic
135C2×#106C22×C311yescharacteristic
136C2×#106C22×C312yes—
137C2×#106C22×C311yescharacteristic
138#389—21no—
139#389—21no—
140#389C22×C312yes—
141#389C22×C312yes—
142#380—22no—
143#340C22×C312yes—
144#340C22×C312yes—
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C312yes—
148#336C22×C312yes—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—28no—
151C2×C3×Q8—82no—
152C2×C3×Q8—82no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—81no—
162C22×C3×C4—81no—
163C22×C3×C4—81no—
164C22×C3×C4—81no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×Q812yes—
171C22×C3×C4C2×Q812yes—
172C22×C3×C4C2×Q812yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C3×M16—28no—
176C2×C3×C8—28no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8C2×Q812yes—
184C2×C3×C8C2×Q812yes—
185C2×C3×C8C2×Q814yes—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—82no—
202C3×C4⋊C4—82no—
203C3×C4⋊C4C2×D414yes—
204C3×C4⋊C4C2×D412yes—
205C3×C4⋊C4C2×D412yes—
206C3×C4⋊C4C2×Q814yes—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—42no—
210C3×C42—24no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—42no—
214C22×Q8—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C23×C311yescharacteristic
218C2×M16—22no—
219C2×M16—22no—
220C2×M16C3×Q814yescomplemented
221C22×C8—22no—
222C22×C8—22no—
223C22×C8C3×Q814yescomplemented
224C22×C8C23×C311yescharacteristic
225C22×C8C23×C311yescharacteristic
226C22×C8C23×C312yes—
227C4⋊Q8—28no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230#125—28no—
231#125—24no—
232#125—24no—
233C22⋊Q8—216no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117C3×D414yes—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4C3×D411yescharacteristic
250C2×C4⋊C4C3×D411yescharacteristic
251C2×C4⋊C4C3×D414yes—
252C2×C4⋊C4C3×Q814yescomplemented
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C312yes—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C3×D412yes—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C23×C312yes—
265#107—22no—
266#107—22no—
267#107—24no—
268#106—22no—
269#106—22no—
270#106—24no—
271#95—22no—
272#95—22no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3#12012yes—
278C23×C3C22⋊Q812yes—
279C23×C3C4⋊Q812yes—
280C23×C3C22×D411yescharacteristic
281C3×Q8—88no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—84no—
307C2×C3×C4—84no—
308C2×C3×C4#12012yes—
309C2×C3×C4C22⋊Q814yes—
310C2×C3×C4C22⋊Q814yes—
311C2×C3×C4C22⋊Q812yes—
312C2×C3×C4C4⋊Q814yes—
313C2×C3×C4C4⋊Q814yes—
314C2×C3×C4C4⋊Q812yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×Q811yescharacteristic
318C2×C3×C4C22×Q811yescharacteristic
319C2×C3×C4C22×Q811yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C3×C8—22no—
322C3×C8—22no—
323C3×C8—24no—
324C24C2×C3×D411yescharacteristic
325C2×Q8—28no—
326C2×Q8—82no—
327C2×Q8—82no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D412yes—
344C22×C4C2×C3×D412yes—
345C22×C4C2×C3×Q812yes—
346C22×C4C2×C3×Q812yes—
347C22×C4C2×C3×Q812yes—
348C22×C4C3×#4112yes—
349C22×C4C24×C311yescharacteristic
350M16—28no—
351C2×C8—28no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8C2×C3×Q812yes—
359C2×C8C2×C3×Q812yes—
360C2×C8C2×C3×Q814yes—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4C2×C3×D414yes—
379C4⋊C4C2×C3×D412yes—
380C4⋊C4C2×C3×D412yes—
381C4⋊C4C2×C3×Q814yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C42—24no—
386C42—22no—
387C42—22no—
388C42—42no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3#39514yes—
396C22×C3C2×#12011yescharacteristic
397C22×C3C2×C22⋊Q811yescharacteristic
398C22×C3C2×C4⋊Q811yescharacteristic
399C22×C3#57611yescharacteristic
400C22×C3#57711yescharacteristic
401C22×C3#57711yescharacteristic
402C22×C3#57811yescharacteristic
403C3×C4—44no—
404C3×C4—42no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—81no—
410C3×C4—81no—
411C3×C4#39514yes—
412C3×C4C2×C22⋊Q811yescharacteristic
413C3×C4C2×C22⋊Q811yescharacteristic
414C3×C4C2×C4⋊Q811yescharacteristic
415C3×C4C2×C4⋊Q811yescharacteristic
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23C3×#12012yes—
421C23C3×C22⋊Q812yes—
422C23C3×C4⋊Q812yes—
423C23C22×C3×D411yescharacteristic
424Q8—88no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4C3×#12012yes—
452C2×C4C3×C22⋊Q812yes—
453C2×C4C3×C22⋊Q814yes—
454C2×C4C3×C22⋊Q814yes—
455C2×C4C3×C4⋊Q812yes—
456C2×C4C3×C4⋊Q814yes—
457C2×C4C3×C4⋊Q814yes—
458C2×C4C22×C3×D411yescharacteristic
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C22×C3×Q811yescharacteristic
461C2×C4C22×C3×Q811yescharacteristic
462C2×C4C22×C3×Q811yescharacteristic
463C2×C4C2×C3×#4111yescharacteristic
464C8—22no—
465C8—22no—
466C8—24no—
467C2×C3—24no—
468C2×C3C2×#39511yescharacteristic
469C2×C3#301211yescharacteristic
470C2×C3#301311yescharacteristic
471C2×C3#308111yescharacteristic
472C2×C3#308211yescharacteristic
473C2×C3#316311yescharacteristic
474C2×C3#316411yescharacteristic
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22C3×#39514yes—
482C22C2×C3×#12011yescharacteristic
483C22C2×C3×C22⋊Q811yescharacteristic
484C22C2×C3×C4⋊Q811yescharacteristic
485C22C3×#57611yescharacteristic
486C22C3×#57711yescharacteristic
487C22C3×#57711yescharacteristic
488C22C3×#57811yescharacteristic
489C4—42no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—44no—
495C4—81no—
496C4—81no—
497C4C3×#39514yes—
498C4C2×C3×C22⋊Q811yescharacteristic
499C4C2×C3×C22⋊Q811yescharacteristic
500C4C2×C3×C4⋊Q811yescharacteristic
501C4C2×C3×C4⋊Q811yescharacteristic
502C3#2355711yescharacteristic, direct product factor, Sylow
503C2—24no—
504C2C2×C3×#39511yescharacteristic
505C2C3×#301211yescharacteristic
506C2C3×#301311yescharacteristic
507C2C3×#308111yescharacteristic
508C2C3×#308211yescharacteristic
509C2C3×#316311yescharacteristic
510C2C3×#316411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#2059C2semidirect
6. C3×#2059C2semidirect
7. C3×#2058C2semidirect
8. C3×#2057C2semidirect
12. #23557C3direct (N × H)
50. #2059C2×C3semidirect
51. #2059C2×C3semidirect
52. #2058C2×C3semidirect
53. #2057C2×C3semidirect
63. C2×C3×M16Q8semidirect
66. C22×C3×C8Q8semidirect
95. C2×C3×C4⋊C4Q8semidirect
220. C2×M16C3×Q8semidirect
223. C22×C8C3×Q8semidirect
252. C2×C4⋊C4C3×Q8semidirect
502. C3#23557direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#4282
C2×C3×#4272
C3×#23162
C3×#20592
C3×#20582
C3×#20572
C3×#17362
C2×C3×#3402
C3×#17292
#235573