Finite Group Databasebeta

← C3×#27615 #10646222

Subgroups — C3×#27615 #10646222

Order 768.

744 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2296C211yesmaximal, characteristic
2C3×#2285C211yesmaximal, characteristic
3C3×#2099C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2084C211yesmaximal, characteristic, complemented
6C3×#2083C211yesmaximal, characteristic, complemented
7C3×#1921C212yesmaximal, complemented
8C3×#1919C212yesmaximal, complemented
9C3×#1854C212yesmaximal
10C3×#1825C212yesmaximal
11C3×#1817C211yesmaximal, characteristic
12#27615C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#117—21no—
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C22⋊C8C2212yes—
20C3×#389C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#385—21no—
25C3×#385—21no—
26C3×#384C2211yescharacteristic
27C3×#384C2211yescharacteristic
28C3×#382—21no—
29C3×#380—21no—
30C3×#380C2212yes—
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4—21no—
33C3×C4×C4⋊C4—21no—
34C3×C4×C4⋊C4C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2296C2×C311yescharacteristic
45#2285C2×C311yescharacteristic
46#2099C2×C311yescharacteristic, complemented
47#2099C2×C311yescharacteristic, complemented
48#2084C2×C311yescharacteristic, complemented
49#2083C2×C311yescharacteristic, complemented
50#1921C2×C312yescomplemented
51#1919C2×C312yescomplemented
52#1854C2×C312yes—
53#1825C2×C312yes—
54#1817C2×C311yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×#117—22no—
61C3×#117—22no—
62C3×#117—24no—
63C3×#117—44no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—21no—
82C2×C3×C42—24no—
83C2×C3×C42—21no—
84C2×C3×C42—22no—
85C2×C3×C42D411yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×#106—24no—
91C3×C22⋊C8—24no—
92C3×#95—22no—
93C3×#95—21no—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—22no—
98C2×#117—21no—
99C2×#117C22×C311yescharacteristic
100C2×#117C22×C311yescharacteristic
101C22×C42C22×C311yescharacteristic
102C2×#107C22×C312yes—
103C2×#106C22×C312yes—
104C2×C22⋊C8C22×C312yes—
105#389C22×C311yescharacteristic
106#389C22×C311yescharacteristic
107#389C22×C311yescharacteristic
108#389C22×C311yescharacteristic
109#385—21no—
110#385—21no—
111#384C22×C311yescharacteristic
112#384C22×C311yescharacteristic
113#382—21no—
114#380—21no—
115#380C22×C312yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C22×C312yes—
120#340C22×C312yes—
121#340C22×C312yes—
122#340C22×C312yes—
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#336C22×C311yescharacteristic
127#336C22×C311yescharacteristic
128#336C22×C312yes—
129C24×C3C2×D411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—41no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4114yes—
151C22×C3×C4C2411yescharacteristic
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4#4112yes—
174C3×C4⋊C4#4112yes—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—42no—
178C3×C42—24no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—44no—
184C3×C42—42no—
185C23×C4C3×D411yescharacteristic
186C23×C4C3×D411yescharacteristic
187C23×C4C23×C311yescharacteristic
188C22×C8C23×C312yes—
189C22×C8C23×C312yes—
190#117—22no—
191#117—22no—
192#117—24no—
193#117—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C311yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42C3×D411yescharacteristic
216C2×C42C3×D411yescharacteristic
217C2×C42C23×C311yescharacteristic
218C2×C42C23×C311yescharacteristic
219#107—24no—
220#106—24no—
221C22⋊C8—24no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—22no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3#12312yes—
235C23×C3C22×D411yescharacteristic
236C2×C3×C4—28no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
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—22no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—44no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4#12312yes—
270C2×C3×C4#13512yes—
271C2×C3×C4AGL(1,ℤ/8)12yes—
272C2×C3×C4#13712yes—
273C2×C3×C4C2×#4112yes—
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C42−1+412yes—
277C3×C8—42no—
278C3×C8—42no—
279C24C2×C3×D411yescharacteristic
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C3×#4114yes—
301C22×C4C24×C311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4C3×#4112yes—
324C4⋊C4C3×#4112yes—
325C22⋊C4—24no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—44no—
334C42—42no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3C2×#12311yescharacteristic
343C22×C3#52511yescharacteristic
344C22×C3#53611yescharacteristic
345C22×C3C2×#13511yescharacteristic
346C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C3C2×#13711yescharacteristic
348C22×C3#57511yescharacteristic
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—41no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—81no—
357C3×C4—81no—
358C3×C4—82no—
359C3×C4—82no—
360C3×C4#55212yes—
361C3×C4#55212yes—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23C3×#12312yes—
369C23C22×C3×D411yescharacteristic
370C2×C4—28no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4C3×#12312yes—
404C2×C4C3×#13512yes—
405C2×C4C3×AGL(1,ℤ/8)12yes—
406C2×C4C3×#13712yes—
407C2×C4C2×C3×#4112yes—
408C2×C4C2×C3×#4111yescharacteristic
409C2×C4C2×C3×#4111yescharacteristic
410C2×C4C3×2−1+412yes—
411C8—42no—
412C8—42no—
413C2×C3—42no—
414C2×C3#248411yescharacteristic
415C2×C3#309111yescharacteristic
416C2×C3#309211yescharacteristic
417C2×C3#310711yescharacteristic
418C2×C3#311511yescharacteristic
419C2×C3#322811yescharacteristic
420C2×C3#323711yescharacteristic
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22C2×C3×#12311yescharacteristic
429C22C3×#52511yescharacteristic
430C22C3×#53611yescharacteristic
431C22C2×C3×#13511yescharacteristic
432C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
433C22C2×C3×#13711yescharacteristic
434C22C3×#57511yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4—82no—
446C4C3×#55212yes—
447C4C3×#55212yes—
448C3#2761511yescharacteristic, direct product factor, Sylow
449C2—42no—
450C2C3×#248411yescharacteristic
451C2C3×#309111yescharacteristic
452C2C3×#309211yescharacteristic
453C2C3×#310711yescharacteristic
454C2C3×#311511yescharacteristic
455C2C3×#322811yescharacteristic
456C2C3×#323711yescharacteristic
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×#2099C2semidirect
4. C3×#2099C2semidirect
5. C3×#2084C2semidirect
6. C3×#2083C2semidirect
7. C3×#1921C2semidirect
8. C3×#1919C2semidirect
12. #27615C3direct (N × H)
46. #2099C2×C3semidirect
47. #2099C2×C3semidirect
48. #2084C2×C3semidirect
49. #2083C2×C3semidirect
50. #1921C2×C3semidirect
51. #1919C2×C3semidirect
448. C3#27615direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#22962
C3×#22852
C3×#20992
C3×#20842
C3×#20832
C3×#19212
C3×#19192
C3×#18542
C3×#18252
C3×#18172
#276153