Finite Group Databasebeta

← C3×#21396 #10640003

Subgroups — C3×#21396 #10640003

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C3×C4×#117C211yesmaximal, characteristic, complemented
4C3×#1989C212yesmaximal, complemented
5C3×#1983C212yesmaximal, complemented
6C3×#1875C212yesmaximal
7C3×#1873C212yesmaximal, complemented
8C3×#1763C212yesmaximal, complemented
9C3×#1762C212yesmaximal, complemented
10#21396C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#529—24no—
12C3×#529C418yescomplemented
13C2×C3×#126—21no—
14C2×C3×#125C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×#123—21no—
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×C3×#117C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#432—24no—
26C2×C3×C4⋊C8C2212yes—
27C2×C3×#103C2214yes—
28C2×C3×#102C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×#97C2212yes—
31C3×C4×C4⋊C4C2212yes—
32C3×C4×C4⋊C4C2212yes—
33C3×C4×C22⋊C4—21no—
34C3×C4×C22⋊C4—21no—
35C3×#376—21no—
36C3×C43—21no—
37C3×#340C2214yes—
38C3×#340C2214yes—
39C2×#529C2×C311yescharacteristic
40C2×#432C2×C311yescharacteristic, complemented
41C4×#117C2×C311yescharacteristic, complemented
42#1989C2×C312yescomplemented
43#1983C2×C312yescomplemented
44#1875C2×C312yes—
45#1873C2×C312yescomplemented
46#1763C2×C312yescomplemented
47#1762C2×C312yescomplemented
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—21no—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C22×C3×C8C2314yes—
56C3×#126—28no—
57C3×#125—22no—
58C3×#125C2×C414yes—
59C3×#124—22no—
60C3×#124C2×C414yes—
61C3×#123—28no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8C2×C414yes—
64C3×#121—22no—
65C3×#121C2×C414yes—
66C3×C4×Q8—22no—
67C3×C4×Q8C2×C414yes—
68C3×C4×D4—28no—
69C3×C4×D4—22no—
70C3×C4×D4C2×C414yes—
71C3×#117—28no—
72C3×#117—28no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2314yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—22no—
87C2×C3×C42—21no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42C2×C414yes—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D414yes—
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C3×C4⋊C8—24no—
97C3×#103—28no—
98C3×#102—28no—
99C3×C22⋊C8—24no—
100C3×#97—24no—
101C3×#95—22no—
102C3×#95—22no—
103#529—24no—
104#529C3×C418yescomplemented
105C2×#126—21no—
106C2×#125C22×C311yescharacteristic
107C2×#124C22×C311yescharacteristic
108C2×#123—21no—
109C2×C22⋊Q8C22×C311yescharacteristic
110C2×#121C22×C311yescharacteristic
111C2×C4×Q8C22×C311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4C22×C311yescharacteristic
114C2×#117—21no—
115C2×#117C22×C312yes—
116C22×C42C22×C311yescharacteristic
117#432—24no—
118C2×C4⋊C8C22×C312yes—
119C2×#103C22×C314yes—
120C2×#102C22×C314yes—
121C2×C22⋊C8C22×C312yes—
122C2×#97C22×C312yes—
123C4×C4⋊C4C22×C312yes—
124C4×C4⋊C4C22×C312yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—21no—
128C43—21no—
129#340C22×C314yes—
130#340C22×C314yes—
131C24×C3—21no—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—21no—
134C2×C3×Q8—24no—
135C2×C3×Q8C22×C412yes—
136C2×C3×D4—24no—
137C2×C3×D4—21no—
138C2×C3×D4—24no—
139C2×C3×D4—44no—
140C2×C3×D4C22×C412yes—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—24no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—44no—
150C22×C3×C4C22⋊C414yes—
151C22×C3×C4C22⋊C414yes—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2411yescharacteristic
160C2×C3×C8—24no—
161C2×C3×C8—24no—
162C2×C3×C8—24no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4C22×C412yes—
171C3×C4⋊C4C22×C412yes—
172C3×C4⋊C4#4118yes—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C42—28no—
180C3×C42—24no—
181C3×C42—44no—
182C3×C42—44no—
183C3×C42C22⋊C414yes—
184C3×C42C22⋊C414yes—
185C3×C42C22×C412yes—
186C3×C42C22×C412yes—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189C22×Q8C23×C311yescharacteristic
190C22×D4—21no—
191C22×D4C23×C311yescharacteristic
192C23×C4—21no—
193C23×C4C3×D411yescharacteristic
194C23×C4C3×D411yescharacteristic
195C23×C4C23×C311yescharacteristic
196C22×C8C23×C314yes—
197#126—28no—
198#125—22no—
199#125C2×C3×C414yes—
200#124—22no—
201#124C2×C3×C414yes—
202#123—28no—
203C22⋊Q8—22no—
204C22⋊Q8C2×C3×C414yes—
205#121—22no—
206#121C2×C3×C414yes—
207C4×Q8—22no—
208C4×Q8C2×C3×C414yes—
209C4×D4—28no—
210C4×D4—22no—
211C4×D4C2×C3×C414yes—
212#117—28no—
213#117—28no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C314yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42C2×C3×C414yes—
232C2×C42C3×D411yescharacteristic
233C2×C42C3×D411yescharacteristic
234C2×C42C3×D414yes—
235C2×C42C23×C311yescharacteristic
236C2×C42C23×C311yescharacteristic
237C4⋊C8—24no—
238#103—28no—
239#102—28no—
240C22⋊C8—24no—
241#97—24no—
242#95—22no—
243#95—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3C2×C22⋊C412yes—
252C23×C3C22×D411yescharacteristic
253C3×Q8—24no—
254C3×Q8—42no—
255C3×D4—24no—
256C3×D4—48no—
257C3×D4—42no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—28no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—48no—
272C2×C3×C4—42no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4C2×C22⋊C412yes—
278C2×C3×C4C2×C22⋊C412yes—
279C2×C3×C4C2×C22⋊C412yes—
280C2×C3×C4C2×C22⋊C412yes—
281C2×C3×C4C2×C22⋊C412yes—
282C2×C3×C4C23×C411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4114yes—
285C3×C8—44no—
286C24—21no—
287C24C2×C3×D411yescharacteristic
288C2×Q8—21no—
289C2×Q8—24no—
290C2×Q8C22×C3×C412yes—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4—24no—
294C2×D4—44no—
295C2×D4C22×C3×C412yes—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4C3×C22⋊C414yes—
306C22×C4C3×C22⋊C414yes—
307C22×C4C22×C3×C412yes—
308C22×C4C2×C3×D411yescharacteristic
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D412yes—
314C22×C4C24×C311yescharacteristic
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—24no—
325C4⋊C4C22×C3×C412yes—
326C4⋊C4C22×C3×C412yes—
327C4⋊C4C3×#4118yes—
328C22⋊C4—28no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C42—28no—
335C42—24no—
336C42—44no—
337C42—44no—
338C42C3×C22⋊C414yes—
339C42C3×C22⋊C414yes—
340C42C22×C3×C412yes—
341C42C22×C3×C412yes—
342C42C2×C3×D412yes—
343C42C2×C3×D412yes—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—84no—
352C22×C3C22×C22⋊C411yescharacteristic
353C22×C3#52512yes—
354C22×C3#57512yes—
355C22×C3#57512yes—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—48no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4C4×#4112yes—
366C3×C4#53212yes—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23C2×C3×C22⋊C412yes—
375C23C22×C3×D411yescharacteristic
376Q8—24no—
377Q8—42no—
378D4—24no—
379D4—48no—
380D4—42no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—28no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4C2×C3×C22⋊C412yes—
401C2×C4C2×C3×C22⋊C412yes—
402C2×C4C2×C3×C22⋊C412yes—
403C2×C4C2×C3×C22⋊C412yes—
404C2×C4C2×C3×C22⋊C412yes—
405C2×C4C23×C3×C411yescharacteristic
406C2×C4C22×C3×D411yescharacteristic
407C2×C4C2×C3×#4114yes—
408C8—44no—
409C2×C3—42no—
410C2×C3—82no—
411C2×C3#229711yescharacteristic
412C2×C3#289611yescharacteristic
413C2×C3#289611yescharacteristic
414C2×C3#311114yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22C22×C3×C22⋊C411yescharacteristic
424C22C3×#52512yes—
425C22C3×#57512yes—
426C22C3×#57512yes—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—48no—
436C4C3×C4×#4112yes—
437C4C3×#53212yes—
438C3#2139611yescharacteristic, direct product factor, Sylow
439C2—42no—
440C2—82no—
441C2C3×#229711yescharacteristic
442C2C3×#289611yescharacteristic
443C2C3×#289611yescharacteristic
444C2C3×#311114yes—
subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#432C2semidirect
3. C3×C4×#117C2semidirect
4. C3×#1989C2semidirect
5. C3×#1983C2semidirect
7. C3×#1873C2semidirect
8. C3×#1763C2semidirect
9. C3×#1762C2semidirect
10. #21396C3direct (N × H)
12. C3×#529C4semidirect
40. C2×#432C2×C3semidirect
41. C4×#117C2×C3semidirect
42. #1989C2×C3semidirect
43. #1983C2×C3semidirect
45. #1873C2×C3semidirect
46. #1763C2×C3semidirect
47. #1762C2×C3semidirect
104. #529C3×C4semidirect
438. C3#21396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4322
C3×C4×#1172
C3×#19892
C3×#19832
C3×#18752
C3×#18732
C3×#17632
C3×#17622
#213963