Finite Group Databasebeta

← C3×#21793 #10640400

Subgroups — C3×#21793 #10640400

Order 768.

1882 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C3×#2280C212yesmaximal, complemented
3C3×#1964C218yesmaximal, complemented
4C3×#1781C214yesmaximal, complemented
5#21793C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×D4C411yescharacteristic, complemented
7C23×C3×D4C2212yes—
8C24×C3×C4C2211yescharacteristic
9C2×C3×#121—22no—
10C2×C3×#121—22no—
11C2×C3×#121—22no—
12C2×C3×#121—22no—
13C2×C3×#121C414yescomplemented
14C2×C3×#121C2214yes—
15C2×C3×#117—22no—
16C22×C3×C4⋊C4C411yescharacteristic, complemented
17C22×C3×C22⋊C4C412yescomplemented
18C2×C3×#99C22116yescomplemented
19C3×#381—22no—
20C3×#380—22no—
21C3×#380C2214yescomplemented
22C3×#379C2214yes—
23C3×C4×C22⋊C4—22no—
24C3×C4×C22⋊C4C2214yescomplemented
25C22×#121C2×C311yescharacteristic
26#2280C2×C312yescomplemented
27#1964C2×C318yescomplemented
28#1781C2×C314yescomplemented
29C25×C3C2×C412yes—
30C25×C3C2311yescharacteristic
31C22×C3×D4—24no—
32C22×C3×D4—24no—
33C22×C3×D4—21no—
34C22×C3×D4—22no—
35C22×C3×D4—21no—
36C22×C3×D4—44no—
37C22×C3×D4—48no—
38C22×C3×D4C2×C414yes—
39C22×C3×D4C2×C414yes—
40C22×C3×D4C2314yes—
41C23×C3×C4—44no—
42C23×C3×C4C2×C411yescharacteristic
43C23×C3×C4C2×C412yes—
44C23×C3×C4C2×C411yescharacteristic
45C23×C3×C4C2×C412yes—
46C23×C3×C4D412yes—
47C23×C3×C4D412yes—
48C23×C3×C4D412yes—
49C23×C3×C4D412yes—
50C23×C3×C4C2312yes—
51C3×#121—24no—
52C3×#121—24no—
53C3×#121—24no—
54C3×#121—24no—
55C3×#121—24no—
56C3×#121—24no—
57C3×#121—24no—
58C3×#121—24no—
59C3×#117—48no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4C2×C414yes—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4C2×C418yes—
71C2×C3×C22⋊C4C2318yes—
72C2×C3×C42—24no—
73C3×#99—416no—
74C3×#95—24no—
75C3×#95—24no—
76C23×D4C3×C411yescharacteristic, complemented
77C23×D4C22×C312yes—
78C24×C4C22×C311yescharacteristic
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121C3×C414yescomplemented
84C2×#121C22×C314yes—
85C2×#117—22no—
86C22×C4⋊C4C3×C411yescharacteristic, complemented
87C22×C22⋊C4C3×C412yescomplemented
88C2×#99C22×C3116yescomplemented
89#381—22no—
90#380—22no—
91#380C22×C314yescomplemented
92#379C22×C314yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4C22×C314yescomplemented
95C24×C3—24no—
96C24×C3—22no—
97C24×C3—22no—
98C24×C3—42no—
99C24×C3—42no—
100C24×C3—44no—
101C24×C3—44no—
102C24×C3—44no—
103C24×C3C22×C411yescharacteristic
104C24×C3C22×C414yes—
105C24×C3C22×C411yescharacteristic
106C24×C3C2×D412yes—
107C24×C3C2×D412yes—
108C24×C3C2411yescharacteristic
109C2×C3×D4—28no—
110C2×C3×D4—28no—
111C2×C3×D4—216no—
112C2×C3×D4—24no—
113C2×C3×D4—24no—
114C2×C3×D4—48no—
115C2×C3×D4—44no—
116C2×C3×D4—48no—
117C2×C3×D4—48no—
118C2×C3×D4—416no—
119C2×C3×D4—416no—
120C2×C3×D4—44no—
121C22×C3×C4—28no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—28no—
126C22×C3×C4—44no—
127C22×C3×C4—44no—
128C22×C3×C4—44no—
129C22×C3×C4—44no—
130C22×C3×C4—42no—
131C22×C3×C4—44no—
132C22×C3×C4—44no—
133C22×C3×C4C22⋊C412yes—
134C22×C3×C4C22⋊C411yescharacteristic
135C22×C3×C4C22⋊C412yes—
136C22×C3×C4C22⋊C412yes—
137C22×C3×C4C22⋊C412yes—
138C22×C3×C4C22⋊C411yescharacteristic
139C22×C3×C4C22⋊C411yescharacteristic
140C22×C3×C4C22⋊C412yes—
141C22×C3×C4C22⋊C412yes—
142C22×C3×C4C22⋊C411yescharacteristic
143C22×C3×C4C22×C414yes—
144C22×C3×C4C22×C412yes—
145C22×C3×C4C22×C412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—48no—
153C3×C22⋊C4—28no—
154C3×C22⋊C4—28no—
155C3×C22⋊C4—216no—
156C3×C22⋊C4—48no—
157C3×C22⋊C4—48no—
158C3×C42—48no—
159C25C2×C3×C412yes—
160C25C23×C311yescharacteristic
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—44no—
167C22×D4—48no—
168C22×D4C2×C3×C414yes—
169C22×D4C2×C3×C414yes—
170C22×D4C23×C314yes—
171C23×C4—44no—
172C23×C4C2×C3×C411yescharacteristic
173C23×C4C2×C3×C412yes—
174C23×C4C2×C3×C411yescharacteristic
175C23×C4C2×C3×C412yes—
176C23×C4C3×D412yes—
177C23×C4C3×D412yes—
178C23×C4C3×D412yes—
179C23×C4C3×D412yes—
180C23×C4C23×C312yes—
181#121—24no—
182#121—24no—
183#121—24no—
184#121—24no—
185#121—24no—
186#121—24no—
187#121—24no—
188#121—24no—
189#117—48no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4C2×C3×C414yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4C2×C3×C418yes—
201C2×C22⋊C4C23×C318yes—
202C2×C42—24no—
203#99—416no—
204#95—24no—
205#95—24no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—24no—
211C23×C3—24no—
212C23×C3—24no—
213C23×C3—24no—
214C23×C3—24no—
215C23×C3—28no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—416no—
223C23×C3—44no—
224C23×C3—44no—
225C23×C3—44no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—44no—
229C23×C3—44no—
230C23×C3—44no—
231C23×C3—816no—
232C23×C3C2×C22⋊C412yes—
233C23×C3C2×C22⋊C411yescharacteristic
234C23×C3C2×C22⋊C411yescharacteristic
235C23×C3C23×C411yescharacteristic
236C23×C3C22×D412yes—
237C23×C3H(5,𝔽2)14yes—
238C3×D4—48no—
239C3×D4—416no—
240C3×D4—416no—
241C3×D4—48no—
242C2×C3×C4—24no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—28no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—48no—
262C2×C3×C4—48no—
263C2×C3×C4—48no—
264C2×C3×C4#9914yescomplemented
265C2×C3×C4C2×C22⋊C412yes—
266C2×C3×C4C2×C22⋊C412yes—
267C2×C3×C4C2×C22⋊C412yes—
268C2×C3×C4C2×C22⋊C412yes—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—42no—
273C24—42no—
274C24—44no—
275C24—44no—
276C24—44no—
277C24C22×C3×C411yescharacteristic
278C24C22×C3×C414yes—
279C24C22×C3×C411yescharacteristic
280C24C2×C3×D412yes—
281C24C2×C3×D412yes—
282C24C24×C311yescharacteristic
283C2×D4—28no—
284C2×D4—28no—
285C2×D4—216no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—416no—
293C2×D4—416no—
294C2×D4—44no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4C3×C22⋊C412yes—
308C22×C4C3×C22⋊C411yescharacteristic
309C22×C4C3×C22⋊C412yes—
310C22×C4C3×C22⋊C412yes—
311C22×C4C3×C22⋊C412yes—
312C22×C4C3×C22⋊C411yescharacteristic
313C22×C4C3×C22⋊C411yescharacteristic
314C22×C4C3×C22⋊C412yes—
315C22×C4C3×C22⋊C412yes—
316C22×C4C3×C22⋊C411yescharacteristic
317C22×C4C22×C3×C414yes—
318C22×C4C22×C3×C412yes—
319C22×C4C22×C3×C412yes—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D412yes—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—216no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C42—48no—
333C22×C3—28no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—22no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—44no—
346C22×C3—48no—
347C22×C3—48no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—88no—
353C22×C3—88no—
354C22×C3—88no—
355C22×C3C2×#9912yes—
356C22×C3#41012yes—
357C22×C3C22×C22⋊C411yescharacteristic
358C22×C3#51812yes—
359C22×C3#53514yes—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—44no—
364C3×C4—88no—
365C3×C4C2×#9914yes—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—28no—
376C23—41no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—41no—
382C23—416no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—816no—
392C23C2×C3×C22⋊C412yes—
393C23C2×C3×C22⋊C411yescharacteristic
394C23C2×C3×C22⋊C411yescharacteristic
395C23C23×C3×C411yescharacteristic
396C23C22×C3×D412yes—
397C23C3×H(5,𝔽2)14yes—
398D4—48no—
399D4—416no—
400D4—416no—
401D4—48no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4C3×#9914yescomplemented
425C2×C4C2×C3×C22⋊C412yes—
426C2×C4C2×C3×C22⋊C412yes—
427C2×C4C2×C3×C22⋊C412yes—
428C2×C4C2×C3×C22⋊C412yes—
429C2×C3—24no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—42no—
434C2×C3—84no—
435C2×C3#230911yescharacteristic
436C2×C3C22×#9911yescharacteristic
437C2×C3C2×#41011yescharacteristic
438C22—28no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—44no—
451C22—48no—
452C22—48no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—88no—
458C22—88no—
459C22—88no—
460C22C2×C3×#9912yes—
461C22C3×#41012yes—
462C22C22×C3×C22⋊C411yescharacteristic
463C22C3×#51812yes—
464C22C3×#53514yes—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—44no—
469C4—88no—
470C4C2×C3×#9914yes—
471C3#2179311yescharacteristic, direct product factor, Sylow
472C2—24no—
473C2—22no—
474C2—22no—
475C2—22no—
476C2—42no—
477C2—84no—
478C2C3×#230911yescharacteristic
479C2C22×C3×#9911yescharacteristic
480C2C2×C3×#41011yescharacteristic
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2280C2semidirect
3. C3×#1964C2semidirect
4. C3×#1781C2semidirect
5. #21793C3direct (N × H)
6. C23×C3×D4C4semidirect
13. C2×C3×#121C4semidirect
16. C22×C3×C4⋊C4C4semidirect
17. C22×C3×C22⋊C4C4semidirect
18. C2×C3×#99C22semidirect
21. C3×#380C22semidirect
24. C3×C4×C22⋊C4C22semidirect
26. #2280C2×C3semidirect
27. #1964C2×C3semidirect
28. #1781C2×C3semidirect
76. C23×D4C3×C4semidirect
83. C2×#121C3×C4semidirect
86. C22×C4⋊C4C3×C4semidirect
87. C22×C22⋊C4C3×C4semidirect
88. C2×#99C22×C3semidirect
91. #380C22×C3semidirect
94. C4×C22⋊C4C22×C3semidirect
264. C2×C3×C4#99semidirect
424. C2×C4C3×#99semidirect
471. C3#21793direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C3×#22802
C3×#19642
C3×#17812
#217933