Finite Group Databasebeta

← C3×#27664 #10646271

Subgroups — C3×#27664 #10646271

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#435C211yesmaximal, characteristic
2C2×C3×#427C211yesmaximal, characteristic, complemented
3C3×#2485C212yesmaximal
4C3×#2087C211yesmaximal, characteristic, complemented
5C3×#2086C211yesmaximal, characteristic, complemented
6C3×#2076C211yesmaximal, characteristic, complemented
7C3×#2075C211yesmaximal, characteristic, complemented
8C3×#1943C212yesmaximal, complemented
9C3×#1825C212yesmaximal, complemented
10C3×#1809C212yesmaximal, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#27664C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C3×#435—24no—
18C3×#427—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×#397—22no—
28C3×#394—22no—
29C3×#389C2212yes—
30C3×#389C2212yes—
31C3×#388—22no—
32C3×#387—22no—
33C3×#380C2212yes—
34C3×#380C2212yes—
35C3×#380C2212yes—
36C3×#340C2211yescharacteristic
37C3×#340C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C2×#435C2×C311yescharacteristic
43C2×#427C2×C311yescharacteristic, complemented
44#2485C2×C312yes—
45#2087C2×C311yescharacteristic, complemented
46#2086C2×C311yescharacteristic, complemented
47#2076C2×C311yescharacteristic, complemented
48#2075C2×C311yescharacteristic, complemented
49#1943C2×C312yescomplemented
50#1825C2×C312yescomplemented
51#1809C2×C312yescomplemented
52#1808C2×C311yescharacteristic, complemented
53C22×C3×D4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C2×C3×M16—22no—
57C2×C3×M16—24no—
58C22×C3×C8—24no—
59C22×C3×C8D414yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×#117—24no—
67C3×#117—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C42—22no—
82C2×C3×C42C2311yescharacteristic
83C3×#107—22no—
84C3×#107—22no—
85C3×#107—24no—
86C3×#106—24no—
87C3×C4⋊C8—22no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8—22no—
90C3×#97—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C22×M16C22×C311yescharacteristic
97C23×C8C22×C311yescharacteristic
98C2×C4×D4C22×C311yescharacteristic
99C2×#117C22×C312yes—
100#435—24no—
101#427—24no—
102C2×#107C22×C312yes—
103C2×#107C22×C311yescharacteristic
104C2×#107C22×C311yescharacteristic
105C2×#106C22×C312yes—
106C2×C4⋊C8C22×C311yescharacteristic
107C2×C22⋊C8C22×C311yescharacteristic
108C2×C22⋊C8C22×C311yescharacteristic
109C2×#97C22×C311yescharacteristic
110#397—22no—
111#394—22no—
112#389C22×C312yes—
113#389C22×C312yes—
114#388—22no—
115#387—22no—
116#380C22×C312yes—
117#380C22×C312yes—
118#380C22×C312yes—
119#340C22×C311yescharacteristic
120#340C22×C311yescharacteristic
121#340C22×C312yes—
122#340C22×C312yes—
123#340C22×C312yes—
124#340C22×C312yes—
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D411yescharacteristic
127C2×C3×D4—22no—
128C2×C3×D4—42no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D412yes—
144C22×C3×C4#4112yes—
145C22×C3×C4C2411yescharacteristic
146C3×M16—44no—
147C2×C3×C8—22no—
148C2×C3×C8—28no—
149C2×C3×C8—21no—
150C2×C3×C8—21no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8C2×D412yes—
154C2×C3×C8C2×D412yes—
155C2×C3×C8#4114yes—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—44no—
166C3×C4⋊C4—82no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—44no—
170C3×C22⋊C4—44no—
171C3×C22⋊C4—82no—
172C3×C22⋊C4—82no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42—44no—
176C22×D4C3×D411yescharacteristic
177C23×C4C23×C311yescharacteristic
178C23×C4C23×C311yescharacteristic
179C2×M16—22no—
180C2×M16—24no—
181C22×C8—24no—
182C22×C8C3×D414yes—
183C22×C8C23×C311yescharacteristic
184C22×C8C23×C311yescharacteristic
185C22×C8C23×C311yescharacteristic
186C22×C8C23×C311yescharacteristic
187C4×D4—22no—
188C4×D4—22no—
189#117—24no—
190#117—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C3×D411yescharacteristic
194C2×C4⋊C4C23×C312yes—
195C2×C4⋊C4C23×C312yes—
196C2×C4⋊C4C23×C312yes—
197C2×C4⋊C4C23×C312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C3×D411yescharacteristic
203C2×C22⋊C4C3×D411yescharacteristic
204C2×C42—22no—
205C2×C42C23×C311yescharacteristic
206#107—22no—
207#107—22no—
208#107—24no—
209#106—24no—
210C4⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213#97—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3#12112yes—
229C23×C3C22×D411yescharacteristic
230C3×D4—44no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4#12112yes—
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C2×#4112yes—
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C42−1+412yes—
262C3×C8—22no—
263C3×C8—41no—
264C3×C8—41no—
265C3×C8#12314yes—
266C24C2×C3×D411yescharacteristic
267C24C2×C3×D411yescharacteristic
268C2×D4—22no—
269C2×D4—42no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D411yescharacteristic
282C22×C4C2×C3×D411yescharacteristic
283C22×C4C2×C3×D411yescharacteristic
284C22×C4C2×C3×D412yes—
285C22×C4C3×#4112yes—
286C22×C4C24×C311yescharacteristic
287M16—44no—
288C2×C8—22no—
289C2×C8—28no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8C2×C3×D412yes—
295C2×C8C2×C3×D412yes—
296C2×C8C3×#4114yes—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—44no—
307C4⋊C4—82no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—21no—
315C42—21no—
316C42—44no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3C2×#12111yescharacteristic
328C22×C3#54712yes—
329C22×C3#57512yes—
330C22×C3#57611yescharacteristic
331C22×C3#57811yescharacteristic
332C3×C4—22no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—82no—
337C3×C4—82no—
338C3×C4—82no—
339C3×C4—82no—
340C3×C4C2×#12311yescharacteristic
341C3×C4#53611yescharacteristic
342C3×C4#55212yes—
343C23—22no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23C3×#12112yes—
353C23C22×C3×D411yescharacteristic
354D4—44no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4C3×#12112yes—
382C2×C4C22×C3×D411yescharacteristic
383C2×C4C2×C3×#4112yes—
384C2×C4C2×C3×#4111yescharacteristic
385C2×C4C3×2−1+412yes—
386C8—22no—
387C8—41no—
388C8—41no—
389C8C3×#12314yes—
390C2×C3—42no—
391C2×C3—81no—
392C2×C3#248611yescharacteristic
393C2×C3#305411yescharacteristic
394C2×C3#306111yescharacteristic
395C2×C3#334712yes—
396C2×C3#334912yes—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22C2×C3×#12111yescharacteristic
408C22C3×#54712yes—
409C22C3×#57512yes—
410C22C3×#57611yescharacteristic
411C22C3×#57811yescharacteristic
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—82no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4C2×C3×#12311yescharacteristic
421C4C3×#53611yescharacteristic
422C4C3×#55212yes—
423C3#2766411yescharacteristic, direct product factor, Sylow
424C2—42no—
425C2—81no—
426C2C3×#248611yescharacteristic
427C2C3×#305411yescharacteristic
428C2C3×#306111yescharacteristic
429C2C3×#334712yes—
430C2C3×#334912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#427C2semidirect
4. C3×#2087C2semidirect
5. C3×#2086C2semidirect
6. C3×#2076C2semidirect
7. C3×#2075C2semidirect
8. C3×#1943C2semidirect
9. C3×#1825C2semidirect
10. C3×#1809C2semidirect
11. C3×#1808C2semidirect
12. #27664C3direct (N × H)
43. C2×#427C2×C3semidirect
45. #2087C2×C3semidirect
46. #2086C2×C3semidirect
47. #2076C2×C3semidirect
48. #2075C2×C3semidirect
49. #1943C2×C3semidirect
50. #1825C2×C3semidirect
51. #1809C2×C3semidirect
52. #1808C2×C3semidirect
423. C3#27664direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4352
C2×C3×#4272
C3×#24852
C3×#20872
C3×#20862
C3×#20762
C3×#20752
C3×#19432
C3×#18252
C3×#18092
C3×#18082
#276643