Finite Group Databasebeta

← C3×#27616 #10646223

Subgroups — C3×#27616 #10646223

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C3×#2283C211yesmaximal, characteristic
3C3×#2101C212yesmaximal, complemented
4C3×#2085C212yesmaximal, complemented
5C3×#1921C212yesmaximal, complemented
6C3×#1920C212yesmaximal, complemented
7C3×#1853C211yesmaximal, characteristic
8C3×#1852C211yesmaximal, characteristic
9C3×#1823C211yesmaximal, characteristic
10C3×#1823C211yesmaximal, characteristic
11C3×#1817C211yesmaximal, characteristic
12#27616C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C22⋊C8C2211yescharacteristic
20C2×C3×C22⋊C8C2211yescharacteristic
21C3×#389—21no—
22C3×#389—21no—
23C3×#389C2212yes—
24C3×#389C2212yes—
25C3×#383—21no—
26C3×#382C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C3×#381—21no—
29C3×#380C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C3×#376—21no—
35C2×C3×#95—21no—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#389C2×C311yescharacteristic
45#2283C2×C311yescharacteristic
46#2101C2×C312yescomplemented
47#2085C2×C312yescomplemented
48#1921C2×C312yescomplemented
49#1920C2×C312yescomplemented
50#1853C2×C311yescharacteristic
51#1852C2×C311yescharacteristic
52#1823C2×C311yescharacteristic
53#1823C2×C311yescharacteristic
54#1817C2×C311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—44no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42—21no—
81C2×C3×C42—21no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#106—24no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8—22no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—21no—
93C3×#95—21no—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C42C22×C311yescharacteristic
104C2×#107C22×C312yes—
105C2×#106C22×C312yes—
106C2×C22⋊C8C22×C311yescharacteristic
107C2×C22⋊C8C22×C311yescharacteristic
108#389—21no—
109#389—21no—
110#389C22×C312yes—
111#389C22×C312yes—
112#383—21no—
113#382C22×C311yescharacteristic
114#382C22×C311yescharacteristic
115#381—21no—
116#380C22×C311yescharacteristic
117#380C22×C311yescharacteristic
118C4×C4⋊C4C22×C311yescharacteristic
119C4×C4⋊C4C22×C311yescharacteristic
120C4×C22⋊C4—21no—
121#376—21no—
122C2×#95—21no—
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127#340C22×C312yes—
128#340C22×C312yes—
129#336C22×C312yes—
130#336C22×C312yes—
131C24×C3C2×D411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—41no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4#4114yes—
178C3×C22⋊C4—42no—
179C3×C22⋊C4—42no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—42no—
187C3×C42—42no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C3×D411yescharacteristic
191C23×C4C3×D411yescharacteristic
192C23×C4C23×C311yescharacteristic
193C22×C8C23×C312yes—
194C22×C8C23×C312yes—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C23×C312yes—
204C2×C4⋊C4C23×C311yescharacteristic
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C311yescharacteristic
208C2×C4⋊C4C23×C312yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C3×D411yescharacteristic
216C2×C42C3×D411yescharacteristic
217C2×C42C23×C311yescharacteristic
218C2×C42C23×C311yescharacteristic
219#107—24no—
220#106—24no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3#12312yes—
238C23×C3#13512yes—
239C23×C3AGL(1,ℤ/8)12yes—
240C23×C3C22×D411yescharacteristic
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—44no—
258C2×C3×C4—48no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—84no—
278C2×C3×C4#12312yes—
279C2×C3×C4#13712yes—
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C42−1+412yes—
284C3×C8—42no—
285C3×C8—42no—
286C24C2×C3×D411yescharacteristic
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C2×C3×D411yescharacteristic
308C22×C4C2×C3×D411yescharacteristic
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C3×#4112yes—
313C22×C4C3×#4112yes—
314C22×C4C24×C311yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4C3×#4114yes—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C42—22no—
336C42—22no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—42no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3#48214yes—
350C22×C3C2×#12311yescharacteristic
351C22×C3#52511yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3C2×#13511yescharacteristic
354C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C3C2×#13711yescharacteristic
356C22×C3#57511yescharacteristic
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—44no—
363C3×C4—82no—
364C3×C4—81no—
365C3×C4—81no—
366C3×C4—81no—
367C3×C4—81no—
368C3×C4#55212yes—
369C3×C4#55212yes—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23C3×#12312yes—
375C23C3×#13512yes—
376C23C3×AGL(1,ℤ/8)12yes—
377C23C22×C3×D411yescharacteristic
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—44no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—84no—
415C2×C4C3×#12312yes—
416C2×C4C3×#13712yes—
417C2×C4C2×C3×#4111yescharacteristic
418C2×C4C2×C3×#4111yescharacteristic
419C2×C4C2×C3×#4112yes—
420C2×C4C3×2−1+412yes—
421C8—42no—
422C8—42no—
423C2×C3—24no—
424C2×C3#248411yescharacteristic
425C2×C3C2×#48211yescharacteristic
426C2×C3#309511yescharacteristic
427C2×C3#310811yescharacteristic
428C2×C3#311411yescharacteristic
429C2×C3#323211yescharacteristic
430C2×C3#323811yescharacteristic
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22C3×#48214yes—
438C22C2×C3×#12311yescharacteristic
439C22C3×#52511yescharacteristic
440C22C3×#53611yescharacteristic
441C22C2×C3×#13511yescharacteristic
442C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
443C22C2×C3×#13711yescharacteristic
444C22C3×#57511yescharacteristic
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—41no—
450C4—44no—
451C4—81no—
452C4—81no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4C3×#55212yes—
457C4C3×#55212yes—
458C3#2761611yescharacteristic, direct product factor, Sylow
459C2—24no—
460C2C3×#248411yescharacteristic
461C2C2×C3×#48211yescharacteristic
462C2C3×#309511yescharacteristic
463C2C3×#310811yescharacteristic
464C2C3×#311411yescharacteristic
465C2C3×#323211yescharacteristic
466C2C3×#323811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2101C2semidirect
4. C3×#2085C2semidirect
5. C3×#1921C2semidirect
6. C3×#1920C2semidirect
12. #27616C3direct (N × H)
46. #2101C2×C3semidirect
47. #2085C2×C3semidirect
48. #1921C2×C3semidirect
49. #1920C2×C3semidirect
458. C3#27616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C3×#22832
C3×#21012
C3×#20852
C3×#19212
C3×#19202
C3×#18532
C3×#18522
C3×#18232
C3×#18172
#276163