Finite Group Databasebeta

← C3×#21134 #10639741

Subgroups — C3×#21134 #10639741

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic
2C3×#2319C211yesmaximal, characteristic, complemented
3C3×#2275C211yesmaximal, characteristic, complemented
4C3×#1926C212yesmaximal, complemented
5C3×#1918C212yesmaximal, complemented
6C3×#1879C212yesmaximal
7C3×#1864C212yesmaximal, complemented
8C3×C4×#106C211yesmaximal, characteristic, complemented
9C3×#1762C211yesmaximal, characteristic, complemented
10C3×#1762C211yesmaximal, characteristic, complemented
11C3×#1746C211yesmaximal, characteristic, complemented
12#21134C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×C4×D8C418yescomplemented
18C2×C3×#107C2211yescharacteristic
19C2×C3×#106C2212yes—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×#390C2212yes—
27C3×#389C2212yes—
28C3×#388—22no—
29C3×#381—21no—
30C3×#380—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C4⋊C4C2212yes—
34C3×C4×C22⋊C4—21no—
35C3×C4×C22⋊C4—22no—
36C3×#376C2211yescharacteristic
37C2×C3×#95—21no—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42C3×#336C2212yes—
43C2×C4×D8C2×C311yescharacteristic
44#2319C2×C311yescharacteristic, complemented
45#2275C2×C311yescharacteristic, complemented
46#1926C2×C312yescomplemented
47#1918C2×C312yescomplemented
48#1879C2×C312yes—
49#1864C2×C312yescomplemented
50C4×#106C2×C311yescharacteristic, complemented
51#1762C2×C311yescharacteristic, complemented
52#1762C2×C311yescharacteristic, complemented
53#1746C2×C311yescharacteristic, complemented
54C22×C3×D4C2311yescharacteristic
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—22no—
57C23×C3×C4—21no—
58C23×C3×C4—21no—
59C2×C3×D8—24no—
60C2×C3×D8C2×C414yes—
61C22×C3×C8C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4C2×C414yes—
69C3×C4×D4C2×C414yescomplemented
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—22no—
81C2×C3×C42—21no—
82C2×C3×C42—21no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2311yescharacteristic
85C3×#107C2×C414yescomplemented
86C3×#106—24no—
87C3×#102—24no—
88C3×#102—24no—
89C3×#102C2×C414yes—
90C3×#102C2×C414yescomplemented
91C3×#97—22no—
92C3×C4×C8C2×C414yescomplemented
93C3×#95—22no—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—22no—
97C3×#95D411yescharacteristic
98C3×#95D411yescharacteristic
99C3×#95D412yes—
100C22×D8C22×C311yescharacteristic
101C2×C4×D4C22×C312yes—
102C2×C4×D4C22×C311yescharacteristic
103C2×C4×D4C22×C311yescharacteristic
104C4×D8C3×C418yescomplemented
105C2×#107C22×C311yescharacteristic
106C2×#106C22×C312yes—
107C2×#102C22×C312yes—
108C2×#102C22×C312yes—
109C2×#102C22×C311yescharacteristic
110C2×#102C22×C311yescharacteristic
111C2×#97C22×C311yescharacteristic
112C2×C4×C8C22×C311yescharacteristic
113#390C22×C312yes—
114#389C22×C312yes—
115#388—22no—
116#381—21no—
117#380—21no—
118C4×C4⋊C4C22×C311yescharacteristic
119C4×C4⋊C4C22×C312yes—
120C4×C4⋊C4C22×C311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—22no—
123#376C22×C311yescharacteristic
124C2×#95—21no—
125#340C22×C312yes—
126#340C22×C312yes—
127#340C22×C312yes—
128#336C22×C312yes—
129#336C22×C312yes—
130C24×C3—21no—
131C24×C3—21no—
132C2×C3×D4—22no—
133C2×C3×D4—21no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—21no—
137C2×C3×D4C22×C412yes—
138C2×C3×D4C22×C412yes—
139C2×C3×D4#4114yes—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—28no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—82no—
148C22×C3×C4—82no—
149C22×C3×C4—82no—
150C22×C3×C4—82no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C3×D8—28no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8C22×C412yes—
163C2×C3×C8C22×C412yes—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—82no—
170C3×C4⋊C4C22×C412yes—
171C3×C4⋊C4C22×C412yes—
172C3×C4⋊C4#4114yes—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—82no—
177C3×C22⋊C4—82no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—81no—
181C3×C42—81no—
182C3×C42—82no—
183C3×C42C22×C412yes—
184C3×C42#4112yes—
185C22×D4C23×C311yescharacteristic
186C22×D4C23×C311yescharacteristic
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C2×D8—24no—
191C2×D8C2×C3×C414yes—
192C22×C8C23×C312yes—
193C22×C8C23×C311yescharacteristic
194C22×C8C23×C311yescharacteristic
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4C2×C3×C414yes—
200C4×D4C2×C3×C414yescomplemented
201C2×C4⋊C4—22no—
202C2×C4⋊C4C23×C312yes—
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C311yescharacteristic
205C2×C4⋊C4C23×C312yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C23×C312yes—
215C2×C42C23×C311yescharacteristic
216#107C2×C3×C414yescomplemented
217#106—24no—
218#102—24no—
219#102—24no—
220#102C2×C3×C414yes—
221#102C2×C3×C414yescomplemented
222#97—22no—
223C4×C8C2×C3×C414yescomplemented
224#95—22no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95C3×D411yescharacteristic
229#95C3×D411yescharacteristic
230#95C3×D412yes—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—22no—
234C23×C3—42no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3C22×D411yescharacteristic
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—24no—
246C3×D4—42no—
247C3×D4#11714yes—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—84no—
268C2×C3×C4—84no—
269C2×C3×C4—88no—
270C2×C3×C4C4×D412yes—
271C2×C3×C4C4×D412yes—
272C2×C3×C4C23×C411yescharacteristic
273C2×C3×C4C2×#4112yes—
274C2×C3×C4C2×#4112yes—
275C2×C3×C4C2×#4111yescharacteristic
276C3×C8—22no—
277C3×C8—22no—
278C3×C8—42no—
279C24—21no—
280C24—21no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4C22×C3×C412yes—
287C2×D4C22×C3×C412yes—
288C2×D4C3×#4114yes—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—28no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C2×C3×D412yes—
301C22×C4C2×C3×D412yes—
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C24×C311yescharacteristic
305D8—28no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8C22×C3×C412yes—
312C2×C8C22×C3×C412yes—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—82no—
319C4⋊C4C22×C3×C412yes—
320C4⋊C4C22×C3×C412yes—
321C4⋊C4C3×#4114yes—
322C22⋊C4—28no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—22no—
328C42—22no—
329C42—81no—
330C42—81no—
331C42—82no—
332C42C22×C3×C412yes—
333C42C3×#4112yes—
334C22×C3—24no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3C2×C4×D411yescharacteristic
345C22×C3#52512yes—
346C22×C3#57612yes—
347C22×C3#57711yescharacteristic
348C22×C3#57711yescharacteristic
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—44no—
353C3×C4—42no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—82no—
357C3×C4C2×#11711yescharacteristic
358C3×C4C4×#4111yescharacteristic
359C3×C4#52912yes—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23C22×C3×D411yescharacteristic
371D4—22no—
372D4—22no—
373D4—22no—
374D4—24no—
375D4—42no—
376D4C3×#11714yes—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—88no—
399C2×C4C3×C4×D412yes—
400C2×C4C3×C4×D412yes—
401C2×C4C23×C3×C411yescharacteristic
402C2×C4C2×C3×#4112yes—
403C2×C4C2×C3×#4112yes—
404C2×C4C2×C3×#4111yescharacteristic
405C8—22no—
406C8—22no—
407C8—42no—
408C2×C3—44no—
409C2×C3—82no—
410C2×C3#229011yescharacteristic
411C2×C3#294811yescharacteristic
412C2×C3#295211yescharacteristic
413C2×C3#311812yes—
414C2×C3#311912yes—
415C22—24no—
416C22—42no—
417C22—42no—
418C22—44no—
419C22—44no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—82no—
424C22—82no—
425C22C2×C3×C4×D411yescharacteristic
426C22C3×#52512yes—
427C22C3×#57612yes—
428C22C3×#57711yescharacteristic
429C22C3×#57711yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—44no—
436C4—42no—
437C4—82no—
438C4C2×C3×#11711yescharacteristic
439C4C3×C4×#4111yescharacteristic
440C4C3×#52912yes—
441C3#2113411yescharacteristic, direct product factor, Sylow
442C2—44no—
443C2—82no—
444C2C3×#229011yescharacteristic
445C2C3×#294811yescharacteristic
446C2C3×#295211yescharacteristic
447C2C3×#311812yes—
448C2C3×#311912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2319C2semidirect
3. C3×#2275C2semidirect
4. C3×#1926C2semidirect
5. C3×#1918C2semidirect
7. C3×#1864C2semidirect
8. C3×C4×#106C2semidirect
9. C3×#1762C2semidirect
10. C3×#1762C2semidirect
11. C3×#1746C2semidirect
12. #21134C3direct (N × H)
17. C3×C4×D8C4semidirect
44. #2319C2×C3semidirect
45. #2275C2×C3semidirect
46. #1926C2×C3semidirect
47. #1918C2×C3semidirect
49. #1864C2×C3semidirect
50. C4×#106C2×C3semidirect
51. #1762C2×C3semidirect
52. #1762C2×C3semidirect
53. #1746C2×C3semidirect
69. C3×C4×D4C2×C4semidirect
85. C3×#107C2×C4semidirect
90. C3×#102C2×C4semidirect
92. C3×C4×C8C2×C4semidirect
104. C4×D8C3×C4semidirect
200. C4×D4C2×C3×C4semidirect
216. #107C2×C3×C4semidirect
221. #102C2×C3×C4semidirect
223. C4×C8C2×C3×C4semidirect
441. C3#21134direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
C3×#23192
C3×#22752
C3×#19262
C3×#19182
C3×#18792
C3×#18642
C3×C4×#1062
C3×#17622
C3×#17462
#211343